A FORTIORI LOGIC

 

A Fortiori Logic

 Innovations, History and Assessments

 

 Avi Sion Ph.D.

(C) Copyright Avi Sion, 2013.

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Published by Avi Sion, Geneva, Switzerland.

Dayeinu: al ahat kamah vekamah tovah kefulah umekhupelet laMakom aleinu…

It would have sufficed us [if He had done us any one or more of these many listed kindnesses, yet] how much more good, many times more and still more, did God [blessed be He] bring upon us!

A fortiori discourse in the Passover Haggadah.

(My translation and brackets mine.)

Abstract

A Fortiori Logic: Innovations, History and Assessments, by Avi Sion, is a wide-ranging and in-depth study of a fortiori reasoning, comprising a great many new theoretical insights into such argument, a history of its use and discussion from antiquity to the present day, and critical analyses of the main attempts at its elucidation. Its purpose is nothing less than to lay the foundations for a new branch of logic and greatly develop it; and thus to once and for all dispel the many fallacious ideas circulating regarding the nature of a fortiori reasoning.

The work is divided into three parts. The first part, Formalities, presents the author’s largely original theory of a fortiori argument, in all its forms and varieties. Its four (or eight) principal moods are analyzed in great detail and formally validated, and secondary moods are derived from them. A crescendo argument is distinguished from purely a fortiori argument, and similarly analyzed and validated. These argument forms are clearly distinguished from the pro rata and analogical forms of argument. Moreover, we examine the wide range of a fortiori argument; the possibilities of quantifying it; the formal interrelationships of its various moods; and their relationships to syllogistic and analogical reasoning. Although a fortiori argument is shown to be deductive, inductive forms of it are acknowledged and explained. Although a fortiori argument is essentially ontical in character, more specifically logical-epistemic and ethical-legal variants of it are acknowledged.

The second part of the work, Ancient and Medieval History, looks into use and discussion of a fortiori argument in Greece and Rome, in the Talmud, among post-Talmudic rabbis, and in Christian, Moslem, Chinese and Indian sources. Aristotle’s approach to a fortiori argument is described and evaluated. There is a thorough analysis of the Mishnaic qal vachomer argument, and a reassessment of the dayo principle relating to it, as well as of the Gemara’s later take on these topics. The valuable contribution, much later, by Moshe Chaim Luzzatto is duly acknowledged. Lists are drawn up of the use of a fortiori argument in the Jewish Bible, the Mishna, the works of Plato and Aristotle, the Christian Bible and the Koran; and the specific moods used are identified. Moreover, there is a pilot study of the use of a fortiori argument in the Gemara, with reference to Rodkinson’s partial edition of the Babylonian Talmud, setting detailed methodological guidelines for a fuller study. There is also a novel, detailed study of logic in general in the Torah.

The third part of the present work, Modern and Contemporary Authors, describes and evaluates the work of numerous (some thirty) recent contributors to a fortiori logic, as well as the articles on the subject in certain lexicons. Here, we discover that whereas a few authors in the last century or so made some significant contributions to the field, most of them shot woefully off-target in various ways. The work of each author, whether famous or unknown, is examined in detail in a dedicated chapter, or at least in a section; and his ideas on the subject are carefully weighed. The variety of theories that have been proposed is impressive, and stands witness to the complexity and elusiveness of the subject, and to the crying need for the present critical and integrative study. But whatever the intrinsic value of each work, it must be realized that even errors and lacunae are interesting because they teach us how not to proceed.

This book also contains, in a final appendix, some valuable contributions to general logic, including new analyses of symbolization and axiomatization, existential import, the tetralemma, the Liar paradox and the Russell paradox.

Contents in brief

Abstract 1

Foreword. 8

PART I – FORMALITIES. 10

  1. The standard forms 10
  2. More formalities 35
  3. Still more formalities 54
  4. Apparently variant forms 79
  5. Comparisons and correlations 92

PART II – ANCIENT AND MEDIEVAL HISTORY.. 110

  1. A fortiori in Greece and Rome. 110
  2. A fortiori in the Talmud. 129
  3. In the Talmud, continued. 167
  4. Post-Talmudic rabbis 211
  5. A fortiori in the Christian Bible. 244
  6. Islamic ‘logic’ 268
  7. A fortiori in China and India. 316

PART III – MODERN AND CONTEMPORARY AUTHORS. 326

  1. Moses Mielziner 326
  2. Adolf Schwarz. 336
  3. Saul Lieberman. 341
  4. Louis Jacobs 349
  5. Heinrich Guggenheimer 371
  6. Adin Steinsaltz. 376
  7. Jonathan Cohen. 387
  8. Michael Avraham.. 390
  9. Gabriel Abitbol 396
  10. Hyam Maccoby. 411
  11. Alexander Samely. 420
  12. Lenartowicz and Koszteyn. 442
  13. Abraham, Gabbay and Schild. 448
  14. Stefan Goltzberg. 463
  15. Andrew Schumann. 471
  16. Allen Wiseman. 492
  17. Yisrael Ury. 528
  18. Hubert Marraud. 544
  19. Various other commentaries 560
  20. A fortiori in various lexicons 578
  21. Conclusions and prospects 593

APPENDICES. 600

  1. A fortiori discourse in the Jewish Bible. 600
  2. A fortiori discourse in the Mishna. 605
  3. A fortiori discourse in the two Talmuds 623
  4. A fortiori discourse by Plato and Aristotle. 630
  5. A fortiori discourse in other world literature. 639
  6. Logic in the Torah. 645
  7. Some logic topics of general interest 661

Main References 689

Contents in detail

Abstract 1

Foreword. 8

  1. Innovations 8
  2. History. 9
  3. Assessments 9

PART I – FORMALITIES. 10

1, The standard forms 10

  1. Copulative a fortiori arguments 10
  2. Implicational a fortiori arguments 15
  3. Validations 17
  4. Ranging from zero or less 26
  5. Secondary moods 30

2. More formalities 35

  1. Species and Genera. 35
  2. Proportionality. 37
  3. A crescendo argument 40
  4. Hermeneutics 45
  5. Relative middle terms 47

3. Still more formalities 54

  1. Understanding the laws of thought 54
  2. Quantification. 56
  3. A fortiori through induction. 60
  4. Antithetical items 67
  5. Traductions 72

4. Apparently variant forms 79

  1. Variations in form and content 79
  2. Logical-epistemic a fortiori 81
  3. Ethical-legal a fortiori 83
  4. There are no really hybrid forms 84
  5. Probable inferences 86
  6. Correlating ontical and probabilistic forms 89

5. Comparisons and correlations 92

  1. Analogical argument 92
  2. Is a fortiori argument syllogism?. 98
  3. Correlating arguments 99
  4. Structural comparisons 100
  5. From syllogism to a fortiori argument 101
  6. From a fortiori argument to syllogism.. 105
  7. Reiterating translations 107
  8. Lessons learned. 109

PART II – ANCIENT AND MEDIEVAL HISTORY.. 110

6. A fortiori in Greece and Rome. 110

  1. Aristotle’s observations 110
  2. The Kneales’ list 116
  3. Aristotle in practice. 118
  4. Relation to syllogism.. 120
  5. Cicero. 122
  6. Alexander of Aphrodisias 125
  7. Historical questions 127

7. A fortiori in the Talmud. 129

  1. Brief history of a fortiori 129
  2. A brief course in the relevant logic. 132
  3. A fresh analysis of the Mishna Baba Qama 2:5. 139
  4. A logician’s reading of Numbers 12:14-15. 152
  5. A critique of the Gemara in Baba Qama 25a. 155
  6. A slightly different reading of the Gemara. 165

8. In the Talmud, continued. 167

  1. Natural, conventional or revealed?. 167
  2. Measure for measure. 171
  3. The dayo principle in formal terms 176
  4. The human element 183
  5. Qal vachomer without dayo. 186
  6. Three additional Gemara arguments 190
  7. Assessment of the Talmud’s logic. 202
  8. The syllogistic Midot 207
  9. Historical questions 209

9. Post-Talmudic rabbis 211

  1. Logic and history issues 211
  2. Philo of Alexandria. 212
  3. Sifra. 215
  4. The Korach arguments 218
  5. Saadia Gaon. 221
  6. Rashi and Tosafot 222
  7. Kol zeh assim.. 224
  8. Maimonides 230
  9. More on medieval authors 234
  10. Moshe Chaim Luzzatto. 238
  11. More research is needed. 242

10. A fortiori in the Christian Bible. 244

  1. In the Christian Bible. 244
  2. Jesus of Nazareth. 248
  3. Paul of Tarsus 253
  4. In later Christian discourse. 264
  5. Additional findings 265

11. Islamic ‘logic’ 268

  1. Logic in the Koran. 268
  2. About the Koran. 276
  3. Logic in the hadiths 281
  4. A fortiori in fiqh, based on Hallaq. 285
  5. Other presentations and issues 296
  6. The dayo principle and more. 304
  7. The essence of Islamic discourse. 308

12. A fortiori in China and India. 316

  1. Zen logic in general 316
  2. A fortiori use in Zen. 321
  3. The Indian kaimutika. 323

PART III – MODERN AND CONTEMPORARY AUTHORS. 326

13. Moses Mielziner 326

  1. Description of the argument 326
  2. Structural analyses 328
  3. Concerning the jus talionis 329
  4. Restrictions and refutations 331

14. Adolf Schwarz. 336

  1. Equation to syllogism.. 336
  2. Jacobs’ critique. 337
  3. Kunst’s critique. 338
  4. Wiseman on Schwarz. 339
  5. Why a fortiori is not syllogism.. 340

15. Saul Lieberman. 341

  1. Hermogenes 341
  2. Influences on rabbis 343
  3. Reassessment 345
  4. Cicero. 346

16. Louis Jacobs 349

  1. The simple and complex types 349
  2. Deficiencies in Jacobs’ forms 353
  3. More comments on Jacobs’ work. 357
  4. A more recent contribution. 362

17. Heinrich Guggenheimer 371

  1. Tout un programme. 371
  2. Theory of a fortiori 372
  3. A faulty approach. 374

18. Adin Steinsaltz. 376

  1. Qal vachomer and dayo. 376
  2. A recurrent fallacy. 380
  3. Lack of formalism.. 386

19. Jonathan Cohen. 387

  1. Formula for a fortiori 387
  2. Fallacy of diverse weights 388
  3. No effort of validation. 389

20. Michael Avraham.. 390

  1. Model of a fortiori 390
  2. Outlook on a fortiori 391
  3. On Baba Qama 2:5. 393

21. Gabriel Abitbol 396

  1. Name and functioning. 396
  2. Tabular representation. 398
  3. Treatment of dayo. 402
  4. Refutations 407
  5. Closing remarks 409

22. Hyam Maccoby. 411

  1. Purely a fortiori argument 411
  2. A crescendo argument 413
  3. Baba Qama 25a. 414
  4. Faulty qal vachomer 417

23. Alexander Samely. 420

  1. General definition. 420
  2. Descriptive formula. 422
  3. Three alleged techniques 425
  4. Bava Kamma 25a-b. 427
  5. Samely’s online database. 429
  6. My critical researches 430

24. Lenartowicz and Koszteyn. 442

  1. The form of the argument 442
  2. The dayo principle. 442
  3. Epistemic substitution. 444

25. Abraham, Gabbay and Schild. 448

  1. Their opinion of past work. 448
  2. Their erroneous basic premise. 450
  3. Some errors of logic. 452
  4. Mixing apples and oranges 455
  5. Quid pro quo. 460

26. Stefan Goltzberg. 463

  1. Source of his definition. 463
  2. Soundness of the argument 465
  3. The dayo principle. 467
  4. His “two-dimensional” theory. 469

27. Andrew Schumann. 471

  1. Interpretation of Baba Qama 25a. 471
  2. Syllogism as a fortiori 475
  3. Grandiosity without substance. 478
  4. Logic custom-made. 481
  5. Not logic, but lunacy. 484

28. Allen Wiseman. 492

  1. Definition and Moods 492
  2. Inductive a fortiori 498
  3. Abduction and conduction. 503
  4. Proportional a fortiori 505
  5. The dayo principle. 509
  6. The scope of dayo. 514
  7. Miriam and Aaron. 520
  8. Summing up. 523

29. Yisrael Ury. 528

  1. An ingenious idea. 528
  2. Diagrams for a fortiori argument 530
  3. No a crescendo or dayo. 535
  4. Kol zeh achnis 538

30. Hubert Marraud. 544

  1. Warrants and premises 544
  2. The main form of a fortiori 546
  3. So-called meta-arguments 550
  4. Paulo minor argument 553
  5. Legal a fortiori argument 555

31. Various other commentaries 560

  1. H. S. Hirschfeld. 560
  2. H.W.B. Joseph. 561
  3. Moshe Ostrovsky. 561
  4. Pierre André Lalande. 564
  5. David Daube. 565
  6. Meir Zvi Bergman. 567
  7. Strack and Stemberger 569
  8. Meir Brachfeld. 569
  9. Gary G. Porton. 570
  10. Mordechai Torczyner 573
  11. Ron Villanova. 574
  12. Giovanni Sartor 575
  13. And others still 576

32. A fortiori in various lexicons 578

  1. The Jewish Encyclopedia. 578
  2. Encyclopaedia Judaica. 579
  3. Encyclopedia Talmudit 583
  4. How to define a fortiori 584
  5. Various dictionaries and encyclopedias 585
  6. Wikipedia. 590

33. Conclusions and prospects 593

  1. My past errors and present improvements 593
  2. Historical research into logic. 594
  3. Assessing contemporaries 596
  4. Perspectives 598

APPENDICES. 600

1. A fortiori discourse in the Jewish Bible. 600

2. A fortiori discourse in the Mishna. 605

3. A fortiori discourse in the two Talmuds 623

4. A fortiori discourse by Plato and Aristotle. 630

  1. Plato. 630
  2. Aristotle. 632

5. A fortiori discourse in other world literature. 639

  1. Ancient literature. 640
  2. More recent literature. 642

6. Logic in the Torah. 645

7. Some logic topics of general interest 661

  1. About modern symbolic logic. 661
  2. The triviality of the existential import doctrine. 664
  3. The vanity of the tetralemma. 672
  4. The Liar paradox (redux) 675
  5. The Russell paradox (redux) 679

Main References 689

Please note that the Appendices are integral parts of the present work, to be studied in conjunction with the chapters in which they are mentioned. Also, the footnotes throughout this volume are intended to be read; they often contain important additional information or reflection.

As regards the spelling of foreign words, no great effort has been made here to harmonize it. The same word may have different spellings in different contexts. Very often, the spelling used depends on the spelling others prefer, who are discussed in the given context. The reader is asked to be indulgent in this matter.

Foreword

When I started writing the present work, in late 2010, I thought it would take a dozen pages and a couple of weeks at most to say what I felt the need to say. I had, I believed, said most of what needed to be said in my previous foray in the field of a fortiori logic, in my 1995 study of Judaic logic. But having noticed that some people were still writing on the subject without reference to my work, and to boot were making serious mistakes, I felt the need to show them the errors of their ways. However, as I proceeded in this set task, I found myself more and more involved in its intricacies.

For a start, to be fair the critiques had to be detailed, and show exactly what had been said and where lay the errors and lacunae. Secondly, I kept discovering more and more commentaries which needed to be similarly reviewed and evaluated. Thirdly, it became obvious that I needed to expand my theoretical investigations, to be able to answer various questions these commentaries brought up, consciously or unconsciously. Eventually, I realized that I had to aim for a history of the subject and a survey of more recent contributions to it, to be able to demonstrate precisely who said what first.

Thus, the work ended up taking me three years to complete. Three parts emerged. The first presented my new, much more detailed theory of a fortiori argument. The second part traced the early history of use and discussion of such argument, so far as I could make it out with the resources available to me. The third focused on modern commentaries on the subject. However, these parts did not emerge separately, but repeatedly impinged on each other, so that many chapters or sections had to be written more than once to be adapted to new findings. For this reason, it was impossible to publish any part of the work before it was all done.

It should be stressed that the work did not proceed in the order that the chapters are now set out. Whereas now all commentators are ordered chronologically, I did not comment on their work in their order of appearance in history. It was all a matter of chance encounter and personal mood. Moreover, my theoretical baggage at each stage was different. For this reason, some earlier chapters may appear more perspicacious or analytically cutting than some later ones. I tried, of course, to harmonize things as much as I could; but as the book grew in size, it became more and more unwieldy. No doubt my memory in these later years is not what it was once; so I may have missed some things.

1.    Innovations

The present work is replete with valuable innovations in the field of a fortiori logic, and in other, related subjects. The present, wider ranging research confirms that my past work in this field, in my 1995 book Judaic Logic, was novel and important. But moreover, the present work corrects some inaccuracies in that past work, and greatly enlarges and sharpens our theory of a fortiori argument, so that it may be said to address almost every nook and cranny of the subject. There is not a single topic that I worked on here that did not yield some new insight or new theoretical development in a fortiori logic. This means that the research was certainly worthwhile and interesting; it is not a mere collection and rehashing of old material.

‘Formalities’, part one of the present volume, presents the author’s largely original theory of a fortiori argument, in all its forms and varieties. Its four (or eight) principal moods are analyzed in great detail and formally validated, and secondary moods are derived from them. A crescendo argument is distinguished from purely a fortiori argument, and similarly analyzed and validated. These argument forms are clearly distinguished from the pro rata and analogical forms of argument. Moreover, we examine the wide range of a fortiori argument; the possibilities of quantifying it; the formal interrelationships of its various moods; and their relationships to syllogistic and analogical reasoning. Although a fortiori argument is shown to be deductive, inductive forms of it are acknowledged and explained. Although a fortiori argument is essentially ontical in character, more specifically logical-epistemic and ethical-legal variants of it are acknowledged.

The present work also contains, in a final appendix, valuable innovations relating to certain topics in general logic; namely, symbolization and axiomatization, existential import, the tetralemma, the Liar paradox and the Russell paradox.

2.    History

Logic science, properly conceived, is not just a theoretical enterprise, but also an investigation into the historical roots of the forms of human discourse. The present work on a fortiori logic constitutes an excellent case study of how a particular form of thought is rooted deep in antiquity (in history), and probably much earlier, in language itself (in prehistory), and then gradually develops as awareness of it dawns, expands and intensifies. There is ample evidence that a fortiori discourse existed in very ancient times and in very diverse cultures. A fortiori reasoning was present in early Greek literature (Homer, Aesop), long before Aristotle first discussed it (in his Rhetoric and Topics); and it was present before that in Jewish literature (the Torah and other Biblical books). Aristotle did not invent the a fortiori argument, any more than he invented the syllogism; he ‘merely’ observed, described and explained them, as a botanist might notice and catalogue interesting plants.

‘Ancient and Medieval History’, part two of the present volume, looks into use and discussion of a fortiori argument in Greece and Rome, in the Talmud, among post-Talmudic rabbis, and in Christian, Moslem, Chinese and Indian sources. Aristotle’s approach to a fortiori argument is described and evaluated. There is a thorough analysis of the Mishnaic qal vachomer argument, and a reassessment of the dayo principle relating to it, as well as of the Gemara’s later take on these topics. The valuable contribution, much later, by Moshe Chaim Luzzatto is duly acknowledged. Lists are drawn up of the use of a fortiori argument in the Jewish Bible, the Mishna, the works of Plato and Aristotle, the Christian Bible and the Koran; and the specific moods used are identified. Moreover, there is a pilot study of the use of a fortiori argument in the Gemara, with reference to Rodkinson’s partial edition of the Babylonian Talmud, setting detailed methodological guidelines for a fuller study. There is also a novel, detailed study of logic in general in the Torah.

3.    Assessments

When I started to study a fortiori logic, I was little aware of the number of people who have since the late 19th century attempted to describe and explain this common form of reasoning. The field seemed nearly empty of contributors, a desert yet to be explored. Only little by little did I realize that many people have indeed tried their hand at solving the enigma of a fortiori argument – some, to be sure, more competently than others. It gradually became clear that a survey of existing contributors needed to be made, and their work had to be carefully studied and assessed. Such assessment depended, of course, on the theoretical and historiographical work undertaken earlier. It was interesting to see how many of the contributors studied past work very little before proposing their own ideas. Each apparently thought he was one of the first explorers.

‘Modern and Contemporary Authors’, part three of the present work, describes and evaluates the work of numerous (some thirty) recent contributors to a fortiori logic, as well as the articles on the subject in certain lexicons. Here, we discover that whereas a few authors in the last century or so made some significant contributions to the field, most of them shot woefully off-target in various ways. The work of each author, whether famous or unknown, is examined in detail in a dedicated chapter, or at least in a section; and his ideas on the subject are carefully weighed. The variety of theories that have been proposed is astonishing, and stands witness to the complexity and elusiveness of the subject, and to the crying need for the present critical and integrative study. But whatever the intrinsic value of each work, it must be realized that even errors and lacunae are interesting because they teach us how not to proceed.

PART I – FORMALITIES

1.  The standard forms

The present treatise on a fortiori logic has three purposes: (a) to present recent innovations I have made in the theory of a fortiori argument; (b) to retrace, as much as I can till now, the history of use and discussion of such argument; and (c) to review and evaluate (praise or criticize) ideas concerning such argument by other commentators or logicians. In comparison with the original theory of a fortiori argument presented in my book Judaic Logic over 15 years ago, the updated theory in the present work contains many significant improvements and enlargements. As this updated theory will, naturally, be the standard of judgment of all use and discussion of the argument throughout the present work, the reader is well advised to get acquainted with its main features before proceeding further[1].

1.    Copulative a fortiori arguments

Based on close analysis of a large number of Biblical and Talmudic examples (some known to Jewish tradition and some newly identified by me), as well as examples from everyday discourse, I discovered and proposed in my book Judaic Logic the four valid moods of copulative a fortiori argument listed below.

An a fortiori argument consists of three propositions called the major premise, the minor premise and the conclusion. A copulative such argument is one involving terms. It comprises four terms, which are always symbolized in the same way. The four terms are called the major, the minor, the middle and the subsidiary; and the symbols for them are respectively P, Q, R and S[2]. Other terminology used will be clarified as we proceed.

  1. The positive subjectal {+s} mood:
P is more R than (or as much R as) Q (is R),
and Q is R enough to be S;
therefore, all the more (or equally), P is R enough to be S.

Notice that the valid inference goes ‘from minor to major’; that is, from the minor term (Q) to the major one (P); meaning: from the minor term as subject of ‘R enough to be S’ in the minor premise, to the major term as subject of same in the conclusion. Any attempt to go from major to minor in the same way (i.e. positively) would be invalid inference.

  1. The negative subjectal {–s} mood:
P is more R than (or as much R as) Q (is R),
yet P is R not enough to be S;
therefore, all the more (or equally), Q is R not enough to be S.

Notice that the valid inference goes ‘from major to minor’; that is, from the major term (P) to the minor one (Q); meaning: from the major term as subject of ‘R not enough to be S’ in the minor premise, to the minor term as subject of same in the conclusion. Any attempt to go from minor to major in the same way (i.e. negatively) would be invalid inference.

We can summarize all information about subjectal argument as follows: “Given that P is more R than or as much R as Q is R, it follows that: if Q is R enough to be S, then P is R enough to be S; and if P is R not enough to be S, then Q is R not enough to be S; on the other hand, if Q is R not enough to be S, it does not follow that P is R not enough to be S; and if P is R enough to be S, it does not follow that Q is R enough to be S.” In this summary format, we resort to nesting: the major premise serves as primary antecedent, and the valid minor premises and conclusions appear as consequent conditions and outcomes, while the invalid moods are expressed as non-sequiturs.

For example: granted Jack (P) can run faster (R) than Jill (Q), it follows that: if Jill can run (at a speed of) one mile in under 15 minutes (S), then surely so can Jack; and if he can’t, then neither can she. Needless to say, the conditions are presumed identical in both cases; we are talking of the same course, in the same weather, and so on. If different conditions are intended, the argument may not function correctly. The a fortiori argument is stated categorically only if there are no underlying conditions. Obviously, if there are conditions they ought to be specified, or at least we must ensure they are the same throughout the argument.

  1. The positive predicatal {+p} mood:
More (or as much) R is required to be P than (as) to be Q,
and S is R enough to be P;
therefore, all the more (or equally), S is R enough to be Q.

Notice that the valid inference goes ‘from major to minor’; that is, from the major term (P) to the minor one (Q); meaning: from the major term as predicate of ‘S is R enough to be’ in the minor premise, to the minor term as predicate of same in the conclusion. Any attempt to go from minor to major in the same way (i.e. positively) would be invalid inference.

  1. The negative predicatal {–p} mood:
More (or as much) R is required to be P than (as) to be Q,
yet S is R not enough to be Q;
therefore, all the more (or equally), S is R not enough to be P.

Notice that the valid inference goes ‘from minor to major’; that is, from the minor term (Q) to the major one (P); meaning: from the minor term as predicate of ‘S is R not enough to be’ in the minor premise, to the major term as predicate of same in the conclusion. Any attempt to go from major to minor in the same way (i.e. negatively) would be invalid inference.

We can summarize all information about predicatal argument as follows: “Given that more or as much R is required to be P than to be Q, it follows that: if S is R enough to be P, then S is R enough to be Q; and if S is R not enough to be Q, then S is R not enough to be P; on the other hand, if S is R not enough to be P, it does not follow that S is R not enough to be Q; and if S is R enough to be Q, it does not follow that S is R enough to be P.” In this summary format, we resort to nesting: the major premise serves as primary antecedent, and the valid minor premises and conclusions appear as consequent conditions and outcomes, while the invalid moods are expressed as non-sequiturs.

For example: granted that it takes more strength (R) to lift 50 kilos (P) than 30 (Q): if someone (S) can lift 50 kilos, then surely he can lift 30; and if he can’t lift 30, then he can’t lift 50. Needless to say, the conditions are presumed identical in both cases; we are talking of the same handle, on the same day, and so on. If different conditions are intended, the argument may not function correctly. The a fortiori argument is stated categorically only if there are no underlying conditions. Obviously, if there are conditions they ought to be specified, or at least we must ensure they are the same throughout the argument.

Thus, to summarize, there are four valid moods of copulative a fortiori argument: two subjectal moods, in which the major and minor terms (P and Q) are the logical subjects of the three propositions concerned; and two predicatal moods, in which the major and minor terms (P and Q) are the logical predicates of the three propositions concerned. The major premise is always positive, though it differs in form in subjectal and predicatal arguments. In each of these types, there are two variants: in one, the minor premise and conclusion are positive; and in the other, they are negative. The positive and negative versions in each case are obviously closely related – the minor premise of the one is the negation of the conclusion of the other, and vice versa; that is, each can be used as a reductio ad absurdum for the other.

Note well the order in which the major and minor terms (P and Q) appear in the four moods: in the subjectal moods they are subjects; and in the predicatal ones they are predicates. It follows that in the two subjectal moods, the subsidiary term (S) is a predicate; and in the two predicatal moods, it (S) is a subject. The middle term (R), however, is a predicate in both premises and the conclusion of all the moods, note well. In subjectal moods it is a predicate of the major and minor terms (P and Q); in the predicatal moods it is a predicate of unspecified subjects in the major premise and a predicate of the subsidiary term (S) in the minor premise and conclusion, the subsidiary term being one instance of the unspecified subject-matter of the major premise.

The difference between subjectal and predicatal moods is called a difference of structure. The difference between positive and negative moods is called a difference of polarity. The difference between moods that go “from minor to major” and those that go “from major to minor” is called a difference of orientation. Sometimes this difference of direction is stated in Latin, as “a minori ad majus” and “a majori ad minus[3]. Note that the “from” term may be the minor or major and occurs in the minor premise; and the “to” term is accordingly the major or minor, respectively, and occurs in the conclusion. Notice the variations in orientation in accord with the structure and polarity involved.

In sum, these four valid moods are effectively four distinct figures (and not merely moods) of a fortiori argument, since the placement of their terms differs significantly in each case. This is clearly seen in the following table:

Figure/mood +s –s +p –p
major premise PQR PQR RPQ RPQ
minor premise QRS PRS SRP SRQ
conclusion PRS QRS SRQ SRP

Table 1.1

We shall deal with the validation of all these arguments further on. Meanwhile, the following clarifications should also be kept in mind:

  • The expression “all the more,” and others like it (such as “a fortiori,” “how much more,” and so on), are often used in practice to signal an intention of a fortiori argument. This is useful specifically when the argument is only partly explicit; but when the argument is fully explicit, as shown above, such expression is in fact redundant, and (as we shall see) can even be misleading (suggestive of ‘proportionality’). When the argument is stated in full, it is sufficient to say “therefore” to signal the conclusion; nothing is added by saying “all the more.”

Incidentally, in practice people sometimes reserve “all the more” for argument that goes from minor to major and “all the less” for argument that goes from major to minor; but it is also true that the expression “all the more” (and others like it) is also often used indiscriminately, and this is the way we usually intend it here.

  • The four arguments function just as well if the major term is greater (in respect of the middle term) than the minor term, or if they are equal. Whence, I have inserted in brackets in each mood: an “as much as” alternative clause to “more than” in the major premise, and an “equally” alternative to the traditional expression “all the more” in the conclusion. So though we have four figures, we may say that they contain two moods each, a ‘superior’ and an ‘egalitarian’ one, making a total of eight moods. Egalitarian a fortiori argument is also sometimes called ‘a pari’.
  • Note that for subjectal moods, I have specified the major premise as “P is more R than Q (is R)” – this is done to avoid confusion with a proposition of the form “P is more R than (P is) Q.” If we try using the latter with “P is Q enough to be S” to conclude “P is R enough to be S,” we would have an argument vaguely resembling a fortiori but which is in fact invalid[4]. In the valid form, Rp > Rq; whereas in the fake form Rp > Qp. Watch out for occurrences of this fallacy in common discourse.

The major premise of predicatal argument, i.e. “More R is required to be P than to be Q,” does not have the same potential for ambiguity. Note, however, that it could alternatively be formulated as “To be P requires more R than to be Q (requires R)” – in which form it might be confused with the major premise of subjectal argument, viz. “What is P is more R than what is Q (is R).”[5]

  • The major premise may occasionally in practice be converted – i.e. it may be stated, in subjectal argument, as “Q is less R than P” instead of as “P is more R than Q;” and in predicatal argument, as “Less R is required to be Q than to be P” instead of as “More R is required to be P than to be Q.” The validity of the argument in such cases is not affected, provided the minor premise and conclusion remain the same. Note this proviso well. Very often, such conversion of the major premise confuses people and they erroneously transpose the minor premise and conclusion[6]. Arguments involving such converted major premises, which may be labeled ‘inferior’, should not be counted as distinct moods.
  • In practice, the major premise is very often simply left out. The proponent of a given argument may have it explicitly or tacitly in mind. But he may also be quite unaware of it, in which case it is only we logicians who tell him it is logically present in the background and playing an active role in the inference. This is not something peculiar to a fortiori argument, but is likewise often encountered in syllogism and other forms of argument. It is called enthymemic argument (a mere technical term); you can call it abridged or abbreviated argument, if you like.
  • Concerning the minor premise and conclusion, the phrase “R enough to be” is often left out in practice. This may occur with the major premise absent, so that the middle term (R) is completely unstated (though of course still logically implicit); or it may occur with the major premise present, in which case the mention of the middle term in it is deemed sufficient for the whole argument. When the said phrase is left out, the minor premise and conclusion are usually stated in one if–then proposition: e.g. “If Q is S, then P is S,” which (to repeat) may be combined with an explicit major premise or presented alone.

The fact that often in practice the middle term R is left tacit should not blind us to the fact that it is a sine qua non for successful a fortiori argument. The proposition “P is more R than Q” combined with “Q is S” is logically quite compatible with “P is not S;” or combined with “P is not S” is logically quite compatible with “Q is S.” Similarly, The proposition “More R is required to be P than to be Q” combined with “P is S” is logically quite compatible with “Q is not S;” or combined with “Q is not S” is logically quite compatible with “P is S.” Note this well. Many commentators fail to realize this, or having learned it quickly forget it. Without the relation “R enough to be” in the minor premise, the a fortiori conclusion cannot be drawn and the argument is fallacious.

  • Evidently, the clause “R enough to be” in positive moods, or “R not enough to be,” in negative moods, even if it is not explicitly stated in the minor premise and conclusion, is absolutely essential to a fortiori argument. If there is no intended threshold of R to be attained or surpassed in order for S to be predicated of or to be subject to the major and minor terms, there is no operative a fortiori argument (though there might be some other thought-process, such as mere analogy). This is evident from the fact that, without this crucial clause, we simply cannot validate the argument. Keep that well in mind.

Note that the expression “R not enough to be” can also be stated as “not enough R to be” or “not R enough to be,” without change of meaning. The form “X is not R enough to be Y,” which is used in the minor premise and conclusion of negative subjectal or predicatal arguments, is the most ambiguous, being used for cases where X is not R at all, as well as more obviously to cases where it is R to some insufficient extent. More will be said about this further on.

  • Moreover, the middle term R must remain constant throughout the argument. That is, the middle term R specified in the minor premise must be identical with the one specified in the major premise. This can be seen by an example: although humans are more intelligent than horses, it does not follow that they can run faster than horses! Obviously, we can only speak of the superiority of humans over horses with respect to what was intended, viz. ‘intelligence’ in this case; this does not exclude the possibility that with respect to other attributes, such as leg muscles, horses are superior.

On a formal level, what this means is that if we do not specify or keep in mind the middle term R intended in the major premise, we might easily intend another middle term, say R’, in the minor premise and conclusion; in which case, our reasoning (whether unconsciously or deliberately done) would of course be faulty. This often happens in practice, and is one reason some people doubt the validity of a fortiori argument in general. But the problem here is not with the argument as such, but with the use of two middle terms. If we use, explicitly or implicitly, two middle terms, the argument is of course invalid, for it cannot be validated any longer. We could label such practice ‘the fallacy of two middle terms’ so as to remember to avoid it and not be taken in by it.

  • Any or all of the four terms, P, Q, R, S, may be a compound, i.e. a conjunction of two or more terms. This of course happens in practice often enough.

It should be stressed that, albeit their various formal differences, the four principal forms of copulative a fortiori argument above enumerated truly deserve to be called by one and the same name; they constitute a family of arguments. The positive and negative moods of a given orientation (subjectal or predicatal) are obviously two facets of the same coin. But moreover, notice the similarity between the positive subjectal and negative predicatal moods, and also between the negative subjectal and positive predicatal moods. Note that the former two moods may be characterized as going “from minor to major,” and the latter two as going “from major to minor.” More will be said about this further on.

The positive subjectal mood may be viewed as the prototype of all a fortiori argument, because of its relative simplicity. Many accounts of a fortiori argument tend to mention only this mood; or rather, examples thereof. Nevertheless, this does not mean that the other three copulative moods, or indeed their implicational analogues, can be ignored. They are distinct movements of thought that merit separate attention.

I should also draw your attention to the possibility that the whole subjectal or predicatal a fortiori argument concerns only one subject, as shown next:

When this thing (say, X) is P, it is more R than when it is Q,
and when it is Q, it is enough R to be S;
therefore, when it is P, it is enough R to be S.
More R is required for this thing (say, X) to be P than for it to be Q,
and when it is S, it is R enough to be P;
therefore, when it is S, is R enough to be Q.

We can construct similar negative moods, of course. Notice that I have specified the subject as ‘this thing’ (or X) in both major premises, but these could equally be generalities, i.e. have ‘something, anything’ as their subject. Such single-subject a fortiori arguments are not mere theoretical possibilities, but often occur in practice. Note the conditional form the sentences take; these are really, therefore, cases of implicational argument (see next section). The conditioning may obviously be based on any type of modality – extensional, natural, temporal or spatial.

2.    Implicational a fortiori arguments

In addition to the above four valid copulative moods, I identified in Judaic Logic four comparable ‘implicational’ moods. The first two I called antecedental (instead of subjectal) and the last two I called consequental (instead of predicatal). These four moods have the same figures as the preceding four; but they differ in involving the relation of implication instead of the copulative one, and therefore theses instead of terms as the items under consideration. I list them for you anyway, just to make sure there is no misunderstanding:

  1. The positive antecedental (+a) mood:
P implies more R than (or as much R as) Q (implies R),
and, Q implies enough R to imply S;
therefore, all the more (or equally), P implies enough R to imply S.
  1. The negative antecedental (–a) mood:
P implies more R than (or as much R as) Q (implies R),
yet, P does not imply enough R to imply S;
therefore, all the more (or equally), Q does not imply enough R to imply S.
  1. The positive consequental (+c) mood:
More (or as much) R is required to imply P than to imply Q,
and, S implies enough R to imply P;
therefore, all the more (or equally), S implies enough R to imply Q.
  1. The negative consequental (–c) mood:
More (as much) R is required to imply P than to imply Q,
yet, S does not imply enough R to imply Q;
therefore, all the more (or equally), S does not imply enough R to imply P.

Clearly, mostly similar comments can be made regarding the structures of these additional four valid moods (or eight, if we distinguish between superior and egalitarian moods) as for those preceding them.

In particular note well the fact that the middle thesis (R) is always a consequent (or non-consequent), whereas the other three theses (P, Q and S) have varied roles as antecedents (or non-antecedents) or consequents (or non-consequents) depending on the figure concerned. In antecedental argument, R is (or is not) a consequent of P and Q; while in consequental argument, R is (or is not) a consequent of S. Do not be misled by the fact that R is placed to the left of P and Q in the major premise of consequental a fortiori arguments. The thesis R does not there play the role of antecedent of P and Q (i.e. it does not imply them). The theses P, Q and R are there all consequents of some unstated antecedents; and thesis S is a specified instance of such unstated antecedent (in the positive case) or not so (in the negative case).

Variation of the middle thesis. Concerning the middle thesis R, the sense in which it is quantitatively variable (i.e. that more or less of it can be implied) needs to be clarified. A proposition as such does not have degrees; so it would be incorrect to imagine that the proposition R as a whole has degrees. A thesis (e.g. Rp) is not a quantity, and so cannot be “greater” than another thesis (e.g. Rq). Therefore, when in the major premises of implicational a fortiori argument we say that “more of thesis R” is implied or required, we must refer to a variation in the predicate and/or in the subject within thesis R. This insight can be better understood if we formulate an implicational a fortiori argument in such a way that the categorical propositions inherent in it are made explicit. This can be done with antecedental and consequental arguments of whatever polarity. Consider for instance the following case, which is doubtless the most frequent:

P (= A is p) implies more R (= C is r) than Q (= B is q) does, and
Q (= B is q) implies enough R (= C is r) to imply S (= D is s).
So, P (= A is p) implies enough R (= C is r) to imply S (= D is s).

Here, I have shown each of the four categorical propositions as involving four different subjects (A, B, C, D) with four different predicates (p, q, r, s). The middle thesis R is here taken to mean that ‘C is r’. The variation of R may in this light be understood in various ways. In the most frequent case, the subject C is constant and it is the predicate r within R that is variable, C being rp in thesis Rp and C being rq in thesis Rq (rp > rq). Comparatively rarely, the predicate r is constant and it is the subject C within R that is variable, Cp being r in thesis Rp and Cq being r in thesis Rq (Cp > Cq)[7]. In more complex cases, both the subject C and the predicate r might conceivably vary, Cp being rp in thesis Rp and Cq being rq in thesis Rq. The important point is that the resultant R theses can reasonably be said to satisfy the condition that Rp > Rq.

As regards language, the major and minor theses might in practice be stated in gerundive form, as ‘A being p’ and ‘B being q’, while the subsidiary term might more naturally be stated in the infinitive form, as ‘D to be S’. For the middle thesis, we might say ‘more r in C’ to signify that it is the predicate that varies, or ‘more C to be r’ to signify that it is the subject that varies. Quite often in practice, people do not state the whole middle thesis, but only the most relevant term in it – i.e. the variable predicate (usually) or subject (rarely). Thus, instead of saying in the major premise “implies more R,” they might say “implies more r” or “implies more C”; and likewise, instead of saying in the minor premise and conclusion “implies R enough,” they might say “implies r enough” or “implies C enough.”

Strictly speaking, of course, this is inaccurate, because a lone term cannot be implied (or imply). The logical relation of implication concerns whole theses, never mere terms. But since this confusion occurs in everyday discourse, it is well to be aware of it and to take it into consideration. Thus, when in practice we encounter an a fortiori argument with whole theses as major and minor items, and a lone term as middle item, we should not think that this exemplifies a ‘hybrid’ type of argument which is partly copulative and partly implicational. Formally, such a construct is still implicational argument, except that the middle thesis is not entirely spoken out loud; i.e. either its subject or its predicate is left tacit. In the same way, the subsidiary thesis is sometimes incompletely stated. To validate such partly formulated arguments, we of course need to specify the intended unspoken term(s).

We could in fact say that all a fortiori arguments are tacitly implicational. The thin line between copulative and implicational argument becomes evident when we reword a typical copulative argument in implicational form, as follows:

P (= something being p) implies more R (= r in it) than Q (= something being q) does, and
Q (= something being q) implies enough R (= r in it) to imply S (= it to be s);
therefore, P (= something being p) implies enough R (= r in it) to imply S (= it to be s).

This argument is obviously a special case of the preceding one. Here, instead of four subjects (A, B, C, D), we only have two (or even just one). They are unspecific (i.e. not labeled A and B, as earlier done), in the sense that they each refer to ‘something’ (i.e. anything – the intent is general, not particular) that is solely defined by the predicate initially attached to it (viz. p, q, respectively). The ‘something’ that is intended in P and the ‘something (else)’ intended in Q are here distinct objects, note (although, as we have already seen, they could well in some cases be one and the same subject). Each of them is subject to a different measure or degree of the middle predicate ‘r’ (whence r is ‘in it’). And each of them is or turns out to be subject to the subsidiary predicate ‘s’. The case shown (here again) is the positive antecedental mood; the same can obviously be done with the positive predicatal mood, and with the negative forms of both of these.

Looking back at the way I came upon these various argument forms when I wrote Judaic Logic, I remember first discovering the copulative forms and later, finding them insufficient to account for all examples of a fortiori argument I came across, I developed the implicational forms. In a sense, they were conceived as generalizations of the corresponding copulative forms. Indeed, I overgeneralized a bit, because I did not realize at the time that the notion that a thesis may “imply more” of another thesis is logically untenable. Much later, I started wondering whether ‘hybrid’ arguments signified additional types, besides the copulative and implicational. It is only recently that I better understood the relationships between the various forms of argument as above described. So the present account amends past errors and uncertainties.

I should also here mention the following special case, where the major premise “P implies more A to be B than Q does” means “P implies that a number x of A are B, and Q implies a that number y of A are B, and x > y.” The change in magnitude involved in this case is not in the subject A or the predicate B inherent in the middle thesis, but in the quantifiers of A. So the middle thesis is not, as might be thought, about “how much A is B,” or even “how much B A is,” but about the frequency of occurrence of ‘A being B’. In such case, the proposition could be stated less ambiguously as “P implies more instances of A to be B than Q does.” The frequency involved may be extensional, as here; or it could have to do with another mode of modality, i.e. more often in time or place, or in more circumstances or contexts.

Moreover, though I have here presented the middle thesis R as a single categorical proposition, it should be kept in mind that R could contain a compound thesis, i.e. it could involve a complex set of variable factors.

In conclusion, when in formulating implicational a fortiori argument we refer to the middle thesis ‘R’, the intention is more precisely ‘something in R’, meaning ‘some term(s) in thesis R’ or even ‘some modal qualifier in thesis R’. That is, when we say: ‘implies more R’ or ‘more R is required to imply’ or ‘implies enough R’ – we must be understood to mean: ‘implies more of something in R’ or ‘more of something in R is required to imply’ or ‘implies enough of something in R’, respectively. Though I will continue to use the abridged formulae, these more elaborate formulae will be tacitly intended.

More will, of course, be said about implicational a fortiori argument as we proceed.

3.    Validations

Validation of an argument means to demonstrate its validity. An argument is ‘valid’ if, given its premises, its conclusion logically follows. Otherwise, if the putative conclusion does not follow from the given premises or if its denial follows from them, the argument is ‘invalid’. If the putative conclusion is merely not implied by the given premises, it is called a non sequitur (Latin for ‘it does not follow’); in such case, the contradictory of the putative conclusion is logically as compatible with the given premises as the putative conclusion is. If a contrary or the contradictory of the putative conclusion is positively implied by the given premises, the putative conclusion is called an absurdity (lit. ‘unsound’) or more precisely an antinomy (lit. ‘against the laws’ of thought).

The validity of an argument does not guarantee that its conclusion is true, note well. An argument may be valid even if its premises and conclusion are in fact false. Likewise, the invalidity of an argument does not guarantee that its conclusion is false. An argument may be invalid even if its premises and conclusion are in fact true. The validity (or invalidity) of an argument refers to the logical process, i.e. to the claim that a set of premises of this kind formally implies (or does not imply) a conclusion of that kind.

A material a fortiori argument may be validated simply by showing that it can be credibly cast into any one of the valid moods listed above. If it cannot be fitted into one of these forms, it is invalid – or at least, it is not an a fortiori argument. The validations of the forms of a fortiori argument may be carried out as we will now expound. Invalid forms are forms that cannot be similarly validated. Obviously, material arguments can also be so validated; but the quick way is as just stated to credibly cast them into one of the valid forms. Once the forms are validated by logical science, the material cases that fit into them are universally and forever thereafter also validated.

One way to prove the validity of a new form of deduction is through the intermediary of another, better known, form of deduction. Such derivation is called ‘reduction’. ‘Direct’ reduction is achieved by means of conversions or similar immediate inferences. If the premises of the tested argument imply those of an argument already accepted as valid, and the conclusion of the latter implies that of the former, then the tested argument is shown to be equally valid. ‘Indirect’ reduction, also known as reduction ad absurdum, on the other hand, proceeds by demonstrating that denial of the tested conclusion is inconsistent with some already validated process of reasoning.

It works like this: Suppose A and B are the two (or more) premises of a proposed argument, and C is its putative conclusion. If the C conclusion is correct, this would mean that (A + B) implies C; which means that the conjunction (A + B + not-C) is logically impossible. Let us now hypothetically suppose that C is not a necessary implication in the context of A + B; i.e. that not-C is not impossible in it. In that case, we could combine not-C with one of the premises A or B, without denying the other. But we already know from previous research that, say, (A + not-C) implies not-B; which means that the conjunction (A + not-C + B) is logically impossible. Therefore, we must admit the validity of the newly proposed argument. Note that the two stated conjunctions of three items are identical except for the relative positions (which are logically irrelevant) of the items conjoined.

Analysis of constituents

The validation procedures[8] are accordingly uniform for copulative and implicational a fortiori arguments. They are based on analysis of the meanings of the propositions involved in such argument, i.e. on reduction of these more complex forms to simpler forms more studied and better understood by logicians.

The following are the two main reductions needed for validation of the earlier listed copulative arguments. The major premises (characterized as “commensurative” because they compare measures or degrees) of subjectal and predicatal arguments are always positive and have the following components:

The subjectal major premise, “P is more R than (or as much R as) Q is,” means:

P is R, i.e. P is to a certain measure or degree R (say, Rp);
Q is R, i.e. Q is to a certain measure or degree R (say, Rq);
and Rp is greater than (or equal to) Rq (whence: Rp implies Rq).

The predicatal major premise, “More (or as much) R is required to be P than to be Q,” means:

Only what is at least to a certain measure or degree R (say, Rp) is P;
only what is at least to a certain measure or degree R (say, Rq) is Q;
and Rp is greater than (or equal to) Rq (whence: Rp implies Rq).

We could more briefly write the first two components of the predicatal major premise as exclusive implications: ‘If and only if something is Rp, then it is P’ and ‘If and only if something is Rq, then it is Q’; or more briefly still, as: ‘Iff Rp, then P’ and ‘Iff Rq, then Q’[9]. Note that in my past treatment of the predicatal major premise, in my book Judaic Logic, I did not specify the exclusiveness of these two implications; but their exclusiveness is clearly implied by the word “required.”

The positive minor premises and conclusions (labeled “suffective” because they concern sufficiency) of copulative arguments have the following four components in common. The symbols X and Y here stand for the symbols P or Q and S as appropriate in each mood; that is, we may have “P is R enough to be S,” “Q is R enough to be S,” “S is R enough to be P,” or “S is R enough to be Q.”

A proposition of the form “X is R enough to be Y” means:

X is R, i.e. X is to a certain measure or degree R (say, Rx);
whatever is at least to a certain measure or degree R (say, Ry), is Y, and
whatever is not at least to that measure or degree R (i.e. is not Ry), is not Y;
and Rx is greater than (or equal to) Ry[10] (whence: “Rx implies Ry”[11]).

All this implies that X is Y, of course. We could more briefly write the two middle components of a suffective proposition as: ‘If something is Ry or more, then it is Y’ and ‘If something is not Ry or more, then it is not Y’; and these can be put together in a single proposition: ‘If and only if something is Ry or more, then it is Y’, which can be expressed still more briefly as: ‘Iff ≥ Ry, then Y’.

Note that in my past treatment of suffective propositions, in my book Judaic Logic, I did not specify the third component, which is the inverse of the second component. I did not at the time realize the significance for a fortiori argument of this negative component, i.e. how essential it is to such argument; so this is an important new finding here. Note that since Ry implies Y and not-Ry implies not-Y, we may say that there is a causal relation – more precisely, a necessary and complete causation – between these two items.

It is this feature that gives meaning to the word “enough” (or “sufficiently”) in such propositions. This tells us that X has whatever amount of R it takes to be Y; i.e. that X has at least the amount of R required for Y. It informs us that there is a threshold of R (viz. Ry) as of and above which X is indeed Y, and anywhere before which X is not Y; Rx is then specified as falling on the required side of the known threshold. In some cases, of course, Rx is exactly equal to Ry; in such cases, the proposition would be stated more precisely as: “X is R just enough to be Y.” If it is known that Rx is (not equal to but) greater than Ry, we would say: “X is R more than enough to be Y.” Thus, “enough” means “either just enough or more than enough.” It is also clear from the above definition that another way to say “X is R enough to be Y” is: “X is too much R to be not-Y” (note the negation of the predicate in the latter form).

Although a proposition of the form ‘X is R enough to be Y’ implies that ‘X is R’ and ‘X is Y’ and ‘Rx ≥ Ry’, it does not follow that the latter propositions together imply the former, for it is not always true that there is a threshold value of R (Ry) as of which a subject (such as X) gains access to the predicate Y. Thus, we must know (or at least inductively assume) that ‘Iff ≥ Ry, then Y’ before we can construct a suffective proposition; without that threshold condition for predication, we do not have such a proposition.

The threshold (Ry), though in principle an exact quantity, need not be precisely specified in practice, but can be vaguely intended by saying “the minimum value of R corresponding to Y, whatever it happen to be.” But in any case, note well, if there is a threshold, there has to be a negative as well as a positive side of it. We shall see the full significance of this insight further on, when we examine negative suffective propositions more closely. As regards the negative moods of copulative arguments (which involve such propositions), they can, as already mentioned, be validated by reductio ad absurdum to the corresponding positive moods, without pressing need to interpret their negative propositions.

It should be emphasized that the kind of thinking that makes a fortiori argument possible depends on there being a regular increase or decrease of the middle term, i.e. along the range R. If we came across a subject (X) whose predicate (Y) varies with respect to R in complex ways – unevenly rising and then falling and/or vice versa, or fluctuating from positive to negative and/or vice versa – we would just not use a fortiori argument. Such argument form is too simple to deal with more complex variables. We would normally only use it for continuous ranges; for discontinuous ones, we would resort to more detailed descriptions and perhaps to mathematical formulas.

Note also that ‘X is R enough to be Y’ implies ‘X is Y’ provided R is indeed by itself enough for Y. If R is in fact only part of a set of conditions necessary for Y, then factor R cannot be truthfully said to be ‘enough’ for Y – or, if it happens to be proposed as ‘enough’ for Y, the remaining required factors must at least be tacitly intended. This would mean, effectively, that the proposition ‘X is R enough to be Y’ is not as it appears categorical but in fact conditioned on the tacit factors, or alternatively that the outcome of R is not yet Y but some earlier stage of development than Y. To give an example of this important issue: suppose membership in an exclusive club depends on one’s age, level of income and maybe other criteria. In that event, one might well say, “this man is old enough but not rich enough to be admitted” – and here, obviously, the man being old ‘enough’ does not imply he will be admitted, although he may be put on a waiting list till he gets rich ‘enough’ too. Thus, in common discourse, the word ‘enough’ may not signify full sufficiency but merely a tendency towards it. But in the present treatise, we intend the word ‘enough’ in its strict sense.

The above general form of suffective proposition will of course concretize in different ways according to the orientation of the copulative a fortiori argument under consideration:

In positive subjectal arguments (where P, Q are subjects), it will have the forms “P or Q is R enough to be S,” which mean:

P or Q (as the case may be) is to a certain measure or degree R (say, Rp or Rq, as appropriate);
whatever is at least to a certain measure or degree R (say, Rs) is S and
whatever is not at least to that measure or degree R (i.e. is not Rs) is not S;
and Rp or Rq is greater than or equal to Rs.

In positive predicatal arguments (where P, Q are predicates), it will have the forms “S is R enough to be P or Q,” which mean:

S is to a certain measure or degree R (say, Rs);
whatever is at least to a certain measure or degree R (say, Rp or Rq, as appropriate) is P or Q (as the case may be), and
whatever is not at least to that measure or degree R (i.e. is not Rp or Rq) is not P or Q;
and Rs is greater than or equal to Rp or Rq.

The formal difference between commensurative and suffective propositions ought to be clarified here, as I did not do this in my previous writings on this topic. Although their components are very similar in form, namely comparative and hypothetical propositions, what distinguishes them is that in commensurative forms the terms compared, viz. P and Q, are either both subjects or both predicates, whereas in suffective forms the terms compared, viz. X and Y, are one a subject and the other a predicate. For this reason, we cannot reduce commensuratives to suffectives or vice versa.

Even so, it is well to notice that the major premise of predicatal a fortiori argument, i.e. the commensurative proposition “More (or as much) R is required to be P than to be Q,” is essentially about sufficiency. The word “required” tells us that there is an unstated quantity of R sufficient for P, whereas lacking that quantity, whatever it happen to be, being R does not entail being P; similarly with regard to Q, of course[12]. Thus, this major premise is a comparison between the thresholds for P and Q, telling us that amounts of R enough for Q are not all enough for P. On the other hand, the major premise of subjectal a fortiori arguments makes no mention of sufficiency, merely informing us that P is R and Q is R, and that these two quantities of R are one greater than (or equal to) the other.

All the above comments can be repeated with regard to the propositions involved in implicational a fortiori argument, mutatis mutandis. Briefly put, we can interpret the commensurative major premises of a fortiori arguments as follows.

The antecedental major premise “P implies more R than (or as much R as) Q does” means:

P implies a certain measure or degree of R (say, Rp);
Q implies a certain measure or degree of R (say, Rq);
and Rp is greater than (or equal to) Rq (whence: Rp implies Rq).

The consequental major premise “More (or as much) R is required to imply P than to imply Q” means:

Only what implies at least a certain measure or degree of R (say, Rp) implies P;
only what implies at least a certain measure or degree of R (say, Rq) implies Q;
and Rp is greater than (or equal to) Rq (whence: Rp implies Rq).

The suffective propositions which are used as minor premises and conclusions of a fortiori arguments can be interpreted as follows. Let us first look at the general positive form, “X implies R enough to imply Y;” this means:

X implies to a certain measure or degree R (say, Rx);
whatever implies at least to a certain measure or degree R (say, Ry) implies Y, and
whatever does not imply at least to that measure or degree R (i.e. does not imply Ry) does not imply Y;
and Rx is greater than or equal to Ry.

Notice that in the negative third clause of this definition, I have opted for the minimalist supposition. This choice seems sufficient to make the intended point, viz. that “without the power to imply at least Ry, Y does not follow.” I could of course have opted for the more emphatic interpretation, viz. “whatever implies less than that measure or degree R (i.e. implies not-Ry), implies not-Y,” but this would limit the application of the form considerably and unnecessarily. It could be that someone, or myself at a later date, considers the more emphatic option more appropriate; but until some specific reason is found to do so, we are wise to opt for the minimalist position. From the point of view of validation of a fortiori argument, both options are acceptable, because in both cases (as we shall presently see) the third and fourth clauses of the minor premise pass over intact into the conclusion.[13]

The above general form of suffective proposition will of course concretize in different ways according to the orientation of the implicational a fortiori argument under consideration:

In positive antecedental arguments (where P, Q are antecedents), it will have the forms “P or Q implies R enough to imply S,” which mean:

P or Q (as the case may be) implies to a certain measure or degree R (say, Rp or Rq, as appropriate);
whatever implies at least to a certain measure or degree R (say, Rs) implies S and
whatever does not imply at least to that measure or degree R (i.e. does not imply Rs) does not imply S;
and Rp or Rq is greater than or equal to Rs.

In positive consequental arguments (where P, Q are consequents), it will have the forms “S implies R enough to imply P or Q,” which mean:

S implies to a certain measure or degree R (say, Rs);
whatever implies at least to a certain measure or degree R (say, Rp or Rq, as appropriate) implies P or Q (as the case may be), and
whatever does not imply at least to that measure or degree R (i.e. does not imply Rp or Rq) does not imply P or Q;
and Rs is greater than or equal to Rp or Rq.

For the rest, what was said earlier for copulatives may be adapted to implicationals.

As regards the production of commensurative and suffective propositions, the following should be said. How are they produced, one might ask? That is, how do we get to know them in the first place? The answer is very simple and obvious. The above stated components of commensurative or suffective propositions may be viewed as the premises of the productive arguments giving rise to them. That is to say, the simpler forms, which we have above identified as implied in and together defining these more complex forms, may be presented as premises of arguments whose conclusions are commensurative or suffective propositions. Note this well, for here we have numerous new arguments for formal logic to list as such. There is, to be sure, a bit of circularity in claiming such arguments. However, though that may be true at the most formal level, at more concrete levels such arguments are quite useful.

Validation procedures

We are now in a position to examine a fortiori argument for purposes of validation. What must be understood is that the middle term (R) of copulative argument is its essential element. Being the subject or predicate of the three other terms (the major term P, the minor term Q, and the subsidiary term S), the middle term underlies, is present in, all of them. Similarly, of course, implicational argument hinges on the middle thesis. We can say that a fortiori argument is principally about the middle item, and only incidentally about the other three items; it is the core or center of gravity of the whole argument; it is the common ground and intermediary of the three other items.

What a fortiori argument does is to relate together three values of the middle item R (here symbolized by Rp, Rq and Rs) found in relation to the other three items and thus representing them. The middle item of a fortiori argument is always something that varies quantitatively, in measure or degree – and the argument constitutes a comparison and hierarchical ordering of its different values (which are given in relation to the three other items). The truth of all this can be easily seen with reference to the following diagram, where quantities of R on the right are greater than quantities of R on the left.

Diagram 1.1
Diagram 1.1

That, then, is the essence of a fortiori argument: it is a comparison between the various quantities (measures or degrees) of the middle item (term or thesis) that are copulatively or implicationally involved in the other three items (as subjects or predicates, or antecedents or consequents, of it, as the case may be). We can thus present the quantitative core of the validations very simply as follows, with reference to the comparative propositions implied in the premises and conclusions. Here, as always, ≥ means ‘is greater than or equal to’ and < means ‘is less than’[14]:

Structure Subjectal or antecedental Predicatal or consequental
Polarity positive negative positive negative
Major premise Rp ≥ Rq Rp ≥ Rq Rp ≥ Rq Rp ≥ Rq
Minor premise Rq ≥ Rs Rp < Rs Rs ≥ Rp Rs < Rq
Conclusion So, Rp ≥ Rs So, Rq < Rs So, Rs ≥ Rq So, Rs < Rp

Table 1.2

Note that the egalitarian positive subjectal (or antecedental) conclusion Rp = Rs can only be drawn from the premises Rp = Rq and Rq = Rs. Likewise, the egalitarian positive predicatal (or consequental) conclusion Rs = Rq can only be drawn from the premises Rs = Rp and Rp = Rq. In all other positive arguments, the conclusions would be Rp > Rs or Rs > Rq (as the case may be), even if one of the premises concerned involves an equation. It follows that the egalitarian negative argument of subjectal form has premises Rp ≥ Rq and Rp ≠ Rs and conclusion Rq ≠ Rs; while that of predicatal form has premises Rp ≥ Rq and Rs ≠ Rq and conclusion Rs ≠ Rp.

Another way to illustrate the quantitative aspect of a fortiori argument is by means of bar charts, as in the diagram below. Given that Rp is greater than (or equal to) Rq, there are three possible positions for Rs: in (a) Rs is greater than (or equal to) Rp and therefore than (or to) Rq; in (b) Rs is smaller than (or equal to) Rq and therefore than (or to) Rp; and in (c) Rs is in between Rp and Rq, in which case no conclusion can be drawn. Chart (a) can be used to illustrate the positive predicatal and negative subjectal moods, and chart (b) the positive subjectal and negative predicatal moods, while chart (c) can be used to explain invalid arguments.

Diagram 1.2
Diagram 1.2

In addition to the quantitative arguments above tabulated[15], we only need to select certain clauses from our premises to derive our conclusions, as follows (check and see for yourself):

  • The conclusion of a positive subjectal argument, namely the positive suffective proposition “P is R enough to be S,” is composed of four clauses:
P is to a certain measure or degree R (say, Rp);
whatever is at least to a certain measure or degree R (say, Rs), is S;
whatever is not at least to that measure or degree R (i.e. is not Rs), is not S;
and Rp is greater than (or equal to) Rs.

In this case, the four components are obtained as follows: the first from the major premise, the second and third from the minor premise, and the fourth from the tabulated quantitative argument which is drawn from both premises. Here, note well, the “enough R” condition of the conclusion (implied in its second and third components) comes from the minor premise, because it concerns the subsidiary term (S). Here, then, the crucial threshold value of R is Rs, i.e. the minimum value of R needed to be S; knowing that Rq equals or exceeds Rs, we can predict that Rp does so too.

  • The conclusion of a positive predicatal argument, namely the positive suffective proposition “S is R enough to be Q,” is composed of four clauses:
S is to a certain measure or degree R (say, Rs);
whatever is at least to a certain measure or degree R (say, Rq), is Q;
whatever is not at least to that measure or degree R (i.e. is not Rq), is not Q;
and Rs is greater than (or equal to) Rq.

In this case, the four components are obtained as follows: the first from the minor premise, the second and third from the major premise, and the fourth from the tabulated quantitative argument which is drawn from both premises. Here, note well, the “enough R” condition of the conclusion (implied in its second and third components) comes from the major premise, because it concerns the minor term (Q). Here, then, the crucial threshold value of R is Rq, i.e. the minimum value of R needed to be Q; knowing that Rp equals or exceeds Rq, we can predict that Rs does so too.

Note that in both the above moods, the conclusion of the a fortiori argument comes solely and entirely from the two premises together (not separately). It is true that the premises contain more information than the conclusion does; but that only means that not all the information in them is used. This does not signify redundancies in the premises, because their form is essential to intuitive human understanding of the argument, whose conclusion has similar form to the minor premise.

The corresponding negative moods are most easily validated by reductio ad absurdum. We say: suppose the putative conclusion is denied, then combining such denial with the same major premise we would obtain a denial of the given minor premise; this being absurd, the putative conclusion must be valid.

More briefly put, the positive conclusions are composed of the following elements drawn from the respective premises: in subjectal argument, “P is Rp, what is Rs is S and what is not Rs is not S, and Rp ≥ Rs;” and in predicatal argument, “S is Rs, what is Rq is Q and what is not Rq is not Q, and Rs ≥ Rq.” The corresponding negative conclusions imply that one or more of these four elements is denied.

It is worth here stressing the utility of the threshold condition, i.e. the implication of the minor premise that there is a threshold value of R (say, Rt), which has to be reached or surpassed before a subject X can accede to a predicate Y (i.e. Rx must be ≥ Rt which is ≥ Ry).

  • In positive subjectal argument, the threshold of the minor premise and thence of the conclusion means that not all R are S (since some things are not Rs). Clearly, if all R were S, then we could from the major premise ‘P is more R than Q’ (which implies that ‘P is R’ and ‘Q is R’), without recourse to the simplified minor premise ‘Q is S’, obtain the conclusion that ‘P is S’ (and even that ‘Q is S’)!
  • In positive predicatal argument, one of the thresholds of the major premise and thence of the conclusion means that not all R are Q (since some things are not Rq). Clearly, if all R were Q, then we could from the major premise ‘More R is required to be P than to be Q’ (which implies that ‘R is required to be P’, and thence that ‘all P are R’[16]), together with the simplified minor premise ‘S is P’, obtain (via the intermediate conclusion ‘S is R’) the conclusion that ‘S is Q’!

In both these eventualities, the argument would be merely syllogistic, and not function like an a fortiori argument. Thus, the threshold condition is essential to the formation of a truly a fortiori argument; it is not something that can be ignored or discarded. Many people think that a fortiori argument can be formulated without this crucial condition, but that is a grave error on their part.

The same validation work can be easily done with implicational arguments, mutatis mutandis. We have thus formally and indubitably demonstrated all the said moods of a fortiori argument to be valid. As regards invalid a fortiori arguments, the following can be said. If the major item P is not identical in the major premise and in the minor premise or conclusion (so that there are effectively two major items), and/or if the minor item Q is not identical in the major premise and in the minor premise or conclusion (so that there are effectively two minor items), and/or if the middle item R is not identical in the major premise, the minor premise and the conclusion (so that there are effectively two or three middle items), and/or if the subsidiary item S is not identical in the minor premise and the conclusion (so that there are effectively two subsidiary items) – in any such cases, there is illicit process. Needless to say, “identical” here refers to identity not only in the words used, but also in their intentions; we are sometimes able to formulate two terms in such a way as to make them seem the same superficially, although in fact they are not the same deeper down[17].

Likewise, if an item or a proposition is negative where it should be positive or vice versa – here again, we have fallacious reasoning. Although all such deviations from the established norms are obviously invalid, since we cannot formally validate them, they are often tried by people in practice, so it is worth keeping them in mind.

Identification in practice. We have so far theoretically described and validated a fortiori arguments. But the reader should also develop the ability to recognize such arguments when they occur in practice, in written text or oral discourse. The following are a few useful pointers. A fortiori argument is usually signaled by some distinctive word or phrase like “a fortiori” or “all the more/less,” or “so much (the) more/less,” or more rhetorically: “how much (the) more/less?!” Such signals are of course helpful, though they do not always occur (and moreover, they are sometimes used misleadingly, when there is no a fortiori argument in fact). Sometimes, we can guess that an a fortiori argument is involved, by noticing the use of an expression like “enough” or “sufficiently.” But sometimes, there is no verbal indicator at all, and we can only determine the a fortiori form of the argument at hand by examining its content.

Very often, the major premise remains unstated, though it can be readily formulated in the light of the minor premise and conclusion. Very often, too, the middle term is left tacit, in the major premise or in the minor premise or in the conclusion, or even throughout the argument; in such cases, we have to guess at the underlying intent of the argument’s author. All we are given, in very many cases, is an if–then statement with three terms; and often the ‘if’ and ‘then’ operators are missing too! There is nevertheless usually enough information for us to reconstruct the intended a fortiori argument, assuming some such argument is indeed intended (i.e. we must be careful not to artificially ‘read in’ the argument for our own purposes).

The following indices permit us to determine the exact mood of copulative argument. Find the term (S) common to both propositions (the premise and conclusion), and see whether it stands as subject or predicate. The positive subjectal form appears as: “Q is S; therefore, P is S;” and the negative subjectal form appears as: “P is not S; therefore, Q is not S.” Notice here that S (the common term) is a predicate, and P and Q (the other two terms) are subjects. The positive predicatal form appears as: “S is P; therefore, S is Q;” and the negative predicatal form appears as: “S is not Q; therefore, S is not P.” Notice that here S (the common term) is a subject, and P and Q (the other two terms) are predicates. Similarly for implicational arguments, except that “implies” appears instead of “is.”

Of course, not even all the details given in the preceding paragraph may appear. For example, instead of “Q is S; therefore, P is S,” the speaker may say “Q is S: all the more P!” But we can easily add the missing clause “is S” that makes the consequent (conclusion) a mirror image of the antecedent (minor premise). We must then look for a middle term R, such that “P is more R than Q” is true (or at least somewhat credible), and also such that “Q is R enough to be S” is true, and therefore “P is R enough to be S” is likewise true – and we have reconstructed the intended a fortiori argument.

Obviously, a proposition of the form “X is Y” does not, strictly speaking, imply one of the form “X is R enough to be Y” – that is, the mere fact that X is Y does not indicate that there is a threshold of R that needs to be crossed for X to be Y. Nevertheless, we often inductively infer the latter from the former by reasoning that if there indeed is an a fortiori argument there must indeed be such a threshold condition for the predication. Thus, we construct the more complex premise from the simpler given, thinking “well, if X is Y, it must have been R enough to be Y!” This concerns the minor premise; as regards the conclusion, we deduce the simpler proposition from the more complex.

It should be stressed that the term common to the two given propositions is in some cases the middle term (R), rather than the subsidiary term (S). An example of that would be the sentence: “Q is bad enough; imagine what P would be!” Here, the common term “bad” is of course the middle term (as the expression “enough” indicates); and no subsidiary term is mentioned, though one can guess what it might be. A fuller statement of the minor premise and conclusion would thus be: “if Q is bad (R) enough to be avoided (S), then all the more P is bad (R) enough to be avoided (S).”

Of course, though we may manage to fully reconstruct the intended a fortiori argument, it may yet be found invalid – e.g. if, as sometimes happens, the roles of P and Q are reversed; but this is another issue, of course. That is: first find out what form the author’s intended argument has; then judge whether it is objectively valid or not. Also, do not confuse the issues of validity and truth: the argument may be well-formed, and yet be wrong due to its reliance on a false premise or other.

4.    Ranging from zero or less

An observed practice

Often, in practice (e.g. in the Talmud), we find a fortiori arguments stated in the following discursive form:

If Q, which is not R, is S, then (all the more) P, which is R, is S.”

Many people get confused by this construction, and fail to understand the nature of a fortiori argument because of it. To put such an argument in standard positive subjectal form, and thus validate it, we must first realize that the antecedent proposition is the minor premise and the consequent one is the conclusion. Then we must see that the major premise is also present by the mention of Q being not R and P being R[18]. This tells us that R is the middle term, ranging from zero to some higher quantity. Whence we can formulate the major premise as “P (for which R > 0) is more R than Q (for which R = 0).” The minor premise can now be more precisely stated as “Q is R enough to be S;” and the conclusion likewise as “P is R enough to be S.”

We can proceed in the same way to deal with a negative subjectal argument which is stated in the form:

If P, which is R, is not S, then (all the more) Q, which is not R, is not S.”

Similarly, we can readily standardize a positive predicatal argument that appears in the form:

If S is P, even though P requires R, then (all the more) S is Q, since Q does not require R.”

Or a negative predicatal argument that appears in the form:

If S is not Q, even though Q does not require R, then (all the more) S is not P, since P requires R.

The clauses “P requires R” and “Q does not require R” used here should be understood to more precisely mean, respectively: “some amount of R is required, for something to be P” and “no amount of R is required, for something to be Q.”

And similarly with the implicational equivalents of these four copulative arguments. In short, do not be confounded by the varying ways that a fortiori argument appears in practice in human discourse, but always be ready to reword it in standard form. Once we have mastered the formalities, no argument looks intractable.

It should be obvious that the four discursive forms we have just listed are merely special cases of another four, more broadly applicable and also often occurring in practice, namely, respectively:

If Q, which is less R, is S, then P, which is more R, is S.”

If P, which is more R, is not S, then Q, which is less R, is not S.”

If S is P, even though P requires more R, it follows that S is Q, since Q requires less R.”

If S is not Q, even though Q requires less R, it follows that S is not P, since P requires more R.”

These four statements are ways we often briefly articulate our a fortiori thoughts. The first two statements allude to subjectal argument. Their common major premise is “P is more R than Q is,” and their minor premises and conclusion are: in the positive case, “if Q is (R enough to be) S, then P is (R enough to be) S;” and in the negative case, “if P is not (R enough to be) S, then Q is not (R enough to be) S.” The second two statements allude to predicatal argument. Their common major premise is “More R is required to be P than to be Q,” and their minor premises and conclusion are: in the positive case, “if S is (R enough to be) P, then S is (R enough to be) Q;” and in the negative case, “if S is not (R enough to be) Q, then S is not (R enough to be) P.”

Clearly, “is R” and “is not R” are special cases of “is more R” and “is less R,” respectively; and “requires R” and “does not require R” are special cases of “requires more R” and “requires less R,” respectively. What the above observations mean is that we can, in theory as well as in practice, count the negation of the middle term R as a limiting or special case of R, i.e. as simply the value of R equal to zero in the range of possible values of R! Upon reflection, it occurs to me that the middle term R may even have negative values! For example, “Jack’s financial situation is better than Jill’s” may be true because Jack has a few dollars in the bank whereas Jill has debts; indeed, both of them may have debts, though his are less than hers. So R may in principle range anywhere from minus to plus infinity, without affecting the said forms of a fortiori argument.

What this insight implies is that, in the context of a fortiori logic, if something is not R (i.e. is zero R or less than that), it is still formally counted as something that is R, with however the understanding that in its case R ≤ 0. That is to say, for our purposes here, granting that R is broad enough to include not-R, it follows that everything is R and nothing is not-R!

How can this be? It must be understood that when we pass over from the logic of R exclusive of not-R, to that of R inclusive of not-R, the meaning of R is subtly changed. Instead of R meaning ‘being R’ (i.e. belonging to class R), it now means ‘having to do with R’ (i.e. merely pertaining to R). We can rightly say that not-R pertains to R, even though not-R is not in a strict sense R. Another way to put it is to say that R as against not-R is denotative, whereas R including not-R is connotative[19]. More will be said on this issue further on, when we deal in a more general way with relative terms.

Although in principle an (inclusive) range R may have any value from minus infinity through zero to plus infinity, in practice it may be more limited. To be in accord with the law of the excluded middle, the range must include, as well as at least one positive value R > 0, the null value R = 0 or a negative value R < 0. An example of a limited range is that of temperature: although we can imagine temperatures to be infinitely cold, physicists have discovered through experiment that the minimum temperature in nature is -273°C (on this basis the Celsius scale was replaced by the Kelvin scale, in which this minimum is 0°K); similarly, we can expect there to be a maximum temperature in nature, even if we do not know its magnitude.

Note well that I did not invent this single-range artifice, but merely noticed its use in a fortiori practice and adopted it for theoretical purposes. This convention is, as we shall now see, a very important new finding and idea, which has important consequences for a fortiori logic, greatly simplifying it. It allows us, notably, to more precisely and positively interpret the negative forms of the commensurative and suffective propositions used in a fortiori argument. However, as we shall see, it is easier to apply to copulative reasoning than to implicational.

Implications of the commensurative forms

First, let us deal with copulative forms. The subjectal major premise “P is more R than Q (is R)” obviously implies both that “P is R (to some unspecified measure or degree)” and that “Q is R (to some unspecified measure or degree),” where P, Q are either designated individual things or they are classes (in which case the meaning is “all P are R” and “all Q are R”). As just pointed out, we can and often do use this form when the middle term R has a range of values from some negative lower limit (known or unknown, stated or unstated) or even from minus infinity, through zero, to some positive upper limit (known or unknown, stated or unstated) or even to plus infinity. That is to say, the possible values R may be a boundless range from negative infinity to positive infinity, or any more limited range in between (such as entirely positive or from zero upward).

Additionally, of course, the subjectal major premise implies that Rp (the value of R for P) is greater than Rq (the value of R for Q). It does not matter whether P is positive, zero or negative, and whether Q is positive, zero or negative, provided that the quantity Rp is superior to the quantity Rq. In the special case where Rq is zero, Rp must be positive; and in the special case where Rp is zero, Rq must be negative. In the special case where Rp and Rq are both zero, or both have the same positive or negative value of R, the proposition must be changed to the egalitarian form “P is as much R as Q (is R).” In cases where Rp is smaller than Rq, the form of course becomes “P is less R than Q (is R).”

Let us now consider the denial of the form “P is more R than Q (is R).” What is the meaning of the negative form “P is not more R than Q (is R),” in the light of the above insights? In the past, before I realized that the values of R for P and/or Q may be zero (or negative), I would have said that such zero (or negative) values are absent from the positive form and therefore implicit in the negative form. However, now that these values are perceived (or conceived, for the purposes of a fortiori logic) as possibilities within the positive form, the negative form acquires a much more specific meaning, namely the disjunction “P is R less than or as much as Q (is R).” Such disjunctive major premises often, of course, occur in practice.

That is to say, the three positive commensurative forms: “P is more R than Q (is R),” “P is less R than Q (is R)” and “P is as much R as Q (is R)” are the exhaustive repertoire of such propositions, so that if any one of them is denied, one of the other two must be true. In other words, in this context, negative propositions are redundant since their meanings can be fully represented by positive ones! This greatly simplifies our formal work in this field.

Similarly, of course, the predicatal major premise “More R is required to be P than to be Q” implies (in an extensional perspective) both that “some R are P,” i.e. that some things that are R to a sufficient degree are also P; and that “some R are Q,” i.e. that some things that are R to a sufficient degree are also Q. Moreover, it is implied that Rp is greater than Rq. Here again, note well, the form is to be taken in a wide sense, allowing in principle for any values of R, positive, zero or negative, though in practice a narrower range may be tacitly or specifically intended. And here again, the negative form, “More R is not required to be P than to be Q,” is redundant, because it just means: “Less or as much R is required to be P than/as to be Q.”

We can therefore define all copulative forms of commensurative proposition as follows, irrespective of the values of Rp and Rq (i.e. be they positive, zero or negative):

The subjectal forms:

P is to a certain measure or degree R (say, Rp);
Q is to a certain measure or degree R (say, Rq);
and Rp is greater than, equal to or lesser than Rq.

And the predicatal forms[20]:

What is to a certain measure or degree R (say, Rp), is P;
what is to a certain measure or degree R (say, Rq), is Q;
and Rp is greater than, equal to or lesser than Rq.

As can be seen from their above definitions, the subjectal commensurative propositions “P is more R than Q is” and “Q is less R than P is” are each other’s converse; we can convert either to the other, without loss of information; similarly, “P is as R as Q is” and “Q is as R as P is” are equivalent, and so are compounds of the said forms. Likewise, the predicatal form “more R is required to be P than to be Q” is convertible to “less R is required to be Q than to be P,” and vice versa; and similarly with the egalitarian and compound forms.

Similar interpretations can be made with regard to the commensurative major premises of implicational a fortiori arguments. Simply put: for the definitions, instead of saying “is” or “to be” in relation to the terms P, Q, and R, we would say “implies” or “to imply” in relation to the theses P, Q, and R.

Thus, the antecedental forms signify:

P implies to a certain measure or degree R (say, Rp);
Q implies to a certain measure or degree R (say, Rq);
and Rp is greater than, equal to or lesser than Rq.

The positive form refers to: “P implies more R than Q does,” and the negation of that means: “P implies less R than or as much R as Q does.”

And the consequental forms signify:

What implies to a certain measure or degree R (say, Rp), implies P;
what implies to a certain measure or degree R (say, Rq), implies Q;
and Rp is greater than, equal to or lesser than Rq.

The positive form refers to: “More R is required to imply P than to imply Q,” and the negation of that means: “Less or as much R is required to imply P than/as to imply Q.”

Implications of the suffective forms

First, let us deal with copulative forms. The positive form of suffective proposition, “X is R enough to be Y,” used in the minor premise and conclusion of a fortiori argument, implies both that “X is R” and that “X is Y.” In the subjectal form, X stands for P or Q (as the case may be) and Y for S; and in the predicatal form, X stands for S and Y for P or Q (as the case may be). Here again, the middle term R may conceivably be positive, zero or negative. The important thing to keep in mind is that there is a threshold value of R as of and above which X is Y, and below which X is not Y. This means that the negative form “X is R not enough to be Y”[21] implies both that “X is R” (whether R is greater than, equal to or less than zero) and that “X is not Y”[22].

As already said, the generic positive suffective form, “X is R enough to be Y” can be defined by means of four simpler propositions as follows:

X is to a certain measure or degree R (say, Rx);
whatever is to a certain measure or degree R (say, Ry), is Y, and
whatever is not to that measure or degree R (i.e. is not Ry), is not Y;
and Rx is greater than (or equal to) Ry.

What does denial of this collection of propositions mean, specifically? We can say that “X is Rx” remains true no matter what, because (as above explained) X like everything else is necessarily R if we define R broadly enough to include not R; and because Rx is by definition the value of R for X, whatever that happen to be. Similarly, Ry is by definition the quantity of R enough for Y, whatever that happen to be, so there is no sense in denying that “Ry is Y;” and likewise, “not-Ry is not Y” is not open to doubt. Therefore, the only way that the collection as a whole can be denied is by denying its last clause, viz. “Rx ≥ Ry,” i.e. by saying that “Rx < Ry;” and this makes sense, because it is the same as saying that “X is not Y”[23]. Thus, the negative form, “X is R not enough to be Y” can also be defined in a positive manner, as follows:

X is to a certain measure or degree R (say, Rx);
whatever is to a certain measure or degree R (say, Ry), is Y, and
whatever is not to that measure or degree R (i.e. is not Ry), is not Y;
and Rx is less than Ry.

That is to say, the difference between the positive and the negative forms is that the former has Rx greater than or equal to Ry, whereas the latter has Rx less than Ry. That’s all. Note this well. Thus, other ways to say: “X is not enough R to be Y” would be: “X is less than enough R to be Y” or “X is too little R to be Y.”

Note well again that the positive form “X is R enough to be Y” implies “X is Y,” and its negation “X is not enough R to be Y” implies “X is not Y.” It follows from this that “X is R enough to be Y” is equivalent to “X is not enough R to be not Y,” and “X is R enough to be not Y” is equivalent to “X is not enough R to be Y.” These are equations we will find good use for further on.

Note also that although “X is R enough to be Y” implies that “X is R” and “X is Y,” it does not follow that given the latter two propositions we have enough information to construct the former one; we additionally need to know that “X is not Y” below some value of R, because this makes possible the statement that in the case of X, the amount of R is enough for it to be Y. Similarly, although “X is R not enough to be Y” implies that “X is R” and “X is not Y,” it does not follow that given the latter two propositions we have enough information to construct the former one; we additionally need to know that “X is Y” as of and above some value of R, because this makes possible the statement that in the case of X, the amount of R is not enough for it to be Y.

We can therefore define all copulative forms of suffective proposition, irrespective of the values of Rx and Ry (i.e. be they positive, zero or negative), as follows:

The subjectal forms:

P or Q (as the case may be) is to a certain measure or degree R (say, Rp or Rq, as appropriate);
whatever is to a certain measure or degree R (say, Rs), is S and
whatever is not to that measure or degree R (i.e. is not Rs), is not S;
and Rp or Rq ≥ Rs (positive form), or Rp or Rq < Rs (negative form).

And the predicatal forms:

S is to a certain measure or degree R (say, Rs);
whatever is to a certain measure or degree R (say, Rp or Rq, as appropriate), is P or Q (as the case may be), and
whatever is not to that measure or degree R (i.e. is not Rp or Rq), is not P or Q;
and Rs ≥ Rp or Rq (positive form), or Rs < Rp or Rq (negative form).

Conversion of suffective propositions is not possible. This can be ascertained by examination of their defining implications. Try relating, for instance, a subjectal form “X is R enough to be Y” to a predicatal form “Y is R enough to be X.” Since the underlying if–then components cannot be converted, nor can their suffective compounds.

Similar interpretations can be made with regard to the suffective minor premises and conclusions of implicational a fortiori arguments. Simply put: for the definitions, instead of saying “is” or “to be” in relation to the terms P, Q, and R, we would say “implies” or “to imply” in relation to the theses P, Q, and R.

5.    Secondary moods

From the preceding interpretations we can derive a number of useful arguments. Such arguments may be considered as belonging to the family of ‘a fortiori’, although they are not among the four regular copulative moods. They can be labeled as ‘secondary’ a fortiori moods, as against the ‘primary’ moods that usually define the argument form for us. They are validable, note well, based on the understanding that the value of R may range from minus infinity through zero to plus infinity[24].

Producing a commensurative proposition. The first two arguments are distinctive in that their premises are both suffective and their conclusion is commensurative. These arguments describe for us how we might occasionally produce the major premises of primary arguments. Consider the following argument, for a start:

P is R enough to be S, and
Q is R not enough to be S.
Therefore, P is more R than Q (is R).

This shows us how a subjectal commensurative proposition “P is more R than Q (is R)” can be constructed (i.e. deduced) from two suffective propositions. The positive premise tells us that the value of R for P is equal to or greater than that for S; and the negative one tells us that the value of R for Q is less than that for S. Since Rp ≥ Rs and Rq < Rs, it follows that Rp > Rq. We also know from the premises that what is P is Rp and what is Q is Rq. Whence the conclusion: P is more R than Q. Similarly, consider the following argument:

S is R not enough to be P, and
S is R enough to be Q.
Therefore, more R is required to be P than to be Q.

This shows us how a predicatal commensurative proposition “More R is required to be P than to be Q” can be constructed (i.e. deduced) from two suffective propositions. The positive premise tells us that the value of R for S is equal to or greater than that for Q; and the negative one tells us that the value of R for S is less than that for P. Since Rs ≥ Rq and Rs < Rp, it follows that Rp > Rq. We also know from the premises that what is Rp is P and what is Rq is Q. Whence the conclusion: More R is required to be P than to be Q.

It should be emphasized that the above two arguments are not the only ways we can produce a subjectal or predicatal commensurative proposition. We can always produce such propositions with reference to their formal definitions. As we have seen, a subjectal commensurate, “P is more R than (or as much R as) Q is,” is composed of the three elements: “P is R, i.e. P is to a certain measure or degree R (say, Rp); Q is R, i.e. Q is to a certain measure or degree R (say, Rq); and Rp is greater than (or equal to) Rq (whence: Rp implies Rq).” A predicatal commensurate, “More (or as much) R is required to be P than to be Q,” is composed of the three elements: “Only what is at least to a certain measure or degree R (say, Rp) is P; only what is at least to a certain measure or degree R (say, Rq) is Q; and Rp is greater than (or equal to) Rq (whence: Rp implies Rq).”

That is, in general, to produce a commensurate proposition, we need only to supply its constituent parts. That is, of course, also logical argument: the said components are the premises and the composite commensurative proposition is the conclusion. The same, of course, can be said regarding the production of suffective from their constituent parts: that is also logical argument.

The above listed two secondary moods of a fortiori argument are only special cases of such production. Similar constructions are of course possible with regard to implicational propositions:

P implies R enough to imply S, and
Q implies R, but not enough to imply S.
Therefore, P implies more R than Q does.
S implies R, but not enough to imply P, and
S implies R enough to imply Q.
Therefore, more R is required to imply P than to imply Q.

As explained further on, we must here be extra careful, as the exact location of the negation in the proposition may affect its meaning and logic. That is to say, it is more difficult with implicationals, than it was with copulatives, to put positives, zeros and negatives in the same basket.

Using a negative commensurative major premise. From each of the above two secondary moods, we can derive two more by reductio ad absurdum. These four resemble primary moods, in that their major premises are commensurative and their minor premise and conclusion are suffective, but note well that they differ in that all have a negative major premise.

  • Looking at the above subjectal argument: if the positive conclusion is denied and the positive major premise is retained, then the negative minor premise must be denied (so we have a negative and two positive propositions); the result is a mood apparently (since from P to Q) from major to minor (more on this in a moment).
P is not more R than Q (is R),
and P is R enough to be S.
Therefore, Q is R enough to be S.
  • Looking again at the above subjectal argument: if the positive conclusion is denied and the negative minor premise is retained, then the positive major premise must be denied (so we have three negative propositions); the result is a mood apparently (since from Q to P) from minor to major (more on this in a moment).
P is not more R than Q (is R),
and Q is R not enough to be S.
Therefore, P is R not enough to be S.

However, these two moods are not as new as they might seem. For their negative major premise, “P is not more R than Q,” can be restated in positive terms as: “P is less R than or as much R as Q.” That is, each negative mood refers to two positive moods, according as the major premise is in fact egalitarian or inferior. In the special case where the major premise is the egalitarian “P is as much R as Q,” the two arguments would be, respectively, positive or negative subjectal a pari arguments. While in the special case where the major premise is the inferior “P is less R than Q,” we could convert it to “Q is more R than P,” and the two arguments would then be, respectively, positive or negative subjectal superior arguments. The difference between these two cases, note well, is that when the major premise is egalitarian, the terms P and Q can still be called the major and minor, whereas when the major premise is inferior, Q refers to the major term and P to the minor term (so the symbols would have to be switched). So really, the above arguments are not deeply new, but only superficially so.

  • Looking at the above predicatal argument: if the positive conclusion is denied and the positive minor premise is retained, then the negative major premise must be denied (so we have a negative and two positive propositions); the result is a mood apparently (since from Q to P) from minor to major (more on this in a moment). Note that the conclusion would still be valid in the special case where the negative major premise means “As much R is required to be P as to be Q,” since then this would be positive predicatal a pari argument (which can, as we have seen, go from minor to major).
More R is not required to be P than to be Q,
and S is R enough to be Q.
Therefore, S is R enough to be P.
  • Looking again at the above predicatal argument: if the positive conclusion is denied and the negative major premise is retained, then the positive minor premise must be denied (so we have three negative propositions); the result is a mood apparently (since from P to Q) from major to minor (more on this in a moment).
More R is not required to be P than to be Q,
and S is R not enough to be P.
Therefore, S is R not enough to be Q.

However, here again, these two moods are not as new as they might seem. For their negative major premise, “More R is not required to be P than to be Q,” can be restated in positive terms as: “Less R or as much R is required to be P than/as to be Q.” That is, each negative mood refers to two positive moods, according as the major premise is in fact egalitarian or inferior. In the special case where the major premise is the egalitarian “As much R is required to be P as to be Q,” the two arguments would be, respectively, positive or negative predicatal a pari arguments. While in the special case where the major premise is the inferior “Less R is required to be P than to be Q,” we could convert it to “More R is required to be Q than to be P,” and the two arguments would then be, respectively, positive or negative predicatal superior arguments. The difference between these two cases, note well, is that when the major premise is egalitarian, the terms P and Q can still be called the major and minor, whereas when the major premise is inferior, Q refers to the major term and P to the minor term (so the symbols would have to be switched). So really, the above arguments are not deeply new, but only superficially so.

Thus, the above listed four secondary moods of a fortiori argument with a negative commensurative major premise’ can be directly reduced to primary forms of the argument. Nevertheless, even though they teach us nothing very new, they are still worth explicitly listing to draw attention to them. The moods shown are copulative arguments. Analogous moods can be formulated for implicational argument.

The above listings are obviously exhaustive, since all formal possibilities are accounted for.

Negative items

The valid primary and secondary moods are all formulated with positive terms or theses (P, Q, R, S). As regards moods involving the negations of some or all of these items, it is obvious that if we substitute not-P for P, and/or not-Q for Q, and/or not-R for R, and/or not-S for S, throughout a given primary or secondary argument, the validity of the argument is in no way affected[25]. Every symbol (P, Q, R, S) is intended broadly enough to apply to any items, whether positive or negative, so switching its polarity throughout an argument has no effect on validity.

Difficulty arises only when such switching occurs in only part of an argument, or when two arguments are compared which have some item(s) of opposite polarity. In such cases, it is wise to tread very carefully, and not draw hasty conclusions. However, most such cases can be solved without too much trouble, as we shall discover in the next section. This is especially true as regards copulative arguments; implicational ones require additional reflection[26].

Arguments in tandem.

In an appendix to my Judaic Logic[27], I note that subjectal and predicatal (or antecedental and consequental) a-fortiori arguments are sometimes found in tandem, forming a sorites, so that the conclusion of one implicitly serves as minor premise in the other. For instances:

Positive subjectal followed by positive predicatal:

A is more R than B,

and B is R enough to be C;

so, A is R enough to be C (this conclusion becomes the minor premise of the next argument).

More R is required to be C than to be D,

and A is R enough to be C (this premise being the conclusion of the preceding argument);

therefore, A is R enough to be D.

Positive predicatal followed by positive subjectal:

More R is required to be A than to be B,

and C is R enough to be A;

so, C is R enough to be B (this conclusion becomes the minor premise of the next argument).

D is more R than C,

and C is R enough to be B (this premise being the conclusion of the preceding argument);

therefore, D is R enough to be B.

I wish to add here that, frankly, I do not remember if I ever saw a specific case. I may have just been assuming the occurrence of this phenomenon offhand. Rather my point was, I would say, that such conjunctions of related a fortiori arguments are conceivable. The thing to keep in mind is that the two a fortiori arguments need not be contiguous, in the discourse of a person or group, or in a document such as the Talmud. One argument may occur in one time or place, and the other in a completely different time or place. If you see things this way, you understand that there is some probability that the conclusion of one argument might be used as the premise of another. This happens in all knowledge all the time, but people pay little attention to it. It is bound to happen sooner or later, because no conclusion is ever left standing without being re-used in other arguments. Otherwise, why bother with it?

It also now occurs to me that the two arguments forming a sorites need not have the same middle term R. Let R1 be the first middle term, and R2 the second. We could equally well reason with mixed middle terms, as follows:

Positive subjectal followed by positive predicatal:

A is more R1 than B,

and B is R1 enough to be C;

so, A is R1 enough to be C (this implies that A is C).

More R2 is required to be C than to be D (this implies that what is C is R2),

and A is R2 enough to be C (this follows from ‘A is C’ and ‘C is R2’, which together imply that A is R2);

therefore, A is R2 enough to be D.

Positive predicatal followed by positive subjectal:

More R1 is required to be A than to be B,

and C is R1 enough to be A,

so, C is R1 enough to be B (this implies that C is B).

D is more R2 than C (this implies that C is R2),

and C is R2 enough to be B (this follows from ‘C is R2’ and ‘C is B’);

therefore, D is R2 enough to be B.

Needless to say, we can predict other examples of sorites by involving other combinations of moods. For instance, the major premise “A is more R than B” might be converted to “B is less R than A,” and so change the character of the first sorites, and so forth.

2.  More formalities

1.    Species and Genera

Since, as shown earlier, the propositions used in a fortiori argument can be reduced to simpler forms, it follows that we can formally combine a fortiori argument with syllogism in certain ways. Consider first the positive subjectal mood:

P is more R than (or as much R as) Q (is R),
and, Q is R enough to be S;
therefore, all the more (or equally), P is R enough to be S.

Suppose that the given subsidiary term S refers to a species of a certain genus G. In that case, S is a subclass of G, i.e. ‘All S are G’ is true. The minor premise “Q is R enough to be S” of our a fortiori argument implies that “Q is S.” By a first figure syllogism, we can infer that “Q is G.” It follows that “Q is R enough to be G” is also true. If we now use this as our minor premise in lieu of the preceding, we can infer that “P is R enough to be G,” since the major premise has remained unchanged. We can thus say, with reference to the subsidiary term: if a positive subjectal a fortiori argument is true of a species (S), then it is equally true of any genus of it (G).[28]

It follows from this, by reductio ad absurdum, that: if a negative subjectal a fortiori argument is true of a genus (G), then it is equally true of any species of it (S). That is to say: given (with the same major premise) that “If P is R not enough to be G, then Q is R not enough to be G,” and “All S are G,” we can infer that “If P is R not enough to be S, then Q is R not enough to be S.” (For otherwise, if Q were R enough to be S, then P would be R enough to be S and therefore G.)

The same can be done with positive predicatal a fortiori argument, as follows.

More (or as much) R is required to be P than to be Q,
and, G is R enough to be P;
therefore, all the more (or equally), G is R enough to be Q.

Suppose that the given subsidiary term G refers to a genus of a certain species S. In that case, S is a subclass of G, i.e. ‘All S are G’ is true. The minor premise “G is R enough to be P” of our a fortiori argument implies that “G is P” (i.e. “all G are P”). By a first figure syllogism, we can infer that “S is P.” It follows that “S is R enough to be P” is also true. If we now use this as our minor premise in lieu of the preceding, we can infer that “S is R enough to be Q,” since the major premise has remained unchanged. We can thus say, with reference to the subsidiary term: if a positive predicatal a fortiori argument is true of a genus (G), then it is equally true of any species of it (S).[29]

It follows from this, by reductio ad absurdum, that: if a negative predicatal a fortiori argument is true of a species (S), then it is equally true of any genus of it (G). That is to say: given (with the same major premise) that “If S is R not enough to be Q, then S is R not enough to be P,” and “All S are G,” we can infer that “If G is R not enough to be Q, then G is R not enough to be P.” (For otherwise, if G were R enough to be P, then G and therefore S would be R enough to be Q.)

So much for copulative a fortiori argument. The same can be done with implicational a fortiori argument, except that we would here use hypothetical instead of categorical syllogism. That is, instead of “All S are G” (where S and G are terms), we would say “If S then, G” (where S and G are theses). Although such combinations of a fortiori argument and syllogism are simple and obvious, they are nevertheless sometimes useful and so well to keep in mind.

A case in point is the a fortiori argument suggested in Num. 12:14, viz. “If her father had but spit in her face, should she not hide in shame seven days? Let her be shut up without the camp seven days, and after that she shall be brought in again.” If we look closely, we see that the subsidiary term in the minor premise “hide in shame seven days” is not identical (as it should be) to the one in the conclusion “shut up without the camp seven days;” in which case the inference is invalid. We can, however, draw a valid conclusion by using the common property (i.e. genus) of these two terms, i.e. what they both imply, viz. “isolation for seven days.” To justify such syllogistic interference, we need the above formal treatment.

Now, it should be obvious that having in our example concluded with “isolation for seven days,” we cannot take that generic conclusion as justification for a more specific conclusion such as “shut up without the camp seven days” which does not exactly correspond to the given species “hide in shame seven days.” This would be fallacious reasoning. In other words, the new “shut up without the camp” specification cannot be claimed to be a conclusion of a fortiori reasoning, but must be regarded as a specification applied to the generic conclusion “isolation for seven days” after the a fortiori argument proper, through some other act of deductive or inductive reasoning or other justification. In this example, the justification would be that God[30] or the Torah has so decreed.

This can be stated in formal terms as follows. Consider a positive subjectal argument, to begin with. As we have just seen, if the minor premise affirms the minor term Q to have species S1 as predicate, then the conclusion must likewise affirm the major term P to have species S1 as predicate, or if we wish it may (via syllogism) affirm the major term P to have some genus G of S1 as predicate. We cannot, however, conclude with affirmation of predication for some separate species of G, say S2[31] – at least not through our a fortiori argument, though we might arrive at S2 by some other process of reasoning thereafter.

Regarding negative subjectal argument, this rule means (notice the changed positions of P and Q here) that if the minor premise denies the major term P to have genus G1 as predicate, then the conclusion must likewise deny the minor term Q to have genus G1 as predicate, or if we wish it may (via syllogism) deny the minor term Q to have some species S of G1 as predicate. We cannot, however, conclude with denial of predication for some separate genus of S, say G2[32] – at least not through our a fortiori argument, though we might arrive at G2 by some other process of reasoning thereafter.

In predicatal argument, the equivalent fallacy would consist in passing from a given genus to a not-given other genus. Thus, in the positive mood of predicatal argument, we might start from given genus G1, saying “G1 is R enough to be P;” from that, given that S is a known species of G1, we can legitimately infer (as above shown, through syllogism) that “S is R enough to be Q;” but then we swerve off and illicitly claim that “G2 is R enough to be Q,” where G2 is another genus of S (though not a species of G1).

Similarly, in the negative mood of predicatal argument, it would be fallacious to pass from a given species to a not-given other species. That is, starting with “S1 is R not enough to be Q;” we can, given that G is a known genus of S1, legitimately infer (as above shown) that “G is R not enough to be P;” but we cannot likewise infer that “S2 is R not enough to be P,” where S2 is another species of G (though not a genus of S1).

It should be added that the fallacious reasoning above described is not uncommon. To sum up, we have above, knowing that all S are G, established four rules of transmission of a fortiori argument from species to genera or vice versa:

  1. If a positive subjectal argument is true of a certain subsidiary predicate (S), then it is also true of any genus of it (G).
  2. If a negative subjectal argument is true of a certain subsidiary predicate (G), then it is also true of any species of it (S).
  3. If a positive predicatal argument is true of a certain subsidiary subject (G), then it is also true of any species of it (S).
  4. If a negative predicatal argument is true of a certain subsidiary subject (S), then it is also true of any genus of it (G).

And likewise, mutatis mutandis, with respect to implicational arguments.

We should also be aware that the middle term R may subtly differ in meaning in relation to the major and minor terms P and Q. In some cases, the meaning of R is identical; but in some cases, the R in relation to P (Rp) and the R in relation to Q (Rq) have the abstraction R in common, but they are each specifically relative to the term they concern. In effect, Rp and Rq are two species of the genus R. The argument, if properly constructed, remains nonetheless valid, because all that matters for its validity is that Rp ≥ Rq and this quantitative condition is here satisfied.

To give an example: “John loves his dog more than Jill does,” could mean that Jill loves John’s dog less than he does (in which case, the middle term is “loves John’s dog”), or it could mean that Jill loves her own dog less than John loves his own dog (in which case, the middle term is the more abstract “loves his/her own dog,” or even just “loves some dog”). In the latter case, the minor premise and conclusion would be: “Given that Jill loves her own dog enough to be classed as an animal lover, it follows that John loves his own dog enough to be classed as an animal lover.” This example is positive subjectal. We may similarly construct examples for negative subjectal, predicatal and implicational arguments with this feature.

2.    Proportionality

This section and the next are very important and should be read carefully.

Closely related to the issue of species and genera, is that of ‘proportionality’. Often, rather than species and genera, what is involved are different degrees or measures of the same term. Thus, in positive subjectal or negative predicatal argument, the distinct species S1 and S2 (neither of which is included in the other, though they have a genus in common) would appear as different degrees or measures of the same genus G; similarly, in positive predicatal or negative subjectal argument, species S would appear as a single degree or measure of two distinct genera, G1 and G2 (neither of which includes the other, though they have a species in common).

For example, in the argument given in Num. 12:14 (see 2.4 below), though the minor premise specifies the quantity “seven days,” we might be tempted (by considerations of proportionality, say) to conclude with another quantity like “fourteen days;” but such reasoning (without additional premises), as we have just shown, would be formally invalid. In purely a fortiori argument, the conclusion can never produce a different quality or quantity than the one given in the minor premise; this is a hard and fast rule based on strictly logical considerations.

Argument a crescendo. A fortiori argument with a ‘proportional’ conclusion is, in itself, by itself, fallacious. The copulative variant has, at least on the surface, the following four forms. The positive subjectal mood resembles that of regular a fortiori, except that, whereas the minor premise predicates a subsidiary term (S) of the minor term (Q), the conclusion predicates a greater subsidiary term (more than S) of the major term (P). It goes like this:

P is more R than Q (is R),
and Q is R enough to be S;
therefore, P is R enough to be more than S.

To avoid confusion between the subsidiary term S in a general sense and its values in the minor premise and conclusion, think of ‘S’ in the former as ‘Sq’ and ‘more than S’ in the latter as ‘more than Sq’ or as ‘Sp’. The corresponding negative subjectal mood has the same major premise, but as usual has the denial of the above conclusion as minor premise and the denial of the above minor premise as conclusion. That is, it argues: Since P is R not enough to be more than S, it follows that Q is R not enough to be S.

The positive predicatal mood resembles that of regular a fortiori, except that, whereas the minor premise predicates the major term (P) of a subsidiary term (S), the conclusion predicates the minor term of a lesser subsidiary term (less than S). It goes like this:

More R is required to be P than to be Q,
and S is R enough to be P;
therefore, less than S is R enough to be Q.

To avoid confusion between the subsidiary term S in a general sense and its values in the minor premise and conclusion, think of ‘S’ in the former as ‘Sp’ and ‘more than S’ in the latter as ‘less than Sp’ or as ‘Sq’. The corresponding negative predicatal mood has the same major premise, but as usual has the denial of the above conclusion as minor premise and the denial of the above minor premise as conclusion. That is, it argues: Since less than S is R not enough to be Q, it follows that S is R not enough to be P.

We could alternatively have, for the positive subjectal mood, ‘less than S’ in the minor premise and ‘S’ in the conclusion; and for the negative subjectal mood, ‘S’ in the minor premise and ‘less than S’ in the conclusion. And likewise, for the positive predicatal mood, ‘more than S’ in the minor premise and ‘S’ in the conclusion; and for the negative predicatal mood, ‘S’ in the minor premise and ‘more than S’ in the conclusion. What matters is that the relative magnitudes be as stated.

In practice, the subsidiary term in the minor premise would always be labeled ‘S’ and the subsidiary term in the conclusion would accordingly be labeled ‘more than S’ or ‘less than S’, as the case may be – for the simple reason that we normally know the minor premise before we get to know the conclusion. I have chosen the terminology above to stress that the negative moods are reducible ad absurdum to the positive ones.

We can likewise construct implicational forms. As these various forms show, ‘proportional’ a fortiori argument is based on the notion that if P is more R than Q is, or more R is required to be P than to be Q, then necessarily a larger amount of S will correspond to P than to Q. But in fact there is no such necessity; it may occasionally be true, but there is no logical reason why it should be. Such arguments, unlike regular a fortiori, simply cannot be validated as they stand.

Looking at the positive subjectal form of ‘proportional’ a fortiori argument, which is prototypical, it is evident that we can equally well refer to it as argument a crescendo (this being a name I invented in the course of my research, having found it useful). This name can be extended to all the other forms[33]. The advantage of such renaming is that it verbally completely distinguishes such argument from strict a fortiori.

Argument pro rata. Argument a crescendo (i.e. ‘proportional’ a fortiori) should not be confused with argument by proportion, which we can refer to as argument pro rata (this Latin name being already well established in the English language), this being understood to mean “at the same rate.” Such argument concerns concomitant variations between two variables, and may be formulated as follows:

Y varies in proportion to X. Therefore:
given that: if X = x, then Y = y,
it follows that: if X = more (or less) than x, then Y = more (or less) than y.

An example of it is Aristotle’s statement: “Every good quality of the soul, the higher it is in degree, so much more useful it is” (Politics 7:1), which intends the argument: given that a certain quality of the soul is good, it is useful; if it is improved, it is still more useful. In practice, pro rata argument is often expressed in the form: “the more X, the more Y; and (by implication) the less X, the less Y.” Note that two variants (which mutually imply each other) are possible: one with “more” and one with “less” – that is, the argument can go either way, increasing or decreasing the quantities involved.

The statement “Y varies in proportion to X” is not an argument but a mere proposition, reflecting some generalized empirical observations or a more theoretical finding. The above argument includes this proposition as its major premise, but requires an additional minor premise (“if X = x, then Y = y”) to draw the conclusion (“if X = more/less than x, then Y = more/less than y”). The conclusion mirrors the minor premise in form, but its content is intentionally different. The quantities involved do not stay the same, but increase or decrease (as the case may be).

Notice that a pro rata argument has no middle term, unlike an a fortiori one. A pro rata argument is thus more akin to apodosis than to syllogism. Its major premise sets a broad principle, of which the minor premise and conclusion are two applications. The argument involved is thus simply inference of one quantity from another within the stated principle. If we found that contrary to expectations X and Y do not vary concomitantly as above implied, we would simply deny the major premise. In other words, this argument is essentially positive in form. A negative mood of it (with the same major premise and denials of the previous conclusion and minor premise) would not make much sense, since its minor premise and conclusion would be in conflict with its major premise.

The above formulas are at least true in cases of direct proportionality; in cases of inverse proportionality, the language would be: “the more X, the less Y; and (by implication) the less X, the more Y;” and the argument would have the following form:

Y varies in inverse proportion to X. Therefore:
given that: if X = x, then Y = y,
it follows that: if X = more (or less) than x, then Y = less (or more) than y.

And of course, in more scientific contexts, we may have access to a more or less complex mathematical formula – say Y = f(X), where f refers to some function – that allows precise calculation of the proportion involved. In other words, the validity of pro rata argument is not always obvious and straightforward, but depends on our having a clear and reliable knowledge of the concomitant variation of the values of the terms X and Y. Given such knowledge, we can logically justify drawing the said conclusions from the said premises. Lacking it, we are in a quandary.

As its name implies, pro rata argument signifies that there is (if only approximately) some constant rate in the relative fluctuations in value of the variables concerned. The variables X and Y may be said to be proportional if X/Y = a constant, or inversely proportional if XY = a constant. In the exact sciences, of course, such a constant is a precisely measurable quantity; but in everyday pro rata discourse, the underlying ‘constant’ is usually a vague quantity, perhaps a rough range of possible values.

Proportionality or inverse proportionality as just defined, which can be represented by a straight line graph, and even when the graphical representation is more curved (e.g. exponential), may be characterized as simple. It becomes complex, when there are ups and downs in the relation of the two variables, i.e. when an increase in X may sometimes imply an increase in Y and sometimes a decrease in Y, it is obviously not appropriate to formulate the matter in the way of a standard pro rata argument. In such cases, we would just say: “the values of X and Y can be correlated in accord with such and such a formula,” and then use the formula to calculate inferred quantities.

Proportionality may be continuous or not. Sometimes, there is proportionality of sorts, but it comes in slices: e.g. from X = 0 to 1, Y = k; from X = 1 to 2, Y = k +1; etc. That is, to each range of values for X, there corresponds a certain value of Y, and the two quantities go increasing (or decreasing, as the case may be). Such proportionality is compatible with pro rata argument. For this reason, it is wise to put the word ‘proportionality’ in inverted commas, so as to remember that it does not always imply continuity.

Note too that proportionality may be natural or conventional. An example of the latter would be a price list: bus fares for children under 16, $1; for adults 16+, $2. However, beware in such case of frequent exceptions or reversals: e.g. unemployed and pensioners, $1. In such cases, any pro rata argument must be stated conditionally: the bus fares are ‘proportional’ to age, provided the adults are not unemployed or pensioners.

It should also be reminded that proportionality (or its inverse), simple or complex, may or may not be indicative of a causal relation (in the various senses of that term). Two variables may vary concomitantly by virtue of being effects of common causes, in which case we refer to parallelism between them, or the one may cause or be caused by the other. Also, of course, such parallelism or causality may be unconditional or conditional. In such cases as it is unconditional, no more need be said. But in such cases as it is conditional, the condition(s) should ideally be clearly stated, although often they are not.

Pro rata argument may occur in discourse independently of a fortiori argument, or in conjunction with such argument. In any case, it should not be confused with a fortiori argument: they are clearly different forms of reasoning. Pro rata involves only two terms, or more precisely two values (or more) of two variables; whereas a fortiori involves four distinct terms, which play very different roles in the argument. Pro rata and a fortiori are both analogical arguments of sorts, but the former is much simpler than the latter.

3.    A crescendo argument

Having thus clarified the differences between (regular) a fortiori argument, a crescendo argument (i.e. ‘proportional’ a fortiori) and pro rata argument, it is obvious that these three types of argument should not be confused, although many people tend to do so. Although these arguments are far from the same in form and in validity, such people wrongly identify pro rata argument and a crescendo argument with a fortiori argument. The reason for such confusion may be that argument a crescendo appears to be a combination of argument a fortiori and pro rata argument. This could be expressed as a formula:

A crescendo = a fortiori cum pro rata.

Though the latter two forms of argument may occur coincidentally (i.e. they may happen to be both true in certain material cases), it does not follow that they formally necessitate or imply each other. But, we may ask, do they together imply a crescendo argument? That is to say, is argument a crescendo valid in cases where both argument a fortiori and argument pro rata happen to be true? To answer this question, consider a regular positive subjectal a fortiori argument:

P is more R than Q (is R),
and Q is R enough to be S;
therefore, P is R enough to be S.

It informs us that the quantity of R corresponding to P (Rp) is greater than the quantity of R corresponding to Q (Rq), and then argues: since the latter quantity (Rq) is big enough to imply Q to be S, then the former quantity (Rp) must be big enough to imply P to be S. It does not tell us that the subsidiary term S is a variable quantity; S is here clearly intended to have one and the same value in the minor premise and conclusion. And if S happens to be a variable quantity, we cannot automatically suppose that the variation of S is tied to that of R, so that S for P (Sp) is necessarily greater than S for Q (Sq), in concomitance with the variation of Rp with Rq.

Under what conditions, then, can we obtain the conceivable a crescendo conclusion “P is R enough to be more than S” from the above a fortiori premises? That is, what additional information do we need to transform the above valid a fortiori argument into a valid a crescendo argument? Or to put it another way again: having already come to the a fortiori conclusion “P is R enough to be S (i.e. Sq),” how can we proceed one step further and obtain the a crescendo conclusion “P is R enough to be more than S (i.e. Sp)”?

The answer is, of course, that we must obtain the additional information required to construct the following pro rata argument:

If, moreover, (for things that are both R and S,) we find that:
S varies in proportion to R, then:
knowing from the above minor premise that: if R = Rq, then S = Sq,
it follows in the conclusion that: if R = more than Rq = Rp, then S = more than Sq = Sp.

Note well the stipulation “for things that are both R and S.” I have put this precondition in brackets, because it is in fact redundant, since as we saw earlier the minor premise of the a fortiori argument implies anyway that not all things that are R are S, but only those things that have a certain threshold value of R or more of it are S. We should not think of S varying with R as a general proposition applicable to all R (implying that all R are S[34]), but remain aware that this concomitant variation occurs specifically in the range of R where the threshold for S has indeed been attained or surpassed (i.e. where the “R enough to be S” condition is indeed satisfied).

If we know (by induction or deduction from other information) that the major premise of our above pro rata component is true (i.e. that S varies in proportion to R), we can infer its minor premise (viz. if R = Rq, then S = Sq) from the minor premise of the a fortiori component and thence draw the conclusion that “If the value of R for P is Rp (> Rq), then the value of S for P is Sp (> Sq).” This is assuming, as earlier specified, that the proportionality proposed in the major premise is direct and simple.

The desired a crescendo conclusion, viz. “P is R (Rp) enough to be more than Sq (Sp)” can then be confidently drawn. That is, the intermediate conclusions, i.e. the a fortiori conclusion (P, being more than Rq, is R enough to be Sq) and the pro rata conclusion (If R = Rp, then S = Sp) together imply the final, a crescendo conclusion (P, which is Rp, is R enough to be Sp). Note well that Sp, here, means nothing more precise than ‘more than Sq’. Though given a distinct symbol, it is not an exact number, unless we are able to calculate it precisely through some sort of mathematical formula.

Note that we have here taken variations in the value of S to be concurrent with variations in the value of the middle term R. Very often, though, a crescendo thinking is based on the assumption that the values of S are proportional to those of the major and minor terms, P and Q. These two views are not necessarily in conflict, though the former (which we have adopted) is the more essential and more generally applicable. P and Q may be single values, or they may be variable over time – so long as their relative magnitudes or degrees are as specified in the major premise, i.e. that P is always more (more R, to be exact, though R is often left tacit) than Q. Thus, S may well be vaguely thought of as proportional to P or Q, although more precisely perceived the proportionality of S is in fact to R, the common factor of P and Q in relation to which P is greater than Q.

We have thus shown that, under certain circumstances, the formula “a fortiori + pro rata = a crescendo” is indeed true. That is to say, although the putative a crescendo conclusion is not per se valid, it can in some cases be valid if it so happen that the a fortiori conclusion from the same premises can be taken a step further by means of an appropriate argument pro rata. This has just been demonstrated for positive subjectal a crescendo.

A similar two-step argument can, of course, be formulated for positive predicatal a crescendo. In this case, we use the following combination of a fortiori and pro rata arguments:

More R is required to be P than to be Q,
and S is R enough to be P;
therefore, S is R enough to be Q.
If, moreover, (for things that are both R and P or Q,) we find that:
R varies in proportion to S, then:
knowing from the above minor premise that: if S = Sp, then R = Rp,
it follows in the conclusion that: if S = less than Sp = Sq, then R = less than Rp = Rq.

Note well the stipulation “for things that are both R and P or Q.” I have put this precondition in brackets, because it is in fact redundant, since as we saw earlier the major premise of the a fortiori argument implies anyway that not all things that are R are P and not all things that are R are Q, but only those things that have certain threshold values of R or more of it are P or Q. We should not think of R varying with S as a general proposition applicable to all S (implying that all S are R[35]), but remain aware that this concomitant variation occurs (at least) specifically in the range of R where the thresholds for P and Q have indeed been attained or surpassed (i.e. where the “R enough to be P” and “R enough to be Q” conditions are indeed satisfied).

If we know (by induction or deduction from other information) that the major premise of this pro rata component is true (i.e. that R varies in proportion to S), we can infer its minor premise (viz. if S = Sp, then R = Rp) from the minor premise of the a fortiori component and thence draw the conclusion that “If the value of S for Q is Sq (< Sp), then the value of R for Q is Rq (< Rp).” This is assuming, as earlier specified, that the proportionality proposed in the major premise is direct and simple.

The desired a crescendo conclusion, viz. “Less than Sp (Sq) is R (Rq) enough to be Q” can thence be confidently drawn. That is, the intermediate conclusions, i.e. the a fortiori conclusion (Sp, being more than Rq, is R enough to be Q) and the pro rata conclusion (If S=Sq, then R=Rq) together imply the final, a crescendo conclusion (Sq, which is Rq, is R enough to be Q). Note well that Sq, here, means nothing more precise than ‘less than Sp’. Though given a distinct symbol, it is not an exact number, unless we are able to calculate it precisely through some sort of mathematical formula.

Thus, if we can provide an appropriate pro rata argument, we can credibly transform an a fortiori conclusion into an a crescendo conclusion, viz. in the case of positive subjectal argument, “P is R enough to be more than S,” and in the case of positive predicatal argument, “Less than S is R enough to be Q.”

As regards the corresponding subjectal or predicatal negative a crescendo arguments, they would consist of a negative a fortiori combined with the same positive argument pro rata. Referring to the above described positive arguments, keeping the major premise and additional premise about proportionality constant, if we deny the conclusion, we must deny the minor premise, as follows:

P is more R than Q (is R),
and P is R not enough to be more than S,
and S varies in proportion to R;
therefore, Q is R not enough to be S.
More R is required to be P than to be Q,
and less than S is R not enough to be Q,
and R varies in proportion to S;
therefore, S is R not enough to be P.

This is how we would derive the negative moods from the positive ones. But granting that the subsidiary term in the minor premise is thought of first, before the subsidiary term in the conclusion, it is more accurate to present these two arguments independently in the following revised forms:

P is more R than Q (is R),
and P is R not enough to be S,
and S varies in proportion to R;
therefore, Q is R not enough to be less than S.
More R is required to be P than to be Q,
and S is R not enough to be Q,
and R varies in proportion to S;
therefore, more than S is R not enough to be P.

I hope the reader is not confused by this revision. Although the negative forms are validated by reduction ad absurdum to the positive ones, viewed as forms in their own right they would be rather worded as just shown. The comparisons between the subsidiary term in the minor premise and that in the conclusion remain the same – i.e. ‘more than S > S’ and ‘S > less than S’ mean the same, and ‘less than S < S’ and ‘S < more than S’ mean the same. Notice that the revised negative subjectal mood goes from S to less than S, like the earlier positive predicatal mood, while the revised negative predicatal mood goes from S to more than S, like the earlier positive subjectal mood. Similar developments to all those above for copulative argument are possible in relation to implicational argument.

Note well that the relation between R and S changes direction, according as our reasoning is subjectal (or antecedental) or predicatal (or consequental) in form. That is to say, whereas in subjectal argument S varies with R, in predicatal argument R varies with S. That is because, in positive subjectal argument, validation relies on the fact that Rq implies Rs; whereas, in positive predicatal argument, it relies on the fact that Rs implies Rp. In both cases, the R value of the subject precedes the R value of the predicate; but in the former, the subject is Q and the predicate is S, while in the latter, the subject is S while the predicate is P. It is because the subject has a certain value of R that it can be attributed the predicate, for which that value of R happens to be a precondition.

Thus, when reasoning a crescendo in the positive subjectal form, we reason from Rq to Sq, and from Rp to Sp; i.e. when R = Rq, then S = Sq, etc. Whereas, when reasoning a crescendo in the positive predicatal form, we reason from Sp to Rp, and from Sq to Rq, i.e. when S = Sp, then R = Rp, etc. Whence, we must reverse the order of dependence between the middle and subsidiary terms, according as we reason this way or that way, to make possible validation of the argument. Of course, if in a given case the required pro rata argument as above specified is not applicable, then the a fortiori argument (whether subjectal or predicatal, as the case may be) cannot be turned into a valid a crescendo one. All this is equally true of negative arguments, of course, as already made clear.

Validation. It is important to stress that the validity of a crescendo argument depends, as we have above clearly shown, on both its a fortiori constituent and its pro rata constituent. A crescendo is neither equivalent to the former nor equivalent to the latter, but emerges from the two together. The a fortiori element is only able to produce the conclusion “P is R enough to be S” (in the positive subjectal case) or “S is R enough to be Q” (in the positive predicatal case). The pro rata element is only able to produce the conclusion that the S is, parallel to R, greater for P than for Q, or lesser for Q than for P. The a crescendo conclusion is a merger of these two partial conclusions, namely (respectively) “P is R enough to be more than S” or “Less than S is R enough to be Q.” Thus neither element is logically redundant; both are necessary to obtain the final conclusion.

Looking at the above descriptions of a crescendo argument, we see that, while the pro rata conclusion is partly based on information provided by the minor premise of the a fortiori argument, it could conceivably be built up without the latter, since it does not use all the information in it. However, although mere pro rata argument[36] no doubt exists, it remains true that the pro rata constituent could not by itself produce the stated final a crescendo conclusion, since the latter proposition is of suffective form; so the a fortiori constituent is also indispensable. Clearly, the final conclusion is made up of elements derived from both types of arguments; i.e. its semantic charge comes from all three premises. To repeat, then, the a crescendo argument cannot be identified with either constituent alone, but requires both to proceed successfully.

What we have done above is to formally demonstrate that, although drawing a ‘proportional’ conclusion from the premises of a valid a fortiori argument is not unconditionally valid, it is also not unconditionally invalid. Such a conclusion is in principle invalid, but it may exceptionally, under specifiable appropriate conditions, be valid. Formally, all depends on whether a pro rata argument can be truthfully proposed in addition to the purely a fortiori argument. In other words, to draw a valid a crescendo conclusion, the premises of a valid a fortiori argument do not suffice; but if they are combined with the fitting premises of a valid pro rata argument, as above detailed, such a conclusion can indeed be formally justified.

Of course, as with all deduction, even if in a given case the inferential process we propose is ideally of valid form, we must also make sure that the premises it involves are indeed true, i.e. that the content of the argument is credibly grounded in fact. Very often, in a crescendo argument, the process is convincing, but the major premise of the implicit pro rata argument is of doubtful truth; this is obviously something to be careful about. Merely declaring a certain proportionality to be true does not make it true – we have to justify all our premises, as well as their logical power to together produce the putative conclusion.

Another way to stress this is to remind that the concluding predicate Sp of positive subjectal a crescendo argument means nothing more than ‘more than Sq’, and the concluding subject Sq of positive predicatal a crescendo argument means nothing more than ‘less than Sp’. These concluding subsidiary terms are not exact numbers, though in theory they might be exact if we happen to have a precise mathematical formula for their calculation – viz. S = f(R) in the case of subjectal argument, or R = f(S) in the case of predicatal argument. In most cases in practice, however, we do not have such a formula, and the terms we use are correspondingly vague and tentative. It is important to remember this.

Sometimes, unfortunately, rhetoric comes into play here, and albeit the lack of mathematical proof, the conclusion is made to seem more precise than deductive logic allows. We could at best refer to such conclusions as intuitively reasonable, or as inductive hypotheses, partly but not wholly sustained by the data in the premises; but we must realize and acknowledge that they are not deductive certainties. Otherwise, we would be engaged in misleading sophistry. Thus, it is important to keep in mind that, while we have shown that a crescendo argument is in principle, i.e. under ideal conditions, valid – it does follow that every a crescendo argument put forward in practice, i.e. in everyday discourse, is valid. It is potentially valid, but not necessarily actually valid. We have to carefully scrutinize each case.

Thus, to summarize: the expression ‘proportional’ a fortiori argument may be intended in a pejorative sense, as referring to argument that unjustifiably draws a proportional conclusion from only two a fortiori premises; or it may be intended to refer to valid a crescendo argument, consisting of three premises, viz. two a fortiori and one pro rata, yielding a justifiably proportional conclusion. A fortiori argument per se, as such, in itself, by itself, is not proportional; such argument may be verbally distinguished as purely a fortiori. However, when combined with pro rata argument, a proportional conclusion is justified, and we had best in such case speak distinctively of a crescendo argument, or as it is often called in a non-pejorative sense ‘proportional’ a fortiori argument.

Thus, a crescendo argument may be viewed as a special case of a fortiori argument; and it is fair to say that the field of a fortiori logic also deals with a crescendo argument. But strictly spoken, ‘a fortiori’ should be reserved for ‘non-proportional’ argument, and ‘a crescendo’ preferred for ‘proportional’ argument. The former is pure because the major and minor premise suffice for the conclusion; whereas the latter is a compound argument, comprising a pure a fortiori argument combined with a mere pro rata argument. Although people often appear to draw an a crescendo conclusion from a fortiori premises, such inference in fact relies on an unspoken additional pro rata premise, and so is not purely a fortiori.

In any case, it is useful to remember the formula: a crescendo equals a fortiori plus pro rata. This means that if you come across an a crescendo argument that looks valid, you can be sure that underlying it are a valid a fortiori argument and a valid pro rata argument. Inversely, if one and/or the other of the latter arguments cannot be upheld, then the former cannot either.

Alternative presentation. We have thus far considered a crescendo argument as a special case of a fortiori argument where purely a fortiori argument is combined with pro rata argument. Another way we might look upon the relationship between these arguments is to say that all a fortiori argument is a crescendo argument, while purely a fortiori argument is a special case where the pro rata argument involves a fixed quantity instead of a variable, i.e. where the ‘proportionality’ involved is a constant. That is to say, we could regard the general forms of a fortiori argument to be the following (for examples, with regard to positive subjectal and positive predicatal moods):

P is more R than Q (is R),
and Q is R enough to be more than S,
and S is constant (pure) or varies in proportion to R (a crescendo);
therefore, P is R enough to be S, or more than S (as the case may be).
More R is required to be P than to be Q,
and S is R enough to be P,
and S is constant (pure) or R varies in proportion to S (a crescendo);
therefore, S, or less than S (as the case may be), is R enough to be Q.

Note how the additional premise about proportionality is now so broadly stated that it includes both the cases of purely a fortiori argument (where S is constant) and those of a crescendo argument (where S varies in magnitude or degree). The advantage of this approach is that it goes to show that purely a fortiori argument and a crescendo argument are essentially two special cases of the same general form, and so that they can legitimately be referred collectively as forms of ‘a fortiori argument’ in the largest sense.

However, it must be emphasized that this joint formulation is just an ex post facto way of looking at things, a perspective. It remains true, as initially stated, that a fortiori argument is essentially pure, since it can be validated without reference to an additional premise about proportionality. In this more accurate perspective, a crescendo argument is an amplification of a fortiori argument, taking its ‘equal’ conclusion and enriching it by turning it into a ‘proportional’ one (by means of a pro rata argument). When we do not have any additional premise about proportionality, we may logically assume that the subsidiary item remains constant, since that is the minimum assumption of any a fortiori reasoning process. In effect, ‘non-proportionality’ is the default character of a fortiori argument.

4.    Hermeneutics

We also need to deal with the important issue of hermeneutics. When someone formulates an argument a crescendo, that person presumably believes that its constituent a fortiori argument and pro rata argument both have true premises and are correctly combined to yield a valid and therefore true conclusion, even if all these factors are not explicitly laid out and confirmed. However, how can other persons know what he (or she) had in mind if some of these relevant factors are left tacit? This is the question of hermeneutics – how are we to interpret and judge an incompletely detailed a crescendo argument presented by someone else?

When judging a given concrete argument: (a) if it is formulated in such a way that only an a fortiori conclusion is claimed, we need only test whether the conclusion follows a fortiori from the premises (assuming them true); but (b) if it is formulated in such a way that an a crescendo (i.e. proportional) conclusion is claimed, we must needs test both whether the premises yield an a fortiori conclusion and whether an additional argument pro rata can be presented that makes possible the transformation of the latter conclusion into an a crescendo one, and whether the proportionality is vague or precisely calculable (and ultimately, of course, whether all these premises are true).

Without such additional information on proportionality, if the a fortiori premises are presented (explicitly or by implication) as by themselves yielding an a crescendo conclusion, the argument must obviously be declared invalid. Moreover, even if the need for an additional premise regarding proportionality is admitted, without a precise formula for the proportionality, an exact conclusion cannot legitimately be claimed. We have to assess from the author’s words how well he understands the conditions for valid a fortiori or a crescendo argument, and whether the author considers that he has the required information at hand. If it seems likely that the author is well aware that a fortiori argument cannot logically by itself yield an a crescendo conclusion, and that a precise conclusion requires more information than a vague one, and is tacitly intending the required underlying pro rata argument or mathematical formula, his argument could be considered valid.

Of course, in interpreting a text or speech, we cannot estimate with certainty what its author’s tacit intentions are or are not. We may be able to guess at the author’s general logical knowhow from the context, and give him the benefit of the doubt in the case at hand. Or we may prefer to be strict, and demand explicit evidence that the author intends an argument pro rata in the case at hand, even if he knows the rule (since, after all, people do make mistakes). Thus, validation or invalidation often depends on the general credibility of the author of the text or speech, and on the severity of the interpreter and judge.

To put the problem in more concrete terms: when we read an ancient or modern text, by Aristotle, in the Tanakh, by a Talmudic sage, in the Christian Bible, or the Muslim Koran, or wherever, wherein the author seems to draw an a crescendo (i.e. ‘proportional’) conclusion from a fortiori premises, how should we react? We could possibly say: since the author of this argument has not justified his conclusion by explicitly proposing an appropriate accompanying pro rata argument, we must declare his reasoning fallacious. But this seems a bit rigid and lacking in subtlety, for logicians well know that discourse in practice is rarely if ever fully explicit. Our judgment in each case must clearly hinge on the wider context of the particular statement.

If we know from pronouncements elsewhere of that particular author that he has demonstrated clear understanding of the difference between a fortiori argument and a crescendo argument – i.e. that the former per se cannot yield a proportional conclusion unless it is backed up by an appropriate pro rata argument – then we can reasonably assume that in this particular case, though the author has not explicitly formulated the needed argument pro rata, he left out the missing pieces merely for brevity’s sake. We can in such case give the author the benefit of the doubt and accept his a crescendo conclusion. Of course, even if he has in other contexts demonstrated his theoretical knowledge, or at least his intuitive rationality, it is still possible that in this particular case he unthinkingly made an error of form and/or content; so we can never be absolutely sure. Nevertheless, even if the hypothesis that he knowingly drew an a crescendo conclusion from a fortiori premises cannot definitely be proved, it has inductive support in his overall logical behavior patterns.

A contrario, if we find that there is no reliable evidence that this author has mentally grasped the difference between a fortiori argument and a crescendo argument, we should certainly consider all his a crescendo arguments as fallacious reasoning. This is true even if we find him sometimes drawing a valid a fortiori conclusion and sometimes drawing a doubtful a crescendo one, for he may be drawing different conclusions from similar premises as convenient to his discursive purposes, or without rhyme or reason, and not because of any awareness that there are precise rules to follow. We may also reasonably reject an a crescendo conclusion of his, if we find that the author has elsewhere in his works denied the truth of the proportionality (i.e. the major premise of the pro rata argument) which would be needed to justify this particular a crescendo conclusion.

Thus, judging a concrete a crescendo argument which is not entirely explicit as valid or invalid is not an easy matter. Of course, if the underlying purely a fortiori argument is formally invalid and/or one or both of its premises is/are untrue, or likewise if the required argument pro rata is obviously inappropriate in form and/or content, we can reject that particular a crescendo argument. But obviously such rejection does not always prove the author to be ignorant of the conditions under which an a fortiori argument may yield an a crescendo conclusion. The author may well have in all sincerity believed the implied proportionality to be true, even if we disagree with him and can prove him wrong. In such cases, it is not the inferential process we are attacking, but some premise(s).

In any case, what we must avoid doing is getting entangled in superficial verbal considerations. Usually, people who engage in a fortiori or a crescendo reasoning do so without explicitly labeling their argument as this or that in form. Sometimes, they call their argument ‘a fortiori’, or they use the words ‘a fortiori’ or some similar expression (‘all the more’, ‘how much more’, etc.) within the argument to signal its logical intent. But in the latter case, they make no verbal distinction between a fortiori and a crescendo: firstly, because the latter expression is new (my own invention) and they have no distinctive label for it; and secondly, because the issue of proportionality is vague and uncertain in most people’s mind, if at all present.

What this implies is that we cannot reject an argument as invalid just because it has a fortiori premises and an a crescendo conclusion. Such an argument may indeed be fallacious, or it may merely be an incomplete statement of a valid argument (furthermore, in the latter case, the content may be true or false, of course). We cannot simply refer to the fact that it has not been labeled at all or that it has been labeled incorrectly. We must, as above explained, look into the matter more deeply and try to determine the actual intentions of the argument’s author, even if they are tacit, and judge the matter in all fairness.

A classic illustration of a crescendo argument, and of the hermeneutic difficulties that surround such argument, is the Talmudic reading of Numbers 12:14 in Baba Qama 25a. Without here going into all the details of this example, which are dealt with in the appropriate chapter further on (7.4), I will here just describe the reasoning involved. The Torah passage reads: “If her father had but spit in her face, should she not hide in shame seven days? Let her be shut up without the camp seven days, and after that she shall be brought in again.” This may be construed as a valid positive subjectal a fortiori argument as follows:

Causing Divine disapproval (P) is a greater offense (R) than causing paternal disapproval (Q).
Causing paternal disapproval (Q) is offensive (R) enough to merit isolation for seven days (S).
Therefore, causing Divine disapproval (P) is offensive (R) enough to merit isolation for seven days (S).

Note that the purely a fortiori conclusion is seven days, since this is the number of days given in the minor premise. The Gemara in BQ 25a (or more precisely, a baraita it quotes[37]), on the other hand, advocates an a crescendo conclusion, namely: “causing Divine disapproval (P) is offensive (R) enough to merit isolation for fourteen days (more than S).” This suggests that the author of this ‘proportional’ conclusion has in mind, consciously or not, the following pro rata argument:

Granting the general principle that the punishment must vary in proportion to the offense, then:
knowing from the above minor premise that: if the offense is paternal disapproval, then the punishment is seven days isolation,
it follows with regard to the conclusion that: if the offense is Divine disapproval (a greater offense), then the punishment has to be fourteen days isolation (a greater punishment).

That is to say, in order to logically end up with the Gemara’s a crescendo conclusion (fourteen days) we have to assume a general principle of proportionality between punishment and offense. Such a principle indeed exists in Jewish tradition – it is the principle of measure for measure (midah keneged midah). The hermeneutic issue here is whether the author of the a crescendo conclusion (i.e. of the Gemara, or of the baraita it relies on) can be reasonably assumed to have reasoned thus (i.e. by means of an argument pro rata) – or whether he believed the a crescendo conclusion to proceed directly from the a fortiori premises, without need of the assistance of the principle of measure for measure.

Another issue in hermeneutics that needs underlining is the issue of the exactitude of the quantity specified in the a crescendo argument conclusion. As we have seen, the formal conclusion is essentially rather vague – that is, the concluding predicate Sp of positive subjectal a crescendo argument means nothing more than ‘more than Sq’, and the concluding subject Sq of positive predicatal a crescendo argument means nothing more than ‘less than Sp’. In most discourse, the subsidiary term used in the conclusion of an a crescendo argument is accordingly vague. But in some cases, a rather precise quantity is proposed (for example, in the above Talmudic illustration, precisely 14 days are specified).

The questions then arise: on the basis of what precise information did the speaker arrive at this specific numerical result? Is he claiming to have a mathematical formula that makes possible its calculation, or at least a generally accepted conventional table? If so, what is it and how reliable is it (merely probable or sure)? Or is he making an unsubstantiated claim, giving the misleading impression that a vague a crescendo argument (or even purely a fortiori argument) can yield such a quantitatively precise conclusion? Is his discourse scientific or rhetorical? Here again, it is only by careful examination of the larger context that we can decide what the speaker consciously or subconsciously intended.

5.    Relative middle terms

We cannot fully understand the practice of a fortiori argument without consideration of relative middle terms. Two middle terms R1 and R2 may be said to be relative (or antiparallel), if ‘more R1’ is equivalent to ‘less R2’, and vice-versa. Examples of such terms abound: much and little, long and short, big and small, far and near, hard and soft, heavy and light, stringent and lenient, good and bad, beautiful and ugly, hot and cold, rich and poor, and so forth. In principle, any term that varies quantitatively (in magnitude, in direction, in measure or degree of any sort) may give rise to a relative term, although we do not commonly construct relative terms without necessity.

Let us first consider commensurative propositions with relative terms. The two subjectal forms “A is more R1 than B is” and “B is more R2 than A is” may be taken to imply each other, i.e. are equivalent. Such propositions are said to be each other’s reverse (note the reversion of roles of A and B in them). For example, if the relative terms are ‘long’ and ‘short’, then if A is longer than B, it follows that B is shorter than A, and vice versa. Similarly, the predicatal commensurative proposition “More R1 is required to be A than to be B” may be reverted to “More R2 is required to be B than to be A,” and vice versa (again note the reversion of roles of A and B). For example, using the same relative terms: if more length is needed to be A than to be B, it follows that more shortness is needed to be B than to be A, and vice versa.

The formal difference between conversion and reversion is that, in conversion, the major term remains major (i.e. the more), and the minor remains minor (i.e. the less), and the middle term remains unchanged; whereas in reversion, the major term becomes the new minor, and the minor term becomes the new major, and the middle term is replaced by its relative. However, on closer scrutiny we realize that the converse and the reverse of a commensurative proposition are effectively the same. This is obvious, since they are both implicants of the same form. For instance, in the case of the subjectal form “A is more R1 than B,” its converse “B is less R1 than A,” and its reverse “B is more R2 than A,” and indeed the converse of the latter “A is less R2 than B,” are equivalent to it and to each other. Similarly for the corresponding predicatal forms.

Relative terms usually evolve from absolute terms. That is to say: initially, the terms R1 and R2 (e.g. much and little, or whatever) are intended absolutely, so that what is R1 is greater than what is R2. They are conceived as separated at some conventional cut-off value (say, v), such that what is more than v is R1 (e.g. much) and what is less than v is R2 (e.g. little). Then, when we realize that this dividing line v is rather conventional, and may in practice be fuzzy rather than precise[38], the terms are made relative, i.e. such that the whole range of values under consideration may be viewed as R1 in one direction and as R2 in the opposite direction. In one direction, the values of R1 increase and those of R2 decrease, and in the other direction the opposite occurs. Neither direction is formally preferable to the other. For this reason, such terms may be characterized as antiparallel.

The propositions “A is more R than B is” and “A is less R than B is” cannot both be true at once, but they can both be false. There is an alternative to them, viz. “A is as much R as B.” Note that this third form is applicable to any equal quantity of R in A and B, just as the other two forms are applicable to any unequal quantities. Likewise, when dealing with relative middle terms R1 and R2, we must take into consideration the three alternatives: “A is more R1 (and less R2) than B is,” and “A is less R1 (and more R2) than B is,” and “A is as much R1 (and as much R2) as B is.” These three forms are mutually exclusive, and usually but not always exhaustive.

This brings us to the issue of negative forms. In most cases, the three forms just mentioned are exhaustive, which means that the denial of any two of them implies the affirmation of the third. However, this is not always true. It is quite possible for A to be neither more R nor less R than B, nor as much R as B, for the simple reason that the whole concept of R is not applicable to A or to B. For example, though all objects extended in physical or mental space[39] may be said to be bigger or smaller or equal in size, such characterizations are inapplicable to spiritual and abstract objects; i.e. the latter must be admitted to be neither bigger nor smaller nor equal in size[40].

Thus, to determine the oppositional relations of given comparative forms, we must first ask whether the concept(s) used as middle term, viz. R (or R1 and R2), is (or are) universally applicable or applicable only within a given sphere. If it is (or they are) universally applicable the said three positive forms (more, less, or equally R – or ditto with R1 and R2) are exhaustive; but if they are applicable only within a circumscribed domain, they may be all three at once denied. Of course, in the latter case, it remains true that the three positive forms are exhaustive contextually, within the sphere of their relevance; so we may continue to think of them as exhaustive provided we keep in mind that this is true conditionally, granting the applicability of the middle term used.

All the above was said for subjectals. It can also be said, mutatis mutandis, for predicatals; and more broadly for implicationals.

Let us now consider the special case of the relativity between a term R and its complement notR. As we saw in the previous chapter (1.4), although these two terms are strictly speaking (by the law of non-contradiction) mutually exclusive, it is possible to conceive of a broader term with the same label ‘R’ which is inclusive of both R in the strict sense and notR, the negation of R in the strict sense. Such broader meaning of R has obviously no negation of its own, note well, since by definition it embraces all conceivable values of the original term R and its negation from plus infinity to minus infinity. However, just as we can construct a broader term R, we can also construct a broader term notR. The latter is not a negation of R in the broader sense, note well, but a term that like it by definition embraces all conceivable values of the original term R and its negation from plus infinity to minus infinity.

Thus, although in their strict senses the terms R and notR are absolutes, and clear contradictories, the broader or looser terms derived from them, also in everyday discourse labeled R and notR, may be viewed as relative terms, which mutually suggest each other, since they both embrace the full range of the strict terms R and notR, although they do so in opposite directions. That is, what is more R is less notR, and vice versa; thus, for instance, “A is more R than B” and “A is less notR than B” are equivalent. For examples: the subjectal forms: “A is more active (R) than B is” and “A is less inactive (notR) than A is;” and likewise, the predicatal forms: “More action (R) is required to be A than to be B” and “Less inaction (notR) is required to be B than to be A.”

As regards eductions, we observed earlier that for any pair of relative terms (R1, R2) the converse and the reverse of a commensurative proposition are effectively the same. This is also true here, with regard to R and notR. For instance, in the case of “A is more R than B,” its converse “B is less R than A,” and its reverse “B is more notR than A,” and indeed the converse of the latter “A is less notR than B,” are four logically equivalent propositions. Similarly for the corresponding predicatal forms.

As regards oppositions, the three sets of propositions “A is more R (less notR) than B is,” “A is less R (more notR) than B is” and “A is as much R (as much notR) as B is,” are not only mutually exclusive but also exhaustive, since here the relative terms are contradictories (so that nothing can be said to lack both R and notR).

The egalitarian forms “A is as much R as B is” and “A is as much notR as B is” are quite compatible; indeed, they imply each other. This may seem odd at first sight, due to thinking in absolute terms. But it is clear that these two propositions do not imply that A and B are both R and notR in absolute terms. They just mean, respectively, that A and B have the same value of R and the same value of notR. And since that value, whatever its magnitude and polarity (positive, zero or negative) is one throughout, the two forms must imply each other.

So for subjectals, and similarly, mutatis mutandis, for predicatals; and more broadly for implicationals.

Some readers may find the above treatment a bit confusing, in view of the different senses of the terms R and notR, as absolute or as relative. For them, I propose a more symbolic treatment, as follows. In this context, let us use the following notation: given the absolute term R and its negation notR, we can conceive of the relative terms R and notR (symbolically distinguished by being underlined).

Whereas the terms R and notR are mutually exclusive, the terms R and notR are inclusive, in the sense that each of them includes both R and notR. Yet R and notR are not identical, because they differ in direction, each being the reverse of the other. That is, whereas R refers to R as a positive quantity and to notR as a zero or negative quantity of R, notR refers to notR as a positive quantity and R as a zero or negative quantity of notR. Thus, R signifies a direction from notR (negative or zero R) to R (positive R), while notR signifies a direction from R (negative or zero notR) to notR (positive notR). We can express these definitions as formulae (where ‘iff’ means ‘if and only if’):

Iff X is R, then R > 0 (i.e. a positive quantity of R).

Iff X is not R, then R ≤ 0 (i.e. a zero or negative quantity of R).

Iff X is not R, then notR > 0 (i.e. a positive quantity of notR).

Iff X is R, then notR ≤ 0 (i.e. a zero or negative quantity of notR).

These formulae imply that ‘R > 0’ = ‘notR ≤ 0’ (since both imply ‘X is R’), and that ‘R ≤ 0’ = ‘notR > 0’ (since both imply ‘X is not R’). Note well that ‘zero R’ and ‘zero notR’ are not the same point, but contradictories, since the former means that R is absent whereas the latter means that R is present. This must be kept in mind to avoid inconsistency. However, the propositional forms involving the terms R and notR being comparative, this issue of ‘zero’ having a different meaning in each of the antiparallel continua has no impact. This will become evident when we consider oppositions and eductions, next.

By definition of ‘more’ and ‘less’, the propositions “A is more R than B” and “B is less R than A” are equivalent (if either is true, so is the other). Likewise, of course, “A is more notR than B” and “B is less notR than A.”

By definition of ‘more’ and ‘less’, the propositions “A is more R than B” and “A is less R than B” are incompatible (only one may be true). Likewise, of course, “A is more notR than B” and “A is less notR than B.”

By definition of R and notR, the propositions “A is more R than B” and “A is less notR than B” are equivalent. Likewise, “A is more notR than B” and “A is less R than B.”

It follows that the propositions “A is more R than B” and “A is more notR than B” are incompatible. Likewise, of course, “A is less R than B” and “A is less notR than B.”

We might define “A is as much R as B” in relation to the propositions “A is more R than B” and “A is less R than B,” as either implying both (i.e. as their intersection) or as denying both (i.e. as an alternative to them). The latter definition seems best, since in accord with actual practice. Similarly, we may take it that “A is as much notR as B” denies both “A is more notR than B” and “A is less notR than B.”

The propositions “A is as much R as B” and “A is as much notR as B” imply each other. For instance, if A and B are both at (say) R = 50, they are equally R (at +50) and equally notR (at –50). This is true even when R = 0 or when notR = 0, i.e. even though a zero quantity of R is a positive quantity of notR and a zero quantity of notR is a positive quantity of R, because each of the propositions “A is as much R as B” and “A is as much notR as B” refers to only one of the relative terms and anyway does not mention any actual quantity.

Whereas, as we have seen, some relative terms R1 and R2 might be both denied (if there exists things to which neither is applicable), in the case of relative complements R and notR, denial of both is impossible. Thus, here, the propositions “A is more R (= less notR) than B,” “A is less R (= more notR) than B” and “A is as much R (or notR) as B,” are always exhaustive (one of them must be true).

So for subjectals, and similarly, mutatis mutandis, for predicatals; and more broadly for implicationals.

Whereas the oppositional relation between the absolute terms R and notR is that they (when predicated of the same subject) are contradictory – i.e. they are incompatible (cannot both be true) and exhaustive (cannot both be false), the relative terms R and notR behave differently. They appear in comparative propositions only, and in that context may be affirmed together (one being more, the other less, or both as much). However, they cannot be both discarded. The peculiarity of such relative terms is that neither of them has a true negation, since both refer to the same full range of existential possibilities from minus infinity through zero to plus infinity (though in opposite directions). That is to say, each of them includes the whole world, as it were (but with a difference in perspective). Everything (not just some X) can be fitted in the continuum R, and simultaneously everything (not just some X) can be fitted in the continuum notR. For this reason, R and notR are not each other’s negation, note well.

Although our introduction of the underlined symbols R and notR for relative complements, to distinguish them from the absolute terms R and notR, does clarify things somewhat, I will not make further use of them here, for the simple reason that I prefer a logic of ordinary language to symbolic language, and in ordinary language we would signify our intention when it is unclear simply by saying of a given term that it is intended as relative (or inclusive). It is just as easy to mentally or out loud say the word ‘relative’ as to say the word ‘underlined’; and the disadvantage of the latter over the former is that one must still add the thought (in words or wordlessly) ‘and underlined means relative’, so one’s thinking is slowed down!

Thus far, we have compared commensurative propositions with relative terms. Let us now compare suffective propositions with relative terms. Two eductions from suffectives need to be investigated: movement from a positive to a negative form, or vice versa, and movement from a subjectal to a predicatal form, or vice versa. Concerning the said changes of polarity, we can do a good deal; but concerning changes of orientation, little can be done. We shall first deal with copulative forms, then with implicationals.

Copulative forms. First, let us interpret the negative forms. As already seen, the positive form “X is R enough to be Y” implies that “X is R” and “X is Y,” as well as “Rx ≥ Ry,” where R is understood as an inclusive middle term, which includes not only R > 0 but also R = 0 and possibly also R < 0. The negation of this form, i.e. “It is not true that X is R enough to be Y,” may colloquially be loosely expressed as “X is R not enough to be Y” or “X is not enough R to be Y” or “X is not R enough to be Y” or “X is R enough not to be Y,” putting the negation in various positions.

However, to avoid ambiguities, we might prefer to write more precisely “X is R [not-enough] to be Y” or “X is [not-enough] R to be Y” or “X is not [R-enough] to be Y” or “X is not [enough-R] to be Y,” adding hyphens as shown. All these forms are equivalent in that they imply “X is R” and “X is not Y,” as well as “Rx < Ry,” note well. They all are contradictory to the said positive form, although they have in common with it that “X is R” (where R is inclusive of notR, remember), because they imply “Rx < Ry” (instead of “Rx ≥ Ry”) and thence “X is not Y” (instead of “X is Y”).

As regards the form “X is R enough to be [not-Y],” with the negation attached to the predicate, it is obviously incompatible with the form “X is R enough to be Y,” since X cannot be R enough to be both Y and not-Y. But are these forms contrary or contradictory? We might think they could both be false, since they have in common that X is R, and this might be false. However, since R is here intended as an inclusive term, “X is not R” is implicitly included in to “X is R;” so it is useless to focus on this factor. We might alternatively compare the form “X is R enough to be [not-Y]” to the preceding three forms, and think that it is not equivalent to them, since it implies “Rx ≥ Rnot-y” whereas they imply “Rx < Ry.” However, albeit the opposite directions of ≥ and < as well as the different suffixes involved, these forms have in common the implication that “X is not Y.”

Wherever the dividing line along the continuum R for Y or for not-Y, once we know on which side of it a subject X falls, the matter is settled. Everything hinges on the result (Y or not-Y), however the condition for it is expressed in a given suffective proposition. The result is what matters, the rest is history. For this reason, whatever their apparent formal differences, i.e. their differences of wording, all suffective forms that imply “X is Y” are equivalent to each other, and all those that imply “X is not Y” are equivalent to each other.

Consider now the ambiguous form: “X is not R enough to be Y,” when its intent is “X is [notR] enough to be Y” (note well the position of the hyphen). Albeit its negative middle term (notR), this is a positive form, which implies that “X is not R” and “X is Y.” Since notR here is also an inclusive term, it does not contradict R but includes it as a possibility among others, and this form is really equivalent to “X is R enough to be Y.” Therefore, we should be careful not to confuse the form: “X is [notR] enough to be Y” with any of the forms: “X is not R-enough to be Y” or “X is R not-enough to be Y” or “X is not-enough R to be Y” or “X is not enough-R to be Y.” They are in fact contradictories.

These interpretations may well appear confusing, because we would not at first sight consider the forms “X is R enough to be Y” and “X is [notR] enough to be Y” to be equivalent. The reason we would not normally equate them is that the ranges R and notR go in opposite directions (the more R, the less notR, and vice versa), whereas the notion of sufficiency is originally in one direction only. Thus, given that “X is R enough to be Y,” we would not naturally simultaneously think that “X is notR enough to be Y.” We would rather wonder whether X is notR enough to be not-Y, associating the negation of R with the negation of Y. But the answer to that question would have to be that “X is not notR enough to be not-Y,” which can also be stated as “X is notR enough to be Y.”

Let me clarify this further. The expression “enough” originally signifies a minimum value, rather than a maximum. Normally, if we say that as of a certain value of R, there is “enough” for Y, we would not say that below that value there is “enough” for not-Y. However, if pressed to the wall, we are forced to say that since there are logically only two choices, viz. Y or not-Y, we must say that “X is R enough to be not-Y” is contradictory to “X is R enough to be Y.” Similarly, as regards forms with notR. It is interesting to note that, although when dealing with commensuratives the difference in direction between R and notR is significant, because more R (less notR) and more notR (less R) are incompatible, when dealing with suffectives R and notR are effectively interchangeable.

In the case of suffectives, since the issue of direction is absent, the terms R and notR in their inclusive senses are just different labels for the same position on the scale. Whether the scale goes from here to there or from there to here, the de facto intended position remains the same. For this reason, the terms are two sides of the same coin, merely differently labeled. In conclusion, though certain constructions are a bit artificial, from a formal point of view we have to accept them and evaluate them as above suggested, so as to be able to anticipate and deal with all possibilities. The same can be done with predicatal suffectives, and with implicational ones, of course.

This understanding for R and its complement notR can obviously be extended to any two terms, R1 and R2, known to be relative. Given, say, “X is R1 enough to be Y,” or any form which like it implies that “X is R1” and “X is Y,” we can readily educe that “X is R2 enough to be Y,” or any form which like it implies that “X is R2” and “X is Y,” because “X is R1” and “X is R2” are equivalent even though in opposite directions. The moment we introduce in either of these forms a negation that affects the predicate Y, or the relational element “enough,” the proposition may be taken as contradictory to the preceding. If on the other hand we introduce a negation that affects the middle term only, making it notR1 or notR2, the relation of X to Y remains essentially unchanged.

Based on this reasoning, we can always interpret suffective propositions involving one or more negative elements. All we need to do is first decide or determine just what each negation is intended to negate. If a negation is aimed at the middle term, assuming it is inclusive, nothing is essentially changed. All other negations are significant; though of course pairs of them cancel each other out. Thus for examples, “X is notR1 not enough not to be Y” is equivalent to “X is R1 enough to be Y;” whereas “X is notR1 not enough to be Y” is equivalent to “X is R1 enough to be not-Y.” The utility of this kind of inference will be seen when we deal with traductions, in the next chapter (3.5).

Second, let us consider the rewriting of subjectal suffectives as predicatals, or vice versa. If we look back at the definitions of these forms, we can see the difficulties that such pursuits present. Consider the components of a proposition of the copulative form “X is R enough to be Y”:

X is to a certain measure or degree R (say, Rx);
whatever is to a certain measure or degree R (say, Ry), is Y, and
whatever is not to that measure or degree R (i.e. is not Ry), is not Y;
and Rx is greater than (or equal to) Ry (whence: “Rx implies Ry”).

In order to obtain a suffective proposition with Y as subject and X (or even its negation) as predicate, we would need to contrapose the conditional propositions concerning X and Y. But though we can do that for Y, we do not have the necessary material for X. Moreover, the quantitative comparison would no longer be appropriate, anyway. So we cannot, as far as I can see, change the orientation of a suffective proposition.

Implicational forms. Let us first closely examine various negative forms. The positive form “X implies R enough to imply Y” implies that “X implies R” and “X implies Y” and “Rx ≥ Ry,” where R is understood as an inclusive middle thesis, which includes not only R > 0 but also R = 0 and possibly also R < 0. The negation of this form, i.e. “It is not true that X implies R enough to imply Y,” may colloquially be loosely expressed in various ways, but it is important here (more so than in the case of conjunctives) to avoid ambiguities. If we want a negative form to closely fit the positive form in all essential respects, we must ensure that it implies “X implies R” and “X does not imply Y” and “Rx < Ry.” The form “X implies R, but not enough to imply Y” would best fit this bill, suggesting that there is a threshold value of R (say Ry) only as of which Y is implied, but that threshold is not reached and therefore Y is not implied; note that the negation here is primarily focused on the relational expression ‘enough’. We should take this as the contradictory form.

As regards the form “X implies R, enough not to imply Y” – which suggests that there is a threshold value of R (say Ry) as of which Y is not implied, and that threshold is reached – although this form still in fact implies both “X implies R” and “X does not imply Y,” it differs from the preceding in that here “Rx ≥ Ry” (though Ry has a different meaning here, note well); that is to say, this form is essentially positive as regards structure, though contrary to the above positive form. As regards the forms “X does not imply R enough to imply Y” or “X does not imply enough R to imply Y,” which are hardly distinguishable, they are also contrary to the positive form, but in a different manner: neither of the propositions “X implies R” and “X does not imply Y” can be educed from them, although presumably they suggest that there is a threshold value of R (say Ry) as of which Y is implied, and that threshold is not reached, meaning that “Rx < Ry;” note that these forms do not clarify whether “X implies R” or “X does not imply R.”

Consider now the form: “X implies notR enough to imply Y.” We should be careful not to confuse this form with any of the preceding four forms: they are in fact antitheses. Albeit its negative middle thesis (notR), this is a positive form, which implies that “X implies notR” and “X implies Y.” NotR here being presumably an inclusive thesis, it does not contradict R but includes it as a possibility among others; therefore, this form is really equivalent to “X implies R enough to imply Y” – from which it follows that their contradictory forms, “X implies notR, but not enough to imply Y” and “X implies R, but not enough to imply Y,” are equivalent to each other.

As regards the rewriting of antecedental suffectives as consequentals, or vice versa, we can as we did for copulatives (and even more so) safely say that it is not formally feasible.

3.  Still more formalities

1.    Understanding the laws of thought

Many people regard Aristotle’s three ‘laws of thought’ – the laws of identity, of non-contradiction and of the excluded middle[41] – as rigid prejudices. They think these are just conventions, that some moronic old fellow called Aristotle had the bad grace to impose on the rest of us, and that we can just chuck ’em out at will. In each of my past works, I have tried to explain why these are fundamental human insights that cannot under any pretext be discarded. I would like to add a few more explanations in the present work.

The laws of thought must not be thought of as mechanical rules, but as repeated insights of our intelligence. Every ‘application’ of these laws in a new context demands a smart new insight from us. We must in each new context reaffirm these laws, and use them creatively to deal with the complexities of the case at hand.

In a fortiori logic, where new forms are encountered, and new problems need solutions, we can expect our intelligence and creativity to be called upon. We have already come across many contexts where subtlety was required. The distinction between a proposition like ‘X is Y’ and ‘X is R enough to be Y’ was one such context. Another was our development of a distinction between absolute terms (R and notR) and relative terms (R and notR). The laws of thought are ever present in logical discourse, but they must always be understood and adapted in ways that are appropriate to the context at hand – so they are not mechanical laws, but ‘smart laws’.

The laws of thought have to repeatedly be adapted to the increasing complexity of discourse. Originally, no doubt, Aristotle thought of the laws with reference to indefinite propositions, saying that ‘A is B’ and ‘A is not B’ were incompatible (law of non-contradiction) and exhaustive (law of the excluded middle). In this simplest of contexts, these laws implied only two alternatives. However, when Aristotle considered quantified propositions, ‘All A are B’ and ‘Some A are B and some A are not B’ and ‘No A is B’ – he realized that the application in this new context of the very same laws implied three alternatives. From this example, we see that the subtleties of each situation must be taken into consideration to properly ‘apply’ the laws. They are not really ‘applied’; they are intelligently formulated anew as befits the propositional forms under consideration.

We could say that the disjunction “Either ‘A is B’ or ‘A is not B’” refers to an individual subject A, whereas the disjunction “Either ‘All A are B’ or ‘Some A are B and some A are not B’ or ‘No A is B’” refers to a set of things labeled A. But then the question arises: what do we mean when we say that an individual A ‘is B’? Do we mean that A is ‘entirely B’, ‘partly B and partly not B’? Obviously, the mutually exclusive and exhaustive alternatives here would be: “Either ‘A is wholly B’ or ‘A is partly B and partly not B’ or ‘A is not at all B’.” It seems obvious that in most cases ‘A is B’ only intends ‘A is partly B and partly not B’ – for if ‘A is wholly B’, i.e. ‘A is nothing but B’ were intended, why would we bother verbally distinguishing A from B? Well, such tautologies do occur in practice, since we may first think of something as A and then of it as B, and belatedly realize that the two names in fact refer to one and the same thing. But generally we consider that only B is ‘wholly B’, so that if something labeled A is said to have some property labeled B, A may be assumed to be intended as ‘only partly B’.

To give a concrete example: my teacup is white. This is true, even though my teacup is not only colored white, but also has such and such a shape and is made of such and such a material and is usually used to drink tea. Thus, though being this teacup intersects with being white, it does not follow that the identity of this individual teacup is entirely revealed by its white color (which, moreover, could be changed). With regard to classes, even though we may choose to define the class of all A by the attribute B, because B is constant, universal and exclusive to A, it does not follow that A is thenceforth limited to B. B remains one attribute among the many attributes that are observed to occur in things labeled A. Indeed, the class A may have other attributes that are constant, universal and exclusive to it (say C, D, etc.), and yet B alone serves as the definition, perhaps because it intuitively seems most relevant. Thus, to define concept A by predicate B is not intended to limit A to B. If A were indeed limited to B, we would not name them differently.

These thoughts give rise to the logical distinction between ‘difference’ and ‘contradiction’, which calls forth some further use of ad hoc intelligence. When we say that ‘A and B are different’, we mean that these labels refer to two distinct phenomena. We mean that to be A is not the same as to be B, i.e. that B-ness is different from A-ness. It does not follow from this that No A is B. That is to say, even though A is not the same thing as B, it is conceivable that some or all things that are A may yet be B in some way. To say the latter involves no contradiction, note well. Therefore, the laws of non-contradiction and of the excluded middle cannot in this issue be applied naïvely, but only with due regard for the subtleties involved. We must realize that ‘difference’ is not the same as ‘contradiction’. Difference refers to a distinction, whereas contradiction refers to an opposition. Two propositions, say X and Y, may have different forms and yet imply each other. It is also possible, of course, that two propositions may be both different and contradictory.

Another subtlety in the application of the laws of thought is the consideration of tense. Just as ‘A is B’ and ‘A is not-B’ are compatible if they tacitly refer to different places, e.g. if they mean ‘A is B here’ and ‘A is not-B there’, so they are compatible if they tacitly refer to different times, e.g. if they mean ‘A is B now’ and ‘A is not-B then’. Thus, if a proposition is in the past tense and its negation is in the present or future tense, there is no contradiction between them and no exclusion of further alternatives. Likewise, if the two propositions are true at different moments of the past or at different moments of the future, they are logically compatible and inexhaustive.

These matters are further complicated when we take into consideration the various modalities (necessity, actuality, possibility), and still further complicated when we take into consideration the various modes of modality (natural, extensional, logical). I have dealt with these issues in great detail in past works and need not repeat myself here. In the light of considerations of the categories and types of modality, we learn to distinguish factual propositions from epistemic propositions, which qualify our knowledge of fact. In this context, for instances, ‘A is B’ and ‘A seems not provable to be B’, or even ‘A is B’ and ‘A seems provable not to be B’, might be both true.

One of the questions Aristotle made a great effort to answer, and had some difficulty doing, was how to interpret the disjunction: “Either there will be a sea battle tomorrow or there will not be a sea battle tomorrow”[42]. But the solution to the problem is simple enough: if we can truly predict today what will (or will not) happen tomorrow, it implies that tomorrow is already determined at this earlier point in time and that we are able to know the fact; thus, in cases where the fact is not already determined (so that we cannot predict it no matter what), or in cases where it is already inevitable but we have no way to predict the fact, the disjunction obviously cannot be bipolar, and this in no way contravenes the laws of thought. Nothing in the laws of thought allows us to foretell whether or not indeterminism is possible in this world.

As a matter of fact, either now there will be the sea battle tomorrow or now there won’t be one or the issue is still undetermined (three alternatives). As regards our knowledge of it, either now there will be the battle tomorrow and we know it, or now there won’t be and we know it, or now there will be and we don’t know it, or now there won’t be and we don’t know it, or it is still undetermined and so we cannot yet know which way it will go (five alternatives). We could partially formalize this matter by making a distinction between affirming that some event definitely, inevitably ‘will’ happen, and affirming only that it just possibly or even very likely ‘will’; the former is intended in deterministic contexts, whereas the latter is meant when human volition is involved or eventually when natural spontaneity is involved. These alternatives can of course be further multiplied, e.g. by being more specific as regards the predicted time and place tomorrow.

What all this teaches us is that propositions like ‘A is B’ and ‘A is not B’ may contain many tacit elements, which when made explicit may render them compatible and inexhaustive. The existence of more than two alternatives is not evidence against the laws of thought. The laws of thought must always be adapted to the particulars of the case under consideration. Moreover, human insight is required to properly formalize material relations, in a way that keeps our reading in accord with the laws of thought. This is not a mechanical matter and not everyone has the necessary skill.

Another illustration of the need for intelligence and creativity when ‘applying’ the laws of thought is the handling of double paradoxes. A proposition that implies its contradictory is characterized as paradoxical. This is a logical possibility, in that there is a quick way out of such single paradox – we can say that the proposition that implies its contradictory is false, because it leads to a contradiction in knowledge, whereas the proposition that is implied by its contradictory is true, because it does not lead to a contradiction in knowledge. A double paradox, on the other hand, is a logical impossibility; it is something unacceptable to logic, because in such event the proposition and its contradictory both lead to contradiction, and there is no apparent way out of the difficulty. The known double paradoxes are not immediately apparent, and not immediately resolvable. Insights are needed to realize each unsettling paradox, and further reflections and insights to put our minds at rest in relation to it. Such paradoxes are, of course, never real, but always illusory.

Double paradox is very often simply caused by equivocation, i.e. using the same word in two partly or wholly different senses. The way to avoid equivocation is to practice precision and clarity. Consider, for instance, the word “things.” In its primary sense, it refers to objects of thought which are thought to exist; but in its expanded sense, it refers to any objects of thought, including those which are not thought to exist. We need both senses of the term, but clearly the first sense is a species and the second sense is a genus. Thus, when we say “non-things are things” we are not committing a contradiction, because the word “things” means one thing (the narrow sense) in the subject and something else (the wider sense) in the predicate. The narrow sense allows of a contradictory term “non-things;” but the wider sense is exceptional, in that it does not allow of a contradictory term – in this sense, everything is a “thing” and nothing is a “non-thing,” i.e. there is no “non-thing.”

The same can be said regarding the word “existents.” In its primary sense, it refers to actually existing things, as against non-existing things; but in its enlarged sense, it includes non-existing things (i.e. things not existing in the primary sense, but only thought by someone to exist) and it has no contradictory. Such very large terms are, of course, exceptional; the problems they involve do not concern most other terms. Of famous double paradoxes, we can perhaps cite the Barber paradox as one due to equivocation[43]. Many of the famous double paradoxes have more complex causes. See for examples my latest analyses, in Appendix 7.4 of the Liar paradox, and in Appendix 7.5 of the Russell paradox. Such paradoxes often require a lot of ingenuity and logical skill to resolve.

2.    Quantification

Many people, even trained logicians among them, find the field of a fortiori logic difficult because of the variety of issues of quantity in the terms. We have already dealt with the issue of proportionality, and the differences between purely a fortiori argument, pro rata argument and a crescendo argument. Here, we will deal with three distinctions between terms: that between constants and variables; that between individuals and classes; and that between distributive and collective terms.

Terms may be constants or variables

We have earlier presented and validated various moods of a fortiori argument, without specifying whether the terms they involve are constants or variables. The reason is simply that they may be either. Although initially all the terms involved might be thought of as constants, we eventually realize that some or all of them may equally well be variables, provided the patterns of interrelationship between the variables continue to fit in to the formulae required for valid a fortiori argument. That is, provided each value of one variable is related to the corresponding value of the other variable in accord with the said formulae; which simply means that the various values of variables are effectively alternative constants.

This should be obvious. But let us consider an example the following positive subjectal argument:

P is more R than Q,
and Q is R enough to be S;
therefore, P is R enough to be S.

Here, P, Q, R, and S may all at first be assumed to have individual values, i.e. to be constants. But it is conceivable that P and Q may have different values of R at different times or places or in different circumstances or in different instances. That changes nothing in our argument, provided the relation ‘Rp > Rq’ remains unaffected for every pair of values of the variables Rp and Rq. If when variable Rp has value Rp1 and variable Rq has value Rq1, ‘Rp1 > Rq1’, and if the same is true for every other pair of values, like Rp2 and Rq2, Rp3 and Rq3, etc., then the a fortiori argument holds. It is also conceivable that Rp is constant while Rq is a variable, or vice versa, provided ‘Rp > Rq’ remains true. In all such cases, though the a fortiori argument seems like one statement, it may also be viewed as a summary of many similar statements[44].

Again, the predication ‘Q is R’ in the minor premise may intend the term R (which here more precisely means Rq) as a constant or as a variable. In the latter case, so long as every value of Rq – namely, Rq1, Rq2, Rq3, etc. – is sufficient to imply Q to be S (whether S is here a constant, or itself also a variable), the minor premise as a whole remains true. The conclusion then (given also the major premise, of course) naturally follows, whether Rp is a constant or a variable (with values Rp1, Rp2, etc.). Note here again that it is possible for Rq to be a variable while Rp is a constant, or vice versa, or they might both be constants or both variables.

The important thing to note here is that whatever value S has in the minor premise, it must be repeated in the conclusion. Where S is variable in the former, it is identically variable in the latter. The a fortiori argument as such cannot change the value of S in the transition from the minor premise to the conclusion; only the very same value of S can be inferred a fortiori. This issue should not be confused with that of ‘proportionality’ of conclusion, which we dealt with in the preceding chapter (2.2-2.3). As we saw there, the value of S in the conclusion may differ from that in the minor premise, if and only if we have an additional premise capable of justifying such change.

Terms may be individuals or classes

We have so far treated the major and minor terms (P and Q) of subjectal a fortiori arguments as indivisible units. Obviously, P and Q may be named or pointed-to individuals; e.g. Tom is richer than Harry, or this man is richer than that man. Also, obviously, P and Q may be whole classes; e.g. gold is worth more than silver, meaning any amount of gold and any equal amount of silver; or again, humans are more intellectual than other animals, meaning all humans are more intellectual than all other animals. The latter universal major premise could conceivably give rise to an a fortiori argument in which the minor premise is particular (or even singular) and the conclusion is general, note well. This means the following positive and negative forms are valid:

All P are more R than all Q,
and, all or some Q are R enough to be S;
therefore, all P are R enough to be S.
All P are more R than all Q,
yet, all or some P are R not enough to be S;
therefore, no Q is R enough to be S.

That is, given the major premise “All P are more R than all Q,” it follows that “if there is any Q that is R enough to be S, then all P are so too;” and inversely, “if there is any P that is R not enough to be S, then no Q can be so.” Note well, “all P” here refers to each and every P, and “all Q” to any and every Q; consequently, the minor premises might be particular, i.e. refer to some Q or P, or even to just one specified or unspecified Q or P. This is obvious enough.

If the major premise is entirely particular, we can still draw a particular conclusion provided the minor premise is general. That is to say, the following positive and negative forms are also valid:

Some P are more R than some Q,
and, all Q are R enough to be S;
therefore, some P are R enough to be S.
Some P are more R than some Q,
yet, no P is R enough to be S;
therefore, some Q are R not enough to be S.

These moods are valid because there is still in them a guarantee of overlap between the instances referred to in the major and minor premises. Note that if we could pinpoint instances, we could treat particulars as generalities; i.e. the following would be valid moods:

Certain P are more R than certain Q,
and, all or some of those Q are R enough to be S;
therefore, all of these P are R enough to be S.
Certain P are more R than certain Q,
yet, all or some of these P are R not enough to be S;
therefore, none of those Q is R enough to be S.

When we have a mix of quantities in the major premise, we can for the same reason in certain cases draw a valid conclusion as follows:

Some P are more R than all Q,
and, all or some Q are R enough to be S;
therefore, some P are R enough to be S.
Some P are more R than all Q,
yet, no P is R enough to be S;
therefore, no Q is R enough to be S.
All P are more R than some Q,
and, all Q are R enough to be S;
therefore, all P are R enough to be S.
All P are more R than some Q,
yet, all or some P are R not enough to be S;
therefore, some Q are R not enough to be S.

This should cover all cases of subjectal argument. Moods not above mentioned as valid are intended as invalid. All this can be formally demonstrated in the usual manner. Note well that the quantities intended in the major premises are not intended as “one for one” correspondences between P and Q. That is, when we say “All (some) P are more R than all (or some) Q,” we do not specifically mean that “for each instance of P referred to, there is a corresponding instance of Q such that this P is more R than that Q” (more on this topic presently); we are dealing with the classes in bulk (though “one for one” may be applicable in specific cases).

As regards predicatal a fortiori argument, the situation is very different since major and minor terms, P and Q, are predicates and not (as above) subjects, and therefore cannot properly be quantified. On the other hand, we can here quantify the subsidiary term, S, which is here a subject and not (as above) a predicate. The rule in this context is simple enough: the quantity in the conclusion is the same as that in the minor premise. If the minor premise (whether positive or negative) addresses some specified, some unspecified or all instances S, then so will the conclusion do. Again, this is obvious enough:

More R is required to be P than to be Q,
and, all (or some) S are R enough to be P;
therefore, all (or some) S are enough to be Q.
More R is required to be P than to be Q,
yet, all (or some) S are R not enough to be Q;
therefore, all (or some) S are R not enough to be P.

So much for copulative argumentation. Regarding implicational a fortiori argument, the theses involved may in principle contain any quantity or mix of quantities, without affecting the given implications; so there is nothing much to say about it. That is to say, though it is of course conceivable that in some cases the implications originally depend on the quantities involved, it remains true that once we are given the implications as premises the quantities in the theses concerned become irrelevant to the drawing of a conclusion from them. In such cases, quantities count as content, not as form, as regards the a fortiori process as such.[45]

Correspondences between terms. A special case of a fortiori argument that ought to be mentioned is when the terms involved are tied together by some correspondence. For example, the argument “all parents (P) are older (R) than their children (Q); therefore, if the children are old enough to vote, so are their parents” – it is clear that the major and minor terms refer to parents and their respective children, and not just to any children of any parents. With this in mind, we can construct the following valid moods in general terms:

Every P is more R than its corresponding Q,
and, a Q is R enough to be S;
therefore, its corresponding P is R enough to be S.
Every P is more R than its corresponding Q,
and, a P is R not enough to be S;
therefore, its corresponding Q is R not enough to be S.

In the above two subjectal moods, the correspondence is between P and Q. There could also be correspondence between P and Q and the subsidiary term (S), as in the following two predicatal moods.

More R is required (of something, e.g. S) to have a corresponding P than to have a corresponding Q,
and, S is R enough to have its corresponding P;
therefore, S is R enough to have its corresponding Q.
More R is required (of something, e.g. S) to have a corresponding P than to have a corresponding Q,
and, S is R not enough to have its corresponding Q;
therefore, S is R not enough to have its corresponding P.

Sometimes, additionally or alternatively, the middle term (R) is tied in this sense. For example, we may say that Noah was more righteous in his generation than many an absolutely more righteous man was righteous in his own respective generation. Valid moods can be constructed with such a variable middle term if this is done carefully – avoiding the fallacy of two middle terms by making sure the effective middle term throughout the argument is a generality. Clearly, in all such cases, the qualification “its corresponding” must be considered as part of the terms it qualifies, even though it is for emphasis stated outside them.

Needless to say, correspondences between terms are special cases; usually the terms involved are not tied together in this manner.

Terms may be distributive or collective

What has been said above about quantification concerns distributive terms – i.e. cases where the quantity all or some or whatever refers to the instances intended each one singly. The situation is very different where collective terms are intended, i.e. where the quantity is used to signify that a number of specified instances together make up a unit.

The latter situation can cause havoc in a fortiori argument, if misunderstood. Consider the following examples. With terms intended distributively, a fortiori argument is always possible: e.g. “If (any of) these five men are rich enough to pay for this object, then this one (of them) is rich enough.” However, when the intent is collective, we cannot readily infer a fortiori. Given that five designated men together can lift a certain weight, it obviously does not follow that three of them are strong enough to do so; or: given that three designated men together can go through a certain door, it obviously does not follow that five of them are thin enough to do so. Thus, note, we cannot formulate an a fortiori argument in either direction – i.e. neither from five to three nor from three to five.

Clearly, what is involved here is an issue of upper and lower limits, i.e. of maxima and minima. A quantity can be enough, or too much or too little. We can formalize our above examples as follows. Suppose ‘X’ is a designated group of individuals, ‘n’ refers to the number of them involved, and ‘Y’ is a predicate concerning them collectively. Given the lower limit ‘Less than nX together cannot be Y’, it follows that ‘nX together are Y’ does not imply ‘less than nX together are Y’. Similarly, given the upper limit ‘More that nX together cannot be Y’, it follows that ‘nX together are Y’ does not imply ‘more than nX together are Y’. In some cases, both limits are set – neither more nor less than a stipulated quantity is allowed.

Moreover, note that collective terms are very specific in their intents: while these five strong men are able to jointly lift said weight, five (or even more) other (weaker) men may not be able to do so; or again, while these three thin men are able to jointly go through said door, three (or even less) other (fatter) men may not be able to do so.

More formally put, given a limit for ‘nX1’, i.e. a number n of designated individuals labeled X1, it does not follow that the same limit obtains for ‘nX2’, i.e. a number n of other designated individuals labeled X2. Collective terms can be very tricky in yet other ways. For instance, these five men may be able to lift a certain weight together, if they all have a handle to take hold of it, but not if they have a hard time grabbing it. Or these three men may be able to go through this door at once, if they are stacked one on top of the other, but not if they try to simply walk through it simultaneously. Thus, a collective term is not fully defined by numbers (like n) and designated individuals (like these men), but may involve more complex conditions. Note this well.

Also to keep in mind, a collective may have very different properties than its components; and conversely, its components may have very different properties than a collective. If we assume that what is true of some or all of its components is true of a collective, we commit the ‘fallacy of composition’. If we assume that what is true of a collective is true of some or all of its components, we commit the ‘fallacy of division’.

As regards application of the concept of existential import to the commensurative and suffective propositions constituting a fortiori argument, my position is simply that the existence of subjects is generally assumed. See Appendix 7.2 for a fuller presentation of this position.

3.    A fortiori through induction

As we have shown, a fortiori argument is essentially a deductive argument, one that can readily be formalized and validated. There is no denying that. Yet many commentators persist in erroneously regarding it as an essentially inductive argument, no doubt simply because they have not perceived its deductive form. We shall now examine the various ways induction may be involved in the formulation and justification of an a fortiori argument.

Induction in general. First, we must distinguish formal induction from material (or informal) induction. An inductive argument can be considered as part of formal logic if the relation between its premise(s) and conclusion can be expressed in formal terms, i.e. with reference to symbols acting as placeholders for any material terms or propositions. If we cannot treat the inductive argument in purely abstract ways, it constitutes a material (or informal) one. Many people think that only deduction can be treated formally, but the truth is that much inductive reasoning can be formulated and justified in formal terms. There are several types of formal induction.

  1. The most important type of formal induction is factorial induction. It is the most important type, because it is the most distinct from deduction and the most reliable. I have treated this topic in great detail in my work Future Logic (part VI).

Factorial induction may be described in the following general terms: given a premise p, or a set of premises p1 and p2 and…, then: if we have a single necessary implication c, then it is the deductive conclusion; whereas, if we have two or more logically possible implications c1 or c2 or…, and for some reason one of them (say, c1) is logically preferable to the other(s), then it is the adopted inductive conclusion. The latter conclusion may of course change as new data is discovered which changes the premises; or c1 may be found empirically untrue, in which case the next most probable inductive conclusion (say, c2) is selected. The conclusions c1, c2, c3, etc. are called the factors of the premise p, or of the set of premises p1, p2, p3, etc. The most probable conclusion is called the strongest factor.

To understand this process, let us look at the simplest example: generalization of a particular proposition. A deductive argument is distinguished by having a single conclusion regarding certain terms (say, X and Y) from the given premise(s); for example, that ‘All X are Y’ (symbol A) implies ‘Some X are Y’ (symbol I) is a deductive argument (an eduction). On the other hand, an inductive argument has in principle two or more conclusions; for example, ‘Some X are Y’ (I) implies either ‘All X are Y’ (A) or ‘Some X are Y and some X are not Y’ (IO) is (part of) an inductive argument[46].

But that is not all. The argument ‘I implies either A or IO’ gives us a choice of inductive conclusions, but it does not give us any advice as to which of the two conclusions to opt for (or at best, it gives us a 50-50 probability for each conclusion, by virtue of symmetry). However, through the inductive act of generalization, we are able to prefer the conclusion A to the conclusion IO, by arguing that whereas A has the same positive polarity as given in the premise, IO involves an additional negative polarity not at all found in the premise. Thus, A requires less assumption than IO. For, though it is true that both A and IO claim quantitatively new information regarding the whole class, nevertheless IO introduces a claim to negative polarity not at all found in the premise I, while A remains entirely consistent in respect of polarity with the premise I. Thus, given I, alternative A is more probable than IO, because the latter is more speculative than the former.[47]

Thus, though our inductive argument starts by yielding a disjunction of conclusions, what makes it truly useful is that we are able to formally (before reference to any content) select one conclusion in preference to the other. It is the latter feature that allows us to refer to the argument as specifically inductive and distinguish it from deductive argument. An inductive argument allows deductively for two or more conclusions, but additionally (in most cases) provides us some formal reason(s) for preferring one of these conclusions over the others. Such induction is thus not a matter of guesswork or of purely material considerations, but is part of ‘formal logic’ in its own right.

What we have described above is called generalization from I to A. This process depends for its validity, note well, on two premises. First, that I is true; this might be determined empirically (i.e. by observation through the senses, as regards material phenomena, or by observation through the proverbial “mind’s eye,” as regards mental phenomena) or by deduction from previous inductions or deductions. Second, that there is no known evidence for claiming that O is true. Given these two premises together, and not just one of them alone (as some people erroneously think), we can inductively conclude that A is true.

It is obvious that if I concerns positive phenomena and is truly empirically established, it is very unlikely to ever turn out to be untrue. However, since I is usually made up of concepts and not just percepts, it is not inconceivable that it be later found inaccurate and should be abandoned; though, to repeat, that is relatively rare. On the other hand, the non-knowledge of O is not a definitive denial, and may more readily be conceived as being overturned. If upon further scrutiny we discover that in fact ‘Some X are not Y’, we must imperatively correct our previous judgment, and conclude with IO instead of A. This is called particularization. This is also a valid process within formal inductive logic. Particularization does not invalidate generalization, note well, but complements it by ensuring that all judgments are subject to correction if the need arise.

One more thing needs clarifying here: what constitutes inconclusiveness. If we lack, and cannot currently infer, any information regarding the relation of X and Y, so that the disjunction ‘either A or IO or E’ is true, we have effectively no conclusion from any known premises. If now, whether by observation or inference, we discover that I is true, this eliminates the E alternative of that disjunction, and we are left with the narrower disjunction ‘either A or IO’. This is, as we have seen, a conclusion of sorts[48]. Thus, conclusiveness is distinguishable from inconclusiveness with reference to the range of alternatives implied. If the range is unlimited, there is no conclusion; if it is partly limited, we have a set of inductive conclusions; if it is limited to one conclusion, we have either a deduction or a preferred induction.

Another approach to factorial induction is with reference to individual cases. Starting with any particular X, we know from the laws of thought that it is either Y or not-Y. Having found that some other X are Y, and not having found any other X that are not-Y, we can predict with considerable certainty that this here X is Y. This is another way of presenting the above detailed generalization from I to A. Similarly, having found that some other X are not-Y, and not having found any other X that are Y, we can predict with considerable certainty that this here X is not-Y. This is generalization from O to E.

If on the other hand, from the very beginning we find that some X are Y and some X are not Y, we obviously cannot generalize either to A or to E. If we have already generalized to one of the latter universal forms, we must retract and admit IO instead. In that case, what can we predict concerning an individual case of X, about which we have no information as yet as to whether it is Y or not-Y? The way to resolve this issue is to compare the frequencies of cases of X that are known to be Y and cases of X that are known to be not-Y. If it looks as if the frequencies are about the same on both sides of the contingency, the probability that this here X is Y and the probability that it is not-Y are even, and no inductive conclusion can be drawn; since the factors have the same weight, they remain formally indistinguishable. But if one side is more frequent than the other, we can identify it as the strongest factor. That is, we can conclude: ‘this X is most probably Y and less probably not-Y’. In some contexts, though not all, the probabilities involved can be precisely quantified.

The process just described may be described as statistical induction, because it relies on approximate or precise enumeration of known cases. If most (i.e. more than half of the) known instances of X are Y, then this unknown instance of X may be assumed to be Y. If most known instances of X are not-Y, then this unknown instance of X may be assumed to be not-Y. If the known instances of X are evenly distributed, with as many Y as not-Y, then this X being Y and this X being not-Y are equally probable, which means nothing definite can be said about the relation of X and Y or not-Y. Probabilities are of course subject to change as more instances of X are otherwise identified (empirically or by other arguments) to be Y or to be not-Y, so that our preferred conclusion may need updating.

In this light, generalization may be viewed as the special case where all known instances have the same polarity, so that we may assume the universal proposition A or (as the case may be) E to be true. We can also view the statistical treatment of contingencies, i.e. the movement from ‘Most known instances’ to ‘Most instances’ as a sort of generalization, in that we reiterate information available for known instances to unknown instances.

  1. The second most important type of formal induction is adduction, also known as ‘the scientific method’. This form of reasoning is an offshoot of the logic of hypotheticals, in contrast to the preceding form, which is an offshoot of the logic of categoricals. The following are the two valid moods of apodosis, or hypothetical deduction:
If P, then Q; If P, then Q;
and P; and not-Q;
therefore, Q therefore, not-P

These moods are respectively called ‘affirming the antecedent’ (modus ponens) and ‘denying the consequent’ (modus tollens). These moods are used, in adduction, to connect new theories to given data or to make new predictions from old theories and, respectively, to exclude new theories unconnected to the data or reject old theories with false predictions. They are valid because the major premise, ‘If P, then Q’ (i.e. if the antecedent P is true, then the consequent Q is true), means that ‘the theses P and not-Q cannot both be true’. From which it follows that if P is true, not-Q must be false; and if not-Q is true, P must be false. The following the two moods are, on the other hand, invalid:

If P, then Q; If P, then Q;
and Q; and not-P;
therefore, P therefore, not-Q

Given the major premise, we cannot infer that P is true from the fact that Q is true, or that Q is false from the fact that P is false. These moods are invalid as apodoses, i.e. as deductive arguments. But they still have some utility as inductive arguments. We can say regarding the positive mood that the evident truth of Q in the context of ‘If P, then Q’ gives some probability to the hypothesis that P is true; for it could be that Q is true because P is true. Likewise, we can say regarding the negative mood that the evident falsehood of P in the context of ‘If P, then Q’ gives some improbability to the hypothesis that Q is true; for it could be that P is false because Q is false.

Such arguments are of course very tenuous taken individually. But if we have many, and still more, cases of evidence like Q confirming hypothesis P, thesis P becomes more and more credible. Likewise, if we have many, and still more, cases of evidence like not-P undermining hypothesis Q, thesis Q becomes more and more incredible. Thus, these forms of argument are used in adduction, respectively, to strengthen or weaken existing theories. Strengthening and weakening do not, of course, mean proving or disproving; they just refer to cumulative credibility.

All such cumulative credibility can fall apart instantly, if even a single instance is found that belies the thesis concerned. That is, if P is repeatedly confirmed through evidence like Q, and an instance of Q is found which implies not-P, the hypothesis that P is true must be rejected. Likewise, if Q is repeatedly undermined through evidence like not-P, and an instance of not-P is found which implies Q, the hypothesis that Q is false must be rejected. How definitive such theory ‘rejection’ is depends on how empirical the reason for it is. If the reason is truly empirical, the rejection is definitive; whereas, if the reason is itself merely theoretical, the rejection is more mooted.

Such thoughts can be compared to generalization and particularization. When we repeatedly find evidence that confirms our hypothesis, and find no evidence to the contrary, we generalize and assume all future evidence will go the same way. When and if we eventually encounter evidence to the contrary, we particularize that generality, and adopt a more contingent stance, if not the very opposite idea. Here again, given a contingency, we may still for statistical reasons prefer a thesis over its negation, or we may find the evidence evenly distributed.

Adduction also, like generalization-particularization, uses disjunctive argument. There may be, and usually there are, two or more hypotheses that could explain the accumulated evidence. In such case, we compare the probabilities for each of the alternative hypotheses, and we opt for the strongest. Other considerations may affect this judgment, such as the simplicity or complexity of a hypothesis; such considerations may be viewed as adding or subtracting credibility to the hypothesis, i.e. as increasing or diminishing its net probability. Thus, the disjunction of possible explanations of varying probability may be said to be a list of the factors of the data to be explained, ordered according to their relative strengths. So adduction may be viewed as a special case of factorial induction.

It is also true that we may regard generalization and particularization as special applications of theory confirmation and elimination, i.e. of adduction. When we generalize from I to A, say, we are effectively regarding the instances subsumed by I as confirmations of the hypothesis A. When we particularize, we reject hypothesis A, and opt for the more empirical thesis IO. We then look towards the relative frequencies of the two polarities, and formulate a new hypothesis that ‘Most X are Y’ or that ‘Most X are not Y’ or that ‘As many X are Y as are not-Y’, as the case may be. This in turn might be modified, as new data comes in. Thus, adduction and generalization-particularization are very closely related reasoning processes.

In this context, quite parenthetically, I would like to say a word or two about the term ‘abduction’, which many people confuse with and prefer to the term ‘adduction’. The term ‘abduction’ refers to the assumption of a hypothesis that explains available evidence[49]. From this definition it is clear that the term refers to a positive argument: If P, then Q; and Q; therefore, P. There is notably no mention here of the possible negative aspect, i.e. the possibility that hypothesis P might entail some false prediction, such as Q2, so that it is eliminated by the argument: If P, then Q2; and not-Q2; therefore, not-P. Nor is the competitive aspect brought out, i.e. that P may be supplanted by some other hypothesis P2 which is found more probable.

Abduction is thus very naïve guesswork. Adduction, on the other hand, refers to the full range of the scientific method: repeatedly testing a hypothesis by means of new evidence, which may turn out to be counter-evidence, and comparing competing hypothesis, to opt for the most likely, as above described. Thus, the two terms should not be confused.

  1. The third most important type of formal induction is induction based on deduction. What this refers to is the following: suppose we have a duly validated deductive argument, say ‘P and Q implies R’. Then we can logically rely on the arguments: ‘If P and probably Q, then equally probably R’, or ‘If probably P and Q, then equally probably R’, or again ‘If probably P and probably Q, then probably R’. In these arguments, the degrees of probability of the premises are passed on to the conclusion, by multiplication. Thus, if one of the premises is 100% probable and the other 50%, the conclusion is 50% probable; whereas if both premises are 50% probable, the conclusion is only 25% probable. This is reasoning mathematically[50].

The important point to note is the deductive underpinning of such induction. The inductive form apes a deductive form. If the deductive form intended is valid, then the inductive form is a credible induction. But if the deductive form intended is not valid, then the inductive form is not a credible induction. The credibility of the induction is based on the credibility of the deduction it imitates. Thus, to give an example: given that ‘All Y are Z and this X is Y, therefore this X is Z’ is deductively valid, we can refer to a parallel inductive argument with, say, ‘this X is probably Y’ as the minor premise and ‘this X is probably Z’ as the conclusion. But, given the same premises, we cannot conclude ‘this X is probably not Z’ because such inductive conclusion has no deductive underpinning. It may well be true for other reasons that ‘this X is probably not Z’ – but this is not an inductive conclusion from the said premises.

Thus, an ‘induction based on deduction’ is very different from an unsupported statement of logical possibility. It is a type of formal induction, which relies for the credibility of its conclusion on the conclusion having a form that is validated by deduction. The relation between premises and conclusion is thus warranted. The implication involved is not merely probable but certain. The process remains certified, even if the premises and therefore the conclusion are only probable. In truth, most arguments we encounter in practice are of this sort, since most of our material knowledge is somewhat open to doubt. Our deductions naturally pass the doubtfulness of the premises onto the conclusion. But the process of deduction is unaffected: once shown logically necessary, it is reliable, no matter what the reliability of the propositions involved.

  1. The last and most unreliable type of ‘induction’ is precisely inductive inference that relies on a doubtful deductive process. Here, the basis is only that ‘P and Q probably implies R’ (rather than ‘P and Q implies R’) – so that, whether the premises are certain or only probable, the conclusion is at best only probable, if at all credible. This may be called induction in only a very loose sense of the term, since by definition we have no formal justification for drawing the putative conclusion from the given premises. Does this happen? Yes, it does. And people indulge in it because it seems to them ‘better than nothing’.

Someone may have information that suggests he might be able to formulate a certain deductive argument, but as yet he does not have enough information to actually formulate that argument – so he refers to a merely ‘probable implication’. There is some truth in that since, indeed, the argument might eventually crystalize; but until it does, the expectation that it might must be regarded with the utmost skepticism. A ‘probable implication’ is, strictly speaking, no implication at all. Thus, though we have premises, we have no real process through which to elicit a conclusion from them. The conclusion’s ‘probability’ is thus not much greater than the ‘probability’ of anything we imagine offhand.

This is comparable to having a few pieces of a jigsaw puzzle and trying to guess what the picture it comes from might be. It is in such contexts that the word ‘abduction’ is most appropriately used. And it is in such contexts that the skills of statisticians are called for, to find ways to select the most credible hypothesis from miniscule indices. The validity of such ‘inference’ is obviously very relative; in absolute terms, it is of doubtful worth.

It should be stated that the basis of all inference is analogy and disanalogy, although of course there are more sophisticated and reliable inferences and more simplistic and unreliable ones. Our deductive and inductive capacities all depend on, among other cognitive processes, the cognition of similarities and differences. If appearances did not seem ‘similar’ in various respects and ‘different’ in others, we could not perceive or conceive any continuity in identity in any individual object across time, or be able to class it with some other objects and differentiate it from yet other objects.

Everything would seem completely different or completely the same. We could not claim any individual subjects to think or talk about, nor any abstractions to predicate of them. We would have no particular propositions to generalize, and all the more no general propositions. Nothing could be related in any way, positively or negatively, to anything else. There could be no analogy and no disanalogy. Abstract theories could not be formed or tested, since nothing could be claimed repetitive and distinguishable. In short, any attempt to deny human ability to cognize similarities and differences, and thence deductive and inductive processes, is a self-defeating proposition.

Induction in a fortiori logic. Now, let us illustrate the above four types of induction with reference to a fortiori argument. Consider an a fortiori argument, say the positive subjectal mood:

P is more R than Q,
and Q is R enough to be S;
therefore, P is R enough to be S.

Put it this form, the argument takes for granted that the premises are true, so that the truth of the conclusion follows necessarily. However, in practice, more often than not, there is some degree of uncertainty in the premises, which is transmitted to the conclusion. Thus, insofar as the conclusion has some material uncertainty, it may be said to be somewhat ‘inductive’. But this induction is quite significant and reliable insofar as the deductive process above described is certain, note well. The following is a general statement of the corresponding inductive argument:

P is probably (x%) more R than Q,
and Q is probably (y%) R enough to be S;
therefore, P is probably (z%) R enough to be S.

The validity of this argument as induction is entirely based on the validity of the deductive argument it is modeled on. It is not an arbitrary construct designed to give an illusion of inference – it is inference. The probability of the conclusion (z%) is presumably a product of the probabilities of the premises (x% and y%), which may each range from > 0% to 100%; thus, we can say: z/100 = x/100 * y/100. When x% and y% both equal 100%, so does z. If x and/or y = 0% (i.e. if a premise is false), z = 0% (which means not that the conclusion is proved false, but that it cannot be determined from these premises).

We may be able to pinpoint more precisely where the uncertainties in the premises and conclusion lie, by looking at the components that make up each proposition. Thus, as we have seen, the major premise “P is more R than Q” is composed of three propositions, to wit: P is Rp, Q is Rq, and Rp > Rq. If P and Q are individuals, we may be assuming by generalization that they are all the time respectively Rp and Rq; or that the R value of P (Rp) is always greater than the R value of Q (Rq). If P and Q are classes, we may have generalized their stated relation from some instances to all instances. Similarly with regard to the minor premise: as we have seen, it is composed of four propositions, viz. Q is Rq, Whatever is at least Rs is S, Whatever is not at least Rs is not S, and Rq ≥ Rs. Here again, generalization may be involved in the claim that Q is Rq, whether Q is an individual or a class; furthermore, the two clauses relating Rs to S positively and negatively, and the clause comparing Rq and Rs all assume generalities.

Thus, beneath the surface discourse, we very probably rely on a number of inductions, which may have proceeded directly by generalization from experience, or indirectly by deduction from such generalities, or via adductive reasoning, or by means of induction based on deduction, or even by means of induction not based on deduction.

We can very well illustrate ‘induction not based on deduction’ with reference to a fortiori argument as follows. Instead of the above positive subjectal forms, someone might propose the following arguments as a deductive a fortiori (on the left) and the corresponding inductive a fortiori (on the right):

P is more R than Q, P is probably more R than Q,
and Q is S; and Q is probably S;
therefore, P is S. therefore, P is probably S.

Notice that the clause “R enough to be” is missing in both the minor premise and conclusion of these two arguments. This admittedly often occurs in practice, although the missing clause may usually be taken as tacitly intended. If it is not tacitly intended, but omitted out of ignorance of its being a requisite for validity, or as a sophistic attempt to mislead, the argument is inevitably deductively invalid – that is to say, there is no way for us to formally prove that the putative conclusion follows from the given premises. The process might superficially seem somewhat credible, because its minor premise and conclusion partly resemble the minor premise and conclusion of valid a fortiori argument. But this is of course an illusion; in fact, the conclusion is not implied by the given premises.

If the alleged deductive underpinning of the inductive argument is invalid, then the inductive argument is of course all the more so. Nevertheless, some people look upon such arguments as inchoate a fortiori, attempts at argument which might eventually crystalize into a fortiori form, at least inductive and then maybe even deductive. But we must keep in mind that the term ‘inductive’ here has its most disreputable sense. It does not refer to any actual inference, but only to an imaginary ‘possible’ inference.

The excuse often given in such circumstances is that it is ‘material’ or ‘informal’ inference. But what does this mean? The suggestion is that we are able to intuit logical relations ad hoc, without reference to logical principles. I agree we are able to do so; indeed, I believe all general logical principles are derived from particular logical intuitions. However, most appeals to ‘material implication’ are in fact admissions of the speaker’s inability to elucidate the ‘formal implication’ underlying it – i.e. they are excuses for ignorance or laziness. For someone else, more skilled in logic and informed regarding the subject-matter, may very often be able to state the formal sources of the material intuition – or alternatively to demonstrate it to be invalid.

While a fortiori argument is often best characterized as inductive rather than deductive, for the above stated reasons, a crescendo argument is very often best so characterized, because of the additional problems involved in affirming proportionality, and all the more so in affirming some quantitatively specific proportionality. As we have seen, an a crescendo argument consists of an a fortiori argument combined with a pro rata argument, the two together yielding a ‘proportional’ conclusion. For instance, the positive subjectal form is:

P is more R than Q (is R),
and Q is R enough to be S,
and S varies in proportion to R;
therefore, P is R enough to be more than S.

As can be seen, the third premise, viz. “S varies in proportion to R” is rather vague, and can only justify the vague conclusion that P is not merely R enough to be S, but even R enough to more than S. Furthermore, the conclusion is at best probable, since the additional premise about proportionality is, of course, normally known by induction, and so is only probably true (say, to degree g%) rather than absolutely certain. Thus, here our formula for calculating the probability of the conclusion would be z/100 = x/100 * y/100 * g/100, where symbols x, y and z have the meanings already above assigned them.

The inductive status of a crescendo argument needs to be further emphasized when the additional premise is not merely that “S varies in proportion to R,” but more specifically that S varies in proportion to R “in accord with such and such formula, say S = f(R),” when the formula concerned is natural (rather than conventional). The formula allows us to calculate a precise value for Sp (i.e. S in relation to P) given a precise value for Sq (S in relation to Q). Such a formula is of course conceivable, and we are often able to produce one. But of course, such a formula is not easy to establish with certainty (unless merely conventional, as happens – e.g. a price list for products of different sizes).

Very often, scientific experiments are necessary. We extrapolate from a number of correspondences between the values of the variables concerned. We propose a summary formula that allows us to predict untested values from the pattern suggested by past events. Thus, the formula constitutes a scientific hypothesis, which is relied on until and unless we come across some serious hitch in its application. Alternatively, the formula might be deduced from more general principles, which have earned our respect over time. But even then, it is still inductive, insofar as the general principles it is based on are themselves products of induction. Thus, in any event, the mathematical formula referred to must be admitted to be probable rather than certain.

Thus, we must remain aware that a crescendo argument is even more likely, than purely a fortiori argument, to deserve to be viewed as inductive. Needless to say, the same can be said for all forms of a crescendo argument, besides the example given above. Nevertheless, in practice, many a crescendo arguments may be viewed as effectively deductive, because the proportionality claimed seems pretty obvious and straightforward – all the more so if the putative conclusion is vague rather than precise. Their validity is so probable that we may take it for granted.

But very often, in everyday discourse, we come across a crescendo claims that are nowhere substantiated, and even that would be hard to substantiate if we tried. It is best to say in such cases that the speaker is proposing an inductive a crescendo argument, meaning that he is not so much deducing the conclusion by a crescendo from established premises but proposing his conclusion in the framework of a possible a crescendo development. That is, he is not so much inferring the putative conclusion as proposing it, although his proposition is not entirely arbitrary but comes embedded within a larger context.

4.    Antithetical items

  1. We saw earlier that arguments with antithetical middle items, such as: “If Q, which is not R, is S, then, all the more, P, which is R, is S,” or “If S is P, though P requires R, then all the more S is Q, which does not require R,” or their negative equivalents, or their implicational equivalents, all of which often occur in practice[51], can readily be assimilated into standard forms by viewing their middle items, R and notR, as inhabiting a common continuum, which is labeled R (or notR, as convenient). In this perspective, the absolute terms R and notR are both seamlessly included in a more expansive relative term R (or notR), and a standard form of a fortiori argument (or similarly, of a crescendo argument) can be formulated instead of the more awkward (i.e. difficult to formally validate) formula commonly given.

In this way, the standard forms are brought to bear to validate the said non-standard forms, which (to repeat) are often used in common discourse. However, it should be made clear that the standard forms we thus construct are based on the information provided in the said non-standard forms. That is to say, it is thanks to the information in the latter that we can put together the major premise of the former, which makes possible inference of the conclusion from the minor premise. In subjectal argument, given that P is R and Q is not R, we can say obviously that P (for which R > 0, i.e. a positive quantity of R) is more R than Q (for which R ≤ 0, i.e. a zero or negative quantity of R) is. Again, in predicatal argument, given that R is required to be P and R is not required to be Q, we can say obviously that More R is required to be P (for which R > 0) than to be Q (for which R ≤ 0).

Thus, there are forms of a fortiori argument (and similarly, a crescendo argument) which explicitly or implicitly involve a middle item R with contradictory or contrary values in relation to the major and minor terms. Such arguments can be formally validated through standardization, by replacing the absolute items R and notR with an all-encompassing but relative middle item R (or notR, as convenient), which ranges in value from any negative or zero absolute value (= notR) to any positive absolute value (= R). Note well, the validity of such arguments is not affected by the occurrence of antithetical middle items.

Let us now pursue this theme of antithetical terms or theses further, and investigate its application to major and minor items and/or to the subsidiary items.

  1. A fortiori argument (whether pure or a crescendo) is sometimes developed in relation to antithetical major and minor items. Antithetical means contrary or contradictory. In copulative reasoning, the positive subjectal form of such argument is as follows:
X (P) is more R than not-X (Q),
and not-X (Q) is R enough to be S;
therefore, X (P) is R enough to be S.

Clearly, X and not-X are here both R, with different values of R, the value of the first being superior to that of the second. We can obviously construct a similar negative subjectal argument. We can likewise formulate a positive predicatal argument, as follows:

More R is required to be X (P) than to be not-X (Q),
and S is R enough to be X (P);
therefore, S is R enough to be not-X (Q).

And we can obviously construct a similar negative predicatal argument. Needless to say, in all these argument forms, X and not-X could just as well switch roles and be Q and P, respectively. The fact that the major and minor terms (whether X and not-X, or not-X and X) are contradictory does not affect the validity of the argument, since it has standard form. It is just a special application of a fortiori, occasionally found in practice[52]. Similar comments can be made with respect to implicational reasoning.

The important thing to note here is that the validity of the standard forms of argument is not affected by their involving antithetical major and minor items, for the simple reason that in each mood the major premise (presumably given, explicitly or implicitly) formally allows us to draw the conclusion from the minor premise (and third premise, if any). We do not have to prove anything new in relation to the antitheses – all the information we need is already given in the premises.

  1. A crescendo argument (unlike purely a fortiori argument) sometimes concerns antithetical subsidiary items. Antithetical means contrary or contradictory. In copulative reasoning, the positive subjectal form of such argument is as follows:
P is more R than Q,
and Q is R enough to be not-Y (S1),
and S varies in proportion to R;
therefore, P is R enough to be Y (S2).

Clearly, not-Y and Y are here two different values within a common continuum called S, one being labeled S1 and the other S2. Assuming direct proportionality of S to R, then S1 < S2, since the argument is from minor to major. Thus, the negative not-Y is the lesser value S1, corresponding to S ≤ 0, and the positive Y is the greater value S2, i.e. S > 0. In cases of inverse proportionality of S to R, the positive Y would be S1 and the negative not-Y would be S2. We could of course switch the roles of Y and not-Y (i.e. reason from Y to not-Y while S is directly proportional to R, or reason from not-Y to Y while S is inversely proportional to R) provided the meanings of these terms were such that S1 < S2. I have chosen the above order of presentation as more natural, taking Y as something more positively S and not-Y as something more negatively S. But these are just symbols, and the reverse order may be more accurate in some cases.

We can obviously construct similar negative subjectal arguments. We can likewise formulate a positive predicatal argument, as follows:

More R is required to be P than to be Q,
and Y (S1) is R enough to be P,
and R varies in proportion to S;
therefore, not-Y (S2) is R enough to be Q.

Here again, Y and not-Y are two different values within the continuum S, one being labeled S1 and the other S2. Assuming direct proportionality of R to S, then S1 > S2, since the argument is from major to minor. Thus, the positive Y is the greater value S1, i.e. S > 0. the negative not-Y is the lesser value S2, corresponding to S = 0. In cases of inverse proportionality of R to S, the negative not-Y would be S1 and the positive Y would be S2. The reverse order of Y and not-Y is here again conceivable, needless to say. And we can obviously construct similar negative predicatal arguments. Similar comments can be made with respect to implicational reasoning.

Now, the big question here is: are these arguments formally valid? That is to say, does the putative conclusion of each follow from the given premises? The answer is obviously: no – although it may be that with additional information these conclusions might well be logically justified. The answer is obviously ‘no’, because the third premise, which tells us about the proportionality of S to R or of R to S, only allows us a vague conclusion, namely that the subsidiary term in the conclusion is more than the subsidiary term in the minor premise, in the case of minor-to-major arguments (namely, the positive subjectal and negative predicatal moods), and that the subsidiary term in the conclusion is less than the subsidiary term in the minor premise, in the case of major-to-minor arguments (namely, the positive predicatal and negative subjectal moods).

If the given subsidiary term is the negative not-Y and the conclusion is supposed to be more than not-Y, it does not necessarily follow that the concluding subsidiary term is the positive Y. For if in a given case not-Y happens to refer to a specific negative value in the continuum S (i.e. S < 0) which happens to be well below zero, then ‘more than not-Y’ cannot be assumed to indeed refer to Y (i.e. S > 0), for it might conceivably still refer to another specific negative value in the continuum S (i.e. S ≤ 0), which is closer to zero but still not above zero. Likewise, if the given subsidiary term is the positive Y and the conclusion is supposed to be less than Y, it does not necessarily follow that the concluding subsidiary term is the negative not-Y. For if in a given case Y happens to refer to a specific positive value in the continuum S (i.e. S > 0) which happens to be well above zero, then ‘less than Y’ cannot be assumed to indeed refer to not-Y (i.e. S ≤ 0), for it might conceivably still refer to another specific positive value in the continuum S (i.e. S > 0), which is closer to zero but still above zero.

It is only when Y and not-Y do not admit of degrees, i.e. when things are either precisely Y or precisely not-Y (S being binary, either = 1 or = 0), than we can take ‘more than not-Y’ to mean ‘Y’ and ‘less than Y’ to mean ‘not-Y’. This happens, but is not a general truth. Therefore, though the moods above described with antithetical subsidiary terms are in exceptional cases indeed valid, they are not universally valid, so we must be careful when we reason in this way that we do not rush to judgment, but ponder the matter carefully.

To clarify this further: if we say that Q or P “is R enough to be” Y (or not-Y), do we mean every value of Y (or not-Y), or some value of Y (or not-Y)? Obviously, the latter. Therefore, even if our explicit reference to Y (or not-Y) is generic, it is tacitly specific – and we cannot tell from such a vague statement what the precise specification is. The only exception to this rule would be when we know for a fact that ‘Y’ has only one possible value (say, 1) and ‘not-Y’ has only one possible value (say, 0) – i.e. when they form a binary pair.

In cases where ‘Y’ and ‘not-Y’ refer to ranges of values of S, i.e. to S > 0 and S ≤ 0 respectively, we would need a precise mathematical formula, viz. S = f(R) or R = f(S), as the case may be, to draw the desired conclusion (i.e. one with an antithetical subsidiary term). That is, to know that ‘more than not-Y’ here means ‘Y’ we must be able to calculate by how many degrees of S the inferred ‘more than not-Y’ is more than the given ‘not-Y’. Or, to know that ‘less than Y’ here means ‘not-Y’ we must be able to calculate by how many degrees of S the inferred ‘less than Y’ is less than the given ‘Y’.

Note that the problem here is not specific to antithetical terms, but applies to any specific terms. Given only a vague premise about proportionality, i.e. “S varies in proportion to R” or “R varies in proportion to S” (as the case may be), we can only come to a vague conclusion. Even if the minor premise contains a very specific subsidiary term, the a crescendo conclusion cannot produce an equally specific subsidiary term, but only a term that is vaguely ‘more than’ or ‘less than’ the one given in the premise. An antithetical subsidiary term being a specific term, this rule applies equally to it.

Of course, if we can amplify our premise about proportionality with a formula that allows us to precisely calculate the concluding subsidiary term from the given subsidiary term, the problem dissolves and we can produce an exact conclusion – which means, in some cases, an antithetical subsidiary term. Though such a formula is conceivable and often available in scientific discourse, it is rarely available in everyday discourse. In the latter, we usually have no practical means of producing such a formula, and must rely more on intuitive understanding to decide whether the putative conclusion, involving an antithetical subsidiary term, seems reasonable or not. Thus, what shall we declare at the last? Shall we say such arguments are valid or invalid? I would suggest we declare them in principle invalid, while admitting that they are in practice often accepted as reasonable[53].

It is clear that to draw the desired conclusions we strictly need more precise information than the vague proportionality given in the above premises. However, being human, we function in practice in a more permissive mode, and grant many inferences on the basis of mere intuitions of proportionality. In such cases, if the inference of a positive from a negative or the inference of a negative from a positive seems reasonable to everyone, we might accept the argument as close enough to valid. Obviously, conclusions based on such relatively lenient standards are inductive rather than deductive. They are probable rather than certain, accepted as true until and unless some objection to them be found.

The logician’s role is not to inhibit human knowledge if it does not meet strict ideal standards, but to help practitioners approach these standards as much as possible in each given context. We have to remain lucid at all times, and be aware of our limitations – but this should not stop us from proceeding apace with our pursuit of knowledge, which is a necessity for us as living beings. On this basis, while I have above demonstrated that deduction of an antithetical subsidiary term on the basis of a vague premise of proportionality is strictly speaking invalid, I would recommend that we in practice, in most cases, adopt a lenient attitude towards such inference.

Its result must, of course, be recognized as inductive rather than deductive. If the argument is based on public information, it is in principle subject to revision when and if new information arises. However, very often we have no means of scientific verification, so there is no likelihood that we shall change our mind that way. In such cases, we rely on a primary insight. A new such insight might conceivably arise in a wider context of knowledge and replace the former. In that event, of course, we would naturally revise our a crescendo argument. Thus, even an argument based on mere insight can in principle be reviewed. So it is fair to call such arguments inductive, and view them as more than just speculative pronouncements.

  1. Let us now consider a crescendo arguments (not purely a fortiori arguments, note) which involve both antithetical major and minor items and antithetical subsidiary items, in tandem. Such arguments may be characterized as a contrario a crescendo arguments, because the polarities of both the terms in minor premise are inverted in the conclusion, i.e. the subject and predicate in the former are negated in the latter.

Such arguments do occur in common discourse, although very rarely. For instance, R. Hananiah ben Gamaliel says, in Mishna Makkoth 3:15: “If in one transgression a transgressor forfeits his soul, how much more should one who performs one precept have his soul granted him!” Here, ‘transgression’ is replaced by ‘performing precepts’ and ‘forfeiting of one’s soul’ is replaced by ‘being granted one’s soul’. If vice causes death, then virtue causes life. Sound reasonable. But is it? Does the first sentence logically imply the second?

The question posed is whether such arguments are valid. Consider for instances the following most typical forms (the first positive subjectal and the second positive predicatal):

X (P) is more R than not-X (Q),
and not-X (Q) is R enough to be not-Y (S1),
and S varies in proportion to R;
therefore, X (P) is R enough to be Y (S2).
More R is required to be (X) P than to be not-X (Q),
and Y (S1) is R enough to be X (P),
and R varies in proportion to S;
therefore, not-Y (S2) is R enough to be not-X (Q).

What distinguishes such arguments is that they compound the feature of antithetical major and minor terms and that of antithetical subsidiary terms. The expression a contrario is appropriately used in relation to such a crescendo arguments, because the movement of thought involved in them is obviously that of inversion. In our two samples, this means:

If not-X, then not-Y; therefore, if X, then Y.

If Y, then X; therefore, if not-Y, then not-X.

However, we know that inversion is not a valid process of eduction (i.e. immediate inference), even if many people wrongly imagine it to be. That is to say, the proposition ‘if not-X, then not-Y’ does not formally imply the proposition ‘if X, then Y’; these propositions are admittedly sometimes true together, but just as often one is true and the other false. This can be proven very precisely[54]. Similarly, of course, for the other pair, ‘if Y, then X’ and ‘if not-Y, then not-X’.

Admittedly, someone who formulates an argument like those above described is not relying on mere immediate inference, but on a complex a crescendo argument. Well, we have already in (b) above examined arguments with antithetical major and minor terms and found them formally valid, since the major premise guarantees the process. We have also in (c) above examined arguments with antithetical subsidiary terms and found them in principle invalid without more precise information regarding the proportionality, although often in practice accepted as effectively valid merely on the basis of intuitive understanding.

On the basis of these earlier findings, we should be able to readily determine the validity or invalidity of the more complicated a contrario forms. As regards the minor and major terms, they present no problem of validity here again, since the major premise explicitly refers to them. As regards the subsidiary terms, the movement of thought is strictly invalid if we lack a precise formula, although reasonable enough in most cases commonly encountered. However, we cannot resolve the issue of a contrario argument simply by this extrapolation, because here the problem of antithetical items is compounded.

That is, in a contrario a crescendo argument we are dealing with two changes of polarity in tandem, from negatives to positives (in our first sample, from not-X to X as well as from not-Y to Y), or from positives to negatives (in our second sample, from Y to not-Y as well as from X to not-X). Thus, here the difficulty is much greater: we need a formula that can express the parallelism between the X and Y values, not merely in a general way, but in a way so precise that it can pinpoint for us that the positives (X and Y) will occur together and the negatives (not-X and not-Y) will occur together[55].

This is surgical precision that cannot simply be assumed offhand. Of course, it may happen that the terms behave in such an orderly manner, in lockstep, as tied couples. But how would we express the underlying determinism in a further premise? Perhaps in a statement like “Change from not-X to X and change from not-Y to Y are linked” (for the first mood) or “Change from Y to not-Y and change from X to not-X are linked” (for the second mood). But how would we know this? Presumably by observing the concomitant variation of the two terms. But then, if we already know that X is coupled with Y or that not-Y is coupled with not-X, what need have we of the proposed a crescendo argument? Does its conclusion teach us anything more?

Conversely, if we do feel the a crescendo argument to be useful, is it not because we do not have the said information on coupling, and seek to obtain it by deduction? I submit that the only way we could express the premise about coupling would be to adduce the conclusion! That is to say, the needed information is so precise that it cannot be specified indirectly. It can only be stated. But if it is stated, this is tantamount to granting the putative conclusion as a premise. We would be begging the question, engaging in circular argument.

I admit I could be wrong in this reasoning, but I think it is best to adopt it, until if ever someone proposes some credible additional premise that would formally guarantee the conclusion without repeating it. In sum, I prefer as a precaution to declare the proposed argument invalid. This does not mean that, in the above described a contrario arguments, the minor premise and conclusion cannot be true together, but it does mean that we cannot deduce the latter from the former, even granting the said major premise and third premise. There is a non-sequitur, but no antinomy.[56]

Thus, argument with antithetical minor and major items and antithetical subsidiary items is best avoided and regarded as rhetorical. Such argument appears reasonable to some people because they see it as mere inversion; but as we have seen, mere inversion is not valid inference, anyway. All the more is it not valid in the more complicated context of a crescendo argument.

5.    Traductions

In the course of interpreting numerous Mishnaic and Talmudic a fortiori arguments, I noticed that there were often two or more ways a given argument could be interpreted to the same effect. I therefore resolved to try and justify such correspondences in formal terms, to facilitate such reinterpretation in the future. I have called the establishment of such rewriting of arguments in alternate forms – ‘traductions’ (which in French means ‘translations’). Traduction should not be identified with reduction, the validation of arguments, since both the source and target arguments here are already known to be valid forms.

The purpose of traduction is more material – it is simply deriving from an argument, of one form, another argument, of a different form; that is, it is merely a different verbal expression of the same thought. The value of studying such changes of wording is that it helps one find the interpretation of a given material argument that is formally closest to the given speech or text, rather than imposing a more common form on it. In this way one demonstrates full understanding of the original movement of thought.

More specifically, the utility of traduction is that it allows us to pass from a positive to a negative form, or vice versa; or from a ‘minor to major’ to a ‘major to minor’ form, or vice versa; or from a subjectal (or antecedental) to a predicatal (or consequental) form, or vice versa – ideally, without loss of information. Correlations between copulative and implicational forms (some of which were dealt with earlier) can also of course be used for purposes of traduction. As we shall see, some traductions are of logical significance, others are more akin to linguistic manipulations.

We shall deal with copulative arguments, and assume offhand that the same can be done with implicational ones (the reader is invited to verify this, as an exercise). Regarding the former, knowing there are four moods of primary copulative a fortiori argument, we shall need to investigate the following possible traductions: the uniformly subjectal or predicatal, and mixtures of subjectal and predicatal. That is, from +s to –s, and vice versa; from +p to –p, and vice versa; from +p to +s and –s, and from –p to +s and –s; and finally from +s to +p and –p, and from –s to +p and –p.

Let us first consider the uniform traductions. The first type is the purely subjectal one: (1a) from +s to –s:

Positive subjectal (minor to major) Negative subjectal (major to minor)
Given that P is more R1 than Q is, it follows that: Given that Q is more R2 than P is, it follows that
if Q is R1 enough to be S, if Q is R2 not enough to be not-S,
then P is R1 enough to be S. then P is R2 not enough to be not-S.

This is a logical process and one not hard to prove. The two major premises imply each other, given that their middle terms R1, R2 are relative. The minor premises likewise imply each other because they both imply that Q is S, and both the ranges R1 and R2 are fully inclusive, though in opposite directions. Similarly, the conclusions imply each other, and that P is S. Thus, albeit apparent differences in middle terms and in the polarities of their minor premises and conclusions and their subsidiary terms, these two a fortiori arguments are formally equivalent.

Compare for example the following two arguments: “given P is longer than Q, and Q is long enough to be S, then P is long enough to be S” and “given Q is shorter than P, and Q is not short enough to be not-S, then P is not short enough to be not-S.” Clearly, if Q is not short enough to be not-S, then it must be long enough to be S; and if Q is long enough to be S, then it cannot be short enough to be not-S; and the conclusions follow.

Note well that both arguments are subjectal, and one is positive and goes from minor to major and the other is negative and goes from major to minor. Keep in mind, also, the special case where R1 is some concept R, and R2 its antithesis not-R; this sometimes occurs. Needless to say, if we substituted not-S for S in the first argument, and put S instead of not-S in the second, we would have another pair of equivalent arguments.

A likewise easily proved corollary of the above traduction is (1b) from –s to +s:

Negative subjectal (major to minor) Positive subjectal (minor to major)
Given that P is more R1 than Q is, it follows that: Given that Q is more R2 than P is, it follows that
if P is R1 not enough to be S, if P is R2 enough to be not-S,
then Q is R1 not enough to be S. then Q is R2 enough to be not-S.

The second sort of uniform traduction is the purely predicatal one: (2a) from +p to –p:

Positive predicatal (major to minor) Negative predicatal (minor to major)
Given that more R1 is required to be P than to be Q, it follows that: Given that more R2 is required to be Q than to be P, it follows that:
if S is R1 enough to be P, if S is R2 not enough to be not-P,
then S is R1 enough to be Q. then S is R2 not enough to be not-Q.

This is also a logical process and one easy enough to prove. The two major premises imply each other, given that their middle terms R1, R2 are relative. The minor premises likewise imply each other because they both imply that S is P, and both the ranges R1 and R2 are fully inclusive, though in opposite directions. Similarly, the conclusions imply each other, and that S is Q. Thus, albeit apparent differences in middle terms and in the polarities of their minor premises and conclusions and their major and minor terms, these two a fortiori arguments are formally equivalent.

Compare for example the following two arguments: “given more strength is required to be P than to be Q, and S is strong enough to be P, then S is strong enough to be Q” and “given more weakness is required to be Q than to be P, and S is weak not enough to be not-P, then S is weak not enough to be not-Q.” Clearly, if S is not weak enough to be not-P, then it must be strong enough to be P; and if S is strong enough to be P, then it cannot be weak enough to be not-P; and the conclusions follow.

Note well that both arguments are predicatal, and one is positive and goes from major to minor and the other is negative and goes from minor to major. Needless to say, we could equally well have made the argument about R1 negative and that about R2 positive[57]. Keep in mind, also, the special case where R1 is some concept R, and R2 its antithesis not-R; this sometimes occurs.

A likewise easily proved corollary of the above traduction is (2b) from –p to +p:

Negative predicatal (minor to major) Positive predicatal (major to minor)
Given that more R1 is required to be P than to be Q, it follows that: Given that more R2 is required to be Q than to be P, it follows that:
if S is R1 not enough to be Q, if S is R2 enough to be not-Q,
then S is R1 not enough to be P. then S is R2 enough to be not-P.

More complex are the mixed traductions, aimed at correlation of predicatal and subjectal forms of a fortiori argument. It is not easy, if not impossible, to effect such changes of form, because in subjectal arguments the minor premises and conclusions have P or Q as subjects and S as predicate, whereas in predicatal ones they have S as subject and P or Q as predicates, and we cannot simply convert one form to the other. However, such changes of form can be effected in a more convoluted manner, by constructing new items from the given elements of the argument.

Some such processes are of logical interest. But in many cases the change is rather artificial, in the sense that the underlying logical form of the argument is not really changed but only superficially made to appear to have been changed; that is, though the explicit wording looks different, the implicit thought is unchanged. Although such processing is thus a bit make believe from a logical point of view, it is still useful in that we can by this means verbally reproduce someone’s actual thought process.[58]

The processes that go from predicatal arguments to subjectal ones are based on a fusion of the given middle term R with the relational concept of its being (or not being) required for some result, yielding either the new positive middle term “demanding of R” or the negative relative term “undemanding of R” as appropriate. We have in all four such traductions to consider.

(3a and 3b) From +p to –s and +s: starting with the following positive predicatal (major to minor) argument:

Given that more R is required to be P than to be Q, it follows that:
if S is R enough (for S) to be P,
then S is R enough (for S) to be Q.

…we can construct the following two subjectal ones, whose equivalence is evidenced by their having the same net implications, viz. that S is P and Q:

Negative subjectal (major to minor) Positive subjectal (minor to major)
Given that P is more [demanding of R] than Q is, it follows that: Given that Q is more [undemanding of R] than P is, it follows that:
if P is [demanding of R] not enough to prevent [S (from being P)], if P is [undemanding of R] enough for [S (to be P)],
then Q is [demanding of R] not enough to prevent [S (from being Q)]. then Q is [undemanding of R] enough for [S (to be Q)].

(3c and 3d) From –p to +s and –s: similarly, starting with the following negative predicatal (minor to major) argument:

Given that more R is required to be P than to be Q, it follows that:
if S is R not enough (for S) to be Q,
then S is R not enough (for S) to be P.

…we can construct the following two subjectal ones, whose equivalence is evidenced by their having the same net implications, viz. that S is not-Q and not-P:

Positive subjectal (minor to major) Negative subjectal (major to minor)
Given that P is more [demanding of R] than Q is, it follows that: Given that Q is more [undemanding of R] than P is, it follows that:
if Q is [demanding of R] enough to prevent [S (from being Q)], if Q is [undemanding of R] not enough for [S (to be Q)],
then P is [demanding of R] enough to prevent [S (from being P)]. then P is [undemanding of R] not enough for [S (to be P)].

Note that the move from +p to –s feels more natural than that from +p to +s, because in the former case, even though the minor premise and conclusion change polarity,  the polarity of the middle term and the movement from major to minor remain the same. For the same reasons, the move from –p to +s feels more natural than that from –p to –s. This is why I have listed the traductions in that order.

The reason I have characterized such traductions as artificial, i.e. as more verbal than logical, is partly of course due to their having recourse to a new middle term, viz. “demanding of R” (or “undemanding of R”), which has a formal element hidden in it, viz. the fact of requirement (or its lack) that is in fact indicative of predicatal argument. But there is a second, ultimately more important reason: it is that the subsidiary item is not really the same in minor premise and conclusion. Although the original subsidiary term S is still present and stands in the foreground at an explicit level in the derived arguments, at a more implicit level we have to specify its being the subject of the subject of the proposition. That is to say, though we say “for S” or “to prevent S,” by the term “S” here we really mean the proposition “S is P” or “S is Q” (as appropriate).

As we all know by now, an a fortiori argument is formally invalid if the subsidiary item is not exactly the same in minor premise and conclusion. So we must regard such traductions as being, strictly speaking, misleading. But as is evident the derived arguments do carry some conviction! Why so? The reason they do so is that they are only apparently subjectal, at the surface level of their wording. In fact they are, at a deeper level, as regards their logical form, still very much predicatal, since the idea of requirement is inherent in their middle terms, viz. “demanding of R” (or “undemanding of R”).

We could perhaps remedy the said fault, of the derived arguments having tacitly unequal subsidiary terms, by resorting to a more abstract subsidiary item. That is to say, instead of specifically saying (or even thinking) “S is P” or “S is Q” – we would state more vaguely “the subject (of the whole suffective proposition concerned, i.e. P or Q as the case may be) is predicated of S.” This would indeed considerably reinforce the subjectal appearance of the derived arguments. But I maintain that these arguments would not be fully understood and believed if we did not have in mind the underlying predicatal discourse.

Consider, for instance, the first of the listed four traductions, viz. “from +p to –s.” Note that both the arguments involved are from major to minor, and that both imply that S is P and Q. The middle term of the first is R, but the middle term of the second is “demanding of R.” The major and minor terms P and Q remain the same, but the given subsidiary term S is amplified by a P or Q predicate (as appropriate) in the derived argument (though this may not be stated out loud, and only tacitly intended). Clearly, although the first argument is predicatal and positive, while the second is subjectal and negative, they tell us exactly the same thing. The latter argument is just as predicatal in essence as the former; such traduction is just a change of wording. But, to repeat, it is still useful sometimes.

For example, consider the positive predicatal argument: “given that more money (R) is needed to buy a car (P) than to buy a bicycle (Q), it follows that if $1000 (S) is enough money (R) for a car (P), then it (S) is enough for a bicycle (Q).” This can be restated in the following negative subjectal form: “given that a car purchase (P) calls for more funds (new R) than a bicycle purchase (Q) does, it follows that if a car purchase (P) calls for funds (new R) not large enough that $1000 cannot effect it (S, in relation to P), then a bicycle purchase calls for funds not large enough that $1000 cannot effect it (S, in relation to Q).” We can similarly explain and exemplify the other three traductions; the reader should perhaps do that as an exercise.

The processes that go from subjectal arguments to predicatal ones are based on making a very abstract subsidiary term, “the subject concerned,” out of the subjects, P and Q; and fabricating two new major and minor items, “P is S” and “Q is S” (or their negations), out of the original major, minor and subsidiary terms. The middle term used here is R (or any relative of it, such as its negation[59]). We have in all four such traductions to consider.

From +s to –p and +p: starting with the following positive subjectal (minor to major) argument:

Given that P is more R than Q is, it follows that:
if Q is R enough (for Q) to be S,
then P is R enough (for P) to be S.

…we can construct the following two predicatal ones, whose equivalence is evidenced by their having the same net implications, viz. that Q and P are S:

Negative predicatal (minor to major) Positive predicatal (major to minor)
Given that more R is required for [P to be S] than for [Q to be S], it follows that: Given that more not-R is required for [Q to be S] than for [P to be S], it follows that:
if [the subject concerned (i.e. Q)] is R not enough for [Q not to be S], if [the subject concerned (i.e. Q)] is not-R enough for [Q to be S],
then [the subject concerned (i.e. P)] is R not enough for [P not to be S]. then [the subject concerned (i.e. P)] is not-R enough for [P to be S].

From –s to +p and –p: starting with the following negative subjectal (major to minor) argument:

Given that P is more R than Q is, it follows that:
if P is R not enough (for P) to be S,
then Q is R not enough (for Q) to be S.

…we can construct the following two predicatal ones, whose equivalence is evidenced by their having the same net implications, viz. that P and Q are not-S:

Positive predicatal (major to minor) Negative predicatal (minor to major)
Given that more not-R is required for [P not to be S] than for [Q not to be S], it follows that: Given that more not-R is required for [Q to be S] than for [P to be S], it follows that:
if [the subject concerned (i.e. P)] is not-R enough for [P not to be S], if [the subject concerned (i.e. P)] is not-R not enough for [P to be S],
then [the subject concerned (i.e. Q)] is not-R enough for [Q not to be S]. then [the subject concerned (i.e. Q)] is not-R not enough for [Q to be S].

Note that the move from +s to –p feels more natural than that from +s to +p, because in the former case, even though the minor premise and conclusion change polarity,  the polarity of the middle term and the movement from major to minor remain the same. For the same reasons, the move from –s to +p feels more natural than that from –s to –p. This is why I have listed the traductions in that order.

We can strongly criticize this set of four traductions as we did the preceding set, and even more so. The major premises of the derived arguments are credible enough, merely regrouping information already present in the original arguments. The problem lies rather in the new minor premises and conclusions. What is problematic in the latter is not the middle term R (or its relative, not-R) or the predicated items, viz. “P is S” and “Q is S” (or their negations), which are clearly intended in the given arguments – the problem lies in the new subsidiary term.

This very abstract term, viz. “the subject concerned,” is intended as a single substitute for the two original subjects, P and Q. We need a single term in this role, because an a fortiori argument cannot have more than one subsidiary. If there are two subsidiaries, the argument becomes invalid. So a vague term is introduced, “the subject concerned,” which tacitly refers to a term (P or Q) which is present elsewhere in the same suffective proposition, namely within a new predicated item, viz. “P is S” and “Q is S” (or their negations), as the case may be.

This verbal artifice allows us to make predicatal arguments out of subjectal ones. But of course, it is a bit of a sleight of hand, because we cannot really understand the abstract term without mentally referring to the P or Q it stands for. Therefore, the tacitly intended P or Q remains the effective subject of the proposition concerned, even if we have hidden it away. So we must admit either that the argument is fallacious (having two subsidiary terms) or that it is not what it seems. That is to say, in the latter case, though we have reformulated the given subjectal argument in such a way that it now looks like a predicatal argument, the thought process involved is still really subjectal. Verbally, on the surface of thought, the argument may seem predicatal, but logically, in the depth of thought, it is quite subjectal. Although such rewording is theoretically a dead end, it can still as earlier indicated be useful for the practical purpose of interpretation.

To analyze one of the listed four traductions, consider for instance the first, viz. “from +s to –p.” Note that both the arguments involved are from minor to major, and that both imply that P and Q are S. The middle term of the first is R, but the middle term of the second is “R in the subject.” The new subsidiary term is the very abstract “there,” and the new minor and major items are “Q is not S” and “P is not S,” which contain the original minor and major terms (Q and P), respectively, to which the original subsidiary term (S) is negatively predicated. Since the new middle term is not enough to entail these negative items, it follows that Q and P are both S.

For example, consider the positive subjectal argument: “given that selling a car (P) generates more income (R) than selling a bicycle (Q) does, it follows that if selling a bicycle (Q) generates income (R) enough to buy a new suit (S), then selling a car (Q) generates income (R) enough to buy a new suit (S).” This can be restated in the following negative subjectal form: “given that more income generation is required for [a car sale to enable a suit purchase], than for [a bicycle sale to do so], it follows that if the subject concerned (here, sale of a bicycle) generated income not enough for [a bicycle sale not to enable purchase of a new suit], then the subject concerned (here, sale of a car) generated income not enough for [a car sale not to enable purchase of a new suit].” We can similarly explain and exemplify the other three traductions; the reader should perhaps do that as an exercise.

To conclude this topic: uniform traductions (purely subjectal or purely predicatal ones) may be qualified as logical processes, whereas mixed traductions (from predicatal forms to subjectal ones, or vice versa) are rather verbal than truly logical.

Mongrel arguments. In this context, somewhat incidentally, I would like to draw attention to a mistake I have often found myself making when attempting to interpret examples of a fortiori argument in formal terms, i.e. when trying to fit them into some standard form. What happens is that we formulate an argument of mixed form; that is, mixing a subjectal major premise with a minor premise and a conclusion of predicatal form, or mixing a predicatal major premise with a minor premise and a conclusion of subjectal form. This produces arguments like the following positive copulative moods:

P is more R than Q is,
and S is R enough to be P;
therefore, S is R enough to be Q.
More R is required to be P than to be Q,
and Q is R enough to be S;
therefore, P is R enough to be S.

Such mixtures are best described as mongrels. The problem with them is not so much the order of the major and minor terms in the minor premise and conclusion, i.e. whether P is inferred from Q or Q is inferred from P; for the order could be changed. The problem is that the major and minor terms are subjects in the major premise and then predicates in the next two propositions, or predicates in the major premise and then subjects in the next two propositions. This is a problem, because it makes validation of these arguments impossible. Moreover, such arguments feel unnatural and unconvincing.

Note that the major premise of the first argument may also be stated as “More R is involved in being P than in being Q,” which gives it a more predicatal air. Similarly, the major premise of the second argument may also be stated as “(to be) P requires more R than (to be) Q does,” which gives it a more subjectal air. But such verbal reconstructions do not affect the essence of the matter. The conceptual difference between subjectal and predicatal argument is clear-cut, and the two forms should not be confused or mixed. As well, very often when one does this, the terms, though closely related, are not quite the same in the major premise on the one hand and in the minor premise and conclusion on the other hand. One should always make sure the terms are identical.

4.  Apparently variant forms

The four copulative and four implicational moods of a fortiori argument described earlier should be viewed as representative of this form of argument, but obviously not as limiting its precise possible contents. They are theoretical models, by way of which we can test whether cases encountered in practice are ‘true to form’, i.e. valid, or not so. For in both types we are concerned with a broader range of propositional forms than may appear at first sight. We shall here describe in some detail some of the variations on the two theoretical themes that we may encounter in practice, and then we will enquire as to whether or when mixtures of them are conceivable[60].

1.    Variations in form and content

In copulative arguments. I have called the first four moods ‘copulative’ because they involve categorical relations indicated by the copula ‘is’ (or ‘to be’). But it should be clear that they could equally well involve other categorical relations; also, negative polarity may be involved and non-actual modalities (can, must, and different probabilities in between) of various modes (de dicto or various types of de re). To give an example: “If this man can run two miles so fast, he can surely run one mile just as fast” (positive predicatal) may be counted as a copulative argument; the effective copula (the relation between terms) here is ‘run’ and the modality (qualifying the relation) is ‘can’, the terms being ‘this man’ and ‘a distance of one or two miles in a given lapse of time’. Moreover, past, present or future tenses may be involved, in various combinations, provided the major premise justifies it. For example: “If a man is that strong when old, he was surely as strong or stronger when younger.”

The verb typically used to relate subjects and predicates in copulative a fortiori arguments is “is” or “to be.” This can be taken very broadly to refer to any classification. But in practice, most verbs can be used here: to have (some quality or entity), to do (some action or go through some process), or whatever, with any object or complement, provided the statement can credibly be recast in the standard form (this process is called permutation[61]). This is true not only of a fortiori argument, but equally of syllogism and other forms of argument; so it requires no special dispensation. For example: “she sings Mozart well” can be recast as “she is a [good Mozart singer].” Sometimes, permutation is formally not possible, or at least not without careful consideration; for instances, the relations of ‘becoming’ and ‘making’ (or ‘causing’) cannot always be permuted.

One or more of the verbs involved may be of negative polarity. Be especially careful when one term is negated and another is posited, for this can confuse. In such cases, i.e. when in doubt, we can ensure that the argument is true to form (i.e. valid) by obverting the predicate(s) concerned. Obversion is permutation of the negation, passing it over from the copula to the predicate. For examples: instead of “is required not to be P” we would read “is required to be nonP;” or instead of “enough not to be S,” read “enough to be nonS.” The middle term (R) may likewise be negative in form, provided it is consistently so throughout the argument. However, if the major premise is negative, as in “P is not more R than Q” or “More R is not required to be P than to be Q,” no such obversion of R is acceptable, although we may be able to convert the comparison involved from “more” to “less” (though in such case check carefully that the minor premise and conclusion are true to form).

Natural, temporal or spatial modality may be introduced in a fortiori argument; i.e. the predications involved may be modal and not merely actual. For example, in the major premise “More R is required to be able to be P than to be able to be Q” (and similarly in the minor premise and conclusion). In such case, I would say that the modality has to be looked on as part of the effective term. In our example, the effective major and minor terms are not really P and Q, but “able to be P” and “able to be Q.” That is, here too permutation of sorts is used to verify that the inference is true to form. So the natural modality is operative here rather as in extensional conditional propositions, than as in categoricals[62].

Logical and epistemic modalities, as well as ethical and legal modalities, are considered separately further on.

In implicational arguments. Similarly, though I have called the second set of four moods ‘implicational’, the relation ‘implies’ involved in them should not be taken in a limited sense, with reference only to logical implication. For it is obvious that, if the moods are valid for that mode, similar moods can be constructed and validated for other modes of conditioning, such as the extensional or the natural (to name two examples)[63]. Indeed, we can apply them more broadly still to a wide range causal propositions (concerning causation, volition or influence, notably). Thus, all sorts of relational expressions might appear in practice in lieu of ‘implies’ provided such a link is ultimately subsumed.

In implicational a fortiori argument, the items P, Q, R and S stand for theses instead of terms. It is clear that any categorical relation may be involved in these clauses – whether the copula ‘is’ or any other, whether positive or negative, whether actual or modal – and indeed ultimately any non-categorical relation. The proviso is that the claimed relations between the clauses and the middle thesis R be indeed applicable (which is not always the case, of course). The logical (‘de dicto’) relation of implication is the basic bond in such argument, but this may be replaced by any ‘de re’ relation that suggests it – such as natural, temporal, spatial or extensional modes of conditioning, or more broadly by the same various modes of causation (including logical causation, of course), and more broadly still (though in such cases the underlying bond becomes more tenuous, a probability rather than a certainty of sequence) by volition or influence.

The following is a sample of thoroughly causal a fortiori (positive antecedental). The important thing to realize here is that the a fortiori argument per se has nothing to do with causality. It takes the truth of the premises for granted and merely tells us the conclusion from them, on the basis of given quantitative relations to the middle term or thesis. It is an a fortiori argument, and not a causal argument.

P causes more R than Q does,
and, Q causes enough R to cause S;
therefore, P causes enough R to cause S.

To give an example: “The car’s good looks generate more sales than its technical features do; and, its technical features generate enough sales to keep the company afloat; therefore, its good looks generate enough sales to keep the company afloat.” Here, the causal relations of generation and maintenance (keeping) replace the logical relation of implication.

Transformations. We may also note in this context that often (though not always) the same a fortiori argument can at will be credibly worded either in copulative form or in implicational form. If intelligently articulated, such transformations do not vitiate the argument. Consider for instance the following argument:

A being C implies more E in it than B being D does,
and, B being D implies enough E in it to imply it to be F;
therefore, A being C implies enough E in it to imply it to be F.

Here, we have two subjects A and B (which may be the same subject, in some cases) with four different predicates C, D, E, F, brought together in truly implicational form. Notice that the middle term is “E in it” – i.e. it refers the predicate E to a corresponding subject, and not to just-any subject. In other words, it signifies the effective middle thesis to be variously “B is E” or “A is E,” as the case may be. The argument can obviously be restated in truly copulative form, as follows:

AC is more E than a BD is,
and, BD is E enough to be F;
therefore, AC is enough E to be F.

The terms AC and BD refer respectively to “A when it is C” and “B when it is D.” This is valid transformation, provided the middle item E suggests a thesis in the implicational form (as above clarified) and a term in the copulative form. As we saw earlier, a middle thesis per se, being a proposition, cannot vary; so that when we say that more or less of it is implied, we always have in mind something within it that varies – usually a term (though not always). So, when we transform the implicational form into a copulative one, we have to identify precisely which content of the middle thesis to use as our middle term. We could similarly, of course, transform the copulative argument into the implicational one, if we proceed carefully.

In practice, it does not matter so much exactly how we word our argument, in implicational or copulative form, provided it ends up matching a valid form. The human brain is very clever and able to assimilate large variations in wording with little difficulty (though it can also, of course, be misled). So, we should not view the division too rigidly. However, such transformations are not always possible: we may have difficulty restructuring the middle item, or at least some information might be lost or might have to be added in the process. So we are justified in regarding copulative and implicational species of a fortiori argument as essentially distinct, even if in some special cases they can be transformed into each other.

2.    Logical-epistemic a fortiori

We need to distinguish between purely ‘ontical’ (or de re) a fortiori argument and more ‘logical-epistemic’ (or de dicto) ones. The adjective ‘ontical’[64] (from Gk. ontos, meaning ‘existence’) applies to the objects of ontology, the study of being, just as the adjective ‘epistemic’ (from Gk. episteme, meaning ‘knowledge’) applies to the objects of epistemology, the study of knowing. Ontical thus characterizes the things we allegedly know, whereas epistemic characterizes our alleged knowledge of them. Clearly, these terms are relative, in that something epistemic may be intended ontically inasmuch as it exists too.

A logical-epistemic a fortiori argument is one applying logical and/or epistemic qualifications to some relatively ontical information. A ‘logical’ qualification logically evaluates the proposed information in a given context of knowledge: it may logically evaluate a term as conceivable, significant, clear, precise, well-defined, and so forth, to various degrees, or the same in negative connotation; or it may evaluate a proposition through a modality with degrees[65] like probable, confirmed, evident, consistent, true, or their negations. An ‘epistemic’ qualification concerns the state of belief, opinion or knowledge of the speaker rather than the content spoken of or its purely logical evaluation; this refers to characterizations like credible, reliable, believable, understandable, to varying degrees, and their negative equivalents.

The distinction can be tested as follows: E.g. for ‘credible’ when we ask ‘to whom?’, we can answer ‘to this person’, or ‘to most people’, or ‘to everyone’, signifying that the issue is relatively subjective; whereas for ‘probable’, we would refer to a more objective issue, such as how often similar subjects have the same predicate.

A logical or epistemic thesis, then, is one which predicates such a logical or epistemic term to an ontical term or thesis. E.g. ‘term X is vague’, ‘thesis X is probable’ are logical propositions, ‘term X is generally understood’, ‘thesis X is widely believed’ are epistemic propositions. Of course, logical and epistemic propositions are in a sense themselves ontical; but they are always relative to information which is more ontical.

The following are examples of purely copulative logical-epistemic a fortiori argument, the first being subjectal and the second predicatal:

Term P is ‘better defined’ (R) than term Q is,
and, term Q is well defined (R) enough to be ‘comprehensible’ (S); therefore,
all the more, term P is well defined (R) enough to be comprehensible (S).
‘Better definition’ (R) is required of a term to ‘pinpoint its instances’ (P) than to ‘be comprehensible’ (Q),
and, term S is well defined (R) enough to pinpoint its instances (P); therefore,
all the more, term S is well defined (R) enough to be comprehensible (Q).

Note that both samples involve only terms (i.e. they are not hybrid) and both have as their middle term R the logical qualification ‘well defined’. In the subjectal example, R characterizes the terms P and Q, whereas in the predicatal example it characterizes the term S. In the subjectal example, the subsidiary term S is ‘comprehensible’, an epistemic qualification suitably related to R, and its major and minor terms P and Q are ontical (at least, relative to the two other terms). In the predicatal example, the major and minor terms P and Q are logical (‘pinpoint its instances’) or epistemic (‘comprehensible’) qualifications suitably related to R, while the subsidiary term S is (at least relatively) ontical.

The following are examples of purely implicational logical-epistemic a fortiori argument, the first being antecedental and the second consequental:

Thesis P implies more ‘correct predictions’ (R) than thesis Q is,
and, Q implies correct predictions (R) enough to imply that ‘thesis A is probably true’ (S); therefore,
all the more, P implies correct predictions (R) enough to imply that thesis A is probably true (S).
More ‘correct predictions’ (R) are required to imply ‘thesis A probably true’ (P) than to imply ‘thesis B probably true’ (Q),
and, thesis S implies correct predictions (R) enough to imply that thesis A is probably true (P); therefore,
all the more, S implies correct predictions (R) enough imply that thesis B is probably true (Q).

Note that both samples involve only theses (i.e. they are not hybrid) and both have as their middle thesis R the logical proposition that ‘many of its predictions are correct’[66]. In the antecedental example, R characterizes the theses P and Q, whereas in the consequental example it characterizes the thesis S. In the antecedental example, the subsidiary thesis S is the logical proposition, suitably related to R, that ‘thesis A is probably true’, while the theses P and Q are (at least relatively) ontical. Here, the probability of Q due to correct prediction is declared in the minor premise high enough to imply A probable; therefore, given the major premise, the same can be concluded with regard to P and A. In the consequental example, the theses P and Q are the logical propositions, suitably related to R, that ‘thesis A is probably true’ and ‘thesis B is probably true’, respectively, while the subsidiary thesis S is (at least relatively) ontical. Here, the probability of S due to correct prediction is declared in the minor premise high enough to imply A probable; therefore, given the major premise, the same can be concluded with regard to S and B.

Though all the above examples are positive, we can easily construct similar arguments in negative form. In all of them, the logical-epistemic middle item (R) may be viewed as the basis of the deduction, and the suitably related logical-epistemic subsidiary item (S) or major and minor items (P and Q) may be viewed as the goal of the deduction; the remaining item(s) usually have ontical content, though they may in special cases (when that is what is discussed) be logical or epistemic too.[67]

These four samples make clear that logical-epistemic a fortiori arguments function like purely ontical a fortiori argument; there is nothing special about them, other than the logical-epistemic nature of some of the items involved. Nevertheless, such arguments seem rare; or at least, I find it difficult to formulate many examples of them. The matter gets more complicated when we, further on, look into ‘hybrid’ a fortiori arguments, which seem to involve mixtures of terms and theses.

3.    Ethical-legal a fortiori

In my book Judaic Logic[68], I showed that, although the eight moods of a fortiori argument listed earlier are formulated very generically, they can be adapted to ethical or legal a fortiori argumentation. Generally, the middle item R may be any quantitative factor shared in some way by the other three items. In ethical or legal argument, this common thread will be specifically an ethical/legal characterization, or a proposition involving such characterization, by which I mean expressions like desirable, advantageous, useful, valuable, good, moral, ethical, legal, obligatory, demanding, important, stringent, and so on – and their negative versions – all of which, note well, have degrees. Coupled with that, either the subsidiary item or the major and minor items must refer to a physical, mental or spiritual action or event related to the ethical-legal qualification; for examples, as something desirable is sought after, or something good is preferred. The remaining item(s) are not ethical-legal in content.

A fortiori arguments involving such ethical or legal expressions must be examined and evaluated carefully, because these characterizations are rather vague and complex. One can easily err using them if one does not take pains to clarify just what they are intended to mean in each case. Consider, for instance, the following subjectal argument:

P is more valuable (R) than Q,
and, Q is valuable (R) enough to make A imperative (S);
therefore, all the more, P is valuable (R) enough to make A imperative (S).

This argument can be interpreted and rewritten as follows:

  • Major premise means: ‘P does more to produce some value R than Q does’,

which in turn means:

‘P produces R to degree RP’, and ‘Q produces R to degree RQ’, and

‘RP is greater than RQ’ – whence, ‘if RP then RQ’.

  • Minor premise means: ‘Q produces R to degree RQ’, and

‘if RQ then S (= the term ‘makes A is imperative’)’.

  • Conclusion means: ‘P produces R to degree RP’ (given), and

‘if RP then S’ (since RP implies RQ, and RQ implies S).

Alternatively, it might be read and rendered negatively, as follows:

  • Major premise means: ‘nonP does more to inhibit some value R than nonQ does’,

which in turn means:

‘nonP inhibits R to degree nonRnonP’, and ‘nonQ inhibits R to degree nonRnonQ’ , and

‘nonRnonP is greater than nonRnonQ’ – whence, ‘if nonRnonP then nonRnonQ’.

  • Minor premise means: ‘nonQ inhibits R to degree nonRnonQ’, and

‘if nonRnonQ then S (= the term ‘makes A is imperative’)’.

  • Conclusion means: ‘nonP inhibits R to degree nonRnonP’ (given), and

‘if nonRnonP then S’ (since nonRP implies nonRQ, and nonRQ implies S).

Sometimes, both these interpretations are intended together. P and Q are two values; and S is some trait or behavior that is being recommended, say. The important factor here is of course the middle term R, which is implicit in the expression ‘valuable’. What does it mean to be more or less valuable, or valuable enough? This has to refer to some causal concept – namely, the positive concept of production and/or the negative concept of inhibition. Where did R come from? It is implicit in the concept of value that something is valuable relative to some standard of value – call it R. So ‘valuable’ means valuable in the pursuit of (say) R.

What does ‘makes A is imperative’ (S) mean? It means that A is absolutely necessary for some unstated goal – or more probably for the ultimate goal here sought, namely R. However, note well, the necessity of A here referred to does not play any part in the actual a fortiori inference. The subsidiary item here is really not just A but the whole clause S (i.e. ‘makes A is imperative’). Another such term like ‘makes A allowed’ or even ‘makes A not imperative’ or ‘makes A forbidden’ could equally well have occurred in that position without affecting the argument as a whole. Clearly, then, the conclusion can be formally inferred from the given premises, so the a fortiori argument as a whole is valid.

Of course, many questions can be asked about how we come to know the premises in the first place. The hierarchy of values P and Q proposed in the major premise has to be justified; and why the minor value Q implies the imperativeness (or whatever) of ‘A’ is not here explained (but taken for granted at the outset). The scale of values on which P and Q are measured could be a merely subjective scale, or one based on biological considerations, or again one based on spiritual ones. ‘A’ might for instance be a cause of Q, P and/or R, though need not be. But these issues stand outside the a fortiori reasoning as such. The a fortiori argument as such does not need more information than the said premises give to draw the said conclusion – provided that the message of each premise and of the conclusion are well understood.

Let’s look at another sample, for instance the predicatal argument:

More ‘virtue’ (R) is required to be (or have or do) P than to be (or have or do) Q,
and, S is virtuous enough to be (or have or do) P;
therefore, all the more, S is virtuous enough to be (or have or do) Q.

In this case, S refers to a person supposedly, and P and Q to character traits, or maybe behavior patterns, which require different degrees of ‘virtue’ (by S) to achieve. Here, the middle term ‘virtue’ has to be understood in a sufficiently uniform manner that the inference becomes possible. Obviously, if it means something different in each proposition – say, courage in one and perseverance in another – we cannot logically draw the conclusion from the premises. Here again, then, caution is called for.

Apart from these words of warning, much the same can be said for ethical-legal a fortiori as was said regarding logical-epistemic a fortiori, so I won’t repeat myself here.

4.    There are no really hybrid forms

I have already shown that my inventory of copulative and implicational a fortiori arguments is in principle exhaustive[69], i.e. that ‘hybrid’ arguments are formally non-existent even if we often in everyday discourse seem to make use of them. The main reason given was that a standalone term cannot imply or be implied by a whole proposition. Terms can only be subjects or predicates; only theses can be antecedents or consequents.

This is true notwithstanding the fact, which we admitted, that since a thesis as such cannot have degrees like a term, the middle thesis of implicational arguments must be examined carefully, to determine what it is in it that is variable (i.e. more, equal or less, or sufficient or insufficient). The variable factor may be a subject or a predicate or a quantity or a modality, or a compound of such elements.

Thus, we can safely say that, formally speaking, there are no hybrid a fortiori argument. There are in principle no partly copulative and partly implicational a fortiori arguments. The four items P, Q, R, S of such arguments are necessarily either all terms (i.e. the main constituents of propositions) or all theses (i.e. propositions of whatever form, constituted by terms). Even if in everyday speech we often give the impression that terms and theses can be mixed indiscriminately, there is always some unspoken intent that explains the illusion. Some commentators have nevertheless tried, wittingly or unwittingly, to propose hybrid forms like the following:

P is more R than Q is,
and, Q is R enough to imply S;
therefore, P is R enough to imply S.

In the above ‘mostly subjectal’ example, S seems to be a consequent of Q and P, although they seem to be subjects of predicate R. The solution may be that S is in fact a term, and what is thought of as implied is the thesis ‘it (i.e. the subject Q or P, as appropriate) is S’. Alternatively, if S is in fact a thesis, it contains ‘it’ (which refers to Q or P, as appropriate) as subject and some additional term (here tacit) as predicate.

More R is required to be P than to be Q,
and, S implies R enough to be P;
therefore, S implies R enough to be Q.

In the above ‘mostly predicatal’ example, S is both antecedent and subject, since it both implies R and is P and Q. Here, the solution may be that R is in fact a term, and by ‘S implies R’ is meant simply ‘S is R’. Alternatively, if R is in fact a thesis, the thought may be that some proposition of which S is the subject (and whose predicate is here tacit) implies R.

P implies more R than Q (implies R),
and, Q implies R enough to be S;
therefore, P implies R enough to be S.

In the above ‘mostly antecedental’ example, P and Q seem to be both antecedents and subjects, since they both imply R and are S. The solution here may be that P, Q and R are indeed theses, and ‘to be S’ is intended to mean ‘to imply it (i.e. the subject, here tacit, of thesis Q or P, as appropriate) to be S’.

More R is required to imply P than to imply Q,
and, S is R enough to imply P;
therefore, S is R enough to imply Q.

In the above ‘mostly consequental’ example, S is both subject and antecedent, since it both is R and implies P and Q. Here, the solution may be that R is in fact a thesis, and by ‘S is R’ is meant ‘S implies R’; or maybe, ‘the subject (here tacit) of S has the predicate given (here tacitly) in R’. Alternatively, if R is in fact a term, ‘S is R’ might signify ‘the subject (here tacit) of S is R’.

On the surface, the above four examples may seem conceivable, because we are dealing in symbols. But if we examine them more closely we find that appearance misleading. For it is a rule of logic that the same item cannot at once be a term and a thesis, as occurs in all of the above proposed moods. So these hybrids are not valid forms, strictly speaking. In each of them, some intent has been left tacit or some verbal or conceptual confusion occurred in the formulation. Nevertheless, it should be kept in mind that in practice we often do so word our sentences as to give the impression that we are mixing copulative and implicational clauses. This is occasionally confusing, but not always.

Let us analyze some more specific cases where confusion or doubt might occur in practice. These are mostly logical-epistemic or ethical-legal arguments that look partly implicational but are in fact wholly copulative. The reason such hybrid-looking arguments arise is that in them a thesis may actually function as (a) a subject-term or (b) a predicate-term.

(a)        In the propositions “X is probable” or “X is desirable,” where ‘X’ is a thesis, say ‘that A is B’, and ‘probable’ or ‘desirable’ is a predicate, thesis ‘X’ may be said to function effectively as a term (a subject), because it is taken as a unitary whole rather than as composed of parts.

For example, consider the a fortiori argument “Given that ‘A is B’ is more probable than that ‘C is D’, it follows that if ‘C is D’ is probable enough to be relied on, then ‘A is B’ is probable enough to be relied on.” We might here think that since ‘A is B’ and ‘C is D’ are theses (the major and minor, respectively), the argument is implicational. On the other hand, since ‘probable’ and ‘relied on’ are terms (the middle and subsidiary, respectively), the argument seems copulative. The solution is not that the argument is hybrid, but that the major and minor theses are in this context intended as terms – i.e. they are the subjects for which the middle and subsidiary terms are predicates. Thus, the form of the argument is really subjectal, and not antecedental or hybrid.

The following is an example of predicatal form with similar effect. “More satisfaction of inductive criteria (R) is needed to adopt a thesis (P) than to merely conceive it possible (Q); and, thesis S satisfies inductive criteria (R) enough to be adopted (P); therefore, thesis S satisfies inductive criteria (R) enough to be conceivable (Q).” Here, although S is a thesis (say, ‘that A is B’), it functions in the present context as a term (a subject), for which R, Q and P are indeed predicates. So, the form of the argument is really predicatal, and not consequental or hybrid.

(b)        Again, looking the propositions “X makes Y probable” or “X makes Y desirable,” where ‘X’ is a term, and ‘Y’ is a term or a thesis, say ‘that A is B’, and ‘probable’ or ‘desirable’ is a predicate, we are tempted to view the relation ‘makes’ as equivalent to an implication (which it indeed implies) and the combination ‘Y is probable’ or ‘Y is desirable’ as an implied thesis, in which case the given proposition as a whole seems to be implicational. However, because X is a subject-term (noun), we have to look upon ‘makes’ as a mere copula (verb) and upon the thesis made, i.e. ‘Y is probable’ or ‘Y is desirable’, as a predicate-term (object).

An example of this would be the following argument: “Term P is more well-defined (R) than term Q; and, term Q is well-defined (R) enough to ‘make term or thesis A conceivable or credible’ (S); therefore, term P is well-defined (R) enough to ‘make term or thesis A conceivable or credible’ (S).” This argument might be interpreted as partly copulative (since P, Q, and R are terms) and partly implicational (since S seems to refer to an implication, i.e. a thesis). But in fact it is wholly copulative, because S is a term, i.e. the clause ‘makes term or thesis A conceivable or credible’ must be taken as a unit and not be cut up. This example is thus subjectal.

A similar predicatal example would be the following: “More precision of definition (R) is required to ‘make term or thesis A comprehensible’ (P) than to ‘make term or thesis B comprehensible’ (Q); and, term S is precisely defined (R) enough to make A comprehensible (P); therefore, term S is precisely defined (R) enough to make B comprehensible (Q).” Here, the argument might be interpreted as partly copulative (since S and R are terms) and partly implicational (since P and Q seem to refer to implications, i.e. theses). But in fact it is wholly copulative, because P and Q are terms, i.e. the clauses ‘make term or thesis A/B comprehensible’ must be taken as units and not be cut up.

All the above examples involve logical-epistemic qualifications. We can similarly construct hybrid-looking arguments with ethical-legal qualifications. E.g. “That ‘A be B’ (P) is more desirable (R) than that ‘C be D’ (Q); and, Q is desirable (R) enough to be pursued regularly (S); therefore, P is desirable (R) enough to be pursued regularly (S).”

In conclusion, hybrid a fortiori argument do not really exist: when they do seem to occur, as they often enough do in logical-epistemic or ethical-legal contexts, it is due to some thesis being taken as a whole, i.e. as effectively a term.

5.    Probable inferences

Very often in practice, though the given argument somehow seems to be an a fortiori, it is really not one at all. We may upon closer scrutiny decide that it is more precisely a hypothetical syllogism or an apodosis. Very often we are misled by expressions like ‘all the more’ indicative of a fortiori argument being inappropriately used in other forms of argument. Inversely, an argument may on the surface not look like an a fortiori at all, but really be one deeper down. Caution is always called for in interpreting arguments. We have to ask what form the underlying reasoning takes, irrespective of the wording used. In some cases, of course, no reasoning is at all intended; yet some people might assume an a fortiori argument to be intended, because a comparison or a threshold is mentioned. We have to always ask how the speaker intends his statement to be taken.

As just stated, some arguments do not immediately appear to be in standard a fortiori format, although one senses that there is an a fortiori ‘flavor’ to them. Consider the following arguments: Are these a fortiori in nature or something else? How are they to be validated?

Copulative form (X, Y, Z are terms):

X more often occurs in Y than in Z; therefore:
If X is found in Z, it is probably also in Y (positive mood), and
If X is not found in Y, it is probably also not in Z (negative mood).

Implicational form (X, Y, Z are theses):

X more often occurs in conjunction with Y than with Z; therefore:
If X is found in conjunction with Z, it is probably also with Y (positive mood), and
If X is not found in conjunction with Y, it is probably also not with Z (negative mood).

These closely resemble a fortiori argumentation. There are copulative and implicational forms (four in all), the former involving terms and the latter theses. In each case, the first proposition is the major premise, and the if–then propositions which follow it contain a minor premise (the antecedent) and a conclusion (the consequent). There is a positive and a negative mood, the positive one being minor to major and the negative one major to minor. However, these arguments as they stand are obviously not in standard form. They need to be reformulated to conform.

If such argument is to be viewed as a variant of a fortiori, the middle term has to be “the probability of occurrence,” while the subsidiary term has to be “the actuality of occurrence.” The major premise, which tells us that “X is in/with Y” occurs more frequently than “X is in/with Z,” means that the former is more probable than the latter. The minor premise, which tells us that “X is in/with Z” has occurred, or that “X is in/with Y” has not occurred, refers to the actuality of occurrence or lack of it. And the conclusion predicts that “X is in/with Y” has probably also occurred, or respectively that “X is in/with Z” has probably also not occurred, again with reference to the actuality or inactuality of occurrence. We can thus reformulate the arguments as follows to bring out their ‘a fortiori’ aspect more clearly:

Positive mood (copulative [in] or implicational [with]):

‘X is in/with Y’ is more probable than ‘X is in/with Z’, and
‘X is in/with Z’ was probable enough to actually occur (at a certain time);
therefore: ‘X is in/with Y’ is probable enough to actually have occurred or to later occur.

Negative mood (copulative [in] or implicational [with]):

‘X is in/with Y’ is more probable than ‘X is in/with Z’, and
‘X is in/with Y’ was not probable enough to actually occur (by a certain time);
therefore: ‘X is in/with Z’ is not probable enough to actually have occurred or to later occur.

Note the introduction, in this improved formulation, of the crucial notion of sufficiency (“enough”) or its absence, in accord with standard a fortiori format. So we can say that we here indeed have a fortiori arguments. The major and minor items P and Q are in this case the theses “X is in/with Y” and “X is in/with Z,” respectively. The middle and subsidiary items R and S are the terms “probably” and “actually occurs.” So the a fortiori argument involved, mixing theses and terms, is a hybrid-seeming one (although strictly-speaking it is wholly copulative, the theses in it being taken as terms). It is a logical a fortiori argument, probability and actuality being modalities.

Note well that the prediction of the conclusion should not be taken as a certainty. The argument makes no pretense to yield anything more than a probable conclusion, the degree of probability being that specified – clearly or vaguely – in the major premise. Though presented as a sort of deduction, the argument is essentially inductive. It could well be that the situation in fact, on the ground, is the opposite of what the argument predicts. Nevertheless, if the only information we have at our disposal is that given in the argument, it is reasonable to adopt the conclusion’s prediction as our ‘best bet’. We have more rational basis for expecting the outcome that the conclusion predicts than we have for expecting the contradictory outcome.

Certainty from mere probability. I would like to draw attention in the present context to the fallacy inherent in certain probabilistic a fortiori arguments, namely those that seem to infer a certainty from a mere probability. The following two arguments, one positive and one negative, illustrate this pitfall:

Thesis P is more probable (R) than thesis Q,
and, thesis Q is probable (R) enough to imply thesis S;
therefore, thesis P is probable (R) enough to imply thesis S.
Thesis P is more probable (R) than thesis Q,
and, thesis Q is probable (R) enough to deny thesis S;
therefore, thesis P is probable (R) enough to deny thesis S.

In these hybrid-looking arguments, the items P, Q and S are theses and R is a logical-epistemic term (it is logical if ‘probable’ here means ‘demonstrably likely to be true’, but epistemic if it merely means ‘believed by many people’). As we have seen, this apparent mix is not necessarily a problem, because theses may in such contexts be intended as (i.e. effectively function as) terms. However, in these two particular cases, the mix is a problem, because the subsidiary item (S) is definitely implied (or denied, i.e. its negation is implied). Since the implication (or denial) is quite intentional, it cannot be written-off as a badly-worded predication.

At first sight, the proposed argument may seem meaningful and credible; but upon closer scrutiny it is found fallacious. The main reason why it is fallacious is that in logic theory no propositions literally imply others when they (the implying ones) are more or less probable. In deductive logic, either a proposition X (Q or P in our example) implies another Y (S or notS, here) or it does not – there is no such thing as X implying Y if X is probable to a sufficient degree, and X not implying Y if X is not probable to that degree. Even in inductive logic, such a concept is unknown – there is only the concept of transmission of probability, i.e. if X implies Y, then increasing the probability of X being true increases that of Y being true.

As for degrees of implication, they are formally conceivable; but given that ‘X probably implies Y’, it does not follow that ‘if X is probable to some high degree it implies Y to be certain’. Rather, probable implication is to be treated as a weakened form of implication, meaning that whereas the form ‘X fully implies Y’ transmits the high probability of X to Y (and in the limiting case, if X is certain, then Y is also certain), the form ‘X only probably implies Y’ transmits only a fraction of X’s probability on to Y (i.e. here, if X is probable, then Y is ‘probably probable’). This can be expressed quantitatively: if X implies Y with probability m%, say; and X is itself only probable to degree n%, say; then the resulting probability of Y is only m% of n%.[70]

It should be added that it makes no difference whether the hybrid-seeming a fortiori argument involves an implication or a denial. It is fallacious either way. The principle of adduction that “no amount of right prediction ever definitely proves a hypothesis, but all it takes is a single wrong prediction to disprove it” has no relevance in the present context. Here, whether S is implied or denied the argument is invalid, because a probability cannot imply a certainty, whether positive or negative.

The lesson these examples teach us is that if we use a logical-epistemic middle term like ‘probable’, then we must also have a logical-epistemic term like ‘reliable’ contained in one or more of the other items of the a fortiori argument. Such terms occur together, not by chance but because their meanings have some rational relation (as probability rating is related to reliability). We cannot combine the middle term ‘probable’ with an assertion of the subsidiary item’s implication or denial. There is in fact no logical discourse corresponding to that schema. It is artificial and conceptually faulty, for the reason already stated that a certainty cannot be implied by a mere probability.

6.    Correlating ontical and probabilistic forms

Having examined the general forms of ontical a fortiori argument and various cases of more specifically logical-epistemic a fortiori argument, the question arises: can logical-epistemic arguments be constructed from ontical ones and/or vice versa? This question immediately comes to mind when we read Aristotle’s descriptions of a fortiori argument, of which the following are some extracts:

Rhetoric, book II, chapter 23:

“…if a quality does not in fact exist where it is more likely to exist, it clearly does not exist where it is less likely. Again, … if the less likely thing is true, the more likely thing is true also.”

Topics, book II, chapter 10:

“If one predicate be attributed to two subjects; then supposing it does not belong to the subject to which it is the more likely to belong, neither does it belong where it is less likely to belong; while if it does belong where it is less likely to belong, then it belongs as well where it is more likely.”

Here, Aristotle’s emphasis is clearly ‘epistemological’, since he repeatedly uses the word ‘likely’ as his middle term, yet judging by the examples he there gives the underlying subject-matter is arguably rather ‘ontological’. This suggests that there are natural bridges between the ontical and logical-epistemic expressions of a fortiori argument. Let us look into the matter with reference to one of Aristotle’s own examples, namely:

A man is more likely to strike his neighbors than to strike his father:
if a man strikes his father,
then he is likely to strike his neighbors too.

This example is clearly intended as logical-epistemic, since it uses the relative likelihood of events to achieve its inference. But one senses that underlying it is another, more ontical argument, such as the following (others could of course be suggested):

More antisocial attitude is required to strike one’s father than to strike one’s neighbor:
if a man is antisocial enough to strike his father,
then he is antisocial enough to strike his neighbor.

Aristotle’s logical-epistemic wording does not reveal to us precisely why the concluding event (man striking neighbors) is more likely than the given event (man striking father), whereas my proposed ontical wording attempts to explain these events and their connection through some psychological attribute (being antisocial) of the subject (a man). Aristotle’s effective middle term is a vague, unexplained ‘likelihood’ – whereas my ontical middle term (antisocial mentality) is more specifically informative as to the causes (different degrees of antisocial mentality) of the events (striking father or neighbor). One finds Aristotle’s argument convincing especially because one (consciously or unconsciously) assumes that there are ontical reasons (such as those I propose) behind the probabilities he declares.

Thus, our first question arises: can we always, in formal terms, similarly infer an underlying ontical a fortiori argument from a given logical-epistemic (probabilistic) one? The answer, I would say, is that we cannot formally infer one, but we can hope to construct one that would seemingly fit the bill, i.e. explain the predicated likelihoods by means of some material property or properties. That is to say, with reference to the following forms, given the one on the left we may, using our knowledge and intelligence to propose an appropriate middle term R, construct the one on the right:

Given probabilistic argument Constructed ontical a fortiori argument
‘S is P’ is more likely than ‘S is Q’: More R is required to be P than to be Q;
if ‘S is P’ occurs, and, S is R enough to be P;
then ‘S is Q’ is likely to occur too. therefore, S is R enough to be Q.

This reconstruction seems reasonable, at least where the original middle term is the degree of ‘likelihood’. But let us look into it more deeply. The given argument compares the likelihood of two events (theses) ‘S is P’ and ‘S is Q’ and tells us that if the more likely one indeed occurs then the less likely one is likely to occur too. Note well: it gives no guarantees as to this outcome, its conclusion being only probable though the minor premise is actual. Our proposed construct introduces a new term R that was not given in the original argument. R serves as middle term of our a fortiori, relative to which the predicates P and Q are compared in the major premise. R is a predicate of S. If the magnitude of R in S is large enough, then S is Q; and if its magnitude is even larger, then S is P. Whence, if S is P, it has to be Q.

Note that the conclusion ‘S is Q’ here is definite – it is not a mere probability as before. However, the proposed construct as a whole certainly cannot be inferred from the given argument. We can only posit our construct in the way of an inductive hypothesis that is hopefully fitting (if we have thought about it sufficiently), but which may turn out upon further experience and reflection to be inadequate (in which case it must be adapted or abandoned). So our new conclusion is not as sure as it appears. Still, once we have a seemingly credible construct, we can claim it (on inductive, not deductive grounds – to repeat) to be the underlying ontical explanation of the given logical-epistemic argument.

Can such ontical explanation be provided for all logical-epistemic a fortiori arguments, or only for middle terms like ‘likelihood’? I would offhand answer yes, arguing that we never use logical-epistemic characterizations entirely without reference to more ontical characteristics. That is, if we ask ourselves why we think a term or thesis deserves logical or epistemic evaluation X, we will argue the point ultimately with reference to some sort of more ontical information. Of course, we may have some such explanation in mind, but be unable to clearly put it in so many words, so this is difficult to prove in every case. Also of course, I am generalizing, since I cannot foresee all cases – so I may be found wrong eventually.

Now, let us turn the initial question around, and ask the reciprocal question: given an ontical a fortiori argument, can we formally derive from it a corresponding logical-epistemic argument (meaning, at least, a probabilistic argument similar to Aristotle’s)? And if so, we might additionally ask, is there great utility in doing so – or is valuable information lost in the process?

If we reflect a moment, it is clear that behind my contention that underlying Aristotle’s probabilistic argument there must be a more ontical argument that explains it – was the thought that Aristotle was really thinking in terms of the ontical argument even if he only verbalized a probabilistic one. So in fact the mental process we were looking for was in the reverse direction: not from logical-epistemic to ontical, but rather from ontical to logical-epistemic. We were not so much asking what ontical information can be drawn from Aristotle’s probabilistic argument (not a lot, as we have just seen), but what ontical argument Aristotle had in mind even as he spoke in probabilistic terminology. We want to retrace his thought process from the pre-verbal ontical thought to its verbal probabilistic expression.

Consider therefore the following pair of arguments, this time the one on the left being a given ontical a fortiori argument and the one on the right a proposed probabilistic construct:

Given ontical a fortiori argument Constructed probabilistic argument
More R is required to be P than to be Q; ‘S is P’ is more likely than ‘S is Q’:
and, S is R enough to be P; if ‘S is P’ occurs,
therefore, S is R enough to be Q. then ‘S is Q’ is likely to occur too.

If we examine these arguments carefully, we see that the latter cannot be inferred from the former. Of course, all information concerning R is lost in transition. But moreover, we have no basis for believing the major premise that ‘S is P’ is more likely than ‘S is Q’; for, given that ‘More R is required to be P than to be Q’, it could still be true that ‘S is P’ is less likely (i.e. occurs less frequently) than ‘S is Q’. The two minor premises are in agreement that ‘S is P’; but, whereas the conclusion of the given a fortiori is definite that ‘S is Q’, the conclusion of the construct is that ‘S is Q’ is merely probable. Thus, not only does the proposed construct’s major premise not follow from the given major premise, but the conclusion of the construct is less informative and sure than that of the original argument.

So, there is in fact no justification for supposing that an ontical a fortiori argument gives rise to a probabilistic argument as above proposed. The ontical argument does not formally tell us anything about the likelihood of the events it discusses. If such likelihood is asserted in an analogous probabilistic argument, it is new information (just as the middle term R was new information, in the opposite direction), which must be separately justified or admitted as a hypothesis to be assessed inductively (e.g. we would have to ask in Aristotle’s above example whether it is empirically true that men strike their neighbors more often than they strike their fathers). Moreover, to repeat, the proposed new argument involves loss of information (about R) and has a less certain conclusion (about S being Q).

So, if we suppose that Aristotle really had an ontical argument in mind when he formulated his probabilistic one, we may say that such discourse on his part was inaccurate and wasteful. Conversely, granting that he meant only what he said, we could read more into it provided we realize that such interpretation on our part is not deductive inference but inductive hypothesis. In short, the relation between ontical and logical-epistemic a fortiori arguments can be described as hermeneutical rather than strictly logical. We often in practice do blithely hop from ontical to probabilistic form or vice versa – but we ought to be careful doing so, because formal analysis shows that it is not always licit. In logic, even the word ‘likely’ means something specific and cannot be used at will.

5.  Comparisons and correlations

We need to be very clear about the differences between a fortiori argument and other forms of argument, with which a fortiori argument is often compared and even confused. Many commentators have wrongly characterized a fortiori argument as analogical argument[71], and many more as syllogism, and they need to be corrected once and for all. We shall begin our comparison and correlation of argument forms with regard to argument by analogy, and then deal with syllogism.

1.    Analogical argument

Just what form does argument by analogy have, and how does it differ from a fortiori argument? Can either of these forms be reduced to the other?

The forms of analogy. Our first task is to formalize analogical argument and identify the conditions of its validity. Qualitative analogical argument, like pure a fortiori argument, consists of four terms, which we may label P, Q, R, S, and refer to as the major, minor, middle and subsidiary terms as before, although here without implying that the major term is greater in any way than the minor. The argument may then take the following four copulative forms:

  1. The positive subjectal mood. Given that subject P is similar to subject Q with respect to predicate R, and that Q is S, it follows that P is S. We may analyze this argument step by step as follows:

Major premise: P and Q are alike in that both of them have R.

This implies both ‘P is R’ and ‘Q is R’, and is implied by them together.

Minor premise: Q is S.

The term S may of course be any predicate; although in legalistic reasoning, it is usually a legal predicate, like ‘imperative’, ‘forbidden’, ‘permitted’, or ‘exempted’.

Intermediate conclusion and further premise: All R are S.

This proposition is obtained from the preceding two as follows. Given that Q is S and Q is R, it follows by a substitutive third figure syllogism that there is an R which is S, i.e. that ‘some R are S’. This particular conclusion is then generalized to ‘All R are S’, provided of course we have no counter-evidence. If we can, from whatever source, adduce evidence that some R (other than Q) are not S, then of course we cannot logically claim that all R are S. Thus, this stage of the argument by analogy is partly deductive and partly inductive.

Final conclusion: P is S.

This conclusion is derived syllogistically from All R are S and P is R.

If the middle term R is known and specified, the analogy between P and Q will be characterized as ‘complex’; if R is unknown, or vaguely known but unspecified, the analogy between P and Q will be characterized as ‘simple’. In complex analogy, the middle term R is clearly present; but in simple analogy, it is tacit. In complex analogy, the similarity between P and Q is indirectly established, being manifestly due to their having some known feature R in common; whereas in simple analogy, the similarity between them is effectively directly intuited, and R is merely some indefinite thing assumed to underlie it, so that in the absence of additional information we are content define it as ‘whatever it is that P and Q have in common’.

Needless to say, the above argument would be equally valid going from P to Q. I have here presented it as going from Q to P to facilitate comparison and contrast to a fortiori argument, which topic will be dealt with further on.[72]

Quantification. Let us next consider the issue of quantity of the terms, which is not dealt with in the above prototype.

In the singular version of this argument, the major premise is ‘This P is R and this Q is R’, where ‘this’ refers to two different individuals. The minor premise is ‘This Q is S’, where ‘this Q’ refers to the same individual as ‘this Q’ in the major premise does. From the minor premise and part of the major premise we infer (by syllogism 3/RRI[73]) that there is an R which is S, i.e. that some R are S – and this is generalized to all R are S, assuming (unless or until evidence to the contrary is found) there is no R which is not S. From the generality thus obtained and the rest of the major premise, viz. this P is R, we infer (by syllogism 1/ARR) the conclusion ‘This P is S’, where ‘this P’ refers to the same individual as ‘this P’ in the major premise does.

In the corresponding general version of the argument, the major premise is ‘All P are R and all Q are R’ and the minor premise is ‘All Q are S’. From the minor premise and part of the major premise we infer (by syllogism 3/AAI) that some R are S – and this is generalized to all R are S, assuming (unless or until evidence to the contrary is found) there is no R which is not S. From the generality thus obtained and the rest of the major premise, viz. All P are R, we infer (by syllogism 1/AAA) the conclusion ‘All P are S’. Note that the minor premise must here be general, because if only some Q are S, i.e. if some Q are not S, then, if all Q are R, it follows that some R are not S (by 3/OAO), and we cannot generalize to all R are S; and if only some Q are R, we have no valid syllogism to infer even that some R are S.

As regards the quantity of P and Q, there is much leeway. It suffices for the major premise to specify only that some Q are R; because, even if some Q are not R, we can still with all Q are S infer that some R are S (3/AII), and proceed with the same generalization and conclusion. Likewise, the major premise may be particular with respect to P, provided the conclusion follows suit; for, even if some P are not R, we can still from some P are R and all R are S conclude with some P are S (1/AII). Needless to say, we can substitute negative terms (e.g. not-S for S) throughout the argument, without affecting its validity.

It is inductive argument. Thus, more briefly put, the said analogical argument has the following form: Given that P and Q are alike in having R, and that Q is S, it follows that P is S. The validation of this argument is given in our above analysis of it. What we see there is that the argument as a whole is not entirely deductive, but partly inductive, since the general proposition ‘All R are S’ that it depends on is obtained by generalization.

Thus, it may well happen that, given the same major premise, we find (empirically or through some other reasoning process) that Q is S but P is not S. This just tells us that the generalization to ‘All R are S’ was in this case not appropriate – it does not put analogical argument as such in doubt. Such cases might be characterized as ‘denials of analogy’ or ‘non-analogies’. Note also that if ‘All R are S’ is already given, so that the said generalization is not needed, then the argument as a whole is not analogical, but entirely syllogistic; i.e. it is: All R are S and P is R, therefore P is S. Thus, analogy as such is inherently inductive. And obviously, simple analogy is more inductive than complex, since less is clearly known and sure in the former than in the latter.

It is interesting in passing to relate this argument form to the rabbinical hermeneutic principles. The second rule of R. Ishmael, the principle of gezerah shavah, which is based on the terms having some Biblical wording or intent in common, may be said to constitute simple analogy. This is because (evident) same wording, or (assumed) same ‘intent’ of different wordings, do not provide a sufficiently explicit predicate (R) in common to the subjects compared (P and Q). Words are explicit, but they are incidental to what they verbalize; therefore, the assumption that the Torah intends them as significant enough to justify an inference is open to debate[74].

The same can be said of the twelfth rule of R. Ishmael, which refers to contextual inferences (meinyano, misofo, and the like): such reasoning is simple analogy. However, the third rule of R. Ishmael, the principle of binyan av, falls squarely under the heading of complex analogy. In fact, our above description of complex analogy is an exact description of binyan av reasoning. When the rabbis want to extend the scope of a Torah law (S), they show that some new subject (P) has some feature (R) in common with the Torah-given subject (Q), and assuming that this feature is the reason for the law (this assumption constitutes a generalization, even if it superficially may seem to be a direct insight), they carry the law over from the given case to the unspecified case.

Other moods. The above, prototypical mood was positive subjectal. Let us now consider the other possible forms of analogical argument.

  1. The negative subjectal mood. Given that subject P is similar to subject Q with respect to predicate R, and that P is not S, it follows that Q is not S. This mood follows from the positive mood by reductio ad absurdum: given the major premise, if Q were S, then P would be S; but P is not S is a given; therefore, Q is not S. This argument is of course just as inductive as the one it is derived from; it is not deductive.
  2. The positive predicatal mood. Given that predicate P is similar to predicate Q in relation to subject R, and that S is P, it follows that S is Q. We may analyze this argument step by step as follows:

Major premise: P and Q are alike in that R has both of them.

This implies both ‘R is P’ and ‘R is Q’, and is implied by them together.

Minor premise: S is P.

Intermediate conclusion and further premise: S is R.

This proposition is obtained from the preceding two as follows. Given that R is P, it follows by conversion that there is a P which is R, i.e. that ‘some P are R’, which is then generalized to ‘all P are R’, provided of course we have no counter-evidence. If we can, from whatever source, adduce evidence that some P are not R, then of course we cannot logically claim that all P are R. Next, using this generality, i.e. ‘all P are R’, coupled with the minor premise ‘S is P’, we infer through first figure syllogism that ‘S is R’. Clearly, here again, this stage of the argument by analogy is partly deductive and partly inductive.

Final conclusion: S is Q.

This conclusion is derived syllogistically from R is Q and S is R.

Note that the generalized proposition here concerns the major and middle terms, whereas in the preceding case it concerned the middle and subsidiary terms. Needless to say, this argument would be equally valid going from Q to P. I have here presented it as going from P to Q to facilitate comparison and contrast to a fortiori argument, which topic will be dealt with further on.

Let us now quantify the argument. In the singular version, the major premise is: this R is both P and Q, and in the general version it is: all R are both P and Q. The accompanying minor premise and conclusion are, in either case: and a certain S is P (or some or all S are P, for that matter); therefore, that S is Q (or some or all S are Q, as the case may be). We could also validate the argument if the major premise is: some R are P and all R are Q; but if only some R are Q, i.e. if some R are not Q, we cannot do so for then the final syllogistic inference would be made impossible[75]. Such argument is clearly inductive, since it relies on generalization. No need for us to further belabor this topic.

  1. The negative predicatal mood. Given that predicate P is similar to predicate Q in relation to subject R, and that S is not Q, it follows that S is not P. This mood follows from the positive mood by reductio ad absurdum: given the major premise, if S were P, then S would be Q; but S is not Q is a given; therefore, S is not P. This argument is of course just as inductive as the one it is derived from; it is not deductive.

We can similarly develop four implicational moods of analogical argument, where P, Q, R, S, symbolize theses instead of terms and they are related through implications rather than through the copula ‘is’. The positive antecedental would read: Given that antecedent P is similar to antecedent Q with respect to consequent R, and that Q implies S, it follows that P implies S. The negative antecedental would read: Given the same major premise, and that P does not imply S, it follows that Q does not imply S. The positive predicatal mood would read: Given that consequent P is similar to consequent Q in relation to antecedent R, and that S implies P, it follows that S implies Q. The negative predicatal mood would read: Given the same major premise, and that S does not imply Q, it follows that S does not imply P. These are, of course, partly inductive arguments since they involve generalizations. Validation of these four moods should proceed in much the same way as that of the four copulative moods.

Quantitative analogy. Analogy may be qualitative or quantitative. The four (or eight) moods of analogical argument above described are the qualitative. In special cases, given the appropriate additional information, they become quantitative.

  1. The positive subjectal mood in such case would read: Given that subject P is greater than subject Q with respect to predicate R, and that Q is S (Sq), it follows that P is proportionately more S (Sp). Obviously, this reasoning depends on an additional (though often tacit) premise that the ratio of Sp to Sq is the same as the ratio of P to Q (with respect to R).

Very often in practice the ratios are not exactly the same, but only roughly the same. Also, the reference to the ratio of P to Q (with respect to R) should perhaps be more precisely expressed as the ratio of Rp to Rq. Note that this argument effectively has five terms instead of only four (since term S splits off into two terms, Sp and Sq). Of course, the additional premise about proportionality is usually known by inductive means. It might initially be assumed, and thereafter found to be untrue or open to doubt. In such event, the argument would cease to be quantitative analogy and would revert to being merely qualitative analogy. Thus, quantitative analogy is inherently even more inductive than qualitative analogy.

Note that the argument here is, briefly put: ‘just as P > Q, so Sp > Sq’. We can similarly argue ‘just as P < Q, so Sp < Sq’, or ‘just as P = Q, so Sp = Sq’. In other words, positive subjectal quantitative analogy may as well be from the inferior to the superior (as in the initial case), from the superior to the inferior, or from equal to equal; it is not restrictive with regard to direction. In this respect, it differs radically from positive subjectal a fortiori argument, which only allows for inference from the inferior to the superior, or from equal to equal, and excludes inference from the superior to the inferior. All this seems obvious intuitively; having validated the qualitative analogy as already shown, all we have left to validate here is the idea of ratios, and that is a function of mathematics.[76]

Similar comments can be made with regard to the other three copulative moods of quantitative analogy, namely:

  1. The negative subjectal mood: Given that subject P is greater than subject Q with respect to predicate R, and that P is not S (Sp), and that the ratio of Sp to Sq is the same as the ratio of P to Q (with respect to R), it follows that Q is not proportionately less S (Sq).

This mood can be validated by reductio ad absurdum to the positive one. Both the major premise (viz. that P > Q, with respect to R) and the additional premise about proportionality (viz. that Sp:Sq = Rp:Rq) remain unchanged. What has ‘changed’ is that the minor premise of the negative mood is the denial of the conclusion of the positive mood, and the conclusion of the negative mood is the denial of the minor premise of the positive mood. Note that here instead of ‘not more S (Sp)’ and ‘not S (Sq)’, I have put ‘not S (Sp)’ and ‘not less S (Sq)’; this is done only to preserve the normal order of thought – it does not affect the argument as such. Here again, needless to say, though the mood shown is based on P > Q, it can easily be reformulated with P < Q or P = Q; this only affects the conclusion’s magnitude (making Sq mean ‘more S’ or ‘equally S’ as appropriate).

  1. The positive predicatal mood: given that predicate P is greater than predicate Q in relation to subject R, and that a certain amount of S (Sp) is P, and that the ratio of Sp to Sq is the same as the ratio of P to Q (in relation to R), it follows that a proportionately lesser amount of S (Sq) is Q.

Here, the argument is essentially that ‘just as P > Q, so Sp > Sq’, i.e. that the amounts of subject S (viz. Sp and Sq) in the minor premise and conclusion differ in accord with the amounts of predicates P and Q (in relation to R). Or maybe we should say that subject R differs in magnitude or degree when its predicate is P (Rp) and when its predicate is Q (Rq), and that subject S differs accordingly (i.e. Sp and Sq differ in the same ratio as Rp to Rq). This is again an inductive argument, and would be equally valid in the forms ‘just as P < Q, so Sp < Sq’, or ‘just as P = Q, so Sp = Sq’.

  1. The negative predicatal mood: given that predicate P is greater than predicate Q in relation to subject R, and that a certain amount of S (Sq) is not Q, and that the ratio of Sp to Sq is the same as the ratio of P to Q (in relation to R), it follows that a proportionately greater amount of S (Sp) is not P.

This mood can be validated by reductio ad absurdum to the positive one. That is, given the same major premise and additional premise about proportionality, we would say: since the lesser amount of S (Sq) is not Q, it must be that the greater amount of S (Sp) is not P. Here again, if the major premise has P < Q or P = Q instead of P > Q, the conclusion follows suit (i.e. Sp < or = instead of > Sq).

We can similarly develop four implicational moods of quantitative analogy. Thus, all eight moods of qualitative analogical argument can be turned into quantitative ones, provided we add additional information attesting to ‘proportionality’.

Face-off with a fortiori. Clearly, while qualitative analogy is somewhat comparable to purely a fortiori argument, quantitative analogy is somewhat comparable to a crescendo argument; but they are still far from the same. Let us first compare and contrast qualitative analogical argument to pure a fortiori argument. For this purpose, let us first focus on the positive subjectal mood, viz.:

P is more R than (or as much R as) Q,
and Q is R enough to be S;
therefore, P is R enough to be S.

Here, as in analogy, the major premise implies that both P and Q are R, but unlike in analogy, it additionally implies that Rp ≥ Rq, i.e. that the quantity of R in P is greater than (or equal to) that in Q. Thus, though we can deduce the major premise of analogical argument from that of a fortiori argument, we cannot reconstruct the major premise of a fortiori argument only from that of analogical argument. Similarly, though the minor premise of a fortiori argument implies that Q is S, and therefore implies the minor premise of analogical argument, the reverse is not true. The difference between the two minor premises is that in a fortiori argument there is the element of sufficiency of R to be S, which is clearly lacking in argument by analogy. For the same reason, although the conclusion of a fortiori argument implies that of analogy, the latter does not by itself enable us to reconstruct the former.

Moreover, even though each of the propositions (the major and minor premises and the conclusion) involved in a fortiori argument implies the corresponding proposition of analogical argument, this does not mean that an a fortiori argument implies an analogical one. For, the a fortiori argument is deductive, i.e. its conclusion follow necessarily from its two premises; whereas, as we have just shown, the argument by analogy, even in its complex form, is inherently inductive, i.e. it requires a generalization of its minor premise to enable us to draw its conclusion. Therefore, even if both arguments may be said to yield a common conclusion, namely ‘P is S’, that conclusion has a very different logical status in the one and in the other.

It follows that we can neither reduce a fortiori argument to argument by analogy, since the latter’s conclusion does not imply the former’s (even though the premises of the former do imply those of the latter), nor can we do the reverse, since the premises of the latter do not imply those of the former (even though the conclusion of the former does imply that of the latter). It does happen that we know enough to form the major premise needed for a fortiori argument, but we do not know enough for its minor premise; or we know enough to form the minor premise needed for a fortiori argument, but we do not know enough for its major premise – in such cases we might have enough information to at least formulate an analogical argument. Thus, sometimes we have more information than we need for an analogy, but not enough for an a fortiori argument – in such cases we can only formulate an analogy.

Therefore, though we can say that a fortiori argument and argument by analogy have some features in common, we must admit that they are logically very distinct forms of argument. This is a formal and undeniable demonstration, once and for all. To repeat: neither argument can be reduced to the other. However, every valid a fortiori argument implies a corresponding argument by analogy involving less information and certainty. The premises of the latter, as we have just seen, lose the quantitative and/or sufficiency factors involved in the former; and the conclusion of the analogical argument is, as a result, both less informative and less sure (being now inductive instead of deductive). But of course, except for the present theoretical clarification, there is in practice no point in resorting to such implication, since the given a fortiori argument is better in all respects.

As regards the opposite direction, it cannot be said that every analogical argument implies a corresponding a fortiori argument. All we can say is that we can, sometimes, when the facts of the case permit it, construct an a fortiori argument which implies the given analogical argument. This is possible if the latter argument has a middle term (R), or an appropriate middle term can be found for it, which can both be used as a continuum of comparison (which, I think, is always possible in practice, although we cannot tell a priori which term is greater than the other) and at the same time serve as the sufficient condition for the subject (Q) to access the predicate (S) in the minor premise (and this is, of course, not always possible in practice). Thus, the construction of a corresponding a fortiori argument from a given analogical argument is not a mechanical matter and cannot always be performed. In effect, when it is found possible, it just means that we should in the first place have resorted to the stronger a fortiori argument yet foolishly opted for the weaker analogical argument.

All that we have said here applies equally well, mutatis mutandis, to the negative subjectal forms of these arguments, and to positive and negative predicatal forms, and again to the four implicational forms. These jobs are left to interested readers.

As regards comparison and contrast between quantitative analogy and a crescendo argument, i.e. ‘proportional’ a fortiori argument, the following need be said. The major premises are the same in both. But the minor premises and conclusions obviously differ, insofar as in quantitative analogy there is no idea of a threshold value of the middle term as there is in a fortiori argument. This explains why the ‘proportionality’ is essentially non-directional in quantitative analogical argument (inference is always possible both from minor to major and from major to minor); whereas it is clearly directional in a fortiori argument (inference is only possible from minor to major in positive subjectal and negative predicatal argument, and from major to minor in negative subjectal and positive predicatal argument).

Note in passing that although quantitative analogy and mere pro rata argument (i.e. used alone, outside of a crescendo argument) are not formally identical the two are effectively the same. Compare for example the following two formulas; clearly, the provisos in them are essentially the same (a concomitant variation between the values of S and the values of R) even if the terms are differently laid out.

Given that P is greater than Q with respect to R, and that Q is S (Sq), it follows that P is proportionately more S (Sp), provided that the ratio of Sp to Sq is the same as the ratio of P to Q (quantitative analogy).

Given that if R has value Rq then S has value Sq, it follows that if R has value more than Rq (say Rp), then S has value more than Sq (say Sp), provided that the values of S vary in proportion to the values of R (pro rata argument).

To conclude: there is, to be sure, an element of ‘analogy’ in all human thinking, including in syllogism and in a fortiori argument, since all abstraction is based on mental acts of comparison and contrast; but to say this loosely is not the same as equating syllogism or a fortiori argument to argument by analogy. When we look into the exact forms of these arguments, we clearly see their significant differences.

2.    Is a fortiori argument syllogism?

The relationship(s) between a fortiori argument and syllogism have been a subject of debate for a long time, with some logicians and commentators equating the two or at least assimilating one to the other, and others denying such correlations. Ignoring for now the historical narrative, let us first focus on the formal issues involved and develop an independent judgment on them. I considered them very briefly in my Judaic Logic[77], saying:

“It could be said that there is an a-fortiori movement of thought inherent in syllogism, inasmuch as we pass from a larger quantity (all) to a lesser (some). But in syllogism, the transition is made possible by means of the relatively incidental extension of the middle term, whereas, as we have seen, in a-fortiori proper, it is the range of values inherent to the middle term which make it possible.”

Let us here look more deeply into the matter, without prejudice. First, it is well to realize that there are variant versions of the thesis of identification between these forms of argument. The most extreme position is of course that syllogism and a fortiori argument are one and the same thing. At the other extreme, all comparison and correlation between the two forms of reasoning might be rejected. But most logicians and commentators, myself included, adopt an median stance.

It should be made clear at the outset that we are here using the word ‘syllogism’ in its strict, Aristotelian sense, which is etymologically composed of ‘syn’ = together and ‘logos’ = thought, and which refers to an argument involving three items disposed in two premises and a conclusion in certain ways. We are not by this word referring more loosely (as some people do) to any form of argument in which an item serves as intermediary, i.e. to ‘mediate inference’. The latter, more generic expression is equally applicable, for instance, to apodosis (i.e. modus ponens or modus tollens), and obviously equally to a fortiori argument.

Quite often, those who try to explain a fortiori argument do so by suggesting that a fortiori is a sort of syllogistic reasoning. As a fortiori has been less studied than syllogism, it is less widely known and understood; so it is natural for people to try to refer it to a more familiar form of reasoning. The way they proceed to do this, however, is (funnily enough) not very logical, since instead of showing that a fortiori argument is or can be reduced to syllogism, they do the opposite – they usually try to show that syllogism can be formulated as a fortiori argument.

Usually, this is done by means of an example (usually, a syllogism in ‘Barbara’ format, 1/AAA). Often, they try to buttress the demonstration by using the words ‘a fortiori’ instead of ‘therefore’ to introduce the syllogistic conclusion, effectively saying: since ‘a fortiori’ means ‘perforce’, and the conclusion of a syllogism follows perforce from its premises, syllogism is comparable to a fortiori argument; this is obviously silly. Rather, it seems to me, what they need to try and do is recast a fortiori argument into syllogistic form. But this is of course more difficult, as it requires a prior clear awareness of the formalities of a fortiori argument, which most people lack.

What are the proponents of the identification thesis claiming, exactly? Some seem to think that all a fortiori argument is syllogism and all syllogistic argument is a fortiori argument. Others seem only to claim that all syllogistic argument is a fortiori. Still others seem to claim the reverse, i.e. that all a fortiori argument is syllogistic. Yet others seem to regard that there are some convergences between the two forms of argument, without going so far as to claim that all cases of the one fall under the other or vice versa. They may speak of possible reduction or analogy, instead of outright equation.

An important issue here is what exactly is meant by equation between two forms distinct enough to be called two. It may be that these forms are logically equivalent, irrespective of superficial verbal differences; i.e. they are poetically different ways to express the exact same logical thought. Alternatively, perhaps, one form can be fully transformed into or wholly reduced to the other, but not vice versa, so that the latter is logically prior to the former; or such transformation or reduction can occur in both directions. This may occur with or without loss of information, i.e. reversibly or not. Or again, the two forms may share some characteristics, i.e. be analogous in some respects, but differ sufficiently to require separate logical treatment.

That is to say, there is a big difference between saying that a fortiori argument ‘is’ or ‘is a special case of’ syllogism and saying that it ‘can be expressed as’ or ‘is reducible to’ syllogism; or vice versa; or both. Thus, these are different sorts and degrees of equation between two kinds of reasoning. To give a familiar example, even though all second and third figure syllogisms can be directly or indirectly reduced to first figure syllogisms, the former remain distinct forms of reasoning and significant in their own right. On the other hand, all cognoscenti agree, many fourth figure syllogisms have no real existence apart from first figure syllogisms, being distinguished only by the order of presentation of the premises and the order of terms in the conclusion.

Before proposing a theory of the precise correlation between the two forms of argument, let us first clarify what we mean by correlation in more formal terms and endow ourselves with the required vocabulary.

3.    Correlating arguments

Let us investigate the possible relations between any two arguments, whatever their forms. An argument consists of one or more premise(s), say p, and a conclusion, say q; if the argument is valid, then p implies q. The given premise(s) p may of course yield more than one conclusion, i.e. q need not be their only conclusion; however, if there are other conclusions, they together with the same premise(s) constitute and may be regarded as other arguments. We will symbolize the two arguments being correlated as p1q1 and p2q2. In all cases, to repeat, p1 implies q1 and p2 implies q2. (I recommend that the reader draw flow charts to illustrate what is described below.)

The two arguments may be said to be ‘implicants’ if their premises mutually imply each other or are identical, i.e. if p1 implies p2 and p2 implies p1; for it follows that p1 also has the conclusion q2, and that p2 also has the conclusion q1. In such case, either the two conclusions, q1 and q2, imply each other, or only one implies the other, or neither implies the other. If q1 and q2 do not mutually imply each other, then though the arguments are implicants they obviously remain logically distinct by virtue of this difference.

  • If neither of q1, q2 implies the other, we have two effectively ‘independent’ arguments with identical or logically similar premises (i.e. premises that imply each other). Neither argument can be logically reduced to the other.
  • If q1 implies q2 (but not vice versa), then the argument p2q2 may be said to be a ‘subaltern’ of p1q1; if the reverse is the case, the result is of course reversed. A subaltern argument is reducible to, i.e. can be validated by, the argument it is subaltern to. For example, the syllogism 1/AAI is a subaltern mood of 1/AAA, since the premises are the same and the conclusion I is a subaltern of the conclusion A.
  • If q1 and q2 imply each other, we may call the two arguments ‘intertwined’. Each is technically reducible to the other. For example, 1/AII and 3/AII are intertwined, being identical except that the minor premise of each is the converse of that of the other. If p1 is formally identical to p2 and q1 to q2, the two arguments can be characterized as ‘same’ or ‘identical’. But if any premise and/or the conclusion is not formally identical in the two arguments, we may not thus fully equate them, even though they are indeed logically closely related. For in such case not only is there some formal difference between them, but that difference may together with some other common premise(s) result in some difference in conclusion.

Now, consider cases where p1 implies p2 but p2 does not imply p1. We may in such cases say that the argument p1q1 ‘implies’ p2q2, but not vice versa. We can infer (via p2) that p1 also has the conclusion q2, but we cannot likewise infer (via p1) that p2 has the conclusion q1 (although that may happen in some cases). As regards the relationships between the conclusions, here again either they imply each other, or only one implies the other, or neither implies the other.

  • If neither of q1, q2 implies the other, the two arguments are clearly independent, even though their premises are somewhat logically related. Neither argument is reducible to the other.
  • If q2 implies q1 (but not vice versa), then the argument p1q1 may be said to be a subaltern of p2q2, because q1 may be viewed as following from p1 through the intermediary of p2 and q2, i.e. p1q1 is directly reducible to p2q2. For example, the syllogism 3/AAI is a subaltern mood of 3/AII or likewise of 3/IAI.
  • If q1 implies q2 (but not vice versa), then the argument p2q2 may be said to be a ‘corollary’ of p1q1. Note well: this does not mean that p2q2 is a subaltern of p1q1 or vice versa; nor can the relation between the arguments be characterized as independent. We cannot here claim to logically reduce either argument to the other. We will encounter many examples of this relationship in the present context.
  • If q1 and q2 imply each other, then the argument p1q1 is a subaltern of p2q2, whereas p2q2 is a corollary of p1q1. The relation between the arguments is not symmetrical, note well, because p1 implies p2 but p2 does not imply p1.

If p2 implies p1 but p1 does not imply p2, the same can be said of their possible relationships in reverse. If neither of p1 and p2 implies the other, they are ‘unrelated’ arguments; of course, they have to be compatible to occur together in a given context or body of knowledge. So much, then, for all the possible correlations between a pair of arguments, we now have a typology and terminology to work with. Note especially the applications of the term ‘corollary’ here introduced.

4.    Structural comparisons

Let us next compare and contrast the structures of syllogism and a fortiori argument. Consider the following typical samples:

a)      A categorical syllogism A copulative a fortiori argument
All Y are Z, and P is more R than Q is,
All X are Y, and, Q is R enough to be S;
therefore: All X are Z. therefore, all the more, P is R enough to be S.

(Here, the items X, Y, Z and P, Q, R, S are terms, note.)

b)     A hypothetical syllogism An implicational a fortiori argument
Y implies Z, P implies more R than Q does,
X implies Y, and, Q implies enough R to imply S;
therefore: X implies Z therefore, all the more, P implies enough R to imply S.

(Here, the items X, Y, Z and P, Q, R, S are theses, note.)

Syllogism refers to inference from one term or thesis (X) to a second (Z) via a third (Y). The third item, known as the middle item, is the intermediary through which the inference is made; and it is found in the two premises, but not in the conclusion. The other two items are found one in each premise and both in the conclusion; they are traditionally called the minor and major item, because valid positive moods of the first figure serve to include a narrower item in a larger (or equal) one – but this initial scenario does not apply to all valid moods (it does not apply to negative moods of the first figure or to moods of the second and third figures), so the words major and minor should not in all cases be taken literally, they are conventional labels. Do not be misled, either, by my adoption of similar terminology for a fortiori argument: the words here differ in meaning.

A fortiori argument also involves three propositions. But the first, called the major premise, differs significantly in form from the second, called the minor premise, and from the conclusion, while the latter two propositions are of the same form. Moreover, these three propositions involve, not just three terms or theses, but four of them. The middle item (R), here, is present (implicitly if not explicitly) in the conclusion as well as in the two premises; it is labeled ‘middle’ because it interrelates the three other items. The major and minor items (P and Q), here, are so named because they refer respectively to a larger and smaller quantity of the middle item; in the limiting cases of equality between them, their quantitative relations become interchangeable, of course. These two items are both present, together with the middle item, in the major premise.

Note well that the minor premise and conclusion do not always contain the same items (as they do in syllogism). In ‘major to minor’ moods the minor premise contains the major item and the conclusion contains the minor item, whereas in ‘minor to major’ moods the minor premise contains the minor item and the conclusion contains the major item. The middle item is, to repeat, found in both propositions. Finally, we here have a fourth item (S), called the subsidiary item, which is present in the minor premise and conclusion, but absent in the major premise.

Syllogism occurs through the inclusion or exclusion between the three classes or theses concerned; whereas a fortiori compares the measures or degrees of the quality or qualification signified by middle item in its relation to the three other items. Both forms of argument, it is true, involve quantity – but they do so with different emphasis and effect.

The quantities (like ‘all’ or ‘some’) involved in categorical syllogism are applicable to the classes concerned as conceptual groupings and not to the individual members they subsume; similar quantifications are involved in hypothetical syllogism with reference to the conditions underlying the theses. The quantities (like ‘more’ or ‘enough’) involved in copulative a fortiori argument relate to a common property of the individual instances of its other terms; or, in implicational a fortiori, of the conditions underlying its other theses. The focus of syllogism is not primarily on quantity, but on essentially qualitative information; in categorical syllogism, it relates to classification; while in hypothetical syllogism, it relates to sequencing. The focus of a fortiori argument is primarily on quantity – it is quantitative ordering of thoughts, entities, qualities or events.

As just explained, the two forms of argument are structurally very different. As I will presently demonstrate, we can – though sometimes in a rather forced manner – derive a fortiori arguments from syllogisms, and vice versa. Nevertheless, these two species of reasoning cannot be considered the same, or one as a subspecies of the other, because, though some of the original meaning is retained when we attempt to transform syllogism into a fortiori argument or vice versa, some important information is lost.

5.    From syllogism to a fortiori argument

Let us now investigate whether syllogism can be reworded as a fortiori argument (yes, it can) and whether such rewording entails any loss of information (yes, it does). We shall begin with a detailed analysis relative to the most typical valid syllogisms, namely the positive first figure moods 1/AAA and 1/AII of categorical syllogism (and their singular equivalent), and later extend our consideration to all other forms. The following translation of such syllogism into a fortiori is proposed. The major premise of the latter is derived from that of the former; the minor premise of the former becomes that of the latter; and the original conclusion is reworded as shown.

a)      Categorical syllogism (given) A fortiori argument (derived)
All Y are Z, and The class Z is bigger than (or as big as) the class Y, and
All (or some) X are Y, The class Y is big enough to include all (or certain) members of the class X,
therefore:
All (or some) X are Z.
therefore: The class Z is big enough to include all (or certain) members of the class X.

We could view this transition from ordinary (Aristotelian) syllogism to a fortiori argument as made through the intermediary of a class-logic syllogism. That is, the major premise “All Y are Z” is first interpreted as “The class of Y is entirely subsumed by the class of Z,” the minor premise likewise as “The class of X is entirely (or partly) subsumed by the class of Y,” and the conclusion as “The class of X is entirely (or partly) subsumed by the class of Z.” Then these three propositions are used to produce the three of the resultant a fortiori argument. This is said in passing.

The resulting a fortiori argument, note, is positive subjectal (minor to major). An example often given to illustrate and justify this operation is: “All men (Y) are mortal (Z) and Socrates (X) is a man, therefore Socrates is mortal” becomes “Since the class of mortals is more extensive than that of men (which is included in it), it follows that if the latter includes Socrates as a member, the former is bound to include him as a member as well.” (People don’t normally think like that, admittedly, but logicians have to sometimes!) But as we shall presently explain, this example is not fully accurate and so a bit misleading.

Note that, in this instance (though not always, as we shall later see), the major term (Z) of the syllogism gives rise to the major term (the class Z) of the a fortiori argument. However, the middle term (Y) of the syllogism gives rise to the minor term (the class Y) of the a fortiori, and the minor term of the syllogism (X) gives rise to the subsidiary term (the class X) of the a fortiori. Thus, do not confuse the appellations of the terms in the two arguments. The middle term of the a fortiori argument is “bigness;” although it is not explicitly given in the syllogism, it is read out of it. Here, “bigness” refers to the extension of each class concerned, i.e. the number of members it includes. We could as well have used the more technical term extensiveness.

Now, the major premise of the a fortiori argument tells us that the term Z is more extensive than the term Y (or only as extensive as Y, if it so happens that All Z are Y is also true). It does not and cannot tell us whether or not Z includes Y, note well. The minor premise and conclusion of the a fortiori argument likewise only tell us that the terms Y and Z respectively are more extensive than (or at least as extensive as) the term X. They do not and cannot tell us that Y and (therefore) Z actually include X; they only affirm in the infinitive that they are big enough to do so, i.e. their inclusion is a logical possibility but not an inferred certainty.

That is to say, the two premises of the derived a fortiori argument do not fully reproduce the information in the given syllogism – they only focus on the relative extensions implied by them, but do not carry over the information about precise inclusions. Consequently, the conclusion of the a fortiori – which likewise contains no actual (only ‘potential’) information about inclusions – cannot be used to infer the conclusion of the syllogism. This means that the given syllogism logically implies the proposed a fortiori argument – but the latter does not logically imply the former! So the derived a fortiori argument is a corollary of the given syllogism.

This correlation becomes more evident if we realize that the a fortiori argument shown here would remain true even if the classes Y and Z were mutually exclusive (!) – provided we knew somehow that their relative sizes (extensions) were as here stated (i.e. such that Z is greater or equal to Y). Even in such case (i.e. that of mutual exclusion), the said minor premise would imply the said conclusion, for neither of these two propositions formally means that the term X is (wholly or partly) actually included by the terms Y and Z – they only tell us that they could be.

As far as the a fortiori argument is concerned, all information about inclusion found in the two premises of the given syllogism is redundant – it is simply not carried over in it; it is effectively lost. The propositional forms used in the a fortiori argument are simply incapable of containing this significant classificatory information, and they have no need of it to deduce the conclusion they yield. Similarly, the latter conclusion (of the a fortiori) has nothing definite to tell us about the inclusion (i.e. of X in Z) that the original syllogistic conclusion refers to, and therefore is unable to reconstruct it.

We could of course have formulated the major premise as “The class Z is bigger than (or as big as) its subclass Y,” and similarly the minor premise as “The class Y is big enough to include all (or certain) members of its subclass X” – but the additional information so tagged on would still play no role in the inference made possible by the a fortiori reasoning as such. It would of course suggest to us that X is indeed included in Y and therefore in Z – but that suggestion would be emerging not from the a fortiori argument itself, but from the syllogism mentally underlying it. The a fortiori argument as such could logically still only yield the said conclusion “The class Z is big enough to include all (or certain) members of the class X,” without specifying that the Xs are indeed included in Z.

It follows that we cannot say that the derived a fortiori argument “is” or “is identical to” the given syllogism. The most we can say is that the former is implicit in the latter; i.e. that it is a part of it or an aspect of it. The derived format cannot logically replace the given format in all respects. Important information is lost in the transition. To return to the above example about Socrates, we now see that the clause about the class of men being included in that of mortals, and the suggestion that Socrates is included in both these classes, are not in fact constituents of the a fortiori argument as such.

As already mentioned, many people affirm that syllogisms can be reworded in a fortiori form, and they give an example or two to prove it, usually a positive first figure mood. But I asked myself two original questions: (a) can all valid moods of the syllogism be likewise recast in a fortiori format, and (b) are all valid forms of a fortiori argument generated by such translations?

To my surprise, the answers to both these questions were found to be: yes, even if some of the processes do seem rather artificial. At the risk of boring the reader stiff (skip it all if you take my word for it), I will now show step by step that all valid moods of syllogism can be recast in a fortiori form; and I will also show that such translations or reinterpretations generate all the varieties of a fortiori argument. Though most of this work was easy enough, some of it was a bit difficult – so it was well worth doing.

We have already dealt with wholly positive first figure syllogism (moods 1/AAA, 1/AII). We can similarly propose a translation of first figure syllogism with a negative major premise and conclusion (moods 1/EAE, 1/EIO), by simply using the term nonZ in place of Z. as follows:

b)     Syllogistic format A fortiori format
No Y are Z (= All Y are nonZ), and The class nonZ is bigger than the class Y, and
All (or some) X are Y, The class Y is big enough to include all (or certain) members of the class X,
therefore: All (or some) X are not Z (= are nonZ). therefore: The class nonZ is big enough to include all (or certain) members of the class X.

Clearly, however, this is still positive a fortiori argument (albeit with a negative term nonZ), and does not correspond to negative a fortiori argument. The question arises, can we generate a negative a fortiori argument from some other syllogism? Yes, but to do so we need to move over to the second figure of syllogism, as follows:

c)      Syllogistic format A fortiori format
All Z are Y, and The class Y is bigger than the class Z, and
All (or some) X are not Y, The class Y is not big enough to include all (or certain) members of the class X,
therefore:
All (or some) X are not Z.
therefore: The class Z is not big enough to include all (or certain) members of the class X.

The derivation might be considered roughly adequate, if we accept that X is not Z (or Y) means that Z (or Y) is not “big enough to include” X. But even if we do so, it is very reluctantly, as we are clearly indulging in an unnatural way of thinking. The above translation concerns the syllogistic moods 2/AEE, 2/AOO; we can do the same for the moods 2/EAE, 2/EIO simply by replacing Z with nonZ, as shown next:

d)     Syllogistic format A fortiori format
No Z are Y (= All Z are nonY), and The class nonY is bigger than the class Z, and
All (or some) X are not Y (= are nonY), The class nonY is not big enough to include all (or certain) members of the class X,
therefore:
All (or some) X are not Z.
therefore: The class Z is not big enough to include all (or certain) members of the class X.

Of course, such a fortiori argument is even more awkward. Still, let us say we have now managed to translate all first and second figure syllogism to positive and negative a fortiori arguments, respectively. What of third figure syllogism – to what would we translate them? Moreover, all the a fortiori forms encountered so far have been subjectal – what of predicatal a fortiori? The answers to these two questions are the same. We can propose the translations of positive and negative moods of third figure syllogism into positive and negative predicatal a fortiori arguments, as follows:

e)      Syllogistic format A fortiori format
All Y are Z, and A bigger class is required to subsume the class Z than to subsume the class Y, and
Some (or all) Y are X, The class X is big enough to include some (or even all) members of the class Y,
therefore:
Some X are Z.
therefore: The class X is big enough to include some members of the class Z.

The above is appropriate for the translation of the mood 3/AII (and likewise its subaltern 3/AAI). For the mood 3/IAI (and its same subaltern 3/AAI), we would have to transpose the premises and convert the particular conclusion.

f)      Syllogistic format A fortiori format
Some (or all) Y are Z, and A bigger class is required to subsume the class X than to subsume the class Y, and
All Y are X, The class Z is big enough to include some (or even all) members of the class Y,
therefore:
Some X are Z.
therefore: The class Z is big enough to include some members of the class X.

The mood 3/OAO (and its subaltern 3/EAO) can be translated as shown next; note the transposition of premises here too.

g)     Syllogistic format A fortiori format
Some (or all) Y are not Z, and A bigger class is required to subsume the class X than to subsume the class Y, and
All Y are X, The class Z is not big enough to include some (or even all) members of the class Y,
therefore:
Some X are not Z.
therefore: The class Z is not big enough to include some members of the class X.

And similarly for the mood 3/EIO (and its subaltern 3/EAO) shown next.

h)     Syllogistic format A fortiori format
No Y are Z (= All Y are nonZ), and A bigger class is required to subsume the class nonZ than to subsume the class Y, and
Some (or all) Y are X (= are not nonX), The class nonX is not big enough to include some (or even all) members of the class Y,
therefore: Some X are not Z (= Some nonZ are not nonX). therefore: The class nonX is not big enough to include some members of the class nonZ.

Note in passing that we could similarly process modal syllogism, the modalities simply passing over from the syllogism to the derived a fortiori argument. We have thus shown that all valid moods of the syllogism can be rephrased (however awkwardly) as a fortiori argument of various sorts. The first figure moods yielded positive subjectal a fortiori; the second figure moods yielded negative subjectal a fortiori; and the third figure moods yielded positive and negative predicatal a fortiori. We have thus also shown that all four forms of a fortiori argument are produced by these processes.

Needless to say (but I will add it anyway, to be foolproof), the middle term of these a fortiori arguments, viz. “big,” is just one possible middle term – most a fortiori arguments we encounter in practice do not use this particular middle term but may use any of countless middle terms. That is to say, although we have generated instances of all sorts of a fortiori, we have certainly not generated all particular instances of a fortiori! Thus, ‘a fortiori argument in general’ must be admitted to be a larger class than ‘a fortiori argument generated from syllogism’ is. Keeping that in mind, we will not overestimate the import of the preceding demonstration.

Our demonstration was made with reference to categorical syllogism and copulative a fortiori – but obviously the same can be done with reference to hypothetical (and likewise, ‘de re’ conditional) syllogism. We can I think take that for granted without worry, so I won’t bore you further with a repetition of all the above work to prove the point. However, it is worth our while looking at just one sample of translation of hypothetical syllogism into a fortiori argument, to highlight and examine the different concepts and language involved in the latter:

Hypothetical syllogism (given) A fortiori argument (derived)
If Y, then Z, and Thesis Z is bigger than thesis Y, and
If X, then Y
(or if X, not-then not Y),
Thesis Y is big enough to include all (or certain) conditions of thesis X,
therefore: If X, then Z
(or if X, not-then not Z).
therefore: Thesis Z is big enough to include all (or certain) conditions of thesis X.

The given propositions can also be written/read respectively as “Y implies Z,” “X implies Y” (or in the weaker case, “X does not imply not Y”), and “X implies Z” (or in the weaker case, “X does not imply not Z”), note. The derived a fortiori argument here is still copulative, note, and not implicational. Its language is clearly similar here to that we used in translating categorical syllogism, except that we say “thesis Z” instead of “class Z,” etc., and we speak of “conditions” instead of “members.”

This change of wording reflects the nature of implication: necessary implication (the stronger variant) means that the consequent occurs under all the conditions applicable to the antecedent, whereas possible implication (the weaker variant) refers to some of these conditions. In logical conditioning, the conditions referred to are internal contexts of knowledge; in ‘de re’ modes of conditioning – i.e. the natural, temporal and spatial modes – they are external circumstances, times or places.

Note well that when we say that “thesis Z is bigger than thesis Y,” we do not mean that the subject of Z is bigger than the subject of Y (taking Z and Y as single categorical propositions here, for the sake of argument). We are not referring to the extensions of the terms involved within the theses, but to the conditions underlying the theses. The proposition “Y implies Z” does not formally exclude the possibility that Y may be singular and Z universal, or vice versa, provided that under all the conditions concerned Y is accompanied by Z. Similarly with the other forms mentioned in the above arguments.

To conclude, it is now formally proved true that syllogism can always be recast in a fortiori form, although it was also evident in the course of our demonstration that such translations are, to varying degrees, very contrived. Maybe logicians occasionally have such convoluted thoughts; but we do not ordinarily think in such awkward ways. Syllogism is quite thinkable and commonly thought by itself, i.e. without need to refer to a fortiori argument as above done. Indeed, syllogism is a simpler and more primitive movement of thought; more easily and widely comprehensible. So we would not in practice rephrase a syllogism as an a fortiori argument. Still, we have answered half the question initially asked.

6.    From a fortiori argument to syllogism

We need now turn to the second part of our initial question: can a fortiori argument be said to be syllogism or at least be expressed as or reduced to syllogism? We can put it more technically, and ask: how can a four-term argument (a fortiori) be reworded as a three-term one (syllogism), with minimal loss of meaning and conviction?

One possible answer is: by ignoring or concealing one of the terms – namely the middle term. We very commonly do leave tacit the middle term in our a fortiori arguments. An argument used by Aristotle that does this is: if even the gods are not omniscient, certainly human beings are not. Here, the middle term is tacitly present in the words ‘even’ and ‘certainly’; often, it is signaled by expressions like ‘enough’ and ‘all the more’. Put in formal terms, this would read: if Q is S enough, then all the more P is so – the a fortiori middle term R being left out entirely. This is a three-term argument – but is it syllogism? Clearly not, since we have no operative syllogistic middle term (S cannot be said to play that role here).

We can in this context, incidentally, pinpoint the usual error of the logicians and commentators who seek to explain a fortiori argument as a sort of syllogism – they wrongly assume that the three terms of such derived syllogism would be the minor, major and subsidiary (P, Q, S), whereas in fact they are the different quantities of the middle term (R) occurring relative to these three terms. This is a subtle distinction that they could not see, because they had not sufficiently analyzed a fortiori argument as such as gravitating around a fourth term, which is often in practice left unstated or so intertwined with the other three that it is almost imperceptible.

To understand how we can transmute a fortiori argument into syllogism, we need to go back to the way a fortiori argument is formally validated – i.e. we need to decorticate its propositions and find out whether the constituents of the given premises justify the constituents of the putative conclusion. To do this we need to look again at Table 1.1 and Diagram 1.1, given in the first chapter.

Once the centrality of the middle term of a fortiori argument is grasped, it is easy for us to formulate its various moods in syllogistic form. We shall now do this with reference to copulative a fortiori argument. The major premise of all four a fortiori arguments can be expressed in the hypothetical form “Rp implies Rq,” which means “the quantity of R corresponding to P implies the quantity of R corresponding to Q.” Rp may be said to imply Rq because a larger quantity implies all lesser quantities; as for example 5 equally implies, 4, 3, 2 or 1 – i.e. you cannot reach the larger amount without passing through the smaller amounts. If you have $5, it is also true that you have $4, $3, etc.

The minor premises and conclusions can similarly, by consideration of their quantitative significances, be expressed as if–then propositions or negations of such, as shown below. Note well that the middle items of the syllogisms produced vary: in cases (a) and (d) the middle theses is Rq, while in cases (b) and (c) it is Rp. Note too that in cases (a) and (b) the premises must be switched to get the stated conclusion; i.e. the major premise becomes the minor and vice versa. Notice, finally, that we obtain valid moods in all three figures of the syllogism, viz. 1/AAA (twice), 2/AOO and 3/OAO, although (predictably, in view of the numbers) we do not produce all the valid moods of syllogism out of those of a fortiori.

a)      Positive subjectal copulative a fortiori Syllogism (1/AAA)
P is more R than (or as much R as) Q is, Rp implies Rq
and, Q is R enough to be S; therefore, Rq implies Rs
all the more (or equally), P is R enough to be S. So, Rp implies Rs
b)     Negative subjectal copulative a fortiori Syllogism (3/OAO)
P is more R than (or as much R as) Q is, Rp implies Rq
yet, P is R not enough to be S; therefore, Rp does not imply Rs
all the more (or equally), Q is R not enough to be S. So, Rq does not imply Rs
c)      Positive predicatal copulative a fortiori Syllogism (1/AAA)
More (or as much) R is required to be P than to be Q, Rp implies Rq
and, S is R enough to be P; therefore, Rs implies Rp
all the more (or equally), S is R enough to be Q. So, Rs implies Rq
d)     Negative predicatal copulative a fortiori Syllogism (2/AOO)
More (or as much) R is required to be P than to be Q, Rp implies Rq
yet, S is R not enough to be Q; therefore, Rs does not imply Rq
all the more (or equally), S is R not enough to be P. So, Rs does not imply Rp

To give an example[78]: “It is psychologically more difficult (R) for a man to strike his father (P) than to strike his neighbors (Q); so, if John (S) was able to strike his father, he is all the more capable of striking his neighbors” (major to minor, positive predicatal) becomes “John’s psychological state is such that he could strike his father, and striking one’s father is more difficult than striking one’s neighbors, therefore John’s psychological state is such that he could strike his neighbors.”

Now, just as we found that in the attempted transformation of syllogism into a fortiori argument there was a loss of information (about inclusions between the classes concerned), so it is with regard to the attempted transformation of a fortiori argument into syllogism some information is inevitably lost in the process. What information is lost? I will now explain, with reference to our earlier analysis of the propositional forms involved in a fortiori argument. Consider for instance process (a), the translation of positive subjectal a fortiori argument into syllogism.

The form “P is more R than Q” tells us more than just “Rp implies Rq” – it also tells us that “P implies Rp” and “Q implies Rq” and “Rp is greater than Rq.” Likewise, the form “Q is R enough to be S” tells us more than just “Rq implies Rs” – it also tells us that “Rs implies S” and “Q implies Rq” and “Rs includes Rq.” Thus, the two premises of the derived syllogism do not carry over all the information that was in the given a fortiori argument, but only parts of it. The conclusion of the derived syllogism, viz. “Rp implies Rs,” is similarly less informative than the conclusion of the given a fortiori argument, which tells us that “Rs implies S” and “P implies Rp” and “Rs includes Rp.” From “Rp implies Rs,” we can infer the element “Rs includes Rp,” but not the elements “Rs implies S” and “P implies Rp.”

So we are unable to logically reconstitute the conclusion of the original a fortiori argument from the conclusion of the derived syllogism – we only have part of its discourse leftover, some of it having been left out on the way. It follows that, though a significant syllogism is formally discernible within the a fortiori argument, that derived syllogism does not store enough information to in turn produce the original a fortiori argument.

It follows that the a fortiori argument logically implies but is not implied by the syllogism shown. That is to say, we cannot claim that the given a fortiori argument “is” or “is identical to” the derived syllogism. The most we can say is that the latter is a corollary of the former: it is implicit in it, a part of it or an aspect of it. But evidently, the latter cannot logically replace the former in all respects. Important information is lost in the transition.

The same can be said with regard to the translation of other forms of a fortiori argument to syllogism, i.e. the processes labeled (b), (c) and (d) above. Though these processes are obviously valid, they do not produce a syllogistic replica of a fortiori argument – only at best a corollary. We could do and say the same for implicational a fortiori argument, but there is no need to repeat ourselves. Suffice it to show and comment on one sample:

Implicational a fortiori argument Syllogism
P implies more R than (or as much R as) Q does, Rp implies Rq
and, Q implies enough R to imply S; therefore, Rq implies Rs
all the more (or equally), P implies enough R to imply S. So, Rp implies Rs

Note that the language of the derived syllogism here is exactly the same as that for copulative a fortiori, since we are still concerned with various degrees of R implying each other. The syllogism derived here – as that derived from copulative a fortiori – is formally hypothetical, not categorical. The items involved in all such derived syllogisms, viz. Rp, Rq and Rs, are the terms or theses “the value of R that P is or implies,” “the value of R that Q is or implies,” and “the value of R that S is or implies,” respectively (using “is” for copulative sources and “implies” for implicational ones). No more need be said.

We have thus demonstrated using formal means that the essence of a fortiori argument can be expressed through syllogism. This is equally true of positive or negative, subjectal or predicatal, copulative or implicational a fortiori. This does not mean that a fortiori argument is the same as syllogism, but it does mean that syllogistic movements of thought are involved in a fortiori reasoning, or (in other words) that a fortiori argument can be partly reduced to and validated by means of syllogism. Even so, as just explained, some information is invariably lost on the way.

7.    Reiterating translations

To complete our analysis of the relationships between a fortiori argument and syllogism we need to examine one more issue. Having found that we can generate a fortiori arguments from syllogisms, and syllogisms from a fortiori arguments – we need to ask the question of reiteration. This does not mean reversibility, since we have already shown that such translations are not reversible – i.e. the arguments generated by such translations are always mere corollaries because some information is always lost in the process. Rather, the question to ask is this: having extracted an a fortiori argument from a syllogism, what syllogism can we in turn extract from that derived a fortiori argument? And conversely, having extracted a syllogism from an a fortiori argument, what a fortiori argument can we in turn extract from that derived syllogism?

Every syllogism implies a corresponding a fortiori argument which does not in turn imply it back – but it does go on to imply some syllogism. Similarly, every a fortiori argument implies a corresponding syllogism which does not in turn imply it back – but it does go on to imply some a fortiori argument. It should be obvious that these pairs of statements are not in contradiction, but I will explain why anyway. Consider the following samples:

a)      Syllogism A fortiori argument
All Y are Z, and The class Z is bigger than (or as big as) the class Y, and
All (or some) X are Y; The class Y is big enough to include all (or certain) members of the class X,
therefore:
All (or some) X are Z.
therefore: The class Z is big enough to include all (or certain) members of the class X.
b)     A fortiori argument Syllogism
P is more R than (or as much R as) Q is, and, Rp implies Rq, and
Q is R enough to be S; therefore, Rq implies Rs;
all the more (or equally), P is R enough to be S. So, Rp implies Rs.

Consider first process (a): if we wanted to draw a syllogism in accord with process (b) out of the a fortiori argument derived from the given syllogism, we would obtain the syllogism shown next, which is considerably different from the original one:

The size of class Z implies the size of class Y,
the size of class Y implies the size of class X;
therefore, the size of class Z implies the size of class X.

I use the word “size” here to avoid using the more barbaric “bigness;” we are of course concerned with extensions, meaning that a bigger size implies a smaller one, as already explained. Note that this new syllogism is not categorical (with “is” relating terms) like the original one, but hypothetical (with “implies” relating theses[79]). If now we compare the information in this derived syllogism to that in the original syllogism, we note that valuable information was lost in the transition: the inclusions of Y in Z and X is Y (and therefore of X in Z) are long gone; instead, all we now discuss are the relative sizes of these classes (irrespective of any subsumption between them). Thus, though we have indeed obtained a new syllogism, it is certainly not the same as the original syllogism but a watered-down corollary of it.

Consider now process (b): if we wanted to draw an a fortiori argument in accord with process (a) out of the syllogism derived from the given a fortiori argument, we would obtain the a fortiori argument shown next, which is considerably different from the original one:

Thesis Rp is bigger than (or as big as) thesis Rq, and
Thesis Rq is big enough to include all the conditions of thesis Rs; therefore,
all the more (or equally), Thesis Rp is big enough to include all the conditions of thesis Rs.

Here, as already explained, sizes (as in “bigger,” “big enough”) refer to numbers of conditions underlying theses, not to numbers of members within classes. Note that the new a fortiori argument is copulative, since it uses “is” rather than “implies” to relate items. Here again, if we compare the information in this derived a fortiori argument to that in the original a fortiori argument, we note that valuable information was lost in the transition: for instance, information we had initially on Q actually being (R enough to be) S is long gone – all we still know about them now is that the value of R corresponding to Q occurs in numerically more conditions than the value of R corresponding to S does. Thus, though we have indeed obtained a new a fortiori argument, it is certainly not the same as the original a fortiori argument but a watered down corollary of it.

Clearly, such reiterations do not produce very interesting results. No doubt if we tried reiterating further, i.e. translating the syllogism and a fortiori argument produced by the above reiterations into new derivatives, we would likewise not produce anything very interesting. But the main point we wished to make – viz. that reiteration should not be confused with reversing – has been convincingly established.

8.    Lessons learned

We can here conclude our research into the relationships between a fortiori argument and syllogism. We looked into this topic because it is one often and hotly debated in literature on a fortiori logic. We decided to carry out a more formal and systematic investigation than past logicians and commentators have done, so as to be able to judge the matter more objectively and definitively. We first looked into the conceivable correlations between any two arguments, and in particular formally defined what we mean by a corollary. Then we compared the structures of syllogism and a fortiori argument, showing the various respects in which they differ significantly.

We then demonstrated that every syllogism contains an implicit a fortiori argument – but we also found that the latter is a mere corollary of the former, information being lost in transition, so that the process of derivation cannot be reversed. We then demonstrated the converse, i.e. that every a fortiori argument contains an implicit syllogism – but we also found that the latter is a mere corollary of the former, information being lost in transition, so that the process of derivation cannot be reversed. Finally, we showed that reiteration of these processes, though logically possible, is not very interesting.

We are now in a position to formulate the following overall lessons learned. Syllogism and a fortiori argument are very different movements of thought. They are structurally different, and each serves a different rational purpose; so they are not equivalent or interchangeable. Although they can be formally interrelated in various ways, they remain logically distinct and irreplaceable in important ways. Syllogism orders terms or theses by reference to the inclusions (or exclusions) between them, while a fortiori argument orders them by reference to the measures or degrees of some property they have (or do not have) in common. Neither function can be substituted for the other. We can (however awkwardly) reword one form of argument into the other; but such translations are not exactly transformations, because significant information cannot be passed on from one form to the other; therefore, neither form of argument can be fully reduced to the other. Thus, both forms are needed by reason to pursue its business; they are complementary instruments of reasoning.

In spite of all that, it should be remembered that a fortiori argument, whether copulative or implicational, is inextricably dependent on hypothetical (rather than categorical) syllogism, as we have shown in detail in the first chapter of the present volume, in the section dealing with validation (1.3). Hypothetical syllogisms (arguments such as if a, then b and if b then c, therefore if a then c) are not by themselves sufficient for validation of a fortiori arguments, since we also need comparative arguments (such as a > b and b > c, therefore a > c); but without them the validations would be impossible. Thus, a fortiori argument is a sort of composite argument, whose components are brought to the surface during the process of validation. Nevertheless, it is a form of argument in its own right, insofar as we are able to reason through it directly, without always having to resort to validation – i.e. it is intuitively credible anyway.

PART II – ANCIENT AND MEDIEVAL HISTORY

6.  A fortiori in Greece and Rome

1.    Aristotle’s observations

Looking at the sayings or writings of ancient Greek philosophers – Thales, Anaximander, Anaximenes, Heraclitus, Pythogoras, Philolaus, Xenophanes, Parmenides, Zeno, Empedocles, Leucippus, Democritus, Anaxagoras, Socrates, Plato, and Aristotle, and their successors – one cannot but be awed by the extraordinary breadth and profundity of their thinking, and their anticipation of many ideas considered important today. For example, I recently realized that Empedocles[80] could be regarded as the precursor of the phenomenological approach, on the basis of his statement: “Think on each thing in the way in which it is manifest.”

It is not surprising, therefore, to find some discussion of a fortiori argument in the works of Aristotle (Greece, 384-322 BCE)[81]. The following quotations from his works (dated c. 350 BCE) seem relevant to our research.[82]

In his Rhetoric 2:23 (i.e. book II, chapter 23), in §4, Aristotle writes:

“Another line of proof is the a fortiori[83]. Thus it may be argued that if even the gods are not omniscient, certainly human beings are not. The principle here is that, if a quality does not in fact exist where it is more likely to exist, it clearly does not exist where it is less likely. Again, the argument that a man who strikes his father also strikes his neighbors follows from the principle that, if the less likely thing is true, the more likely thing is true also; for a man is less likely to strike his father than to strike his neighbors. The argument, then, may run thus. Or it may be urged that, if a thing is not true where it is more likely, it is not true where it is less likely; or that, if it is true where it is less likely, it is true where it is more likely: according as we have to show that a thing is or is not true.”

In this passage, Aristotle shows he considers a fortiori argument as a “line of proof” – by which he presumably means that it is a deductive argument. He marks his understanding of a fortiori argument as going from denial of the ‘more’ to denial of the ‘less’, or from affirmation of the ‘less’ to affirmation of the ‘more’. On this basis, we can say that Aristotle was aware of at least two valid moods: positive argument “from minor to major,” and negative argument “from major to minor,” though he does not use such terminology, but only says: “according as we have to show that a thing is or is not true”[84]. Clearly, therefore, what he has in mind here are positive and negative subjectal arguments. His arguments can be reworded as follows to clarify their standard formats (with the symbols P, Q, R, and S, denoting the major, minor, middle and subsidiary terms, respectively):

His first example is negative subjectal: that the gods are omniscient (P) is more credible (R) than that human beings are so (Q); therefore if the gods’ omniscience is not credible enough to be assumed (S), the omniscience of human beings is not credible enough to be assumed. This illustrates the principle: if a quality in a certain place (P) is more likely to be found (R) than the same quality in another place (Q) is, then if the quality in the first place is not sufficiently likely to be found to be considered as existing in fact (S), it follows that the quality in the second place is not sufficiently likely to be found to be considered as existing in fact (S).

His second example is positive subjectal: a man striking his neighbors (P) is a more likely event (R) than the man striking his father (Q); therefore, if a man striking his father is likely enough to be expected (S), then the man striking his neighbors is likely enough to be expected. This illustrates the principle: if something somewhere (P) is more likely (R) than the same thing elsewhere (Q), then if the latter is likely enough to be declared true (S), it follows that the former is likely enough to be declared true. (To which he adds the negative mood: if the former is not likely enough to be declared true, it follows that the latter is not likely enough to be declared true.[85])

Noteworthy here is Aristotle’s formulation of these a fortiori arguments in logical-epistemic terms, i.e. using a logical middle term (such as ‘likely’) and an epistemic subsidiary term (such as ‘believed’)[86]. His above two examples could of course have been formulated in purely ontical terms, as follows. The gods (P) are more well-endowed (R) than human beings (Q) are; therefore, if the gods are not well-endowed enough to be omniscient (S), then human beings are not well-endowed enough to be omniscient. Or again: striking one’s neighbors (P) generally seems more natural (R) than striking one’s father (Q); therefore, if striking his father seems natural enough to a certain man for him to actually do it (S), then striking his neighbors seems natural enough to him for him to actually do it.[87]

Still in Rhetoric 2:23, Aristotle adds a number of examples of allegedly a pari a fortiori argument. I say allegedly, because the proposed arguments are not complete enough to judge the matter. Note that five of the examples have negative form, while two have positive form. In any case, this serves to show us his awareness of such argument:

“This argument might also be used in a case of parity, as in the lines: Thou hast pity for thy sire, who has lost his sons: Hast none for Oeneus, whose brave son is dead? And, again, ‘if Theseus did no wrong, neither did Paris’; or ‘the sons of Tyndareus did no wrong, neither did Paris’; or ‘if Hector did well to slay Patroclus, Paris did well to slay Achilles’. And ‘if other followers of an art are not bad men, neither are philosophers’. And ‘if generals are not bad men because it often happens that they are condemned to death, neither are sophists’. And the remark that ‘if each individual among you ought to think of his own city’s reputation, you ought all to think of the reputation of Greece as a whole’.”

In his Topics 2:10 (book II, chapter 10), where Aristotle begins with: “Moreover, argue from greater and less degrees…,” I will divide what he thereafter says in three parts for purposes of analysis:

“See whether a greater degree of the predicate follows a greater degree of the subject: e.g. if pleasure be good, see whether also a greater pleasure be a greater good: and if to do a wrong be evil, see whether also to do a greater wrong is a greater evil. Now this rule is of use for both purposes: for if an increase of the accident follows an increase of the subject, as we have said, clearly the accident belongs; while if it does not follow, the accident does not belong. You should establish this by induction.”

This first paragraph, if it is at all related to a fortiori argument, makes clear by implication that Aristotle does not universally approve of a crescendo argument, i.e. of argument resembling a fortiori but having a ‘proportional’ conclusion. He is clearly not saying, for instance, that if pleasure is good it follows deductively that more pleasure is better – he is only saying that the question should be asked and that the answer is to be sought by induction; he explicitly conceives the possibility that it may not follow. This is an important finding concerning Aristotle, considering that (as we shall see) many people who historically came after him did not likewise realize the invalidity of ‘proportional’ a fortiori argument. He goes on:

“If one predicate be attributed to two subjects; then supposing it does not belong to the subject to which it is the more likely to belong, neither does it belong where it is less likely to belong; while if it does belong where it is less likely to belong, then it belongs as well where it is more likely. Again: If two predicates be attributed to one subject, then if the one which is more generally thought to belong does not belong, neither does the one that is less generally thought to belong; or, if the one that is less generally thought to belong does belong, so also does the other. Moreover: If two predicates be attributed to two subjects, then if the one which is more usually thought to belong to the one subject does not belong, neither does the remaining predicate belong to the remaining subject; or, if the one which is less usually thought to belong to the one subject does belong, so too does the remaining predicate to the remaining subject.”

Aristotle here details the positive and negative moods of three seemingly distinct a fortiori arguments. The first concerns two subjects (A, B) with a common predicate (C), and its major premise is: ‘A is C’ (P) is more likely (R) than ‘B is C’ (Q). The second concerns one subject (A) with two predicates (B, C), and its major premise is: ‘A is B’ (P) is more generally thought (R) than ‘A is C’ (Q). The third concerns two subjects (A, B) with two predicates (C, D), and its major premise is: ‘A is B’ (P) is more usually thought (R) than ‘C is D’ (Q). Although the middle term (R) is differently worded in each case, no great significance should be attached to this variation: all three may be taken to mean about the same, say ‘likely’. The subsidiary term (S) may in all cases be regarded as ‘believed’ (or ‘adopted’ or any similarly convenient qualification). In each case, the said major premise is followed by the minor premises and conclusions in the standard forms below:

Given something (P) is more likely (R) than another thing (Q) is, it follows that:
if Q is R enough to be believed (S), then P is R enough to be S;
and if P is R not enough to be S, then Q is R not enough to be S.

Clearly, the three sets of argument of positive and negative forms are effectively one and the same set. They illustrate subjectal a fortiori argument with a logical middle term (e.g. ‘likely’) and an epistemic subsidiary term (e.g. ‘believed’). Note well, though these arguments concern whole propositions (labeled P and Q by me), they are not to be regarded as antecedental since no implication between propositions is suggested. Though the major and minor terms, P and Q, are propositions, each stands in this context as a unitary term, a subject of which may be predicated the said logical and epistemic qualifications. The four terms of the a fortiori argument as such are the two effective subjects P and Q, and the two predicates ‘likely’ (R) and ‘believed’ (S). Aristotle does not seem aware of all that.

The three or four terms mentioned by Aristotle as subjects and predicates (labeled A, B, C and D by me) are terms within the propositions P and Q, and not the terms of the a fortiori argument as such, note well. These terms (A, B, C, D) are thus quite incidental to the argument, which are used to illustrate possible uses of such argument. Aristotle could well have mentioned only one such illustration if his intention was to abstractly describe a fortiori argument as such. It appears, then, that it was not his primary intention to do that here. His primary intention was probably to concretely describe different ways a predicate may be found to belong or not belong to a subject, by using a fortiori argument.

Nonetheless, judging by this second paragraph, which clearly concerns a fortiori argument, we can again say that Aristotle was well aware of the positive and negative subjectal moods. Thus far, however, there is still no evidence of his being aware of predicatal arguments. We might, on a superficial reading, have thought that Aristotle here marks the difference between subjectal and predicatal a fortiori, when he speaks of one predicate for two subjects or two predicates for one subject. He might have been referring, in the first case to the subsidiary term being predicated of the major and minor terms (the subjectal mood), and in the second case to the major and minor terms being predicated of the subsidiary term (the predicatal mood). But when we actually set out his arguments in standard forms, we see clearly that they are all subjectal.

I should also stress that though Aristotle’s arguments in the above paragraph of Topics, as well as in the Rhetoric passage earlier considered, can be cast in standard forms using qualifications like ‘likely’ as middle term and ‘believed’ as subsidiary term, it is obvious that Aristotle himself does not formulate his arguments as clearly. He is not sharply aware of the distinct functions of these two terms (R and S) in his arguments. In fact, he tends to lump them together, i.e. treat them as one and the same. This observation will be further confirmed further on, when we analyze Topics 3:6. Still in Topics 2:10, he goes on:

“Moreover, you can argue from the fact that an attribute belongs, or is generally supposed to belong, in a like degree, in three ways, viz. those described in the last three rules given in regard to a greater degree. For supposing that one predicate belongs, or is supposed to belong, to two subjects in a like degree, then if it does not belong to the one, neither does it belong to the other; while if it belongs to the one, it belongs to the remaining one as well. Or, supposing two predicates to belong in a like degree to the same subject, then, if the one does not belong, neither does the remaining one; while if the one does belong, the remaining one belongs as well. The case is the same also if two predicates belong in a like degree to two subjects; for if the one predicate does not belong to the one subject, neither does the remaining predicate belong to the remaining subject, while if the one predicate does belong to the one subject, the remaining predicate belongs to the remaining subject as well.”

Looking at this third paragraph, we can also say that Aristotle realized that a fortiori inference is also possible between equals, not just from the more to the less or vice versa. And as he points out, in such case the argument can function either way, i.e. from minor to major or from major to minor, whether it is positive or negative. I have in my Judaic Logic account called such argument, in which the major premise is a statement of equality, egalitarian a fortiori; another name for it is a pari.

Apart from that, there is nothing new in this paragraph – it still concerns only subjectal moods. There is still no mention of equivalent predicatal moods (which involve quite different arrangements of terms). Even so, this insight of his has some importance. He also says further on in the Rhetoric chapter above quoted: “This argument [i.e. a fortiori] might also be used in a case of parity.” He again implies as much in Topics 2:11: “You can argue, then, from greater or less or like degrees of truth in the aforesaid number of ways” (italics mine) and elsewhere.

It should be noted that, though Aristotle, as we have seen in Rhetoric 2:23 and in the second passage of Topics 2:10, formulates a fortiori argument primarily in logical-epistemic terms, looking at the third passage of Topics 2:10 it appears that he also conceives of purely ontical a fortiori argument, since he speaks repeatedly of a predicate belonging to a subject, as against being supposed to belong. This is confirmed by some of his a fortiori pronouncements in other contexts; for example in Topics 3:6 (see below), or again in the History of Animals 5:14, where he says: “If a sow be highly fed, it is all the more eager for sexual commerce, whether old or young,” implying that being well-fed physically causes (and not merely implies) a sow to want sex.

It is reasonable to suppose that, though Aristotle only mentions the distinction between a property and a supposed property in the said third passage, which deals with terms of “like degree,” he does not consider this distinction as exclusive to egalitarian a fortiori arguments. For a start, he makes no mention of such exclusiveness; and besides, examples like the one just cited from the History of Animals show that he does not intend it. Thus, this distinction between a property and a supposed property can be fairly applied to non-egalitarian a fortiori arguments too.

In other words, we may say that Aristotle is somewhat aware of purely ontical argument, and does not limit on principle a fortiori to the logical-epistemic variety, even if he appeals to the latter more often (so far). In my theory of a fortiori argument, note, the emphasis is rather on the ontological variety. This does not of course exclude the epistemological variety, which Aristotle seemingly emphasized, nor for that matter the ethical and legalistic variety, which the Rabbis and others have emphasized; I view (and have from the start viewed) all these other varieties as special cases of the primary, ontological variety.

A further thing to notice is the uncharacteristic lack of formalization in Aristotle’s treatment of a fortiori argument. This is no doubt because he mentions such argument in passing, without focusing on it particularly or very deeply. Although he does discuss the argument in relatively abstract terms, as when he says in the Rhetoric passage: “if a thing is not true where it is more likely, it is not true where it is less likely; or … if it is true where it is less likely, it is true where it is more likely,” and not merely through concrete examples (like the man striking his father or neighbors), he does not go one step further as he did with syllogistic reasoning and use symbols (A, B, Γ, Δ) in lieu of terms to list all possible moods of the argument and, most importantly, to formally validate or invalidate them. My theory of a fortiori argument does this crucial job.

In this regard, we should note too that Aristotle does not here (or elsewhere, to my knowledge) formulate any rule of reasoning comparable to the rabbinical dayo principle (which appears on the stage of documented history perhaps some four and a half centuries later[88]) – or more precisely, to the principle of deduction as it applies specifically to a fortiori argument, namely the rule that the subsidiary term (which is a predicate in subjectal argument) must be identical in the minor premise (where it concerns the minor term) and in the conclusion (where it is applied to the major term). Aristotle may well in practice reason correctly in accord with this principle, but he does not explicitly express theoretical awareness of it – unless we count the already mentioned passage: “See whether a greater degree of the predicate follows a greater degree of the subject,” which we interpreted as an effective rejection of a crescendo argument, as intended by him to be an admonishment by him not to always reason proportionately.

Let us now move on and examine a passage of his Topics 3:6 (book III, chapter 6), which again I split up as convenient:

“Moreover you should judge by means of greater or smaller or like degrees: for if some member of another genus exhibit such and such a character in a more marked degree than your object, while no member of that genus exhibits that character at all, then you may take it that neither does the object in question exhibit it; e.g. if some form of knowledge be good in a greater degree than pleasure, while no form of knowledge is good, then you may take it that pleasure is not good either.”

In this first paragraph, Aristotle shows stronger awareness of the middle term of a fortiori argument, namely the “such and such a character” (R) which the “other genus” (i.e. the major term, P) exhibits in a more marked degree than “your object” (i.e. the minor term, Q). Notice, too, that this middle term (R) is definitely ontological, rather than as before epistemological. However, his argument is not very well formulated, in that his major premise states that “some members” of P “exhibit this character,” whereas his minor premise states contradictorily that “no members” of P “exhibit this character.” This confusion is not due to his insertion of quantification issues into the equation, but to his conflation between the middle term (in the major premise) and the subsidiary term (in the minor premise and conclusion). The latter is a not uncommon error of formulation[89]. He goes on:

“Also, you should judge by a smaller or like degree in the same way: for so you will find it possible both to demolish and to establish a view, except that whereas both are possible by means of like degrees, by means of a smaller degree it is possible only to establish, not to overthrow. For if a certain form of capacity be good in a like degree to knowledge, and a certain form of capacity be good, then so also is knowledge; while if no form of capacity be good, then neither is knowledge. If, too, a certain form of capacity be good in a less degree than knowledge, and a certain form of capacity be good, then so also is knowledge; but if no form of capacity be good, there is no necessity that no form of knowledge either should be good. Clearly, then, it is only possible to establish a view by means of a less degree.”

This second paragraph serves to show (only by means of example, but clearly enough) that Aristotle is aware that, even though one may argue positively, from predication of the subsidiary term to the minor term to predication of the subsidiary term to the major term, it does not follow that one may argue negatively, from denial of predication of the subsidiary term to the minor term to denial of predication of the subsidiary term to the major term – except, of course, where the argument is a pari. He here obviously refers specifically to subjectal argument, since in fact (although he makes no remark to that effect) the opposite rule would hold for predicatal argument. His statement of this rule is significant, since he thereby declares a mood invalid, whereas previously he only declared moods valid.

Note however that he does not similarly point out that, though (in subjectal argument) one may argue negatively, from denial of predication of the subsidiary term to the major term to denial of predication of the subsidiary term to the minor term, it does not follow that one may likewise argue positively, from predication of the subsidiary term to the major term to predication of the subsidiary term to the minor term – except, of course, where the argument is a pari. That is, even though he has previously mentioned both positive and negative moods for validation purposes, in the present remark he only mentions a negative mood for invalidation purposes and omits to mention the corresponding positive mood for invalidation purposes.

Moreover, Aristotle’s “invalidation” of a mood of a fortiori argument here is merely intuitive, i.e. a raw rational insight – he does not explain or formally prove the invalidity of the mood in question. He tells us that it is wrong reasoning, but he does not tell us why it is so.

Furthermore, in the example he gives, the major term is “knowledge” and the minor term is an unspecified “capacity,” while the middle and subsidiary terms are “good.” In this passage, then, he again confuses the issue somewhat by contradicting elements of his major premise, viz. “if a certain form of capacity be good [middle term, R] in a like degree to knowledge,” in his minor premise and conclusion, viz. “if no form of capacity be good [subsidiary term, S], then neither is knowledge.”

The error here, as already pointed out, is to use one and the same term (viz. “good,” in this example) both as middle and as subsidiary. For the argument to be consistent and valid, these two must be distinct (the middle term might, say, be “valuable” and the subsidiary term “pursued,” so that the argument reads: if a certain capacity is as valuable as knowledge, it follows that if no capacity is valuable enough to be pursued, then knowledge is not valuable enough to be pursued). Aristotle, then, is apparently not aware of this important rule, i.e. of the need to distinguish the middle and subsidiary terms.

We might more generously see, in Aristotle’s affirmation of something in one premise and negation of it in the other, as a recognition by him of the possibility of using a term so abstractly that both its position (e.g. “good”) and its negation (“not good”) are included in it, as different degrees of it (above zero and zero or less, respectively). Looking at a term R in this way, we can both claim that P is more R than Q, and claim that P and Q are not R at all, without self-contradiction. This seems to be the thought in Aristotle’s head, though he does not (here at least) make any explicit remark to that effect. To be sure, knowing that Aristotle is not prone to self-contradiction, this is a credible hypothesis.

It is worth noting too in this context that, although Aristotle associates a fortiori argument with the idea of greater, lesser or equal degrees, there is no evidence in the above cited passages of any notion of “sufficiency,” i.e. of there being a threshold as of which predication occurs and before which it does not occur. This is an important deficiency in his treatment (if indeed, as I presume, he nowhere else mentions this feature of a fortiori argument). Had he been aware of the “sufficiency” issue (i.e. the need to have enough of the middle term for predication) in a fortiori inference, he would have quickly realized that the middle term mentioned in the major premise cannot reasonably be identical with the predicate inferred from the minor premise to the conclusion.

As we shall see further on, all but one of Aristotle’s many a fortiori arguments in practice are formulated without the crucial feature of “sufficiency” of the middle term for predication. The one exception shows that Aristotle was slightly aware of this feature, but not enough to make it explicit in all his a fortiori discourse, and not enough to take it into consideration in his theorizing.

2.    The Kneales’ list

In their historical opus, The Development of Logic[90], William and Martha Kneale give seven references in Aristotle’s Topics concerning a fortiori argument, namely: “ii. 10 (114b37); iii. 6 (119 b17); iv. 5 (127 b18); v. 8 (137 b14); vi. 7 (145 b34); vii. 1 (152 b6); vii. 3 (154 b4)”[91]. I have above dealt in detail with the first two of these passages (namely, 2:10 and 3:6), which are the most interesting, in that Aristotle is in them effectively teaching us something about a fortiori argument. The remaining passages are less interesting: Aristotle uses rather than discusses a fortiori argument in two of them (namely, 4:6 and 7:3), while the rest (namely, 5:8, 6:7 and 7:1) have nothing to do with such argument but were only apparently listed because they contain a reference to degrees. Only the following two remaining passages, then, concern a fortiori argument:

Topics 4:6 – This chapter contains an a fortiori argument of positive subjectal form:

“On the other hand, the comparison of the genera and of the species one with another is of use: e.g. supposing A and B to have a like claim to be genus, then if one be a genus, so also is the other. Likewise, also, if what has less claim be a genus, so also is what has more claim: e.g. if ‘capacity’ have more claim than ‘virtue’ to be the genus of self-control, and virtue be the genus, so also is capacity. The same observations will apply also in the case of the species. For instance, supposing A and B to have a like claim to be a species of the genus in question, then if the one be a species, so also is the other: and if that which is less generally thought to be so be a species, so also is that which is more generally thought to be so.”

The reasoning here is: Given that A seems more fitting to be a genus (or a species) than B is, it follows that: if B seems so fitting that it may be declared a genus (or a species), then A must also be fitting enough for that; if A and B are equally fitting (parity), then the inference goes both ways. We can distinguish two moods (from minor to major, and a pari), each with two alternative middle terms (one for genus and one for species); but all four arguments have really one and the same thrust.

Topics 7:3 – This chapter contains a very similar a fortiori argument:

“Moreover, look at it from the point of [sic][92] and like degrees, in all the ways in which it is possible to establish a result by comparing two and two together. Thus if A defines a better than B defines [b?] and B is a definition of [b?] so too is A of a. Further, if A’s claim to define a is like B’s to define b, and B defines b, then A too defines a. This examination from the point of view of greater degrees is of no use when a single definition is compared with two things, or two definitions with one thing; for there cannot possibly be one definition of two things or two of the same thing.”

The reasoning here is: Given that A defines ‘a’ more fittingly than B does ‘b’, it follows that if B defines ‘b’ so fittingly that it may be declared the definition, then A defines ‘a’ must also be fitting enough for that; if Aa and Bb are equally fitting (parity), then the inference from Bb to Aa is also valid (more significantly, the reverse inference is also possible now: though Aristotle does not say so, he probably intended it). Here again, note, there is only really one argument, though it is worded in two ways.

We do not learn anything new about a fortiori argument from these two passages; they each give an example of a fortiori argument, rather than a discussion of it. I should perhaps, after all, say a bit more about the three passages listed by the Kneales that do not contain a fortiori arguments. Aristotle seems there and elsewhere[93] to have some beliefs about the degrees of things that I do not entirely agree with.

Consider for instance the following comment drawn from Topics 5:8:

“Next look from the point of view of greater and less degrees… See, for destructive purposes, if P simply fails to be a property of S simply; for then neither will more-P be a property of more-S, nor less-P of less-S, nor most-P of most-S, nor least-P of least-S. … For constructive purposes, on the other hand, see if P simply is a property of S simply: for then more-P also will be a property of more-S, and less-P of less-S, and least-P of least-S, and most-P of most-S.”

What this, and more of the same (which I have left out, for brevity’s sake), suggests is that Aristotle considers concomitant variation to be a universal law. According to him, if S is P, then to every degree of S there corresponds a comparable degree of P, and if such parallel increase and decrease in magnitude does not occur, then S is not P. This is highly to be doubted, in my view. In some cases, the same value of a predicate P is applicable to all values of a subject S. In some cases, a constant subject S has (over time) different degrees of a predicate P. The variations may be inverted, with increase on one side and decrease on the other, or vice versa. Many other complications are conceivable and occur in practice.

As an example of such inference that Aristotle gives us is: “Thus, inasmuch as a higher degree of sensation is a property of a higher degree of life, a lower degree of sensation also would be a property of a lower degree of life, and the highest of the highest and the lowest of the lowest degree, and sensation simply of life simply.” Well, it may be true that degrees of sensation are proportional to degrees of life (whatever that means: presumably complexity of organization?), but I very much doubt that we can universally infer a concordance of lesser degrees from one of a higher degrees, and so on, as he apparently recommends. Perhaps he only means that such concomitant variation is a good working hypothesis, a probability to be verified empirically.

Again, consider the following comment drawn from Topics 4:6:

“Moreover, judge by means of greater and less degrees: in overthrowing a view, see whether the genus admits of a greater degree, whereas neither the species itself does so, nor any term that is called after it… If, therefore, the genus rendered admits of a greater degree, whereas neither the species does so itself nor yet any term called after it, then what has been rendered could not be the genus.”

Let G be a genus and S be a species, or a species of a species. The question here posed is whether G is or is not indeed a genus of S; or conversely, whether S is or is not indeed a species of G. The answer is sought through comparison of changes in magnitude; actually, only increase in magnitude is mentioned, not decrease (no explanation is given for this unreasonable stipulation). It is not clarified what is here increased – it seems to be the degree of G or S itself, rather than of some property thereof. The changes in degree seem to refer to comparisons of instances (extensional mode), rather than to changes over time (temporal mode).

Aristotle reasons syllogistically that if the genus is variable then the species must be variable too. But to my mind this is an error of logic. Surely a variable is a set of constants, in which case a genus may be variable and yet composed of species some or all of which are (different) constants. The error is to treat the predicate ‘variable’ as distributive, whereas it is here intended as collective – it applies to the class as a whole, not necessarily to any of its parts.

Such comments by Aristotle, though not directly relevant to a fortiori argument, have indirect relevance, since belief in the universality of concomitant variation would lead us to automatically draw an a crescendo conclusion from a fortiori premises, whereas in fact an appropriate pro rata argument is a formally required intermediary for such deduction. But as we have earlier seen (in the previous section, in the first passage of Topics 2:10), Aristotle explicitly (though without naming it) presents argument pro rata as inductive rather than deductive. It follows that he cannot (without self-contradiction) have here intended to suggest that pro rata argument always possible, i.e. formally universal for any terms. Thus, a fortiori argument must be distinguished from a crescendo.

About a contrario. In this context, I could additionally point to some of Aristotle’s remarks in his Rhetoric, which give the impression that he advocates a contrario argument, which has some resemblance to a fortiori argument but is really very different. The following passage, drawn from the already mentioned chapter of Rhetoric will illustrate what I mean:

“One line of positive proof is based upon consideration of the opposite of the thing in question. Observe whether that opposite has the opposite quality. If it has not, you refute the original proposition; if it has, you establish it. E.g. ‘Temperance is beneficial; for licentiousness is hurtful’. Or…: ‘If war is the cause of our present troubles, peace is what we need to put things right again’.”

If we read this literally, we would suppose that ‘If all X are Y, then no not-X is Y’. But such inversion, as Aristotle surely well knew, is not universally valid. We can only educe from ‘all X are Y’ (via: ‘all not-Y are not-X’) that ‘some not-X are not Y’; it remains possible that ‘some not-X are Y’. So, we should view his remarks on such arguments as mere observations. They are presented as forms of rhetoric, rather than of logic, so as to point out noncommittally that people do use them, without intent to imply them to be necessarily valid.

The examples he gives seem credible enough, being particular causative arguments. Since licentiousness hurts, we should try temperance to diminish if not remove our pain. Since war causes troubles, we should try peace to diminish if not stop our malaise. These are only probable arguments, however, which do not guarantee that the desired change will occur. They are not, of course, a fortiori arguments, although they have some resemblance.

Compare the commonly used formulation of a fortiori argument: ‘If Q, which is not R, is S, then, all the more, P, which is R, is S’, with the following a contrario statement (which for the sake our present demonstration involves the vaguer term ‘something’ in the places of P and Q): ‘If something which is not-R is S, then something which is R is not-S’ – and it is easy to see the difference. In the former case (i.e. a fortiori), the predicate is S in both the antecedent and consequent, whereas in the latter case (i.e. a contrario), the predicate is S in the premise and not-S in the conclusion. The resemblance is thus quite superficial.

A contrario argument, like a fortiori, can be copulative or implicational. In the former case, it has the form: ‘If X is Y, then not-X is not-Y’; and in the latter case, it has the form: ‘If X implies Y, then not-X implies not-Y’. While such reasoning is sometimes applicable, it is not – to repeat – universally valid.

Finally, let me quote the Kneales’ sole remark about Aristotle in relation to a fortiori argument:

“…the theory of arguments a fortiori, or, as Aristotle says, ‘from the more and the less’. This is a topic to which he refers many times and always in a way which suggests that he thinks of it as a well-recognized theme. It was natural, therefore, that he should wish to incorporate his views on the subject into his later work on logic, and it seems probable that this is what he had in mind when he spoke later of his intention to write on arguments ‘according to quality’ (κατἀ ποιὀτητα).” (Pp. 42-43.)

This comment suggests that Aristotle was rather interested in a fortiori argument and seemingly intended to treat the subject in more detail eventually. The Kneales do not specifically cite the passages in Aristotle’s works they base these remarks on. As already mentioned, they do give a number of references in the Topics, but I do not see that these passages justify the above claims. Not that it matters greatly, but I would have liked to know what the Kneales meant more precisely. Because, judging by the texts analyzed above, Aristotle’s involvement in theoretical a fortiori logic was not very intense.

3.    Aristotle in practice

Let us now take a closer look at Aristotle’s practice of a fortiori argument, which differs considerably from his theoretical treatment. For this purpose, I looked into all instances I could find of Aristotle’s use of the argument[94]. See Appendix 4 for a detailed list of citations[95]. These included 40 occurrences of the fifteen key phrases most often used to signal a fortiori discourse, namely: a fortiori (12), all the more (22), how much more (2), how much less (0), so much more (1), so much less (0), much more (2), much less (1), (how/so) much the more (0), (how/so) much the less (0). Plus 3 occurrences of more widely used character strings, namely: more so (1), less so (0), even more (2), even less (0). Additionally, I referred to the passages in Aristotle’s Rhetoric and Topics found by the Kneales (see previous two sections), which contain numerous a fortiori arguments without use of the key phrases (except once), and found another 37 occurences.

Altogether, I found in Aristotle’s works, 80 cases of a fortiori argument, of which at least 11 were a pari (i.e. involved a major premise with equal major and minor terms). As could be expected, most cases, 48 to be exact, were positive subjectal in form; and indeed, of these 8 could be said to be a crescendo. Without surprise, another 22 cases were found to be negative subjectal. The interesting findings were that 5 cases were positive predicatal and 3 cases were negative predicatal; and moreover that 2 cases were antecedental. What these findings teach us is that, although Aristotle reasoned often enough in subjectal formats, which he mentions in his more theoretical exposés, he also occasionally reasoned in other formats, which he does not consciously distinguish in theoretical contexts.

Aristotle, as everyone knows, was Plato’s star student. Examining the latter’s main works, I found at least 15 instances of a fortiori discourse, 9 of them spoken (if we are to believe Plato) by Socrates, and the rest by others. Of these instances, 9 are positive subjectal in form (and of those, 4 seem to have an a crescendo intent), 1 is negative subjectal, 4 are negative predicatal, and 1 is negative consequental in form. These findings are based on computer searches for specific strings; more cases, involving other wording, may conceivably yet be found. These figures on Plato are also significant, assuming that Aristotle read these works (a fair assumption), since they are additional evidence that Aristotle did not closely examine all the data he had on hand when analyzing a fortiori argument. The corresponding findings for Aristotle are as follows:

Mood of
a fortiori argument
Orientation Number found Of which
a pari
Of which
crescendo
Copulative
Positive subjectal {+s} from minor to major (Q-P) 48 7 8
Negative subjectal (–s) from major to minor (P-Q) 22 4
Positive predicatal {+p} from major to minor (P-Q) 5
Negative predicatal (–p) from minor to major (Q-P) 3
Implicational
Positive antecedental (+a) from minor to major (Q-P) 2
Negative antecedental (–a) from major to minor (P-Q) 0
Positive consequental (+c) from major to minor (P-Q) 0
Negative consequental (–c) from minor to major (Q-P) 0
Totals 80 11 8

Table 6.1

Needless to say, the arguments are here classified on the basis of their apparent forms, without regard to the truth or falsehood of their contents.

As regards Aristotle’s own use of predicatal argument, 1 case occurs in On the Soul, 1 case in Parva Naturalia, 1 case in History of Animals, 1 case in Metaphysics, 2 cases in the Posterior Analytics, and 2 cases in Rhetoric. For example: “But if the Soul does not, in the way suggested [i.e. with different parts of itself acting simultaneously], perceive in one and the same individual time sensibles of the same sense, a fortiori it is not thus that it perceives sensibles of different senses” (Parva Naturalia, 7). This has to be read as a predicatal argument[96], since the subjects of the minor premise and conclusion are one and the same (viz. “the soul”) and their predicates are different (viz. “it perceives sensibles of the same sense” and “it perceives sensibles of different senses”).

Aristotle’s two uses of implicational argument (both positive antecedental) occur in History of Animals; notice that there is no use of negative antecedental or of positive or negative consequental argument. An example is: “Now, as the nature of blood and the nature of the veins have all the appearance of being primitive, we must discuss their properties first of all, and all the more as some previous writers have treated them very unsatisfactorily” (3:2). This has to be read as an implicational argument[97], because in the minor premise and conclusion, the antecedents and consequents contain different subjects and predicates, so that these propositions consist of theses implying theses.

Thus, judging by his extant works, Aristotle did not pay close attention to his own uses, or his teacher’s uses, of a fortiori argument, when discussing this form of reasoning. Had he done so, he would have discovered predicatal argument and implicational argument.

Furthermore, as regards his 8 uses of a crescendo argument (all positive subjectal), it may be supposed that Aristotle uttered them in good faith, i.e. that he believed that in these specific cases proportionality was justified. But he apparently nowhere remarks on the important difference between purely a fortiori argument and the more elaborate a crescendo argument, even though he uses both these types of reasoning. That is to say, he does not formulate a rule comparable to the much later rabbinical “sufficiency (dayo) principle,” according to which (in the simplest reading of it[98]) the conclusion of an a fortiori argument should exactly mirror its minor premise, and not indulge in proportionality (to which we should add: unless, of course, an appropriate pro rata argument can be additionally put forward to justify such proportionality).

It is noteworthy that, in all the instances of a fortiori argument I found in Plato and Aristotle works, only one instance contains the word ‘enough’ or ‘sufficient’. The instance is found in Aristotle’s work and reads: “But since even water by itself alone, that is, when unmixed, will not suffice for food – for anything which is to form a consistency must be corporeal – , it is still much less conceivable that air should be so corporealized [and thus fitted to be food]” (On Sense and the Sensible, 5). This shows that Plato was unaware of this crucial feature of a fortiori argument, and Aristotle was a bit more but still barely aware of it.

Finally, it is interesting to note the following statistics: of the a fortiori arguments used by Aristotle, only 16 are logical-epistemic[99], the remaining 57 being ontical. What this tells us is that the impression given by Rhetoric 2:23 and Topics 2:10 that he regards a fortiori argument as essentially logical-epistemic is belied by his actual practice.

4.    Relation to syllogism

One more important question to ask regarding Aristotle’s theoretical treatment of a fortiori is whether he regarded such argument as capable of identification with syllogism. Wiseman[100] suggests that Aristotle did not make such an equation, saying:

“Interestingly, Aristotle did not consider the a fortiori to be the same as his categorical syllogism; rather, he understands it as an analogic[al] device, unlike what we have encountered in some definitions so far that meant to show it as deductively valid. Perhaps Aristotle was the first to view the a fortiori as an inductive analogy.”

As regards Wiseman’s claim that Aristotle viewed a fortiori as a mere analogical device, I tend not to agree. Wiseman is basing this assumption, I take it, on the first of the above quoted paragraphs in Topics 2:10– which, as already pointed out, is not clearly about a fortiori argument (even though the next paragraph indeed is about it). Aristotle is here neither proposing a necessary deduction (a fortiori or other) nor suggesting a weaker argument by analogy – on the contrary, he is saying one cannot predict which way things will go (“See whether a greater degree of the predicate follows a greater degree of the subject…”) and must resort to induction for the answer. Moreover, if we look at the earlier Rhetoric quotation, a different picture emerges.

As regards the suggestion that the two forms of argument are different, note that Wiseman does not quote Aristotle as saying so; he only theorizes it is so, based on the information available to him. I would certainly lean towards the same assumption, however. It would seem (given his extant works) that Aristotle did not ask himself or try to answer that specific question, about whether a fortiori argument is or is not a sort of syllogistic argument; had he done so, he would surely have stressed the fact explicitly, one way or the other. On the other hand, it could be argued that Aristotle tended to consider syllogism as the essential form of all argument (certainly many people after him seem to have thought he did so) – in which case he would not necessarily think he needed to specifically subsume a fortiori for us.

Consider now an example of a fortiori argument given by Aristotle in Rhetoric 2:23: a man is less likely to strike his father than to strike his neighbors; therefore, if a man strikes his father, he is likely to strike his neighbors too. We see here that Aristotle is aware of the major premise[101], as well as of the minor premise and conclusion. However, he does not discuss the real middle term, which tacitly underlies and would explain and justify the apparent middle term ‘likely’ that he takes for granted. Why is a man more likely to strike his neighbors than his own father? Because it is generally easier, psychologically, socially and ethically to strike one’s neighbors than one’s father. The apparent middle term ‘likely’ is based on an emotional and cultural fact (or at least, the assumption of such a fact).

A fortiori argument usually appears as essentially deductive – in the sense that given the premises we can confidently infer the conclusion – yet in the present example there is clearly a sense that the conclusion is at best probable. Why is that? Because it so happens that the example under scrutiny is about human volition, i.e. something that by nature cannot be predicted with certainty. A man may well generally find it easier to hit neighbors than his own father; but in truth, a man may consider the latter action as more legally permissible, being a private as against public matter, or again, he may out of cowardice hit on his weak old father more readily than he assaults his strong young neighbors.

Such actions are based on personal perceptions or belief systems, and depend on personal inclinations and conscience, and they are ultimately produced by freewill. For this reason, Aristotle indeed had to qualify things as only “likely” throughout his example. But such approximation is not inherent to a fortiori, but a function of the content in this particular sample. If we look at the other example Aristotle gives in the same passage of Rhetoricif even the gods are not omniscient, certainly human beings are not – it is clear that he sees the conclusion as certain[102], and not as a mere rough analogy[103].

We can thus, to conclude, say that since – as far as we know – Aristotle did not fully analyze a fortiori argument, he is not likely to have made a pronouncement as to whether it was the essentially same as syllogism or not; or, for that matter, as to whether it is deductive or merely analogical. The truth is, Aristotle was a genius who ranged far and wide in logic, philosophy and the special sciences, and touched upon a great many subjects, some of which he took time to look into more deeply and systematically, and some of which he only briefly considered in passing. Regarding a fortiori, the latter seems to be applicable. Moreover, of course, Aristotle was human, and however authoritative his viewpoints on many issues, he was not omniscient (as he readily admits in one of the said examples).

Whatever Aristotle may have or not have privately thought on the issue, my own formalization of a fortiori, presented in the preceding chapters, justifies our henceforth definitively adopting the position that Aristotle’s categorical syllogism (and also for that matter hypothetical syllogism, which is very similar in overall form) is very different from copulative (or implicational, as the case may be) a fortiori argument, though the latter is also a form of deduction. Moreover, although we can correlate these two forms of argument in various ways, we cannot formally reduce either of them to the other; they are distinct and relatively independent movements of thought.

5.    Cicero

Marcus Tullius Cicero (Rome, 106-43 BCE), who was an influential philosopher and jurist among many other things, left us some interesting reflections on a fortiori argument in his Topics[104]. Cicero there tells us (this was a year before his death) he composed the book as a commentary to Aristotle’s work with the same name, from memory; but his treatment is distinctive. It seems to have been equally influenced by Aristotle’s Rhetoric (II, 23) and by some later, Stoic texts[105]. Concerning argumentation in general, Cicero has this to say:

“6. Every systematic treatment of argumentation has two branches, one is concerned with invention of arguments and the other with judgment of their validity; Aristotle was the founder of both in my opinion.”

By “invention of arguments” he apparently means formulation of arguments. From his mention here of validation, we see that Cicero’s interest was in logic, and not merely in rhetoric. He discusses in some detail all the arguments he lists, giving examples from Roman law practices. Arguments by comparison (i.e. a fortiori) are classified as arguments “from the things which are in some way closely connected with the subject,” which in turn fall under the heading of arguments “inherent in the nature of the subject.” This teaches us that Cicero looked upon a fortiori argument as essentially ontical, rather than as logical-epistemic. He introduces a fortiori argument in §23 as follows:

“23. All arguments from comparison are valid if they are of the following character: what is valid in the greater should be valid in the less (Quod in re maiore valet, valeat in minori), as for example… Likewise the reverse: what is valid in the less should be valid in the greater (Quod in minori valet, valeat in maiore); the same example may be used if reversed. Likewise, what is valid in one of two equal cases should be valid in the other (Quod in re pari valet valeat in hac quae par est); for example… Equity should prevail, which requires equal laws in equal cases.”

Cicero here apparently lists three varieties of the argument: from major to minor; from minor to major; and from equal to equal. Let us look at the examples here proposes for them. The first example concerns reasoning from major to minor: “since there is no action for regulating boundaries, there should be no action for excluding water in the city.” This argument seems to be a negative subjectal; we can formalize it as follows:

Regulating boundaries (P) is more serious a matter (R) than excluding water in the city (Q) is,
yet, regulating boundaries (P) is not a serious matter (R) enough to justify an action (S);
therefore, excluding water in the city (Q) is not a serious matter (R) enough to justify an action (S).[106]

For reasoning from minor to major, Cicero unfortunately gives no example here, but only says “the same example may be used if reversed.” It is not clear what “reversed” (convertere) here means. It surely does not mean simple conversion, for such argument would obviously be logically invalid[107]. That is, we can reasonably assume he is not suggesting that “since there is no action for excluding water in the city, there should be no action for regulating boundaries” follows from the preceding case. Therefore, he presumably intends a hypothetical contraposition of it: “if there was a possibility of action for excluding water in the city, there would be a possibility of action for regulating boundaries,” which signifies: positive subjectal argument.

The example Cicero adduces for a pari argument is: “since use and warranty run for two years in the case of a farm, the same should be true of a (city) house. But a (city) house is not mentioned in the law, and is included with the other things use of which runs for one year”[108]. It is not clear to me what the intended conclusion is, here. The first sentence seems to conclude with equality; but the second sentence denies the equality. I think that the solution to that problem is simply that Cicero here proposes two a pari arguments, one positive and one negative. The first says hypothetically: “if farm and city house were equal, the law of the former would apply to the latter.” The second says factually: “but since they are not equal, the law of the former does not apply to the latter.”

Thus, to summarize, Cicero seems to have pointed to positive and negative subjectal a fortiori argument, including their a pari versions. What about the positive and negative predicatal moods? I do not think that we can judge on the basis of the examples he gives that Cicero consciously limited a fortiori to the subjectal moods, to the exclusion of the predicatal ones; or for that matter, that he intended to limit it to copulative forms, to the exclusion of implicational ones. He obviously simply stated three directions “from major to minor,” “from minor to major,” and “from equal to equal” – unaware of the distinctions between positive and negative, subjectal and predicatal, or copulative and implicational. In other words, let us not misinterpret his vagueness as an exclusive (or even inclusive) intent.

It seems that some of this ambiguity was corrected by later writers, judging by a maxim claimed by Mielziner to have been in use in 19th century jurisprudence[109]: “Quod in minor valet, valebit in majori; et quod in majori non valet, nec valet in minori” – meaning: “what avails in the less, will avail in the greater; and what will not avail in the greater, will not avail in the less.” The similarity of this statement to Cicero’s is striking, but so is the difference. Here, the minor to major case is consciously positive, since the major to minor case is explicitly negative. The trouble with this more precise later statement, however, is that (if it was intended as exhaustive) it effectively limits a fortiori reasoning to the subjectal mode, to the exclusion of the predicatal mode. But such exclusiveness may have been, and probably was, unintentional.

In fact, Cicero further expounds “the topic of comparison” in §68-71.

“68… a definition and example were given above. Now, I must explain more fully how it is used. To begin with, comparison is made between things which are greater, or less or equal. And in this connexion, the following points are considered: quantity, quality, value, and also a particular relation to certain things.”

He then goes on to clarify each of these considerations with many examples. I will reproduce here one example for each. For “quantity”: “more ‘goods’ are preferred to fewer;” for “quality”: “we prefer… the easy task to the difficult;” for “value”: “we prefer… the stable to the uncertain;” for “relation to other things”: “the interests of leading citizens are of more importance than those of the rest.” Clearly, these considerations refer to possible contents of a fortiori argument: the examples he proposes are sample major premises.

The uniform ‘X is preferred to Y,’ format of his proposed major premises suggests to me that Cicero was only consciously aware of subjectal a fortiori argument; he did not consciously notice (though he might have in practice used) predicatal a fortiori argument. Granting this, it follows that when earlier Cicero referred to inference from major to minor, he did have in mind negative subjectal argument; and therefore for him inference from minor to major meant positive subjectal. Note also that the format is also always copulative, never implicational and the middle term is always ‘preference’ – one thing is preferable to another. This is a limitation which we might excuse by saying that Cicero had in mind disputes between people in front of a court.

We can thus guess the forms of argument Cicero had in mind to have been: given ‘X is better than Y,’ it follows that ‘if Y is good enough for Z, then so is X’ and ‘if X is not good enough for Z, then neither is Y.’ He also says: “70… And just as these are the things which in a comparison are regarded as the better, so the opposites of these are regarded as worse.” What he had in mind here is: since ‘X is better than Y’ is convertible to ‘Y is worse than X,’ it also follows that ‘if X is bad enough for Z, then so is Y’ and ‘if Y is not bad enough for Z, then neither is X.’ Cicero does not say this explicitly, but that is evidently what he means. Note that these alternate arguments are formally the same, i.e. just as subjectal.

Regarding a pari argument, he adds: “71. When equals are compared, there is no superiority or inferiority; everything is on the same plane.” He gives a new example of it: “If helping one’s fellow-citizens with advice and giving them active assistance are to be regarded as equally praiseworthy, then those who give advice and those who defend ought to receive equal glory. But the first statement is true, therefore the conclusion is also.” Now, my impression here is that Cicero is having trouble formulating a sample a pari argument! What he has just put forward is not a fortiori argument, but simply apodosis: ‘If A, then B; but A, therefore B.’

The correct formulation of an a pari argument would be, according to me: ‘X is as good as Y, therefore: if X is good enough for Z, so is Y; and if Y is good enough for Z, so is X; and if either is not good enough for Z, neither is the other.’ Or, to use Cicero’s sample terms: ‘Giving advice and actively assisting are equally praiseworthy, therefore: if either is praiseworthy enough to deserve glory, so is the other; and if either is not praiseworthy enough to deserve glory, neither is the other.’ It seems that Cicero did not fully grasp this form.

Finally, we should note that Cicero does not mention anywhere the principle of deduction for purely a fortiori argument, according to which the subsidiary term should be identical in the conclusion to what it is in the minor premise, and not made ‘proportional’ (in an attempt to reflect the proportion between the major and minor terms). There is accordingly no mention by him of the a crescendo argument, where a ‘proportional’ conclusion is indeed allowed, being made possible by means of an additional premise about concomitant variation.

The rabbinical dayo (sufficiency) principle, being first mentioned in the Mishna Baba Qama 2:5, may be said to have appeared in Jewish legal discourse sometime in 70-135 CE at the latest, this being the period when R. Tarfon (who is mentioned in the said Mishna) was active. This principle, as we shall see, prohibits lawmakers from inferring a greater penalty for a greater crime from a lesser penalty for a lesser crime given in the Torah. I have not found evidence of a similar restriction in Cicero’s Topics. However, Roman law does seem to have generated an apparently similar principle, which reads: “In poenis bensignior est interpretatio facienda,” meaning: in penalties, the more benign interpretation is to be applied[110].

I do not know when this principle first appeared in Roman law. If it was developed before or during Cicero’s time, he would surely have mentioned it somewhere (in his Topics or elsewhere), being an expert in Roman law. If it emerged later, it might still have done so before it made its appearance in Jewish jurisprudence – or it may have come after. This historical question must be resolved by competent historians. In any case, it cannot be said with certainty that the law system where the principle appeared first influenced the law system where it appeared second. There could have been a common inspiration, or an inspiration from one to the other, or the two cultures could have arrived at the same idea independently[111].

To summarize, what is evident is that though Cicero had some knowledge of a fortiori argument, he was not conscious of all its forms (namely, predicatal and implicational forms); also, some of the forms he was conscious of (namely the a pari) he did not quite master. Moreover, the issue of ‘proportionality’ apparently eluded him. Another important observation we must make is there is no evidence of formalization or validation in Cicero’s treatment of the subject, though he mentions the issue of “validity” at the beginning of his book. Thus, we must say that on the whole Cicero did not go much further than Aristotle as regards a fortiori logic. Still, he enriches the field a bit through his more conscious distinction between three variants of a fortiori argument (viz. major to minor, minor to major, and a pari) and his listing of various possible contents (quantity, quality, value and importance).

All this is certainly interesting historically, in that it gives us an idea of the state of knowledge and skill regarding a fortiori argument in Cicero’s lifetime in Rome. Because Cicero was one of the foremost legal thinkers, lawyers and orators of his generation, we can reasonably consider his level as the ‘state of the art’ for his time and place, that is about three centuries after Aristotle in the Greco-Roman world. Needless to say, this is said on the basis of a spot check, and not on the basis of a thorough study of all the relevant literature in that region and period. There may well have been other logicians or rhetoricians who said more on a fortiori argument than we have discovered thus far.

6.    Alexander of Aphrodisias

The Kneales’ account makes no mention of any discussion of a fortiori argument in the Hellenistic world in the centuries between Aristotle and Alexander of Aphrodisias, who was a 3rd century CE Peripatetic philosopher and commentator of Aristotle’s works. In particular, they do not mention Cicero’s contribution to the subject, which we presented in the previous section, even though they do examine his work on other topics. Obviously, then, their silence regarding a fortiori argument should not be interpreted to mean that there was no discussion of the subject; it could well just mean that they did not consider it important enough to mention. Anyway, as regards the said Alexander, the Kneales tell us the following, further to their earlier comments regarding the treatment of a fortiori argument by Aristotle:

“From Alexander’s explanation it appears that an argument of type (5), i.e. κατἀ ποιὀτητα, is an a fortiori argument with a general conditional premiss[112]. His example is:

If that which appears to be more sufficient for happiness is not in fact sufficient, neither is that which appears to be less sufficient.

Health appears to be more sufficient for happiness than wealth and yet is not sufficient.

Therefore wealth is not sufficient for happiness.

The theory of arguments κατἀ ποιὀτητα was probably an attempt to systematize what Aristotle says of a fortiori arguments in various passages of his Topics” (p. 111).

I do not see that this remark tells us much more about Aristotle or about a fortiori argument, but I quote it to be exhaustive. As regards Alexander’s example, I would rephrase it in standard format as follows:

Health (P) is apparently more conducive to happiness (R) than wealth (Q) is.
Health (P) is not conducive to happiness (R) sufficiently to actually produce happiness (S).
Therefore, wealth (Q) is not conducive for happiness (R) sufficiently to actually produce happiness (S).

In this format, it is seen to be a valid negative subjectal (major to minor). Let us analyze Alexander’s statement in detail, now. The Kneales’ remark about this being “an a fortiori argument with a general conditional premiss” refers to the first proposition: “If that which appears to be more sufficient for happiness is not in fact sufficient, neither is that which appears to be less sufficient.” If we look at this proposition, we see that it is only general regarding the major and minor terms P and Q (respectively, “more sufficient for happiness” and “less so”), but not general as regards the middle term R (which is specified as “sufficient for happiness”). Thus, it is only partly general. To be fully general, i.e. effectively a formal statement, the middle term should have been “something.” That is to say, the proposition should have read: “If that which appears to be more sufficient for something is not in fact sufficient, neither is that which appears to be less sufficient.”

In fact, therefore, since it is not “general” enough to be formal, this first proposition is redundant. Alexander’s second and third propositions contain, without need of the initial not-quite-abstract statement, the whole concrete a fortiori argument. The second proposition, “Health appears to be more sufficient for happiness than wealth and yet is not sufficient,” lists both the operative major premise (“Health appears to be more sufficient for happiness than wealth”) and minor premise (“and yet [health] is not sufficient [for happiness]”); and the third proposition (“Therefore wealth is not sufficient for happiness”) concludes the argument. Now, this is a well-constructed a fortiori argument, because it has an explicit middle term (“sufficient for happiness” – meaning, rather, conducive to happiness), relative to which the major and minor terms are compared, and it has two premises and a conclusion, and its minor premise and conclusion contain the idea of sufficiency (in negative form) for a certain result (actual happiness, in this case).

So this is on the whole a good effort by Alexander, although not perfect. The imperfections are (a) the first proposition, which is not general enough to count as a formal statement and therefore redundant (since the next proposition does the job just as well without it); (b) the lumping together of the operative major and minor premises into an apparently single statement (so that the different roles of the conjuncts in it are blurred); and (c) the use of the term “sufficient” in two senses: as ‘conducive ’ and ‘enough (to actualize)’. The latter equivocation causes some confusion in the reading of Alexander’s a fortiori argument, and is indicative of some confusion within him. It is indicative of a commonplace error, which we have already spotted in Aristotle’s treatment – namely, the conflation of the middle and subsidiary terms, the failure to clearly distinguish them in view of their quite distinct roles in the argument.

Thus, all things considered, Alexander’s statement is a well-constructed example of (subjectal) a fortiori argument, showing considerable implicit understanding of the form of inference – but it is not a successful explicit formalization, showing complete understanding. And of course, so far as we can tell from the Kneales’ account, there is no effort at validation. This is all a bit surprising, since Alexander was an Aristotelian, and so presumably well acquainted with Aristotle’s formal methods. We could regard Alexander’s first, “general” proposition as his attempt at validation. He perhaps viewed this statement as justifying the inference from the second proposition to the third (much like in syllogism the general major premise justifies the inference from the minor premise to the conclusion). But though such application of a wider generality gives an impression of validation, it does not in fact constitute validation, since the wider generality remains unproved.

Still, Alexander’s work is an improvement. He places more emphasis than Aristotle seems to have done on ontical a fortiori. He is also more advanced in his clear focus on sufficiency in the example quoted, whereas Aristotle does not use the word in the present context. Of course, several centuries separate the two. Note in passing that in Alexander’s case we are already in Talmudic times (not that I suggest a causal relation between his thought and that of the rabbis – but the parallelism is interesting).

It is (according to Ventura[113]), be it said in passing, to this Alexander that we owe the Greek word logika in the sense of the modern term ‘logic’. Previously, the word had rather the sense of ‘dialectic’ (e.g. as used by Cicero). Aristotle’s word for what we call logic was ‘analytic’; whence the titles of two of his works: Prior Analytics and Posterior Analytics. Alexander also inaugurated the term Organon to refer to a collection of Aristotle’s logical works[114].

As for the Kneales, their failure to analyze the “general conditional premiss” sufficiently to realize its relative informality shows that they did not have an entirely clear idea of what constitutes formalization. For this reason, and because I have in the past found errors in their analyses in other contexts, I do not take for granted their following statement: “The theory of arguments κατἀ ποιὀτητα was probably an attempt to systematize what Aristotle says of a fortiori arguments in various passages of his Topics.” They do not specify which passages. I would want to see these passages for myself before accepting that there is significant “systematization” in them. All we are shown here is a negative subjectal argument; there is no positive subjectal and there are no predicatal forms on display, to convince us that Alexander indeed achieved a systematic understanding. He made a valuable contribution, but I reserve judgment as to its full scope.

7.    Historical questions

What is the precise history of a fortiori argument in ancient Greek, Roman and Hellenistic literature, whether philosophical, religious or secular? This question is always answered briefly and rather vaguely by historians of logic, if at all, because no one has apparently ever systematically researched the answers to it. In fact, this question should be asked for every type of argument, in every culture, if we want to be able to eventually trace the development of reasoning by human beings. But historical research into the a fortiori argument would be a good start, a good model, as it is a rather distinctive form of argument which is used and discussed in the said ancient Western civilizations though not so frequently as to be overwhelming. This is a scientific task, akin to biological research into a particular species of life in a particular environment, and it should be carried out with appropriate rigor and exhaustiveness.

The first step in such research would be collection of all relevant data. This means identifying the precise locations in various extant texts where such argument appears (in full or in part) to be used, and of course registering the argument made there in a data base so that it becomes henceforth readily available for future discussions. The literature[115] to be looked into dates from about the 8th century BCE to about the 5th century CE, in the Greco-Roman world, mainly in the Greek and Latin languages. Apart from actual occurrences of a fortiori argument, abstract discussions relating to the use of such argument must be identified and collected. Discussion of a form of argument signifies a higher degree of logical awareness than mere usage; and any attempts at theory, i.e. to formalize it, to find its varieties and to validate it, signify a higher level still. All these stages in logical awareness should obviously be distinguished, assuming instances of all of them are found.

Once the said raw data is collected, logicians can begin to sift through it and analyze its full significance. We can find out when and where the argument first and subsequently appeared within the period and region studied, and what form it took in each case. We can follow the flowering of varieties of the argument over time and in different places, as practice becomes more sophisticated. We can distinguish the different contexts of usage: poetic, business, legal, philosophical, scientific. We can compare the frequencies of use of such argument in different cultures[116]. We can perhaps trace the travels of the argument from one culture or subculture to another, as it is passed on from one people or social group to another, along trade routes or through various kinds of intellectual influence (for examples, through a philosophical author or a religious holy book). We can hopefully perceive the dawning self-awareness of those using the argument, as they begin to marvel at it, discern its parts and try to understand how it functions.

Clearly, we have here a sketch of a very interesting and enriching research project that someone or some people could and should take up. Similar research should of course also be carried out for other periods of history and regions of the world.

7.  A fortiori in the Talmud

1.    Brief history of a fortiori

There is credible written evidence that a fortiori argument was in use in very early times thanks to the Jewish Bible. Five instances are apparent in the Torah proper (the Five Books of Moses, or Pentateuch) and about forty more are scattered throughout the Nakh (the other books of the Bible). According to Jewish tradition, the Torah dates from about 1300 BCE (the time of the Exodus from Egypt and wanderings in the Sinai desert)[117], and subsequent Biblical books range in age from that time to about the 4th century BCE (the period of the return from Babylon of some of the captives after the destruction of the first Temple). The oldest apparent a fortiori (actually, a crescendo) argument in the Torah is the one formulated in Gen. 4:24 by Lamekh (before the deluge); while the oldest purely a fortiori argument is the one formulated in Gen. 44:8 by Joseph’s brothers (patriarchal era). A fortiori arguments are also found in some of the latest books of the Bible (first exile period).

Of the 46 or so instances of a fortiori argument in the Tanakh (see Appendix 1), at least 10 were known to (i.e. were consciously identified as such by) the rabbis of the Talmud – so it is not surprising that this form of argument came to play such an important role in the development of Jewish law. The qal vachomer argument, as it is called in Hebrew, is mentioned in several lists of Talmudic hermeneutic principles. It is the first rule in the list of 7 attributed to Hillel (the Elder, Babylonia and Eretz Israel, c. 110 BCE-10 CE) and the first rule in the list of 13 attributed to R. Ishmael (ben Elisha, Eretz Israel, 90-135 CE), both of which are given at the beginning of the Sifra (a halakhic midrash, attributed by many to Rab, i.e. Abba Arika, 175–247 CE). It is also found (as rules 5 and 6) in the slightly later list of 32 rules of R. Eliezer b. Jose ha-Gelili (Eretz Israel, ca. 2nd cent. CE)[118], and among the much later 613 rules of the Malbim (Meïr Leibush ben Yechiel Michel Weiser, Ukraine, 1809-1879) in his work Ayelet haShachar, the introduction to his commentary on the Sifra.[119]

As regards historical source, there can be little doubt that the rabbis learned a fortiori argument from its use in the Tanakh – and not (as some commentators have suggested) from surrounding cultures (Greek, Roman, or whatever). We can be sure of that knowing that the Talmudic rabbis’ attention was wholly turned towards Jewish Scriptures and oral tradition; and a fortiori arguments were clearly in use in these sources; and moreover, everyone agrees that the Torah, at least, antedates by several centuries the historical appearance of a fortiori argument in other cultures. This does not, of course, imply that the Greeks and other early users of a fortiori argument learned this form of reasoning from the Torah or other Jewish sources. There is no doubt that a fortiori argument arose independently in different cultures at different times, simply due to its being a natural form of human reasoning[120].

In the lists of Hillel and R. Ishmael, all that is offered is a title or heading: “qal vachomer,” which is variously translated as light and heavy, easy and difficult, lenient and stringent, or minor and major. It should be said that the language of a fortiori argument in the Tanakh, though very varied (but not always distinctive, i.e. not always specifically reserved for such argument), does not include the words qal vachomer. This expression is presumably therefore of rabbinical origin. Two other expressions indicative of a fortiori discourse are also found in rabbinic literature: kol she ken (which seems to be the Hebrew equivalent of ‘all the more so’) and al achat kama vekama (which seems to be the Hebrew equivalent of ‘how much the more’).[121]

The term qal vachomer is somewhat descriptive, in the way of a hint – but note well that it is certainly not a description of a fortiori argument in formal terms, and it does not validate or even discuss the validity of the argument (but, obviously, takes it for granted). The list of R. Eliezer b. Jose ha-Gelili is not much more informative in that respect than those of its predecessors, since it only adds that qal vachomer may be meforash (i.e. explicit) or satum (i.e. implicit)[122]. Other early rabbinic literature does not go much further in elucidating the definition and more theoretical aspects of qal vachomer; it is all taken for granted.

Rather, the form and operation of a fortiori argument are taught through concrete examples. Ten Biblical examples of the argument, four in the Torah and six elsewhere[123], are listed in Genesis Rabbah (in Heb. Bereshith Rabbah), a midrashic work (closed ca. 400-450 CE) attributed by tradition to R. Oshia Rabba (d. ca. 350 CE). This just says: “R. Ishmael taught: [There are] ten a fortiori arguments recorded in the Torah” (92:7), and lists the ten cases without further comment. But of course, the main teaching of such argumentation is through the practice of the rabbis. There are a great many concrete examples of a fortiori reasoning in the Talmud and other rabbinic literature[124], which incidentally serve to clarify the form for future generations.

There is, however, one passage of the Talmud which is very instructive as to how the rabbis theoretically understood the qal vachomer (a fortiori type) argument and the dayo (sufficiency) principle related to it – and that is pp. 24b-25a and further on pp. 25b-26a of the tractate Baba Qama (meaning: ‘first gate’), which is part of the order of Neziqin (‘damages’). For the time being we shall concentrate on this important passage. We shall have occasion further on in the present volume to consider and explore some other significant Talmudic a fortiori arguments.

However, this book makes no claim to constituting an exhaustive study of this subject. Nonetheless, while I must confess being largely ignorant of the ‘Sea of the Talmud’, I believe the present contribution will be found very valuable due to the considerable extent and depth of new logical insight it contains. We shall in the present chapter, further on, describe in detail just what the said passage of the Talmud reveals. But first permit me to prepare you, the reader, with some background information and analysis, so that you come properly armed to the crux of the matter.

The Talmud (meaning: the teaching) in general consists of a series of rabbinical discussions on various legal and other topics stretching over centuries, roughly from about the 1st century BCE to about the 5th century CE. It has two essential components: the first and historically earliest stratum (ca. 200 CE) is the Mishna (meaning: repetition) and the second and later stratum is the Gemara (meaning: completion)[125]. The Gemara is a commentary (in Aramaic) on the Mishna (which is in Hebrew), clarifying, explaining and amplifying it[126].

The compiling and editing of the Mishna (whose participants are known as Tannaim, teachers) is traditionally attributed to R. Yehudah HaNassi (d. 219 CE), while the redaction of the Gemara (whose participants are known as Amoraim, expounders) took more time and was the work of many (until ca. 500 CE). This refers to the main, Babylonian (Bavli) Talmud, with which we are here concerned; there is an earlier, less authoritative compilation known as the Jerusalem (Yerushalmi) – or more precisely put, the Land of Israel[127] – Talmud (closed ca. 350-400 CE).[128]

The genesis of these various documents is an interesting historical issue, which has received much attention over time and more critical attention in modern times. Their redactors are thought to have been numerous and stretched out over centuries[129]. Some of the individuals involved are known by tradition, others remain anonymous. They should not, of course, be viewed as standing outside looking in on the collective discursive process they describe. Some of them were without doubt active or passive contemporary participants in some of the Talmudic discussions they report. But even those who do not fall in the category of eye-witnesses must be considered as effectively participants, albeit sometimes centuries after the fact, since by their selection, ordering and slanting of scattered material, their paraphrases and explanations, not to mention their outright interpolations, they necessarily affect our perceptions of the presumed original discussions. It would be a grave error to regard such redactors as entirely self-effacing, perfectly objective and impartial, contemporary observers and stenographers.

The Mishna and the Gemara[130]were conceived as written records of past and present oral legal (halakhic) and to a lesser extent, non-legal (haggadic) traditions. The rabbis (as we shall here indifferently call all participants) mentioned or implied in them did not all live at the same time and in the same place, note well. Their discussions were rarely face to face, but were brought together in one continuous document by the redactors, who were therefore perforce (albeit often invisibly) themselves important participants in the discussions, by virtue of their work of selection, structuring and commentary. Keep in mind this scattering in time and place of participants, and also the constant presence of the redactors in the background of all discourse[131]. Too often, traditional students of the Talmud approach it naïvely and idealistically as an essentially indivisible unit, somehow transcending time and space, perfectly harmonious.

There were perforce long periods of time when the traditions that were eventually put down in writing were transmitted by word of mouth. It must be considered whether such transmission was always perfect, or whether some elements were lost, transformed or added along the way. While it is true that people in those days were more used to memorizing things than we are today, and that they used various mnemonic devices to do so, one may still reasonably assume that some change in the information occurred over time if only unwittingly. Also, as Louis Jacobs has pointed out[132], in the name of I. H. Weiss, with reference to modern day scholars who are able to recite the whole of the Talmud by heart, it is surely easier to memorize a document one has read than to memorize information never seen in written form. It should also be considered that people naturally vary in intelligence, and students often do not understand all that their teachers do, and indeed sometimes students understand more than their teachers do. In short, the oral tradition should never be looked upon as some static solid phenomenon, but rather as a living mass subject to some change over time.

2.    A brief course in the relevant logic

Before we examine any Talmudic text in detail, we need to briefly clarify the logical point of view on a fortiori argument. This clarification is a necessary propaedeutic, because many of the Talmudists and students of the Talmud who may choose to read this essay are probably not acquainted with any objective analysis of the underlying logic, having only been trained in rabbinical ways, which are rarely very formal. The treatment proposed in the present section is of course minimal – much more can be learned about the a fortiori argument in other chapters of the present volume and in my past work called Judaic Logic.

Formal validation of a fortiori argument. The paradigm of a fortiori argument, the simplest and most commonly used form of it, is the positive subjectal mood[133], in which the major and minor terms (here always labeled P and Q, respectively) are subjects and the middle and subsidiary terms (here always labeled R and S, respectively) are predicates. It proceeds as follows[134]:

P is R more than Q is R (major premise).
Q is R enough to be S (minor premise).
Therefore, P is R enough to be S (conclusion).

An example of such argument would be: “If her father had but spit in her face, should she not hide in shame seven days? Let her be shut up without the camp seven days, and after that she shall be brought in again.” (Num. 12:14). This can be read as: if offending one’s father (Q) is bad (R) enough to deserve seven days isolation (S), then surely offending God (P) is bad (R) enough to deserve seven days isolation (S); the tacit major premise being: offending God (P) is worse (R) than offending one’s father (Q).

This form of argument can be logically validated (briefly put) as follows. The major premise tells us that P and Q are both R, though to different measures or degrees. Let us suppose the measure or degree of R in P is Rp and that of R in Q is Rq – then the major premise tells us that: if P then Rp, and if Q then Rp, and Rp is greater than Rq (which in turn implies: if something is Rp then it is also Rq, since a larger number includes all numbers below it[135]). Similarly, the minor premise tells us that nothing can be S unless it has at least a certain measure or degree of R, call it Rs; this can be stated more formally as: if Rs then S and if not Rs then not S. Obviously, since Q is R, Q has the quantity Rq of R, i.e. if Q, then Rq; but here we learn additionally (from the “enough” clause) that Rq is greater than or equal to Rs, so that if Rq then Rs; whence, the minor premise tells us that if Q then S. The putative conclusion simply brings some of the preceding elements together in a new compound proposition, namely: if P then Rp (from the major premise) and if Rs then S and if not Rs then not S (from the minor premise), and Rp is greater than Rs (since Rp > Rq in the major premise and Rq ≥ Rs in the minor premise), so that if Rp then Rs; whence, if P then S. The conclusion is thus proved by the two premises (together, not separately, as you can see). So the argument as a whole is valid – i.e. it cannot logically be contested.

Having thus validated the positive subjectal mood of a fortiori argument, it is easy to validate the negative subjectal mood by reductio ad absurdum to the former. That is, keeping the former’s major premise: “P is R more than Q is R,” and denying its putative conclusion, i.e. saying: “P is R not enough to be S,” we must now conclude with a denial of its minor premise, i.e. with: “Q is R not enough to be S.” For, if we did not so conclude the negative argument, we would be denying the validity of the positive argument.

We can similarly demonstrate the validity of the positive, and then the negative, predicatal moods of a fortiori argument. In this form, the major, minor and middle terms (P, Q and R) are predicates and the subsidiary term (S) is a subject.

More R is required to be P than to be Q (major premise).
S is R enough to be P (minor premise).
Therefore, S is R enough to be Q (conclusion).

An example of such argument would be: “Behold, the money, which we found in our sacks’ mouths, we brought back unto thee out of the land of Canaan; how then should we steal out of thy lord’s house silver or gold?” (Gen. 44:8). This can be read as: if we (S) are honest (R) enough to return found valuables (P), then surely we (S) are honest (R) enough to not-steal (Q); the tacit major premise being: more honesty (R) is required to return found valuables (P) than to refrain from stealing (Q).

Here the validation proceeds (again briefly put) as follows. The major premise tells us that iff (i.e. if only if) Rp then P, and iff Rq then Q, and Rp is greater than Rq (whence if Rp then Rq). The minor premise tells us additionally that if S then Rs, and (since it is “enough”) Rs is greater than or equal to Rp (whence if Rs then Rp), from which it follows that if S then Rp; and since iff Rp then P, it follows that if S then P. From the preceding givens, we can construct the putative conclusion, using if S then Rs (from the minor premise), and Rs is greater than Rq (from both premises, whence if Rs then Rq); these together imply if S then Rq, and this together with iff Rq then Q (from the major premise) imply if S then Q. The conclusion is thus here again incontrovertibly proved by the two premises jointly. The negative predicatal mood can in turn be validated, using as before the method of reductio ad absurdum. That is, if the major premise remains unchanged and the putative conclusion is denied, then the minor premise will necessarily be denied; but since the minor premise is given and so cannot be denied, it follows that the conclusion cannot be denied.

Notice that the reasoning proceeds from minor to major (i.e. from the minor term (Q) in the minor premise, to the major term (P) in the conclusion) in the positive subjectal mood; from major to minor in the negative subjectal mood; from major to minor in the positive predicatal mood; and from minor to major in the negative predicatal mood. These are valid forms of reasoning. If, on the other hand, we proceeded from major to minor in the positive subjectal mood, from minor to major in the negative subjectal mood; from minor to major in the positive predicatal mood; or from major to minor in the negative predicatal mood – we would be engaged in fallacious reasoning. That is, in the latter four cases, the arguments cannot be validated and their putative conclusions do not logically follow from their given premises. To reason fallaciously is to invite immediate or eventual contradiction.

Note well that each of the four arguments we have just validated contains only four terms, here labeled P, Q, R, and S. Each of these terms appears two or more times in the argument. P and Q appear in the major premise, and in either the minor premise or the conclusion. R appears in both premises and in the conclusion. And S appears in the minor premise and in the conclusion. The argument as a whole may be said to be properly constructed if it has one of these four validated forms and it contains only four terms. Obviously, if any one (or more) of the terms has even slightly different meanings in its various appearances in the argument, the argument cannot truly be said to be properly constructed. It may give the illusion of being a valid a fortiori, but it is not really one. It is fallacious reasoning.

The above described a fortiori arguments, labeled subjectal or predicatal, relate to terms, and may thus be called ‘copulative’. There are similar ‘implicational’ arguments, which relate to theses instead of terms, and so are labeled antecedental or consequental. To give one example of the latter, a positive antecedental argument might look like this:

Ap (A being p) implies Cr (r in C) more than Bq (B being q) does,
and Bq implies Cr enough for Ds (for D to be s);
therefore, Ap implies Cr enough for Ds.

Notice the use of ‘implies’ instead of ‘is’ to correlate the items concerned. I have here presented the theses as explicit propositions ‘A is p’, ‘B is q’, ‘C is r’ and ‘D is s’, although they could equally well be symbolized simply as P, Q, R, and S, respectively. The rules of inference are essentially the same in implicational argument as in copulative argument.

The principle of deduction. This forewarning concerning the uniformity throughout an argument of the terms used may be expressed as a law of logic. It is true not just of a fortiori argument, but of all deductive argument (for instances, syllogism or apodosis). We can call this fundamental rule ‘the principle of deduction’, and state it as: no information may be claimed as a deductive conclusion which is not already given, explicitly or implicitly, in the premise(s). This is a very important principle, which helps us avoid fallacious reasoning. It may be viewed as an aspect of the law of identity, since it enjoins us to acknowledge the information we have, as it is, without fanciful additions. It may also be considered as the fifth law of thought, to underscore the contrast between it and the principle of induction[136], which is the fourth law of thought.

Deduction must never be confused with induction. In inductive reasoning, the conclusion can indeed contain more information than the premises make available; for instance, when we generalize from some cases to all cases, the conclusion is inductively valid provided and so long as no cases are found that belie it. In deductive reasoning, on the other hand, the conclusion must be formally implied by the given premise(s), and no extrapolation from the given data is logically permitted. In induction, the conclusion is tentative, subject to change if additional information is found, even if such new data does not contradict the initial premise(s)[137]. In deduction, on the other hand, the conclusion is sure and immutable, so long as no new data contradicts the initial premise(s).

As regards the terms, if a term used in the conclusion of a deductive argument (such as a fortiori) differs however slightly in meaning or in scope from its meaning or scope in a premise, the conclusion is invalid. No equivocation or ambiguity is allowed. No creativity or extrapolation is allowed. If the terms are not exactly identical throughout the argument, it might still have some inductive value, but as regards its deductive value it has none. This rule of logic, then, we shall here refer to as ‘the principle of deduction’.

The error of ‘proportional’ a fortiori argument. An error many people make when attempting to reason a fortiori is to suppose that the subsidiary term (S) is generally changed in magnitude in proportion (roughly) to the comparison between the major and minor terms (P and Q). The error of such proportional’ a fortiori argument, as we shall henceforth call it, can be formally demonstrated as follows.

Consider the positive subjectal mood we have described above. Suppose instead of arguing as we just did above, we now argue as do the proponents of such fallacious reasoning that: just as ‘P is more R than S’ (major premise), so S in the conclusion (which is about P) should be greater than it is in the minor premise (which is about Q). If we adhered to this ‘reasoning’, we would have two different subsidiary terms, say S1 for the minor premise and S2 for the conclusion, with S2 > S1, perhaps in the same proportion as P is to Q, or more precisely as the R value for P (Rp) is to the R value for Q (Rq), so that S1 and S2 could be referred to more specifically as Sq and Sp. In that case, our argument would read as follows:

P is R more than Q is R (major premise).
Q is R enough to be S1 (minor premise).
Therefore, P is R enough to be S2 (conclusion).

The problem now is that this argument would be difficult to validate, since it contains five terms instead of only four as before. Previously, the value of R sufficient to qualify as S was the same (viz. R ≥ Rs) in the conclusion (for P) as in the minor premise (for Q). Now, we have two threshold values of R for S, say Rs1 (in the minor premise, for Q) and Rs2 (in the conclusion, for P). Clearly, if Rs2 is assumed to be greater than Rs1 (just as Rp is greater than Rq), we cannot conclude that Rp > Rs2, for although we still know that Rp > Rq and Rq ≥ Rs1, we now have: Rp > Rs1 < Rs2, so that the relative sizes of Rp and Rs2 remain undecidable. Furthermore, although previously we inferred the “If Rs then S” component of the conclusion from the minor premise, now we have no basis for the “If Rs2 then S2” component of the conclusion, since our minor premise has a different component “If Rs1 then S1” (and the latter proposition certainly does not formally imply the former).[138]

It follows that the desired conclusion “P is R enough to be S2” of the proposed ‘proportional’ version of a fortiori argument is simply invalid[139]. That is to say, its putative conclusion does not logically follow from its premises. The reason, to repeat, is that we have effectively a new term (S2) in the conclusion that is not explicitly or implicitly given in the premises (where only S1 appears, in the minor premise). Yet deduction can never produce new information of any sort, as we have already emphasized. Many people find this result unpalatable. They refuse to accept that the subsidiary term S has to remain unchanged in the conclusion. They insist on seeing in a fortiori argument a profitable argument, where the value of S (and the underlying Rs) is greater for P than it is for Q. They want to ‘quantify’ the argument more thoroughly than the standard version allows.

We can similarly show that ‘proportionality’ cannot be inferred by positive predicatal a fortiori argument. In such case, the subsidiary term (S) is the subject (instead of the predicate) of the minor premise and conclusion. If that term is different (as S1 and S2) in these two propositions, we again obviously do not have a valid a fortiori argument, since our argument effectively involves five terms instead of four as required. We might have reason to believe or just imagine that the subject (S) is diminished in some sense in proportion to its predicates (greater with P, lesser with Q), but such change real or imagined has nothing to do with the a fortiori argument as such. S may well vary in meaning or scope, but if it does so it is not due to a fortiori argument as such. Formal logic teaches generalities, but this does not mean that it teaches uniformity; it allows for variations in particular cases, even as it identifies properties common to all cases.

People who believe in ‘proportional’ a fortiori argument do not grasp the difference between knowledge by a specific deductive means and knowledge by other means. By purely a fortiori deduction, we can only conclude that P relates to precisely S, just as Q relates to S in the minor premise. But this does not exclude the possibility that by other means, such as observation or induction, or even a subsequent deductive act, we may find out and prove that the value of S relative to Q (S1) and the value of S relative to P (S2) are different. If it so happens that we separately know for a fact that S varies in proportion to the comparison of P and Q through R, we can after the a fortiori deduction further process its conclusion in accord with such additional knowledge[140]. But we cannot claim such further process as part and parcel of the a fortiori argument as such – it simply is not, as already demonstrated in quite formal terms.

Formal logic cuts up our long chains of reasoning into distinguishable units – called arguments – each of which has a particular logic, particular rules it has to abide by. Syllogism has certain rules, a fortiori argument has certain rules, generalization has certain rules, adduction has certain rules, and so on. When such arguments, whether deductive or inductive, and of whatever diverse forms, are joined together to constitute a chain of reasoning (the technical term for which is enthymeme), it may look like the final conclusion is the product of all preceding stages, but in fact it is the product of only the last stage. Each stage has its own conclusion, which then becomes a premise in the next stage. The stages never blend, but remain logically distinct. In this way, we can clearly distinguish the conclusion of a purely a fortiori argument from that of any other argument that may be constructed subsequently using the a fortiori conclusion as a premise.

Some of the people who believe that a fortiori argument yields a ‘proportional’ conclusion are misled by the wording of such conclusion. We say: “since so and so, therefore, all the more, this and that.” The expression “all the more” seems to imply that the conclusion (if it concerns the major term) is quantitatively more than the minor premise (concerning the minor term). Otherwise, what is “more” about it? But the fact is, we use that expression in cases of major to minor, as well as minor to major. Although we can say “how much more” and “how much less,” we rarely use the expression “all the less”[141] to balance “all the more” – the latter is usually used in both contexts. Thus, “all the more” is rather perhaps to be viewed as a statement that the conclusion is more certain than the minor premise[142]. But even though this is often our intention, it is not logically correct. In truth, the conclusion is always (if valid) as certain as the minor premise, neither more nor less. Therefore, we should not take this expression “all the more” too literally – it in fact adds nothing to the usual signals of conclusion like “therefore” or “so.” It is just rhetorical emphasis, or a signal that the form of reasoning is ‘a fortiori’.

The argument a crescendo. Although ‘proportional’ a fortiori argument is not formally valid, it is in truth sometimes valid. It is valid under certain conditions, which we will now proceed to specify. When these conditions are indeed satisfied, we should (I suggest) name the argument differently, and rather speak of ‘a crescendo’ argument’[143], so as to distinguish it from strict ‘a fortiori’ argument. We could also say (based on the common form of the conclusions of both arguments) that ‘a crescendo’ argument is a particular type of a fortiori argument, to be contrasted to the ‘purely a fortiori’ species of a fortiori argument. More precisely, a crescendo argument is a compound of strictly a fortiori argument and ‘pro rata’ argument. It combines premises of both arguments, to yield a special, ‘proportional’ conclusion.

The positive subjectal mood of a crescendo argument has three premises and five terms:

P is more R than Q is R (major premise);
and Q is R enough to be Sq (minor premise);
and S varies in proportion to R (additional premise).
Therefore, P is R enough to be Sp (a crescendo conclusion).

The ‘additional premise’ tells us there is proportionality between S and R. Note that the subsidiary term (Sp) in the conclusion differs from that (Sq) given in the minor premise, although they are two measures or degrees of one thing (S). This mood can be validated as follows:

The purely a fortiori element is:

P is more R than Q is R,
and Q is R enough to be Sq.
(Therefore, P is R enough to be Sq.)

To this must be added on the pro rata element:

Moreover, if we are given that S varies in direct proportion to R, then:
since the above minor premise implies that: if R = Rq, then S = Sq,
it follows that: if R = more than Rq = Rp, then S = more than Sq = Sp.

Whence the a crescendo conclusion is:

Therefore, P is R enough to be Sp.

If the proportion of S to R is direct, then Sp > Sq; but if S is inversely proportional to R, then Sp < Sq. The negative subjectal mood is similar, having the same major and additional premise, except that it has as minor premise “P is R not enough to be Sp” and as a crescendo conclusion “Q is R not enough to be Sq.”

The positive predicatal mood of a crescendo argument has three premises and five terms:

More R is required to be P than to be Q (major premise);
and Sp is R enough to be P (minor premise);
and S varies in proportion to R (additional premise).
Therefore, Sq is R enough to be Q (a crescendo conclusion).

As before, the ‘additional premise’ tells us there is proportionality between S and R. Note that the subsidiary term (Sq) in the conclusion differs from that (Sp) given in the minor premise, although they are two measures or degrees of one thing (S). This mood can be validated as follows:

The purely a fortiori element is:

More R is required to be P than to be Q,
and Sp is R enough to be P.
(Therefore, Sp is R enough to be Q.)

To this must be added on the pro rata element:

Moreover, if we are given that R varies in direct proportion to S, then:
since the above minor premise implies that: if S = Sp, then R = Rp,
it follows that: if S = less than Sp = Sq, then R = less than Rp = Rq.

Whence the a crescendo conclusion is:

therefore, Sq is R enough to be Q.

If the proportion of R to S is direct, then Rq < Rp; but if R inversely proportional to S, then Rq > Rp. The negative predicatal mood is similar, having the same major and additional premise, except that it has as minor premise “Sq is R not enough to be Q” and as a crescendo conclusion “Sp is R not enough to be P.”

In practice, we are more likely to encounter subjectal than predicatal a crescendo arguments, since the subsidiary terms in the former are predicates, whereas those in the latter are subjects, and subjects are difficult to quantify. We can similarly construct four implicational moods of a crescendo argument, although things get more complicated in such cases, because it is not really the middle and subsidiary theses which are being compared but terms within them. These matters are dealt with more thoroughly in earlier chapters, and therefore will not be treated here.

From this formal presentation, we see that purely a fortiori argument and a crescendo argument are quite distinct forms of reasoning. The latter has the same premises as the former, plus an additional premise about proportion, which makes possible the ‘proportional’ conclusion. Without the said ‘additional premise’, i.e. with only the two premises (the major and the minor) of a fortiori argument, we cannot legitimately draw the a crescendo conclusion.

Thus, people who claim to draw a ‘proportional’ conclusion from merely a fortiori premises are engaged in fallacy. They are of course justified to do so, if they explicitly acknowledge, or at least tacitly have in mind, the required additional premise about proportion. But if they are unaware of the need for such additional information, they are definitely reasoning incorrectly. The issue here is not one of names, i.e. whether an argument is called a fortiori or a crescendo or whatever, but one of information on which the inference is based.

To summarize: Formal logic can indubitably validate properly constructed a fortiori argument. The concluding predication (more precisely, the subsidiary item, S) in such cases is identical to that given in the minor premise. It is not some larger or lesser quantity, reflecting the direct or inverse proportion between the major and minor items. Such ‘proportional’ conclusion is formally invalid, if all it is based on are the two premises of a fortiori argument. To draw an a crescendo conclusion, it is necessary to have an additional premise regarding proportionality between the subsidiary and middle items.

Regarding the rabbis’ dayo (sufficiency) principle. It is evident from what we have just seen and said that there is no formal need for a “dayo (sufficiency) principle” to justify a fortiori argument as distinct from a crescendo argument. It is incorrect to conceive, as some commentators do (notably the Gemara, as we shall see), a fortiori argument as a crescendo argument artificially circumvented by the dayo principle; for this would imply that the natural conclusion from the two premises of a fortiori is a crescendo, whereas the truth is that a fortiori premises can only logically yield an a fortiori conclusion. The rule to adopt is that to draw an a crescendo conclusion an additional (i.e. third) premise about proportionality is needed – it is not that proportionality may be assumed (from two premises) unless the proportionality is specifically denied by a dayo objection.

In fact, the dayo principle can conceivably ‘artificially’ (i.e. by Divine fiat or rabbinic convention) restrain only a crescendo argument. In such case, the additional premise about proportion is disregarded, and the conclusion is limited to its a fortiori dimension (where the subsidiary term is identical in the minor premise and conclusion) and denied its a crescendo dimension (where the subsidiary term is greater or lesser in the minor premise than in the conclusion). Obviously, if the premise about proportionality is a natural fact, it cannot logically ever be disregarded; but if that premise is already ‘artificial’ (i.e. a Divine fiat or rabbinic convention), then it can indeed conceivably be disregarded in selected cases. For example, though reward and punishment are usually subject to the principle of ‘measure for measure’, the strict justice of that law might conceivably be discarded in exceptional circumstances in the interest of mercy, and the reward might be greater than it anticipates or the punishment less than it anticipates.

Some commentators (for instance, Maccoby) have equated the dayo principle to the principle of deduction. However, this is inaccurate, for several reasons. For a start, according to logic, as we have seen, an a fortiori argument whose conclusion can be formally validated is necessarily in accord with the principle of deduction. In truth, there is no need to refer to the principle of deduction in order to validate the conclusion – the conclusion is validated by formal means, and the principle of deduction is just an ex post facto observation, a statement of something found in common to all valid arguments. Although useful as a philosophical abstraction and as a teaching tool, it is not necessary for validation purposes.

Nevertheless, if a conclusion was found not to be in accord with the principle of deduction, it could of course be forthwith declared invalid. For the principle of deduction is also reasonable by itself: we obviously cannot produce new information by purely rational means; we must needs get that information from somewhere else, either by deduction from some already established premise(s) or by induction from some empirical data or, perhaps, by more mystical means like revelation, prophecy or meditative insight. So obvious is this caveat that we do not really need to express it as a maxim, though there is no harm in doing so.

For the science of logic, and more broadly for epistemology and ontology, then, a fortiori argument and the ‘limitation’ set upon it by the principle of deduction are (abstract) natural phenomena. The emphasis here is on the word natural. They are neither Divinely-ordained (except insofar as all natural phenomena may be considered by believers to be Divine creations), nor imposed by individual or collective authority, whether religious or secular, rabbinical or academic, nor commonly agreed artificial constructs or arbitrary choices. They are universal rational insights, apodictic tools of pure reason, in accord with the ‘laws of thought’ which serve to optimize our knowledge.

The first three of these laws are that we admit facts as they are (the law of identity), in a consistent manner (the law of non-contradiction) and without leaving out relevant data pro or con (the law of the excluded middle); the fourth is the principle of induction and the fifth is that of deduction.

To repeat: for logic as an independent and impartial scientific enterprise, there is no ambiguity or doubt that an a fortiori argument that is indeed properly constructed, with a conclusion that exactly mirrors the minor premise, is valid reasoning. Given its two premises, its (non-‘proportional’) conclusion follows of necessity; that is to say, if the two premises are admitted as true, the said conclusion must also be admitted as true. Moreover, to obtain an a crescendo conclusion additional information is required; without such information a ‘proportional’ conclusion would be fallacious. A principle of deduction can be formulated to remind people that such new information is not producible ex nihilo; but such a principle is not really needed by the cognoscenti.

This may all seem obvious to many people, but Talmudists or students of the Talmud trained exclusively in the traditional manner may not be aware of it. That is why it was necessary for us here to first clarify the purely logical issues, before we take a look at what the Talmud says. To understand the full significance of what it says and to be able to evaluate its claims, the reader has to have a certain baggage of logical knowledge.

The understanding of qal vachomer as a natural phenomenon of logic seems, explicitly or implicitly, accepted by most commentators. Rabbi Adin Steinsaltz, for instance, in his lexicon of Talmudic hermeneutic principles, describes qal vachomer as “essentially logical reasoning”[144]. Rabbi J. Immanuel Schochet says it more forcefully: “Qal vachomer is a self-evident logical argument”[145]. The equation of the dayo principle to the principle of deduction is also adopted by many commentators, especially logicians. For instance, after quoting the rabbinical statement “it is sufficient if the law in respect of the thing inferred be equivalent to that from which it is derived,” Ventura writes very explicitly: “We are resting here within the limits of formal logic, according to which the conclusion of a syllogism must not be more extensive than its premises”[146].

However, as we shall discover further on, the main reason the proposed equation of the dayo principle to the principle of deduction is ill-advised is that it incorrect. There are indeed applications where the dayo imperative happens to correspond to the principle of deduction; but there are also applications where the two diverge in meaning. Commentators who thought of them as equal only had the former cases in mind when they did so; when we consider the latter cases, we must admit that the two principles are very different.

3.    A fresh analysis of the Mishna Baba Qama 2:5

In the Mishna Baba Qama 2:5, there is a debate between the Sages and R. Tarfon about the concrete issue of the financial liability of the owner of an ox which causes damages by goring on private property. This debate has logical importance, in that it reveals to a considerable extent skills and views of Talmudic rabbis with regard to the a fortiori argument. The Sages consider that he must pay for half the damages, whereas R. Tarfon advocates payment for all the damages[147].

The Sages (hachakhamim) are unnamed rabbis of Mishnaic times (Tannaim) and R. Tarfon is one of their colleagues (of the 3rd generation), who lived in Eretz Israel roughly in the late 1st – early 2nd century CE. We are not told how many were the Sages referred to in this Mishna (presumably there were at least two), nor who they were. The contemporaries of R. Tarfon include R. Eleazar b. Azariah, R. Ishmael b. Elisha, R. Akiva, and R. Jose haGelili; it is conceivable that these are the Sages involved in this debate. They are all big names, note; the latter three, as we have seen, produced hermeneutic principles. R. Tarfon, too, was an important and respected figure. So the debate between them should be viewed as one between equals.[148]

The Mishna (BQ 2:5) is as follows[149]:

“What is meant by ‘ox doing damage on the plaintiff’s premises’? In case of goring, pushing, biting, lying down or kicking, if on public ground the payment is half, but if on the plaintiff’s premises R. Tarfon orders payment in full whereas the Sages order only half damages.

  1. Tarfon there upon said to them: seeing that, while the law was lenient to tooth and foot in the case of public ground allowing total exemption, it was nevertheless strict with them regarding [damage done on] the plaintiff’s premises where it imposed payment in full, in the case of horn, where the law was strict regarding [damage done on] public ground imposing at least the payment of half damages, does it not stand to reason that we should make it equally strict with reference to the plaintiffs premises so as to require compensation in full?

Their answer was: it is quite sufficient that the law in respect of the thing inferred should be equivalent to that from which it is derived: just as for damage done on public ground the compensation [in the case of horn] is half, so also for damage done on the plaintiff’s premises the compensation should not be more than half.

  1. Tarfon, however, rejoined: but neither do I infer horn [doing damage on the plaintiff’s premises] from horn [doing damage on public ground]; I infer horn from foot: seeing that in the case of public ground the law, though lenient with reference to tooth and foot, is nevertheless strict regarding horn, in the case of the plaintiff’s premises, where the law is strict with reference to tooth and foot, does it not stand to reason that we should apply the same strictness to horn?

They, however, still argued: it is quite sufficient if the law in respect of the thing inferred is equivalent to that from which it is derived. Just as for damage done on public ground the compensation [in the case of horn] is half, so also for damage done on the plaintiff’s premises, the compensation should not be more than half.”

This discussion may be paraphrased as follows. Note that only three amounts of compensation for damages are considered as relevant in the present context: nil, half or full; there are no amounts in between or beyond these three, because the Torah never mentions any such other amounts.

(a) R. Tarfon argues that in the case of damages caused by “tooth and foot,” the (Torah based) law was lenient (requiring no payment) if they occurred on public ground and strict (requiring full payment) if they occurred on private ground – “does it not stand to reason that” in the case of damages caused by “horn,” since the (Torah based) law is median (requiring half payment) if they occurred on public ground, then the law (i.e. the rabbis’ ruling in this case) ought to likewise be strict (requiring full payment) if they occurred on private ground? Presented more briefly, and in a nested manner, this first argument reads as follows:

If tooth & foot, then:

if public then lenient, and

if private then strict.

If horn, then:

if public then median, and

if private then strict (R. Tarfon’s putative conclusion).

  1. Tarfon thus advocates full payment for damage on private property. The Sages disagree with him, advocating half payment only, saying “dayo—it is enough.”

(b) R. Tarfon then tries another tack, using the same data in a different order, this time starting from the laws relating to public ground, where that concerning “tooth and foot” is lenient (requiring no payment) and that concerning “horn” is median (requiring half payment), and continuing: “does it not stand to reason that” with regard to private ground, since the law for “tooth and foot” damage is strict (requiring full payment), the law (i.e. the rabbis’ ruling in this case) for “horn” damage ought to likewise be strict (requiring full payment)? Presented more briefly and in a nested manner, this second argument reads as follows:

If public, then:

if tooth & foot then lenient, and

if horn then median.

If private, then:

if tooth & foot then strict, and

if horn then strict (R. Tarfon’s putative conclusion).

  1. Tarfon thus advocates full payment for damage on private property. The Sages disagree with him again, advocating half payment only, saying “dayo—it is enough.”

More precisely, they reply to him both times: “it is quite sufficient that the law in respect of the thing inferred should be equivalent to that from which it is derived” – meaning that only half payment should be required in the case under consideration (viz. damages by “horn” on private grounds). In Hebrew, their words are: כנדון להיות הדין מן לבא דיו (dayo lavo min hadin lihiot kenidon) – whence the name dayo principle[150].

Now, the first thing to notice is that these two arguments of R. Tarfon’s contain the exact same given premises and aim at the exact same conclusion, so that to present them both might seem like mere rhetoric (either to mislead or out of incomprehension). The two sets of four propositions derived from the above two arguments (by removing the nesting) are obviously identical. All he has done is to switch the positions of the terms in the antecedents and transpose premises (ii) and (iii). The logical outcome seems bound to be the same:

  • If tooth & foot and public, then lenient (i).

If tooth & foot and private, then strict (ii).

If horn and public, then median (iii).

If horn and private, then strict (R. Tarfon’s putative conclusion).

  • If public and tooth & foot, then lenient (same as (i)).

If public and horn, then median (same as (iii)).

If private and tooth & foot, then strict (same as (ii)).

If private and horn, then strict (same putative conclusion).

However, as we shall soon realize, the ordering of the terms and propositions does make a significant difference. And we shall see precisely why that is so.

(a) What is R. Tarfon’s logic in the first argument? Well, it seems obvious that he is making some sort of argument by analogy; he is saying (note the identity of the two sentences in italics):

Just as, in one case (that of tooth & foot), damage in the private domain implies more legal liability than damage in the public domain (since strict is more stringent than lenient).
So, in the other case (viz. horn), we can likewise say that damage in the private domain implies more legal liability than damage in the public domain (i.e. given median in the latter, conclude with strict, i.e. full payment, in the former, since strict is more stringent than median).

Just as in one case we pass from lenient to strict, so in the other case we may well pass from median to strict[151]. Of course, as with all analogy, a generalization is involved here from the first case (tooth & foot being more stringent for private than for public) up to “all cases” (i.e. the generality in italics), and then an application of that generality to the second case (horn, thusly concluded to be more stringent for private than for public). But of course, this is an inductive act, since it is not inconceivable that there might be specific reasons why the two cases should behave differently. Nevertheless, if no such specific reasons are found, we might well reason that way. That is to say, R. Tarfon does have a point, because his proposed reasoning can well be upheld as an ordinary analogical argument. This might even be classified under the heading of gezerah shavah or maybe binyan av (the second or third rule in R. Ishmael’s list of thirteen)[152].

The above is a rather intuitive representation of R. Tarfon’s first argument by analogy. Upon reflection, this argument should be classified more precisely as a quantitative analogy or pro rata argument:

The degree of legal liability for damage is ‘proportional’ to the status of the property the damage is made on, with damage in the private domain implying more legal liability than damage in the public domain.
This is true of tooth and foot damage, for which liability is known to be nil (lenient) in the public domain and full (strict) in the private domain.
Therefore, with regard to horn damage, for which liability is known to be half (median) in the public domain, liability may be inferred to be full (strict) in the private domain.

This argument, as can be seen, consists of three propositions: a general major premise, a particular (to tooth and foot) minor premise and a particular (to horn) conclusion. The major premise is, in fact, known by induction – a generalization of the minor premise, for all damage in relation to property status. But once obtained, it serves to justify drawing the conclusion from the minor premise. The pro rata argument as such is essentially deductive, note, even though its major premise is based on an inductive act. But its conclusion is nevertheless a mere rough estimate, since the ‘proportionality’ it is based on is very loosely formulated. Notice how the minor premise goes from zero to 100%, whereas the conclusion goes from 50% to 100%[153].

The Sages, on the other hand, seem to have in mind, instead of this ordinary argument by analogy or pro rata argument, a more elaborate and subtle a fortiori argument of positive subjectal form. They do not explicitly present this argument, note well; but it is suggested in their reactions to their colleague’s challenge. Their thinking can be construed as follows:

Private domain damage (P) implies more legal liability (R) than public domain damage (Q) [as we know by extrapolation from the case of tooth & foot].
For horn, public domain damage (Q) implies legal liability (R) enough to make the payment half (median) (S).
Therefore, for horn, private domain damage (P) implies legal liability (R) enough to make the payment half (median) (S).

We see that the subsidiary term (S) is the same (viz. ‘median’, i.e. half payment) in the Sages’ minor premise and conclusion, in accord with a fortiori logic; and they stress that conclusion in reply to R. Tarfon’s counterarguments by formulating their dayo principle, viz. “it is quite sufficient that the law in respect of the thing inferred should be equivalent to that from which it is derived,” to which they add: “just as for damage [by horn] done on public ground the compensation is half, so also for damage [by horn] done on the plaintiff’s premises the compensation should not be more than half.”[154]

We see also that the major premise of the Sages’ qal vachomer is identical to the statements in italics of R. Tarfon’s argument by analogy, i.e. to the major premise of his pro rata argument. In both R. Tarfon and the Sages’ arguments, this sentence “private damage implies more legal liability than public damage” is based on the same generalization (from tooth & foot, in original premises (i) and (ii), as already seen) and thence applicable to the case under scrutiny (horn, for which proposition (iii) is already given)[155]. So both their arguments are equally based on induction (they disagreeing only as to whether to draw the conclusion (iv) or its contrary).

But the most important thing to note here is that the same premises (viz. (i), (ii) and (iii)) can be used to draw contrary conclusions (viz. full payment vs. half payment, respectively, for damage by horn on private grounds), according as we use a mere analogical or pro rata argument, like R. Tarfon, or a more sophisticated strictly a fortiori argument, like the Sages. This discrepancy obviously requires explanation. Since both arguments are built on the same major premise, produced by the same inductive act of generalization, we cannot explain the difference by referring to the inductive preliminaries.

The way to rationalize the difference is rather to say that the argument by analogy or pro rata is more approximate, being a mere projection of the likely conclusion; whereas the a fortiori argument is more accurate, distilling the precise conclusion inherent in the premises. That is to say, though both arguments use the same preliminary induction, the argument of R. Tarfon is in itself effectively a further act of induction, whereas the argument of the Sages is in itself an act of pure deduction. Thus, the Sages’ conclusion is to be logically preferred to the conclusion proposed by R. Tarfon.

Note well that we have here assumed that R. Tarfon’s first argument was merely analogical/pro rata, and that the Sages proposed a purely a fortiori argument in response to it. It is also possible to imagine that R. Tarfon intended a purely a fortiori argument, but erroneously drew a ‘proportional’ conclusion from it; in which case, the Sages’ dayo objection would have been to reprove him for not knowing or forgetting (or even maybe deliberately ignoring) the principle of deduction, i.e. that such argument can only yield a conclusion of the same magnitude as the minor premise. However, I would not support this alternative hypothesis, which supposes R. Tarfon to have made a serious error of reasoning (or even intentionally engaged in fallacy), because it is too far-fetched. For a start, R. Tarfon is an important player throughout the Mishna, someone with in general proven logical skills; moreover, more favorable readings of this particular argument are available, so we have no reason to assume the worst.

Another possible reading is that R. Tarfon’s first argument was not merely analogical/pro rata but was intended as a crescendo, i.e. as a combination of a fortiori argument with pro rata argument, which can be briefly presented as follows:

Private domain damage (P) implies more legal liability (R) than public domain damage (Q) [as we know by extrapolation from the case of tooth & foot].
For horn, public domain damage (Q) implies legal liability (Rq) enough to make the payment half (median) (Sq).
The payment due (S) is ‘proportional’ to the degree of legal liability (R).
Therefore, for horn, private domain damage (P) implies legal liability (Rp) enough to make the payment full (strict) (Sp = more than Sq).

In that case, the dayo statement by the Sages may be viewed as a rejection of the additional premise about ‘proportionality’ between S (the subsidiary term) and R (the middle term) in the case at hand. That would represent them as saying: while proportionality might seem reasonable in other contexts, in the present situation it ought not to be appealed to, and we must rest content with a purely a fortiori argument. The advantage of this reading is that it conceives R. Tarfon as from the start of the debate resorting to the more sophisticated a fortiori type of argument, even though he conceives it as specifically a crescendo (i.e. as combined with a pro rata premise). The Sages prefer a purely a fortiori conclusion to his more ambitious a crescendo one, perhaps because it is easier to defend (i.e. relies on less assumptions), but more probably for some other motive (as we shall see).

(b) So much for the first argument; now let us examine the second argument. This, as many later commentators noticed, and as we shall now demonstrate, differs significantly from the preceding. The most important difference is that, here, the mere argument by analogy (or argument pro rata, to be more precise), the purely a fortiori argument and the a crescendo argument (i.e. a fortiori and pro rata combo), all three yield the same conclusion. Note this well – it is crucial. The second analogical argument proceeds as follows:

Just as, in one case (that of the public domain), damage by horn implies more legal liability than damage by tooth & foot (since median is more stringent than lenient).
So, in the other case (viz. the private domain), we can likewise say that damage by horn implies more legal liability than damage by tooth & foot (i.e. given strict in the latter, conclude with strict, i.e. full payment, in the former, since strict is ‘more stringent than’ [here, as stringent as[156]] strict).

This argument is, as before, more accurately represented as a pro rata argument:

The degree of legal liability for damage is ‘proportional’ to the intentionality of the cause of damage, with damage by horn implying more legal liability than damage by tooth & foot.
This is true of the public domain, for which liability is known to be nil (lenient) for damage by tooth and foot and half (median) for damage by horn.
Therefore, with regard to the private domain, for which liability is known to be full (strict) for damage by tooth and foot, liability may be inferred to be full (strict) for damage by horn.

This argument visibly consists of three propositions: a general major premise, a particular (to the public domain) minor premise and a particular (to the private domain) conclusion. The major premise is, in fact, inductive – a generalization of the minor premise, for all damage in relation to intentionality (in horn damage the ox intends to hurt or destroy, whereas in tooth and foot damage the negative consequences are incidental or accidental). But once obtained, the major premise serves to justify drawing the conclusion from the minor premise. Here again, the ‘proportionality’ is only rough; but in a different way. Notice how the minor premise goes from 0% to 50%, whereas the conclusion goes from 100% to 100%.

The purely a fortiori reading of this second argument would be as follows:

Horn damage (P) implies more legal liability (R) than tooth & foot damage (Q) [as we know by extrapolation from the case of public domain].
For private domain, tooth & foot damage (Q) implies legal liability (R) enough to make the payment full (strict) (S).
Therefore, for private domain, horn damage (P) implies legal liability (R) enough to make the payment full (strict) (S).

Note that the conclusion would be the same if this argument was constructed as a more elaborate a crescendo argument, i.e. with the additional pro rata premise “The payment due (S) is ‘proportional’ to the degree of legal liability (R).” The latter specification makes no difference here (unlike in the previous case), because (as we are told in the minor premise) the minimum payment is full and (as regards the conclusion) no payment greater than full is admitted (by the Torah or rabbis) as in the realm of possibility anyway. Thus, whether we conceive R. Tarfon’s second argument as purely a fortiori or as a crescendo, its conclusion is the same. Which means that the argument, if it is not analogical/pro rata, is essentially a fortiori rather than a crescendo.

Observe here the great logical skill of R. Tarfon. His initial proposal, as we have seen, was an argument by analogy or pro rata, which the Sages managed to neutralize by means of a logically more powerful a fortiori argument; or alternatively, it was an a crescendo argument that the Sages (for reasons to be determined) limited to purely a fortiori. This time, R. Tarfon takes no chances, as it were, and after judicious reshuffling of the given premises offers an argument which yields the same strict conclusion whether it is read as an argument by analogy (pro rata) or a more elaborate a crescendo – or as a purely a fortiori argument. A brilliant move! It looks like he has now won the debate; but, surprisingly, the Sages again reject his conclusion and insist on a lighter sentence.

Note well why R. Tarfon tried a second argument. Here, the stringency of the target law (viz. horn in the private domain) is equal to (and not, as in his first argument, greater than) the stringency of the source law (viz. tooth & foot in the private domain); i.e. both are here ‘strict’. This makes R. Tarfon’s second argument consistent with a fortiori logic and with the dayo principle that the Sages previously appealed to, since now “the law in respect of the thing inferred” is apparently “equivalent to that from which it is derived.” Yet, the Sages reiterate the dayo principle and thus reject his second try. How can they do so?

What is odd, moreover, is that the Sages answer both of R. Tarfon arguments in exactly the same words, as if they did not notice or grasp the evident differences in his arguments. The following is their identical full reply in both cases:

“It is quite sufficient that the law in respect of the thing inferred should be equivalent to that from which it is derived: just as for damage done on public ground the compensation is half, so also for damage done on the plaintiff’s premises the compensation should not be more than half.”

(אמרו לו דיו לבא מן הדין להיות כנדון מה ברה”ר חצי נזק אף ברשות הניזק חצי נזק)

One might well initially wonder if the Sages did not perchance fail to hear or to understand R. Tarfon’s second argument; or maybe some error occurred during the redaction of the Mishna or some later copying (this sure does look like a ‘copy and paste’ job!). For if the Sages were imputing a failure of dayo to R. Tarfon’s second argument, in the same sense as for the first argument, they would not have again mentioned the previous terms “public ground” for the minor premise and “the plaintiff’s premises” for the conclusion, but instead referred to the new terms “tooth and foot” and “horn.” But of course, we have no reason to distrust the Sages and must therefore assume that they know what they are talking about and mean what they say.

Whence, we must infer that the Sages’ second dayo remark does not mean exactly the same as their first one. In the first instance, their objection to R. Tarfon was apparently that if the argument is construed as strictly a fortiori, the conclusion’s predicate must not surpass the minor premise’s predicate; in this sense, the dayo principle simply corresponds to the principle of deduction, as it naturally applies to purely a fortiori argument. Alternatively, if R. Tarfon’s first argument is construed as pro rata or as a crescendo, the Sages’ first dayo objection can be viewed as rejecting the presumption of ‘proportionality’. However, such readings are obviously inappropriate for the Sages’ dayo objection to R. Tarfon’s second argument, since the latter however construed is fully consistent with the dayo principle in either of these senses.

How the second dayo differs from the first. An explanation we can propose, which seems to correspond to a post-Talmudic traditional explanation[157], is that the Sages are focusing on the generalization that precedes R. Tarfon’s second argument. The major premise of that argument, viz. “Horn damage implies more legal liability than tooth & foot damage” was derived from two propositions, remember, one of which was “In the public domain, horn damage entails half payment” (and the other was “In the public domain, tooth & foot damage entails no payment”). R. Tarfon’s putative conclusion after generalization of this comparison (from the public domain to all domains), and a further deduction (from “In the private domain, tooth & foot damage entails full payment”), was “In the private domain, horn damage entails full payment.” Clearly, in this case, the Sages cannot reject the proposed deduction, since it is faultless however conceived (as analogy/pro rata/a crescendo or even purely a fortiori). What they are saying, rather, is that the predicate of its conclusion cannot exceed the predicate (viz. half payment) of the given premise involving the same subject (viz. horn) on which its major premise was based.

We can test this idea by applying it to R. Tarfon’s first argument. There, the major premise was “Private domain damage implies more legal liability than public domain damage,” and this was based on two propositions, one of which was “For tooth & foot, private domain damage entails full payment” (and the other was “For tooth & foot, public domain damage entails no payment”). R. Tarfon’s putative conclusion after generalization of this comparison (from tooth & foot to all causes), and a further deduction (from “For horn, public domain damage entails half payment”), was “For horn, private domain damage entails full payment.” Clearly, in this case, the Sages cannot object that the predicate of its conclusion exceeds the predicate of the given premise involving the same subject (viz. private domain, though more specifically for tooth & foot) on which its major premise was based, since they are the same (viz. full payment). Their only possible objection is that, conceiving the argument as purely a fortiori, the predicate of the conclusion cannot exceed the predicate (viz. half payment) of the minor premise (i.e. “For horn, public domain damage entails half payment”). Alternatively, conceiving the argument as pro rata or a crescendo, they for some external reason (which we shall look into) reject the implied proportionality.

Thus, the Sages’ second objection may be regarded as introducing an extension of the dayo principle they initially decreed or appealed to, applicable to any generalization preceding purely a fortiori argument (or possibly, pro rata or a crescendo arguments, which as we have seen are preceded by the same generalization). The use and significance of generalization before a fortiori argument (or eventually, other forms of argument) are thereby taken into consideration and emphasized by the Sages. This does not directly concern the a fortiori deduction (or the two other possible arguments), note well, but only concerns an inductive preliminary to such inference. However, without an appropriate major premise, no such argument can be formed; in other words, the argument is effectively blocked from taking shape.

The question arises: how is it possible that by merely reshuffling the given premises we could obtain two different, indeed conflicting, a fortiori (or other) conclusions? The answer is that the two major premises were constructed on the basis of different directions of generalization[158]. In the first argument, the major premise is based entirely on tooth & foot data, and we learn something about horn only in the minor premise. In the second argument, the major premise relies in part on horn data, and the minor premise tells us nothing about horn. Thus, the two preliminary generalizations in fact cover quite different ground. This explains why the two a fortiori processes diverge significantly, even though the original data they were based on was the same.

The first dayo objection by the Sages effectively states that, if R. Tarfon’s first argument is construed as purely a fortiori, the conclusion must logically (i.e. by the principle of deduction) mirror the minor premise; alternatively, construing it as pro rata or a crescendo, the needed ‘proportionality’ is decreed to be forbidden (for some reason yet to be dug up). For the second argument, which has one and the same conclusion however construed (whether a fortiori or other in form), the Sages’ dayo objection cannot in the same manner refer to the minor or additional premise, but must instead refer to the inductive antecedents of the major premise, and constitute a rule that the conclusion cannot exceed in magnitude such antecedents. This explains the Sages’ repetition of the exact same sentence in relation to both of R. Tarfon’s arguments.

A problem and its solution. There is yet one difficulty in our above presentation of the Sages’ second dayo objection that we need to deal with.

As you may recall, the first dialogue between R. Tarfon and the Sages could be described as follows: R. Tarfon proposes an a crescendo argument concluding with full payment for damage by horn on private property, whereas the Sages conclude with half payment through the purely a fortiori argument leftover after his tacit premise of ‘proportionality’ is rejected by their dayo. That is, they effectively say: “The payment due (S) is not ‘proportional’ to the degree of legal liability (R).” Thus, the first exchange remains entirely within the sphere of a fortiori logic, despite the dayo application.

But the second dialogue between these parties cannot likewise be entirely included in the sphere of a fortiori logic, because the final conclusion of the Sages here is not obtained by a fortiori argument. Since the effect of their second dayo objection is to block the formation by generalization of the major premise of R. Tarfon’s second a fortiori argument, it follows that once this objection is admitted his argument cannot proceed at all; for without a general major premise such argument cannot yield, regarding horn damage on private property, a conclusion of half compensation any more than a conclusion of full compensation. Yet the Sages do wish to conclude with half compensation. How can they do so?

The answer to the question is, traditionally, to refer back to the Torah passage on which the argument is based, namely Exodus 21:35: “And if one man’s ox hurt another’s, so that it dieth; then they shall sell the live ox, and divide the price of it; and the dead also they shall divide”. This signifies half compensation for horn damage without specifying the domain (public or private) in which such damage may occur – thus suggesting that the compensation may be the same for both domains. In the above two a fortiori arguments, it has been assumed that the half compensation for horn damage applies to the public domain, and as regards the private domain the compensation is unknown – indeed, the two a fortiori arguments and the objections to them were intended to settle the private domain issue.

This assumption is logically that of R. Tarfon. Although the said Torah passage seems to make no distinction between domains with regard to damage by horn, R. Tarfon suspects that there is a distinction between domains by analogy to the distinction implied by Exodus 22:4 with regard to damage by tooth and foot (since in that context, only the private domain is mentioned[159]). His thinking seems to be that the owner of an ox has additional responsibility if he failed to preempt his animal from trespassing on private property and hurting other animals in there. So he tries to prove this idea using two arguments.

The Sages, for their part, read Exodus 21:35 concerning horn damage as a general statement, which does not distinguish between the public and private domains; and so they resist their colleague’s attempt to particularize it. For them, effectively, what matters is that two oxen belonging to two owners have fought, and one happened to kill the other; it does not matter who started the fight, or where it occurred or which ox killed which – the result is the same: equal division of the remaining assets between the owners, as the Torah prescribes. Effectively, they treat the matter as an accident, where both parties are equally faultless, and the only thing that can be done for them is to divide the leftovers between them.

Clearly, if compensation for horn damage on public grounds could be more than half (i.e. if half meant at least half), R. Tarfon could still (and with more force) obtain his two ‘full compensation’ conclusions (by two purely a fortiori arguments), but the Sages’ two dayo objections would become irrelevant. In that event, the conclusion regarding horn damage would be full compensation on both the public and private domains. But if so, why did the Torah specify half compensation (“division” in two)? Therefore, the compensation must at the outset be only half in at least one domain. That this would be the public domain rather than the private may be supposed by analogy from the case of tooth and foot[160]. This is a role played by the major premise of the first argument. This means that the first argument (or at least, its major premise) is needed before the formulation of the second. They are therefore not independent arguments, but form (in part) a chain of reasoning (a sorites) – and their order of appearance is not as accidental as we might initially have thought.

It should be realized that the assumption that the liability for horn damage on private property is equal to or greater than same on public grounds is not an a priori truth. It is not unthinkable that the liability might be less (i.e. zero) in the former case than in the latter. Someone might, say, have argued that the owner of the private property, whose animal was gored there, was responsible to prevent other people’s oxen from entering his property (e.g. by fencing it off), and therefore does not deserve any compensation! In that case, it would be argued that on public grounds he deserves half compensation because he has no control over the presence of other people’s oxen thereon. In this perspective, the onus would be on the property owner, rather than on the owner of the trespassing ox.

Given this very theoretical scenario, it would no longer be logically acceptable to generalize from the liability for damage by tooth & foot, which is less (zero) on public ground and more (full) on private ground, and to say that liability for damage of any sort (including by horn) is greater in a private domain than in the public domain. However, this scenario is not admitted by the rabbis (I do not know if they even discuss it; probably they do not because it does not look very equitable[161]). Therefore, the said generalization is accepted, and serves to determine the compensation for damage by horn on private property in both arguments. In the first argument, this generalization (from tooth & foot damage to all damage) produces the major premise. In the second argument, it serves only to eliminate in advance the possibility of zero compensation in such circumstance.

Thus, we can interpret the Torah as teaching that compensation for horn damage is generally at least half – and more specifically, no more than half on public grounds and no less than half on private property. Thereafter, the issue debated in the Mishna is whether the latter quantity is, in the last analysis, ‘only half’ or ‘more than half (i.e. full)’ compensation. Both parties in the Mishna take it for granted that the half minimum is a maximum as regards public grounds; but they leave the matter open to debate as regards its value on private property. R. Tarfon tries, in his second argument, to prove that the compensation in such circumstance ought to be full, by comparison to the law relating to tooth & foot damage in the same circumstance. But the Sages, interdict his major premise by saying dayo, in view of the textual data that premise was based on, and thus opt for only half compensation.

Following this dayo, note well, the Sages’ conclusion is not obtained by a modified a fortiori argument, since (as already mentioned) such an argument cannot be formulated without an appropriate major premise, but is obtained by mere elimination. Their form of reasoning here is negative disjunctive apodosis (modus tollens):

The appropriate compensation for horn damage on private property is, according to the Torah, at least (lav davka) half, i.e. either only half or full.
But it cannot be proved to be full (since the major premise of R. Tarfon’s attempt to do so by a fortiori cannot be sustained due to a dayo objection).
Therefore, it must be assumed to be only (davka) half (as the Sages conclude).

It should be said that this reasoning is not purely deductive, but contains an inductive movement of thought – namely, the generalization from the failure to prove full compensation specifically through R. Tarfon’s a fortiori argument in the light of the Sages’ renewed dayo objection to the impossibility henceforth to prove full compensation by any means whatever. This is a reasonable assumption, since we cannot perceive any way that the dayo might be avoided (i.e. a way not based on the given of half compensation for damage by horn on public grounds[162]); but it is still a generalization. Therefore, the apodosis is somewhat inductive; this means that further support for the Sages’ conclusion of only half compensation for damage by horn on private property would be welcome.

Thus, strictly speaking, in the last analysis, although a fortiori argument is attempted in the second dialogue, it is not finally used, but what is instead used and what provides us with the final conclusion is a disjunctive argument.

The essence of the dayo principle. We can thenceforth propose a more inclusive formulation of the Sages’ dayo principle, which merges together the said two different cases, as follows. Whenever (as in the present debate) the same original propositions can, via different directions of preparatory induction and/or via different forms of deduction, construct two or more alternative, equally cogent arguments, the chain of reasoning with the less stringent final result should be preferred. This, I submit, is to date the most accurate, all-inclusive statement of the dayo principle formulated on the basis of this Mishnaic sugya.

In the light of this broader statement of the dayo principle, we can read the two applications given in the present debate as follows. In the first argument, where there was a choice between a pro rata or a crescendo argument with a stringent conclusion, and a purely a fortiori argument with a median conclusion, the Sages chose the latter argument, with the less stringent conclusion, as operative. In the second argument, where all three forms of argument yielded the same stringent conclusion, the Sages referred instead to the preliminary generalization; in this case they found that, since the terms of one of the original propositions generalized into the major premise corresponded to the terms of the putative final conclusion, and the former proposition was less stringent than the latter, one could not, in fact, perform the generalization, but had to rest content with the original proposition’s degree of stringency in the final one.

In the first instance, the dayo principle cannot refer to the inductive antecedent of the argument, because that original proposition does not have the same terms as the final conclusion, however obtained; so we must look at the form of the deductive argument. In the second instance, the dayo principle cannot refer to the deductive argument, since whatever its form it results in the same the final conclusion; so we must look at the preliminary generalization preceding such argument. Thus, one and the same dayo principle guides both of the Sages’ dayo objections. Their teaching can thus be formulated as follows: ‘Given, in a certain context, an array of equally cogent alternative arguments, the one with the less stringent conclusion should be adopted’.

In other words, the dayo principle is a general guideline to opt for the less stringent option whenever inference leaves us a choice. It is a principle of prudence, the underlying motive of which seems to be moral – to avoid any risk of injustice in ethical or legal or religious pronouncements based on inference. We could view this as a guideline of inductive logic, insofar as it is a safeguard against possible human errors of judgment. It is a reasonable injunction, which could be argued (somewhat, though not strictly) to have universal value. But in practice it is probably specific to Judaic logic; it is doubtful that in other religions, let alone in secular ethical or legal contexts, the same restraint on inference is practiced.

An alternative translation of the Sages’ dayo principle that I have seen, “It is sufficient that the derivative equal the source of its derivation,” is to my mind very well put, because it highlights and leaves open the variety of ways that the “derivation” may occur in practice. The dayo principle, as we have seen, does not have one single expression, but is expressed differently in different contexts. The common denominator being apparently an imperative of caution, preventing too ready extrapolation from given Scriptural data. In the last analysis, then, the dayo principle is essentially not a logical principle, but rather a moral one. It is a Torah or rabbinical decree, rather than a law of logic. As such, it may conceivably have other expressions than those here uncovered. For the same reason, it could also be found to have exceptions that do not breach any laws of logic. Traditionally, it is deemed as applicable in particular to qal vachomer argument; but upon reflection, in view of its above stated essential underlying motive or purpose, it is evident that it could equally well in principle apply to other forms of argument. Such issues can only be definitely settled empirically, with reference to the whole Talmudic enterprise and subsequent developments in Jewish law.

Alternative scenarios. Our proposed scenario for the Mishna debate is thus as follows. R. Tarfon starts the discussion by proposing a first argument, whose form may be analogical/pro rata or a crescendo, which concludes with the imperative of full payment in the case of horn damage in the private domain. The Sages, appealing to a dayo principle, interdict the attempted ‘proportionality’ in his argument, thus effectively trumping it with a purely a fortiori argument, which concludes with a ruling of half payment. In response, R. Tarfon proposes a second argument, based on the very same data, which, whether conceived as analogical/pro rata or a crescendo, or as purely a fortiori, yields the very same conclusion, viz. full payment. This time, however, the Sages cannot rebut him by blocking an attempt at ‘proportionality’, since (to repeat) a non-‘proportional’ argument yields the very same conclusion as ‘proportional’ ones. So the Sages are obliged to propose an extension or enlargement of the initial dayo principle that focuses instead on the generalization before deduction. In this way, they again rule half payment.

This scenario is obvious, provided we assume the Sages’ two dayo objections are expressions of a dayo principle. It is also conceivable, however, that they have no such general principle in mind, but merely intend these objections to be ad hoc decisions in the two cases at hand. In that case the dayo principle is a “principle,” not in the strict sense of a universal principle that must be applied in every case of the sort, but in the looser sense of a guiding principle that may on occasion, for a variety of unspecified motives, be applied[163]. In fact, if we look at the Mishna passage in question, we see that nowhere is there any mention of a dayo “principle.” There is just statement “It is quite sufficient that the law in respect of the thing inferred should be equivalent to that from which it is derived,” which was presumably labeled “the dayo principle” by later commentators. This statement could be interpreted equally well as having a general or particular intent.

If we adopt the latter assumption, the scenario for the Mishna debate would be as follows: when R. Tarfon proposes his first argument, whether it is construed as pro rata or a crescendo, the Sages merely refuse his inherent ‘proportional’ premise in this particular case, without implying that they would automatically refuse it in other eventual cases. Similarly, when he proposes his second argument, whether it is construed as pro rata, a crescendo, or purely a fortiori, they merely refuse his preparatory generalization in this particular case, without implying that they would automatically refuse it in other eventual cases. Thus, the Sages might be said to making ‘ad hoc’ dayo objections, rather than appealing to a dayo ‘principle’ in the strict sense. Why would the Sages raise a dayo objection in this particular case, and not raise it in other cases? Conceivably, they perceive some unspecified danger in the present case that may be absent in other cases.

Granting this alternative view of the dayo principle, be it said in passing, there is conceivably no need to mention qal vachomer argument at all in this Mishna debate! In this view, it is possible that neither R. Tarfon nor the Sages intended any genuine a fortiori type of reasoning, but were entirely focused on mere analogy. As we shall see, although the Gemara probably does intend an a crescendo interpretation of the two arguments of R. Tarfon, it is not inconceivable that its author simply had in mind analogical/pro rata argument. Although the expression qal vachomer does appear in the Gemara, it does not necessarily have to be taken as referring to a fortiori or a crescendo argument, but could be read as referring to pro rata. It is anyhow worthwhile stating that another viewpoint is possible, because this allows us to conceptually uncouple the dayo principle from qal vachomer.

But the main value of our proposing alternative scenarios is that these provide us with different explanations of the disagreement between R. Tarfon and the Sages. Where, precisely, did they disagree? Given the primary scenario, where the dayo principle is a hard and fast principle in the eyes of the Sages, the question arises: how come R. Tarfon forgot or did not know or chose to ignore this principle? If the Sages claim it as a Divine decree, i.e. an ancient tradition dating “from the Sinai revelation,” whether inferred from Scripture or orally transmitted, it is unthinkable that a man of R. Tarfon’s caliber would be ignorant of it or refuse to accept it. Thus, the primary scenario contains a difficulty, a kushia.

One possible resolution of this difficulty is to say that the Sages were here legislating, i.e. the dayo principle was here in the process of being decided by the rabbis collectively, there being one dissenting voice, viz. that of R. Tarfon, at least temporarily till the decision was declared law. In that event, the conflict between the two parties dissolves in time. Another possible resolution is to say that the Sages did not intend their dayo statement as a hard and fast principle, but as a loose guideline that they considered ought to be applied in the present context, whereas their colleague R. Tarfon considered it ought not to be applied in the present context. In that event, the two parties agree that the dayo principle is not universal, but merely conditional, and their conflict here is only as to whether or not its actual application is appropriate in the case at hand.

This would explain why R. Tarfon can put forward his first and second arguments failing each time to anticipate that the Sages would disagree with him. He could not offhand be expected to predict what their collective judgment would be, and so proposed his opinion in good faith. That they disagreed with him is not a reflection on his knowledge of Torah or his logical powers; there was place for legitimate dissent. Thus, while the hypothesis that the Sages’ dayo objections signify a hard and fast rule of Sinaitic origin is problematic, there are two viable alternative hypotheses: namely, that the Sages’ dayo objections constituted a general rabbinical ruling in the making; or that they were intended as ad hoc, particular and conditional statements, rather than as reflections of a general unbreakable rule. The problem with the former hypothesis is explaining away R. Tarfon’s implied ignorance or disagreement; this problem is solved satisfactorily with either of the latter two hypotheses.

The Gemara commentary revolves around this issue, since its first and main query is: “Does R. Tarfon really ignore the principle of dayo? Is not dayo of Biblical origin?” The Gemara’s thesis thus seems to be that dayo is a principle of Biblical origin and that therefore R. Tarfon knew about it and essentially agreed with it. We shall presently see where it takes this assumption.

About method. An issue arising from this Mishnaic discussion is whether it is based on revelation or on reason. If we examine R. Tarfon’s discourse, we see that he repeatedly appeals to reason. Twice he says: “does it not stand to reason?” (eino din) and twice he claims to “infer” (edon)[164]. This language (the translations are those in the Soncino edition) suggests he is not appealing to Divine revelation, but to ordinary human reason. And, significantly, the Sages do not oppose him by explicitly claiming that their dayo principle is Divinely-ordained (as the Gemara later claims) and thus overrides his merely rational argument – no, they just affirm and reaffirm it as something intuitively self-evident, on moral if not logical grounds. Thus, from such positive and negative evidence, it is possible to suppose that both R. Tarfon and the Sages regard their methodological means as essentially rational.

Concerning the logical skills of R. Tarfon and the Sages, neither party to the debate commits any error of logic, even though their approaches and opinions differ. All arguments used by them are formally valid. At no stage do the Sages deny R. Tarfon’s reasoning powers or vice versa. The two parties understand each other well and react appropriately. There is no rhetorical manipulation, but logic is used throughout. Nevertheless, a pertinent question to ask is: why did R. Tarfon and the Sages not clarify all the logical issues involved, and leave their successors with unanswered questions? Why, if these people were fully conscious of what they were doing, did they not spell their intentions out clearly to prevent all possible error? The most likely answer is that they functioned ‘intuitively’ (in a pejorative sense of the term), without awareness of all the formalities involved. They were skillful practitioners of logic, but evidently not theoreticians of it. They did not even realize the importance of theory.

4.    A logician’s reading of Numbers 12:14-15

We have thus far analyzed the Mishnaic part of Baba Qama 24b-25a. Before we turn to the corresponding Gemara, it is wise for us – in the way of a preparatory study – to look at a Torah passage which plays an important role in that Gemara, as an illustration of the rabbinical hermeneutic rule of qal vachomer (a fortiori argument) and as a justification of its attendant dayo (sufficiency) principle.

The Torah passage in question is Numbers 12:14-15. The reason why this passage was specifically focused on by the Gemara should be obvious. This is the only a fortiori argument in the whole Tanakh that is both spoken by God and has to do with inferring a penalty for a specific crime. None of the other four a fortiori arguments in the Torah are spoken by God[165]. And of the nine other a fortiori arguments in the Tanakh spoken by God, two do concern punishment for sins but not specifically enough to guide legal judgment[166]. Clearly, the Mishna BQ 2:5 could only be grounded in the Torah through Num. 12:14-15.

Num. 12:14-15 reads: “14. If her father had but spit in her face, should she not hide in shame seven days? Let her be shut up without the camp seven days, and after that she shall be brought in again. 15. And Miriam was shut up without the camp seven days; and the people journeyed not till she was brought in again.” Verse 14 may be construed as a qal vachomer as follows:

Causing Divine disapproval (P) is a greater offense (R) than causing paternal disapproval (Q). (Major premise.)
Causing paternal disapproval (Q) is offensive (R) enough to merit isolation for seven days (S). (Minor premise.)
Therefore, causing Divine disapproval (P) is offensive (R) enough to merit isolation for seven days (S). (Conclusion.)

This argument, as I have here rephrased it a bit, is a valid purely a fortiori of the positive subjectal type (minor to major)[167]. Some interpretation on my part was necessary to formulate it in this standard format[168]. I took the image of her father spitting in her face (12:14) as indicative of “paternal disapproval” caused presumably, by analogy to the context, by some hypothetical misbehavior on her part[169]. Nothing is said here about “Divine disapproval;” this too is inferred by me from the context, viz. Miriam being suddenly afflicted with “leprosy” (12:10) by God, visibly angered (12:9) by her speaking ill of Moses (12:1). The latter is her “offense” in the present situation, this term (or another like it) being needed as middle term of the argument.

The major premise, about causing Divine disapproval being a “more serious” offense than causing paternal disapproval, is an interpolation – it is obviously not given in the text. It is constructed in accord with available materials with the express purpose of making possible the inference of the conclusion from the minor premise. The sentence in the minor premise of “isolation” for seven days due to causing paternal disapproval may be inferred from the phrase “should she not hide in shame seven days?” The corresponding sentence in the putative conclusion of “isolation” for seven days due to causing Divine disapproval may be viewed as an inference made possible by a fortiori reasoning.

With regard to the term “isolation,” the reason I have chosen it is because it is the conceptual common ground between “hiding in shame” and “being shut up without the camp.” But a more critical approach would question this term, because “hiding in shame” is a voluntary act that can be done within the camp, whereas “being shut up without the camp” seems to refer to involuntary imprisonment by the authorities outside the camp. If, however, we stick to the significant distinctions between those two consequences, we cannot claim the alleged purely a fortiori argument to be valid. For, according to strict logic, we cannot have more information in the conclusion of a deductive argument (be it a fortiori, syllogistic or whatever) than was already given in its premise(s).

That is to say, although we can, logically, from “hiding in shame” infer “isolation” (since the former is a species the latter), we cannot thereafter from “isolation” infer “being shut up without the camp” (since the former is a genus of the latter). To do so would be illicit process according to the rules of syllogistic reasoning, i.e. it would be fallacious. It follows that the strictly correct purely a fortiori conclusion is either specifically “she shall hide in shame seven days” or more generically put “she shall suffer isolation seven days.” In any case, then, the sentence “she shall be shut up without the camp seven days” cannot logically be claimed as an a fortiori conclusion, but must be regarded as a separate and additional Divine decree that even if she does not voluntarily hide away, she should be made to do so against her will (i.e. imprisoned).

We might of course alternatively claim that the argument is intended as a crescendo rather than purely a fortiori. That is to say, it may be that the conclusion of “she should be shut up without the camp seven days” is indeed inferred from the minor premise “she would hide in shame seven days” – in ‘proportion’ to the severity of the wrongdoing, comparing that against a father and that against God. For this to be admitted, we must assume a tacit additional premise that enjoins a pro rata relationship between the importance of the victim of wrongdoing (a father, God) and the ensuing punishment on the culprit (voluntary isolation, forced banishment and incarceration).

Another point worth highlighting is the punishment of leprosy. Everyone focuses on Miriam’s punishment of expulsion from the community for a week, but that is surely not her only punishment. She is in the meantime afflicted by God with a frightening disease, whereas the hypothetical daughter who has angered her father does not have an analogous affliction. So the two punishments are not as close to identical as they may seem judging only with reference to the seven days of isolation. Here again, we may doubt the validity of the strictly a fortiori argument. This objection could be countered by pointing out that the father’s spit is the required analogue of leprosy. But of course the two afflictions are of different orders of magnitude; so a doubt remains.

We must therefore here again admit that this difference of punishment between the two cases is not established by the purely a fortiori argument, but by a separate and additional Divine decree. Or, alternatively, by an appropriate a crescendo argument, to which no dayo is thereafter applied. We may also deal with this difficulty by saying that the punishment of leprosy was already a fact, produced by God’s hand, before the a fortiori argument is formulated; whereas the latter only concerns the punishment that is yet to be applied, by human intervention – namely, the seven days’ isolation. Thus, the argument intentionally concerns only the later part of Miriam’s punishment, and cannot be faulted for ignoring the earlier part.

It is perhaps possible to deny that an a fortiori argument of any sort is intended here. We could equally well view the sentence “Let her be shut up without the camp seven days” as an independent decree. But, if so, of what use is the rhetorical exclamation “If her father had but spit in her face, should she not hide in shame seven days?” and moreover how to explain to coincidence of “seven days” isolation in both cases? Some sort of analogy between those two clauses is clearly intended, and the a fortiori or a crescendo argument serves to bind them together convincingly. Thus, although various objections can be raised regarding the a fortiori format or validity of the Torah argument, we can say that all things considered the traditional reading of the text as a qal vachomer is reasonable. This reading can be further justified if it is taken as in some respects a crescendo, and not purely a fortiori.

What, then, is the utility of the clause: “And after that she shall be brought in again”? Notice that it is not mentioned in my above a fortiori construct. Should we simply read it as making explicit something implied in the words “Let her be shut up without the camp seven days”? Well, these words do not strictly imply that after seven days she should be brought back into the camp; it could be that after seven days she is to be released from prison (where she has been “shut up”), but not necessarily brought back from “without the camp.” So the clause in question adds information. At the end of seven days, Miriam is to be both released from jail and from banishment from the tribal camp.

Another possible interpretation of these clauses is to read “Let her be shut up without the camp seven days” as signifying a sentence of at least seven days, while “And after that she shall be brought in again” means that the sentence should not exceed seven days (i.e. “after that” is taken to mean “immediately after that”). They respectively set a minimum and a maximum, so that exactly seven days is imposed. What is clear in any case is that “seven days isolation” is stated and implied in both the proposed minor premise and conclusion; no other quantity, such as fourteen days, is at all mentioned, note well. This is a positive indication that we are indeed dealing essentially with a purely a fortiori argument, since the logical rule of the continuity between the given and inferred information is (to that extent) obeyed.

As we shall see when we turn to the Gemara’s treatment, although there is no explicit mention of fourteen days in the Torah conclusion, it is not unthinkable that fourteen days were implicitly intended (implying an a crescendo argument from seven to fourteen days) but that this harsher sentence was subsequently mitigated (brought back to seven days) by means of an additional Divine decree (the dayo principle, to be exact) which is also left tacit in the Torah. In other words, while the Torah apparently concludes with a seven-day sentence, this could well be a final conclusion (with unreported things happening in between) rather than an immediate one. Nothing stated in the Torah implies this a crescendo reading, but nothing denies it either. So much for our analysis of verse 14.

Let us now briefly look at verse 15: “And Miriam was shut up without the camp seven days; and the people journeyed not till she was brought in again.” The obvious reading of this verse is that it tells us that the sentence in verse 14 was duly executed – Miriam was indeed shut away outside the camp for exactly seven days, after which she was released and returned to the camp, as prescribed. We can also view it as a confirmation of the reasoning in the previous verse – i.e. as a way to tell us that the apparent conclusion was the conclusion Moses’ court adopted and carried out. We shall presently move on, and see how the Gemara variously interpreted or used all this material.

But first let us summarize our findings. Num. 12:14-15 may, with some interpolation and manipulation, be construed as an a fortiori argument of some sort. If this passage of the Torah is indeed a qal vachomer, it is not an entirely explicit (meforash) one, but partly implicit (satum). In some respects, it would be more appropriate to take it as a crescendo, rather than purely a fortiori. It could even be read as not a qal vachomer at all; but some elements of the text would then be difficult to explain.

It is therefore reasonable to read an a fortiori argument into the text, as we have done above and as traditionally done in Judaism. It must however still be stressed that this reading is somewhat forced if taken too strictly, because there are asymmetrical elements in the minor premise and conclusion. We cannot produce a valid purely a fortiori inference without glossing over these technical difficulties. Nevertheless, there is enough underlying symmetry between these elements to suggest a significant overriding a fortiori argument that accords with the logical requirement of continuity (i.e. with the principle of deduction). The elements not explained by a fortiori argument can and must be regarded as separate and additional decrees. Alternatively, they can be explained by means of a crescendo arguments.

In the present section, we have engaged in a frank and free textual analysis of Num. 12:14-15. This was intentionally done from a secular logician’s perspective. We sought to determine objectively (irrespective of its religious charge) just what the text under scrutiny is saying, what its parts are and how they relate to each other, what role they play in the whole statement. Moreover, most importantly, the purpose of this analysis was to find out what relation this passage of the Torah might have to a fortiori argument and the principle of dayo: does the text clearly and indubitably contain that form of argument and its attendant principle, or are we reading them into it? Is the proposed reasoning valid, or is it somewhat forced?

We answered the questions as truthfully as we could, without prejudice pro or con, concluding that, albeit various difficulties, a case could reasonably be made for reading a valid a fortiori argument into the text. These questions all had to be asked and answered before we consider and discuss the Gemara’s exegesis of Num. 12:14-15, because the latter is in some respects surprisingly different from the simple reading. We cannot appreciate the full implications of what it says if we do not have a more impartial, scientific viewpoint to compare it to. What we have been doing so far, then, is just preparing the ground, so as to facilitate and deepen our understanding of the Gemara approach to the qal vachomer argument and the dayo principle when we get to it.

One more point needs to be made here. As earlier said, the reason why the Gemara drew attention in particular to Num. 12:14-15 is simply that this passage is the only one that could possibly be used to ground the Mishna BQ 2:5 in the Torah. However, though as we have been showing Num. 12:14-15 can indeed be used for this purpose, the analogy is not perfect. For whereas the Mishnaic dayo principle concerns inference by a rabbinical court from a law (a penalty for a crime, to be precise) explicit in the Torah to a law not explicit in the Torah (sticking to the same penalty, rather than deciding a proportional penalty), the dayo principle implied (according to most readings) in Num. 12:14-15 relates to an argument whose premises and conclusion are all in the Torah, and moreover it infers the penalty (for Miriam’s lèse-majesté) for the court to execute by derivation from a penalty (for a daughter offending her father) which may be characterized as intuitively-obvious morality or more sociologically as a pre-Torah cultural tradition.

For if we regard (as we could) both penalties (for a daughter and for Miriam) mentioned in Num. 12:14-15 as Divinely decreed, we could not credibly also say that the latter (for Miriam) is inferred a fortiori from the former (for a daughter). So the premise in the Miriam case is not as inherently authoritative as it would need to be to serve as a perfect analogy for the Torah premise in the Mishnaic case. For the essence of the Mishnaic sufficiency principle is that the court must be content with condemning a greater culprit with the same penalty as the Torah condemns a lesser culprit, rather than a proportionately greater penalty, on the grounds that the only penalty explicitly justified in the Torah and thus inferable with certainty is the same penalty. That is, the point of the Mishnaic dayo is that the premise is more authoritative than the conclusion, whereas in the Num. 12:14-15 example this is not exactly the case. What this means is that although the Mishnaic dayo can be somewhat grounded on Num. 12:14-15, such grounding depends on our reading certain aspects of the Mishna into the Torah example. That is to say, the conceptual dependence of the two is mutual rather than unidirectional.

5.    A critique of the Gemara in Baba Qama 25a

As regards the Gemara of the Jerusalem Talmud, all it contains relative to the Mishna Baba Qama 2:5 is a brief comment in the name of R. Yochanan[170] that R. Tarfon advocates full payment for damages in the private domain, whereas the Sages advocate half payment[171]. This is typical of this Talmud, which rarely indulges in discussion[172]. On the other hand, the Gemara of the Babylonian Talmud has quite a bit to say on this topic (see p. 25a there), though perhaps less than could be expected. When exactly that commentary on our Mishna was formulated, and by whom, is not there specified; but keep in mind that the Gemara as a whole was redacted in Babylonia ca. 500 CE, i.e. some three centuries after the Mishna was closed, so these two texts are far from contemporaneous[173]. It begins as follows:

“Does R. Tarfon really ignore the principle of dayo? Is not dayo of Biblical origin? As taught: How does the rule of qal vachomer work? And the Lord said unto Moses: ‘If her father had but spit in her face, should she not be ashamed seven days?’ How much the more so then in the case of divine [reproof] should she be ashamed fourteen days? Yet the number of days remains seven, for it is sufficient if the law in respect of the thing inferred be equivalent to that from which it is derived!”

The a crescendo reading. Reading this passage, it would appear that the Gemara conceives qal vachomer as a crescendo rather than purely a fortiori argument; and the dayo principle as a limitation externally imposed on it. It takes the story of Miriam (i.e. Numbers 12:14-15) as an illustration and justification of its view, claiming that the punishment due to Miriam would be fourteen days by qal vachomer were it not restricted to seven days by the dayo principle. The dayo principle is here formulated exactly as in the Mishna (as “It is sufficient, etc.”); but the rest of the Gemara’s above statement is not found there.

In fact, the Gemara claims that the thesis here presented is a baraita – i.e. a tradition of more authoritative, Tannaic origin, even though it is not part of the Mishna[174]. This is conventionally signaled in the Gemara by the expression ‘as taught’: דתניא (detania)[175]. The baraita may be taken as the Hebrew portion following this, i.e. stretching from “How does the rule of qal vachomer work?” to “…from which it is derived.” Note well that baraita thesis is clearly delimited: the preceding questions posed by the Gemara – viz. “Does R. Tarfon really ignore the principle of dayo? Is not dayo of Biblical origin?” – are not part of it; we shall return to these two questions further on.

As we have shown in our earlier analysis, Num. 12:14-15 could be read as devoid of any argument; but then we would be hard put to explain the function of the first sentence: “If her father had but spit in her face, etc.,” and its relation to the second: “Let her be shut up without the camp, etc.”. It is therefore a reasonable assumption that an argument is indeed intended. This argument can be construed as purely a fortiori; in that event, its conclusion is simply seven days isolation, the same number of days as mentioned in the minor premise; and if the dayo principle have any role to play here it is simply that of the principle of deduction, i.e. a reminder that the conclusion must reflect the minor premise. It is also possible to interpret the argument as a crescendo, as the Gemara proposes to do; in that event, its conclusion is a greater number of days of isolation (say, fourteen days); and the dayo principle plays the crucial role of resetting the number of days to seven.

The latter is a conceivable hypothesis, but by no means a certainty, note well. There is clearly no mention of “fourteen days” in the Torah passage referred to, i.e. no concrete evidence of an a crescendo argument, let alone of a dayo principle which cuts back the fourteen days to seven. The proposed scenario is entirely read into the Biblical text, rather than drawn from it, by the baraita and then the Gemara; it is an interpolation on their part. They are saying: though the Torah does not explicitly mention fourteen days, etc., it tacitly intends them. This is not inconceivable; but it must be admitted to be speculative, since other readings are equally possible.

The baraita apparently proposes to read, not only the particular qal vachomer about Miriam, but qal vachomer in general as a crescendo argument, since it says “How does the rule of qal vachomer work?” rather than “how does the following example of qal vachomer work?” Thus, the Tanna responsible for it may be assumed to believe unconditionally in the ‘proportionality’ of a fortiori argument. Likewise, the Gemara – since it accepts this view without objection or explanation. If it is true that this Gemara (and the baraita it is based on – but I won’t keep mentioning that) regards a fortiori argument to always be a crescendo argument, it is way off course, of course.

As we have seen, as far as formal logic is concerned a fortiori argument is essentially not a crescendo, even though its premises can with the help of an additional premise about proportionality be made to yield an a crescendo conclusion. It is conceivable that the particular argument concerning Miriam is in fact not only a fortiori but a crescendo (assuming the premise of proportionality is tacitly intended, which is a reasonable assumption); but it is certainly not conceivable that all a fortiori arguments are a crescendo. The Gemara’s identification of a fortiori argument with a crescendo is nowhere justified by it. The Gemara has not analyzed a fortiori argument in general and found its logical conclusion to be a crescendo (i.e. ‘proportional’); it merely asserts this to be so in the case at hand and, apparently, in general.

While it is true that, empirically, within the Talmud as well as outside it, convincing examples of seemingly a fortiori argument yielding a (roughly or exactly) proportional conclusion can be adduced, it is also true that examples of a fortiori argument yielding a non-proportional conclusion can be adduced. This needs to be explained – i.e. commentators are duty-bound to account for this variation in behavior, by specifying under what logical conditions a ‘proportional’ conclusion is justified and when it is not justified. The answer to that is (to repeat) that a fortiori argument as such does not have a ‘proportional’ conclusion and that such a conclusion is only logically permissible if an additional premise is put forward that justifies the ‘proportionality’. The Gemara does not demonstrate its awareness of these theoretical conditions, but functions ‘intuitively’. Its thesis is thus essential dogmatic – an argument by authority, rather than through logical justification.

Thus, for the Gemara, or at least this here Gemara, the words “qal vachomer,” or their English equivalent “a fortiori argument,” refer to what we have called a crescendo argument, rather than to purely a fortiori argument. There is nothing wrong with that – except that the Gemara does not demonstrate awareness of alternative hypotheses.

A surprising lacuna. Furthermore, it should imperatively be remarked that the Gemara’s above explanation of the Mishna debate, by means of the Miriam story, is only relevant to the first exchange between R. Tarfon and the Sages; it does not address the issues raised by the second exchange between them.

For in the first exchange, as we have seen, R. Tarfon tries by means of a possible pro rata argument, or alternatively an a crescendo argument (as the Gemara apparently proposes), to justify a ‘proportional’ conclusion (i.e. a conclusion whose predicate is greater than the predicate of the minor premise, in proportion to the relative magnitudes implied in the major premise); and here the Sages’ dayo objection limits the predicate of conclusion to that of the minor premise; so the analogy to the Miriam case is possible. But in the second exchange, the situation is quite different! Here, as we earlier demonstrated, the dayo objection refers, not to the information in the minor premise, but to the information that was generalized into the major premise. That is to say, whereas the first objection is aimed at the attempted pro rata or a crescendo deduction, the second one concerns the inductive preliminary to the attempted pro rata or a fortiori or a crescendo deduction.

The Gemara makes no mention of this crucial distinction between the two cases. It does not anywhere explicitly show that it has noticed that R. Tarfon’s second argument draws the same conclusion whether it is considered as pro rata, a crescendo, or even purely a fortiori, so that it formally does not contravene the Sages’ first objection. The Gemara does not, either, marvel at the fact that the Sages’ second objection is made in exactly the same terms, instead of referring to the actual terms of the new argument of R. Tarfon. It does not remark that the Miriam story (as the Gemara interprets it) is therefore irrelevant to the second case, since it does not resemble it, and some other explanation must be sought for it. This lacuna is of course a serious weakness in the Gemara’s whole hypothesis, since it does not fit in with all the data at hand.

To be sure, the distinction between the two cases does appear in rabbinic literature. This distinction is solidified by means of the labels dayo aresh dina and dayo assof dina given to the two versions of the dayo principle. But I do not think the distinction is Talmudic (certainly, it is absent here, where it is most needed). Rather, it seems to date from much later on (probably to the time of Tosafot). These expressions mean, respectively, applying the dayo “to the first term (or law)” and applying it “to the last term (or law).” In my opinion, assof dina must refer to the dayo used on the first qal vachomer, while aresh dina refers to the dayo used on the second qal vachomer[176].

Be that as it may, what concerns us here is the Gemara, which evidently makes no such distinction (even if later commentators try to ex post facto give the impression that everything they say was tacitly intended in the Gemara). What this inattentiveness of the Gemara means is that even if it manages to prove whatever it is trying to prove (we shall presently see just what) – it will not succeed, since it has not taken into account all the relevant information. Its theory will be too simple, insufficiently broad – inadequate to the task. The Gemara’s failure of observation is of course also not very reassuring.

The claim that dayo is of Biblical origin. Let us now return to the initial questions posed by the Gemara, viz. “Does R. Tarfon really ignore the principle of dayo? Is not dayo of Biblical origin?” (ור”ט לית ליה דיו והא דיו דאורייתא הוא). As already remarked, it is important to notice that these questions are not part of the baraita. They are therefore the Gemara’s own thesis (or an anonymous thesis it defends as its own) – indeed, as we shall see, they are the crux of its commentary. The baraita with the a crescendo reading is relatively a side-issue. What the Gemara is out to prove is that R. Tarfon “does not ignore” the dayo principle, because “it is of Biblical origin.” What is not of Biblical origin may conceivably be unknown to a rabbi of Tarfon’s level; but what is of Biblical origin must be assumed as known by him.

The question of course arises what does “of Biblical origin” (deoraita) here mean exactly? It cannot literally mean that the principle of dayo is explicitly promulgated and explicated in the Torah. Certainly it is nowhere to be found in the Torah passage here referred to, or anywhere else in that document. Thus, this expression can only truly refer to an implicit presence in the Torah. And indeed the Torah passage about Miriam, brought to bear by the Gemara, seems to be indicated by it as the needed source and justification of the principle, rather than as a mere illustration of it. However, as we shall see further on, there is considerable circularity in such a claim. So claiming the dayo principle to have “Biblical origin” is in the final analysis just say-so, i.e. a hypothesis – it does not solidly ground the principle and make it immune to all challenge, as the Gemara is suggesting.

It could well be thought, reading the Mishna, that R. Tarfon was not previously aware of the Sages’ alleged dayo principle, since he did not preempt their two dayo objections. Had he known their thinking beforehand, he would surely not have wasted his time trying out his two arguments, since he would expect them to be summarily rejected by the Sages. Since he did try, and try again, the Sages must have been, in his view, either unearthing some ancient principle unknown to him, or deciding a new principle, or proposing ad hoc decisions. It is this overall reasonable conclusion from the Mishna that the Gemara seeks to combat, with its claim that the dayo principle was of Biblical origin and therefore R. Tarfon must have known it. Note this well.

I do not know why the Gemara is not content with the perfectly legal possibilities that the dayo principle might be either a tradition not known to R. Tarfon, or a new general or particular decision by the Sages (derabbanan). For some reason, it seeks to impose a more fundamentalist agenda, even though the alternative approaches are considered acceptable in other Talmudic contexts. The Gemara does not say why it is here unacceptable for the Sages to have referred to a relatively esoteric tradition or made a collegial ruling (by majority, rov)[177]. It seems that the Gemara is driven by a desire to establish that R. Tarfon and the Sages are more in harmony than they at first seem; but it is not clear why it has chosen the path it has, which is fraught with difficulties.

The claim that dayo is conditional. The Gemara shifts the debate between R. Tarfon and the Sages from one as to if the dayo principle is applicable to one as to when it is applicable. The two parties, according to the Gemara, agree that the dayo principle is “of Biblical origin,” and thus that there is a dayo principle; but they disagree on whether or not it is applicable unconditionally. In this view, whereas the Sages consider the dayo principle as universally applicable, R. Tarfon considers it as only conditionally applicable. Thus, the parties agree in principle, and their disagreement is only in a matter of detail. The Gemara then proceeds to clarify R. Tarfon’s alleged conditions[178]:

“The principle of dayo is ignored by him [R. Tarfon] only when it would defeat the purpose of the a fortiori, but where it does not defeat the purpose of the a fortiori, even he maintains the principle of dayo. In the instance quoted there is no mention made at all of seven days in the case of divine reproof; nevertheless, by the working of the a fortiori, fourteen days may be suggested: there follows, however, the principle of dayo so that the additional seven days are excluded, whilst the original seven are retained. Whereas in the case before us the payment of not less than half damages has been explicitly ordained [in all kinds of grounds]. When therefore an a fortiori is employed, another half-payment is added [for damage on the plaintiff’s premises], making thus the compensation complete. If [however] you apply the principle of dayo, the sole purpose of the a fortiori would thereby be defeated.”

Let us try and understand what the Gemara is saying here. It is proposing a distinction (allegedly by R. Tarfon) between two obscure conditions: when applying the dayo principle “would defeat the purpose of the qal vachomer,” it is not applied; whereas where applying the dayo principle “would not defeat the purpose of the qal vachomer,” it is applied. What does this “defeating the purpose of the a fortiori argument” condition refer to? The Gemara clarifies it by comparing R. Tarfon’s (alleged) different reactions to two cases: that concerning Miriam and the (first) argument in the Mishna (the Gemara has apparently not noticed the second argument at all, remember).

The Gemara here reaffirms its theory that, although the Torah (“the instance quoted” – i.e. Num. 12:14-15) does not mention an initial or an additional seven days[179], “nevertheless, by the working of the a fortiori” (as conceived by the Gemara, meaning a crescendo) fourteen days in all (i.e. seven plus seven) are intended, and the dayo principle serves after that to “exclude” the additional seven days, admitting only the “original” seven days. In this case, then, the dayo principle is to be applied. The Gemara then turns to R. Tarfon’s (first) argument, claiming that in its case the dayo principle is not to be applied. Why? Because “the payment of not less than half damages has been explicitly ordained [in all kinds of grounds].” This is taken by commentators (Rashi is mentioned) to mean that since the Torah does not make a distinction between public and private property when it specifies half liability for damage by horn[180], it may be considered as intending this penalty to be (the minimum[181]) applicable to both locations.

The Gemara goes on to tell us that through “a fortiori” inference “another half-payment is added, making thus the compensation complete.” The implication is that, whereas the Sages would at this stage apply the dayo principle and conclude with only half payment, R. Tarfon (according to the Gemara) considered that doing so would “defeat the purpose of the a fortiori” and he concluded instead with full payment. In the Miriam case, we go from no information to fourteen days and back to seven; so we still end up with new information (seven) after the dayo application to the qal vachomer increase. Whereas in the Mishna case, we go from half to full payment and back to half; so that dayo application here would altogether cancel out the qal vachomer increase. Thus, R. Tarfon is presented by the Gemara as knowing and accepting the dayo principle, but applying it more conditionally than the Sages do[182].

But I would certainly challenge the underlying claim that the a fortiori argument used by R. Tarfon (which concludes with full payment for damage by horn on private property) is “nullified” by the Sages’ objection to it (which limits the payment to half). What is given in the Torah is that such damage (on whatever domain) is liable to half payment. This “half” is indefinite, and must be interpreted as at least half (i.e. a minimum of half, no less than half), which leaves open whether only half (i.e. a maximum of half, no more than half) or full (i.e. more than half) is intended. R. Tarfon’s argues (through a crescendo, i.e. ‘proportional’ a fortiori argument) in favor of the conclusion “full,” whereas the Sages argue (through dayo, or purely a fortiori argument) in favor of the alternative conclusion “only half.” R. Tarfon’s argument is certainly not made logically useless by the Sages’ dismissal of it, but constitutes a needed acknowledgment of one of the two possible interpretations of “half,” just as the Sages’ dayo duly acknowledges the other possibility. If the Mishna had directly interpreted “half” as “only half,” without regard to the possibility of “full,” the interpretation would have seemed unjustified.[183]

An argument ex machina. But let us dig deeper into the alleged conditionality of dayo application. Why, more precisely, does the Gemara’s R. Tarfon consider that applying the dayo principle in the case of the Miriam argument does not “defeat the sole purpose of the a fortiori,” yet would do so in the case of his formally similar (first) argument? What is the significant difference between these two cases? And what sense are we to make of the Gemara’s further explanations, viz.:

“And the Rabbis? — They argue that also in the case of divine [reproof] the minimum of seven days has been decreed in the words: Let her be shut out from the camp seven days. And R. Tarfon? — He maintains that the ruling in the words, ‘Let her be shut out etc.’, is but the result of the application of the principle of dayo [decreasing the number of days to seven]. And the Rabbis? — They argue that this is expressed in the further verse: And Miriam was shut out from the camp. And R. Tarfon? — He maintains that the additional statement was intended to introduce the principle of dayo for general application so that you should not suggest limiting its working only to that case where the dignity of Moses was involved, excluding thus its acceptance for general application: it has therefore been made known to us [by the additional statement] that this is not the case.”[184]

It seems[185] that R. Tarfon’s thought (still according to the Gemara, note well) is that, with regard to Miriam, no part of the penalty for offence against God is explicitly mentioned in the Torah (Num. 12:14-15), so that all fourteen days must be inferred by “a fortiori” (i.e. a crescendo); after which the dayo principle is used to revoke seven of those days, leaving seven. Whereas, in the case of horn damage on private property, the minimum liability of half payment is already explicitly given in the Torah (Ex. 21:35), so that the “a fortiori” (i.e. a crescendo) argument only serves to add on half payment; in which case, applying the dayo principle here would completely nullify the effect of the qal vachomer.

Thus, it is implied, the dayo principle is applicable in the Miriam case, but inappropriate in the case of a goring ox. The Sages (allegedly) then object that the initial seven days are indeed given in the Torah, in the sentence “Let her be shut out from the camp seven days.” To which R. Tarfon (allegedly) retorts that this sentence refers to the dayo principle’s “decreasing the number of days to seven.” The Sages reply that that function is fulfilled by the sentence “And Miriam was shut out from the camp.” To which R. Tarfon retorts that the latter rather has a generalizing function from the present case to all others. As far as I am concerned, most of this explanation by the Gemara is artificial construct and beside the point. It is chicanery, pilpul (in the most pejorative sense of that term).

The claim it makes (on R. Tarfon’s behalf) that all fourteen days for offence against God must be inferred is untrue – for the fourteen days are not inferred from nothing, as it suggests; they are inferred from the seven days for offence against a father. The inference of the conclusion, whether it is a crescendo or purely a fortiori, depends on this minor premise. The seven days for a father are indeed a given minimum, also applicable to God; otherwise, there would be no a crescendo or a fortiori inference at all. The Gemara is claiming an “a fortiori” (i.e. a crescendo) argument to be present in the text, and yet denying the relevance of the textual indicators for such an assumption. Its alleged “a fortiori” argument is therefore injected into the discussion ex machina, out of the blue, without any textual justification whatsoever. This is not logic, but rhetoric.

The situation in the argument about Miriam is thus in fact technically exactly identical to the (first) argument relating to liability for damages by horn in the Mishna. Both arguments do, in fact, have the minor premise needed to draw the conclusion. Whence the Gemara’s concept of “defeating the sole purpose of the a fortiori” is a red herring; it is just a convenient verbal artifice, to give the impression that there is a difference where there is none. The Gemara has evidently tried to entangle us in an imaginary argument. For, always remember, it is the Gemara’s reading which is at stake here, and not R. Tarfon’s actual position as it appears in the Mishna, which is something quite distinct.

The roles of the verses in Num. 12:14-15. What is evident is that neither of the readings of the said Torah portion that the Gemara attributes to R. Tarfon and the Sages fully corresponds to the simple reading (peshat). They are both awkward inventions[186] designed to justify the Gemara’s own strange thesis. The Gemara’s thesis is not something necessary, without which the Mishna is incomprehensible; on the contrary, it clouds the issues and misleads. Whatever the author’s authority, it is unconvincing.

The simple reading of Num. 12:14-15 is, as we saw earlier[187], that the sentence “If her father had but spit in her face, should she not hide in shame seven days?” (first part of v. 14, call it 14a) provides the minor premise of a possible a fortiori argument (whether strict or a crescendo), while the sentence “Let her be shut up without the camp seven days, and after that she shall be brought in again” (second part of v. 14, call it 14b) provides its immediate conclusion. Note well that it is from these two sentences (i.e. v. 14a & 14b) that we in the first place surmise that there is an a fortiori argument in the text; to speak of an a fortiori argument without referring to both these indices would be concept stealing. The further sentence “And Miriam was shut up without the camp seven days; and the people journeyed not till she was brought in again” (v. 15) plays no part in the a fortiori argument as such, but serves to confirm that the sentence was carried out by Moses’ court as prescribed by God.

The Gemara’s R. Tarfon makes no mention of the role of v. 14a in building a qal vachomer, and regards v. 14b as the final conclusion of the argument, after the operation of an entirely tacit a crescendo inference to fourteen days and an also tacit application of dayo back to seven days; as regards v. 15, it effectively plays no role within the argument in his view, having only the function of confirming that the dayo application is a general principle and not an exceptional favor[188]. The Gemara’s Sages, on the other hand, regard v. 14b (not 14a, note well) as the minor premise of the qal vachomer, and v. 15 its final conclusion, after the operation of an a crescendo inference to fourteen days and an application of dayo back to seven days.

Both parties make serious errors. The first of these is that neither of them accounts for v. 14a – why is it mentioned here if as both parties suppose it plays no role? No a fortiori argument can at all be claimed without reference to this information. The R. Tarfon thesis here is largely imaginary, since he ignores the role of v. 14a in justifying a qal vachomer; there is no trace in the Torah text of the a crescendo argument he claims, other than v. 14b. On the basis of only the latter textual given of seven days, he projects into the text a minor premise of seven days, an intermediate a crescendo conclusion of fourteen days and a dayo principle application, yielding a final conclusion of seven days (v. 14b). But if all the textual evidence we rely on is v. 14b, on what basis can we claim any a crescendo reasoning has at all occurred before it, let alone a dayo application, with this verse as the final conclusion? The whole process becomes a patent fabrication.

Nowhere in the proof text, note well, are the words qal vachomer or dayo used, or any verbal signal to the same effect. And this being so, what credence can be assigned to the Gemara’s central claim, viz. that the dayo principle is “of Biblical origin?” It is surely paradoxical that it is able to support this ambitious claim only by means of a very debatable mental projection of information into the Torah, like a magician pulling a rabbit out of a hat after showing us it was empty. This means that the Gemara’s proposed argument in favor of this claim is circular: it assumes X in order to prove X. This is of course made possible through the use of complicated discourse; but the bottom line is still the same.

The Sages’ thesis is a bit more credible in that, even if they also grant no role to v. 14 a, they at least do propose a minor premise (v. 14b), as well as a final conclusion (v. 15). However, it is hard to see how “Let her be shut up without the camp seven days” (v. 14b) could be the minor premise of qal vachomer yielding the conclusion “And Miriam was shut up without the camp seven days” (v. 15)! These two propositions have the same subject (as well as the same explicit predicates), so where is the qal vachomer? Moreover, the Sages thereby subscribe to R. Tarfon’s strange misconception regarding a fortiori argument.

A fortiori argument with a single subject. I am referring here to the bizarre notion that (in the qal vachomer argument under consideration, which is positive subjectal) the subject of the minor premise must be repeated in the conclusion, while the subsidiary terms (i.e. the predicates of these propositions) go from less to more (implicitly). In fact, positive subjectal argument, whether a fortiori or a crescendo, formally has different subjects (the minor and the major terms, respectively) in the minor premise and conclusion (as for the predicate, i.e. the subsidiary term, it remains constant in pure a fortiori, while it increases in a crescendo). There has to be two subjects for the argument to logically function. The bizarre notion in the Gemara of a single subject argument is the reason why both parties in it ignore v. 14a and look for some other proposition to use as minor premise.

It should be stressed that there is no allusion whatsoever to such an idea in the Mishna. The Mishna’s R. Tarfon and Sages manifestly have an entirely different dialogue than the one the Gemara attributes to them. The discussion in the Mishna is much more credible than that in the Gemara. The Gemara makes up this notion solely in order to create a distinction between the Miriam case and the Mishna’s (first) argument. It needs to do this, remember, in order to justify its theory that R. Tarfon and the Sages agree on the dayo principle, although R. Tarfon applies it conditionally whereas the Sages apply it universally. But as we shall demonstrate formally, this notion is logically untenable. Buying the Gemara’s scenario is like buying Brooklyn Bridge from someone who doesn’t own it.

The thesis of R. Tarfon in the Gemara is that, in the Miriam case, we must have a minor premise that offending God (rather than merely one’s father) justifies a minimum of seven days of punishment, in order to be able to infer qal vachomer (i.e. a crescendo) that offending God justifies fourteen days of punishment – just as with regard to an ox, we (allegedly) reason from half liability for damage done on private (rather than public) property to full liability on private property. The Sages do not object to this claim. But this claim is simply not true – there is no such technical requirement for positive subjectal a crescendo (or a fortiori) inference. We can very well, and normally do, reason with a change of subject, i.e. from the penalty for offence to one’s father to that for offence to God, or from the liability for damage on public grounds to that on private grounds. This is precisely the power and utility of a fortiori (and a crescendo) inference.

Moreover, we in fact can, by purely a fortiori argument, infer the needed minor premise about seven days penalty for offending God (from the same penalty for offending one’s father), and likewise the half liability on private property (from the same liability on public property)[189]. One cannot claim an a crescendo argument to be valid without admitting the validity of the purely a fortiori argument (and pro rata argument) underlying it. Obtaining the minor premise demanded by the Gemara’s R. Tarfon is thus not the issue, in either case. The issue is whether such a minor premise will allow us to draw the desired ‘proportional’ conclusion. And the answer to that, as we show further on, is: No!

Furthermore, if we carefully compare the Gemara’s argument here to the first argument laid out in the Mishna, we notice a significant difference. As we just saw, the Gemara concludes with full liability for horn damage on private property on the basis of half liability for horn damage on private property. As earlier explained, it bases this minor premise on the fact that Ex. 21:35 does not make a distinction between public and private property when it prescribes half liability for damage by horn, so that this may be taken as a minimum in either case. Thus, for the Gemara, half liability for horn damage on private property is a Torah given, which does not need to be deduced. On the other hand, in the Mishna, the minor premise of the first argument refers to the public domain rather than to private property.

In his first argument, R. Tarfon argues thus (italics mine): “…in the case of horn, where the law was strict regarding [damage done on] public ground imposing at least the payment of half damages, does it not stand to reason that we should make it equally strict with reference to the plaintiffs premises so as to require compensation in full?” And to justify his second argument he argues thus: “but neither do I infer horn [doing damage on the plaintiff’s premises] from horn [doing damage on public ground]; I infer horn from foot, etc.”[190] Thus, his first argument is clearly intended as an inference from the penalty for horn damage in the public domain (half) to that in the private domain (full). The Gemara’s construct is thus quite different from the Mishna’s, and cannot be rightly said to represent it.

As regards the rule here apparently proposed by the Gemara (which it attributes to R. Tarfon), viz. that the subject must be the same in minor premise and conclusion, as already stated there is no such rule in formal logic for positive subjectal argument[191]. Such argument generally has the minor and major terms as subjects of the minor premise and conclusion respectively, even if the subsidiary term sometimes (as is the case in a crescendo argument) varies in magnitude ‘proportionately’. In the case of a crescendo argument, where the predicate (subsidiary term) changes, there absolutely must be a change of subject, since otherwise we would have no explanation for the change of predicate. That is, we would have no logical argument, but only a very doubtful ‘if–then’ statement. The proposed rule is therefore fanciful nonsense, a dishonest pretext.

We can examine this issue in more formal terms. A positive subjectal a fortiori argument generally has the form: “P is more R than Q is; and Q is R enough to be S; therefore, P is R enough to be S” (two premises, four terms). If the argument is construed as a crescendo, it has the form: “P is more R than Q is; and Q is R enough to be Sq; and S is ‘proportional’ to R; therefore, P is R enough to be Sp” (three premises, five terms). The argument form attributed by the Gemara to R. Tarfon simply has the form: “If X is S1, then X is S2” (where X is the sole subject, and S1 and S2 the subsidiary terms, S2 being greater than S1); that is, in the Miriam sample: “if offending God merits seven days penalty, then offending Him merits fourteen days penalty,” and again in the Mishna’s first dialogue: “If liability for horn damage on private property is half payment, then liability for same on private property is full payment.” This is manifestly not a fortiori or a crescendo argument, but mere if–then assertion; it could conceivably happen to be true, but it is not a valid inference.

It is clear that the latter inference, proposed by the Gemara in the name of R. Tarfon, has no logical leg to stand on. It has no major premise comparing the subjects (P and Q); and no need or possibility of one, since there is only one subject (X). Having no major premise, it has no middle term (R); and therefore no additional premise in which the subsidiary term (S) is presented as ‘proportional’ to it. Thus, no justification or explanation is given why S should go from Sq in the minor premise to Sp in the conclusion. It is therefore not an a fortiori or a crescendo argument in form, even if it is arbitrarily so labeled by the Gemara. You cannot credibly reason a fortiori or a crescendo, or any other way, if you cannot produce the requisite premises. There is no such animal as “argument” ex nihilo.

The Gemara’s proposed if–then statement is certainly not universal, since that would mean that if any subject X has any predicate Y then it has a greater predicate Y+, and if Y+ then Y++, and so forth ad infinitum – which would be an utter absurdity[192]. From this we see that not only has the Gemara’s argument no textual bases (as we saw earlier), but it has no logical standing. There is in fact no “argument,” just arbitrary assertion on the Gemara’s part. For both the Miriam sample and the (first) Mishna sample, the Gemara starts with the convenient premise that “there is a qal vachomer here,” which it considers as given (since it is traditionally assumed present, on the basis of other readings of these texts), and then draws its desired conclusion without recourse to any other proposition, i.e. without premises![193]

If this requirement for a single subject is not a rule of logic, is it perhaps a hermeneutic principle, i.e. a rule prescribed by religion? If so, where (else) is it mentioned in the oral tradition or what proof-text is it drawn from? Is it practiced in other contexts, or only in the present one, where it happens to be oh-so-convenient for the Gemara’s interpretative hypothesis? If it is an established rule, how come the Sages do not agree to it? The answers to these questions are pretty obvious: there is no such hermeneutic rule and no basis for it. It was unconsciously fabricated by the Gemara author in the process of developing the foolish scenario just discussed. It is not a general necessity (or even a possibility, really), but just an ad hoc palliative.

Unfortunately, when people use complex arguments (such as the a fortiori or the a crescendo) without prior theoretical reflection about them, they are more or less bound to eventually try to arbitrarily tailor them to their discursive needs.

To sum up. We have seen that the Gemara introduces a number of innovations relative to the Mishna it comments on. The first we noted was that the Gemara, in the name of an anonymous Tanna, reads the qal vachomer in Num. 12:14-15, and apparently all a fortiori argument in general, as a crescendo argument. Next we noted a surprising lacuna in the Gemara’s treatment, which was that while it dealt with R. Tarfon’s first argument, it completely ignored his second, and failed to notice the curious verbatim repetition in the Sages’ two dayo objections. Third, we showed that the thesis that dayo is “of Biblical origin,” so that R. Tarfon must have been aware of it, was the Gemara’s main goal in the present sugya. In the attempt to flesh out this viewpoint, the Gemara proceeds to portray R. Tarfon as regarding the dayo principle as being applicable only conditionally, in contrast to the universal dayo principle seemingly advocated by the Sages.

To buttress this thesis, the Gemara is forced to resort to an argument ex machina – that is, although vehemently denying the role of both parts of Num. 12:14 in the formation of a qal vachomer, the Gemara’s R. Tarfon nevertheless assumes one (i.e. a phantom a fortiori argument) to be somehow manifest between the lines of the proof-text. Moreover, in order to make a distinction between the Miriam example and the (first) Mishna argument, so as to present the dayo principle as applicable to the former and inapplicable to the latter, the Gemara’s R. Tarfon invents a preposterous rule of inference for qal vachomer, according to which the subject must be the same in the minor premise and the conclusion. In the Miriam example, the absence of a minor premise with the required subject (offending God) means that dayo is applicable, for applying it would not “defeat the purpose of the qal vachomer;” whereas in the (first) Mishna argument, the presence of a minor premise with the required subject (damage by ox on private property) means that dayo is inapplicable, for applying it would “defeat the purpose of the qal vachomer.”

This all looks well and good, if you happen to be sound asleep as the Gemara dishes it out. For the truth is that at this stage the whole structure proposed by the Gemara comes crashing down.

The trouble is, there is no such thing as an a fortiori argument (or a crescendo argument) that takes you from no information to a conclusion, whether maximal or minimal. If the proposed qal vachomer “argument” has no minor premise (since v. 14a is explicitly not admitted as one) and no major premise (since the subject of the conclusion must, according to this theory, be the same in the minor premise as in the conclusion), then there is no argument. You cannot just declare, arbitrarily, that there is an argument, while cheerfully denying that it has any premises. And if you have no argument with a maximum conclusion, then you have no occasion to apply the dayo principle, anyway.

Moreover, there is no such one-subject rule in a fortiori logic; indeed, if such a rule were instituted, the argument would not function, since it would have no major premise, and no major, minor or middle term; consequently, if it was intended as ‘proportional’ (as the Gemara claims), it would imply an inexplicable and absurd increase in magnitude of the subsidiary term. Thus, even if the Gemara’s textually absent argument about Miriam were generously granted as being at least ‘imaginable’ (in the sense that one might today imagine, without any concrete evidence, Mars to be inhabited by little green men), the subsequent demand that a qal vachomer have only one subject would make the proposed solution formally impossible anyway.

The Gemara’s explanation is thus so much smoke in our eyes, a mere charade; it has no substance. We need not, of course, think of the Gemara as engaging in these shenanigans cynically; we can well just assume that the author of this particular commentary was unconscious. In fine, the Gemara’s scenario, in support of its claim that the dayo principle is “of Biblical origin” and so R. Tarfon did not ignore it—is logically unsustainable.

6.    A slightly different reading of the Gemara

As we saw previously, the two arguments featured in Mishna BQ 2:5 may objectively be variously interpreted. R. Tarfon’s first argument may be read as pro rata or as a crescendo, though not as purely a fortiori (since his conclusion is ‘proportional’), while his second argument may be read in all three ways. As regards the Sages’ first dayo objection, if R. Tarfon’s first argument is supposed to be intended as a pure a fortiori, the objection to it would simply be that such argument cannot logically yield a ‘proportional’ conclusion; this reading is very unlikely. Rather, the first dayo objection may be taken as a refusal of the ‘proportionality’ of the pro rata or a crescendo arguments, and possibly the proposal of a purely a fortiori counterargument, i.e. one without a ‘proportional’ conclusion. The Sages’ second dayo objection, on the other hand, cannot have the same intent, since in this case all three forms of argument yield the very same ‘proportional’ conclusion; so it must be aimed at the inductive processes preceding these arguments.

In our above analysis of the corresponding Gemara, we have mostly represented it as conceiving of one possible scenario for both[194] arguments of the Mishna, that of a crescendo argument moderated by a dayo principle. This is the traditional and most probable interpretation, but it should be said that an alternative reading is quite possible. Certainly, the Gemara here does not accept, or even consider, the alternative hypothesis that purely a fortiori argument may be involved in the second argument of R. Tarfon, since it clearly assumes that the conclusion’s predicate is bound to be greater than the minor premise’s predicate. However, it would be quite consistent to suppose that the Gemara is in fact not talking of two a crescendo arguments, but of two analogical/pro rata arguments. There is some uncertainty as to the Gemara’s real intent, since it does not explicitly acknowledge the various alternative hypotheses and eliminate all but one of them for whatever reasons.

Looking at the Mishna and Gemara discourses throughout the Talmud, it is obvious that the people involved use purely a fortiori argument, a crescendo argument, and argument pro rata in various locations. But it is not obvious that there is a clear distinction in their minds between these three forms of argument. It is therefore not impossible that when they say “qal vachomer,” they might indiscriminately mean any of these three forms of argument. It should be clear to the reader that the issue I am raising here is not a verbal one. I am not reproaching the Talmud for using the words “qal vachomer” in a generic or vague sense. I certainly cannot reproach it for not using the expressions ‘a crescendo’ or ‘pro rata’, as against ‘a fortiori’, since these names were not in its vocabulary.

What I am drawing attention to is the Talmud’s failure to demonstrate its theoretical awareness of the difference between the three forms of argument, whatever they are called. How could such awareness be demonstrated? It would have sufficed to state (if only by means of concrete examples, without abstract explanations) that the two premises of a fortiori per se do not allow a ‘proportional’ conclusion to be drawn, but must be combined with a third, pro rata premise for such a conclusion (i.e. a crescendo) to be justified; and that it is also possible to arrive at a ‘proportional’ conclusion without a fortiori reasoning, through merely analogical (i.e. pro rata) reasoning.

That is to say, for instance in the positive subjectal mood, the major premise “P is more R than Q is” and the minor premise “Q is R enough to be S” do not suffice to draw the conclusion “P is R enough to be more than S.” To deduce the latter a crescendo conclusion, an additional premise must be given, which says that “S is proportional to R.” Given all three said premises, we can legitimately conclude that “P is R enough to be (proportionately) more than S;” but without the third one, we can only conclude “P is R enough to be S.” Alternatively, we might infer from “S is in general proportional to R,” combined with “a given value of S is proportional to a given value of R,” that “a greater value of S is proportional to a greater value of R” (this is pro rata without a fortiori).

Thus, although we have taken for granted in our above analysis the traditional view that when the Gemara of Baba Qama 25a speaks of qal vachomer, it is referring to a fortiori argument, i.e. more precisely put to a crescendo argument (since it advocates ‘proportional’ conclusions), it is quite conceivable that it was unconsciously referring to mere pro rata argument. The dayo principle is not something conceptually, even if halakhically, tied to a fortiori (or a crescendo) argument, but could equally well concern pro rata argument (or even other forms of reasoning). And what I have above called the “bizarre notion,” which the Gemara credits to R. Tarfon, that the minor premise and conclusion of a positive subjectal argument must have the same subject for the argument to work, could equally be applied to pro rata argument as to a crescendo, since it is an arbitrary rule of Judaic logic without formal support in generic logic. Therefore, our above analysis of the Gemara would not be greatly affected if we assume it to refer to pro rata instead of to a crescendo argument. This is not a very important issue, but said in passing.

8.  In the Talmud, continued

The present chapter is a continuation of the preceding, aimed at further clarifying some details.

1.    Natural, conventional or revealed?

Our above critique of the Gemara was based to some extent on the assumption that it considers dayo as a principle, which the Sages regard as a hard and fast rule and R. Tarfon views as a conditional rule, depending on whether or not its application “defeats the purpose of the qal vachomer.” But in truth, the idea of dayo as a “principle” may be an interpolation, because the original Aramaic text (viz. “ור”ט לית ליה דיו והא דיו דאורייתא הוא”) does not use the word “principle” in conjunction with the word “dayo.”

The translation given in the Soncino Babylonian Talmud (viz. “Does R. Tarfon really ignore the principle of dayo? Is not dayo of Biblical origin?”) does of course use this word. But if we look at the Talmud Bavli translation (with their running commentary here put in square brackets), viz. “And does R’ Tarfon not subscribe to [the principle of] ‘It is sufficient…’ – Why, [the principle of] ‘It is sufficient…’ is contained in the [Written] Torah, [and R’ Tarfon must therefore certainly accept it!]” – it becomes evident that the word “principle” is an add-on. This of course does not mean that it is unjustified, but it opens possibilities.

If we do accept the translations, it is clear that the word “principle” is here equivocal, anyway – granting that for the Sages it means a universal proposition whereas for R. Tarfon it means a merely conditional one. This equivocation implies that the positions of the two parties are not as harmonious as the Gemara tries to suggest. They do not agree on principle and merely differ on matters of detail, as it were. On one side, there is a hard and fast rule; and on the other, one that is subject to adaptation in different situations. This is a radical difference, which is hardly diminished by assuming the “principle” to be of Biblical origin.

In view of this, it is difficult to guess what might be the Gemara’s purpose in positing that the dayo principle is deoraita (of Biblical origin – as against derabbanan, of rabbinic origin) and is known and essentially accepted by R. Tarfon. Moreover, as we have exposed, the Gemara’s scenario for R. Tarfon’s thesis is forced and untenable, being based on doubtful readings of the Torah and Mishna texts it refers to and, worst of all, on a parody of logic. Certainly, the Gemara’s scenario does not prove the claim of Biblical origin. If anything, that claim is weakened by virtue of having been supported by such rhetoric. But is the claim now disproved, or can it be supported by other means?

The Gemara is, of course, correct is in linking the issue of Biblical origin with that of R. Tarfon’s knowledge and acceptance. If the principle is of Biblical origin – i.e. is given in the Written Torah, or (since it is not manifest in the Pentateuch) at least the Oral Torah – it must be assumed to be known and accepted by him, as well as by the Sages. If he did not know and accept it, but only the Sages did, it cannot be of Biblical origin. However, I do not see how the Gemara can claim a different understanding of the dayo principle of Biblical origin for R. Tarfon than for the Sages. What would be the common factor between their views, which would be a “principle” of Biblical origin? The difference between universal and only-conditional applicability is too radical; these two theses are logically contrary. Their only possible intersection is that valid dayo objections may occur. This is hardly enough to constitute a “principle,” although we might in the limit grant it such status.

On the other hand, it would be quite consistent to say that the Sages and R. Tarfon both believe in a dayo principle of Biblical origin that is only conditionally applicable, but only differ with regard to the precise conditions of its application. Thus, the Biblical origin hypothesis remains conceivable, provided the word “principle” is understood in its softer sense, in such a way that debate is logically possible in particular cases, so that R. Tarfon might win in some cases and the Sages in other cases. The dayo principle would then consist in the bare fact that “some dayo objections are justifiable, though some are not;” and its being of Biblical origin would mean that this vague, contingent prediction was given at Sinai. Such conceivability does not of course prove that this much-reduced dayo principle was indeed of Biblical origin. Nor does it explain why the Gemara tried so hard to establish it as such. But it at least leaves the hypothesis in the running, so long as no other plausible reasons are found to discard it.

As mentioned at the end of our analysis of the Mishna, there are yet other equally viable hypotheses. We can still uphold the conflict between the Sages and R. Tarfon to be one between a hard and fast view of the dayo principle and an only-conditional view of it, provided we do not claim this principle to be of Biblical origin, but only of rabbinic origin (derabbanan). In the latter case, the Sages are collectively in the process of legislating the dayo principle in our Mishna, and though R. Tarfon initially tries to argue against this innovation by means of his two arguments, at the end he is forced to accept the majority decision. This scenario is equally consistent, and to my knowledge the Gemara offers no reason for dismissing it.

In this context, we could suggest that the dayo principle being “of Biblical origin” means, not that is was explicitly mentioned in or logically deduced from the Torah, but simply that something to be found in the Torah inspired the rabbis to formulate and adopt this principle. We might even propose (this is pure speculation on my part) the inspiration to have come specifically from Deuteronomy 4:2[195], which reads: “Ye shall not add unto the word which I command you, neither shall ye diminish from it.” It could well be that the rabbis, consciously or otherwise, saw in this warning of the Torah a justification for the cautiousness called for by their dayo principle. In that event, both R. Tarfon and the Sages obviously agreed regarding the truth of the inspiring Torah passage, but they differed as to how far the inspiration should be allowed to go. The dayo principle is not, in either case, precisely deducible from the said Torah passage, but a relation of sorts between the two can be claimed. The rabbinical principle, however broadly understood, is not in ‘the letter of the law’, but it is surely in ‘the spirit of the law’.

Another possibility is that there is no dayo principle, whether universal or conditional, at all, but each recorded dayo objection stands on its own as an individual rabbinical decree, for whatever reason the rabbis consider fit. This too can be used to explain the disagreements between R. Tarfon and the Sages in a consistent manner. This hypothesis logically differs very little from the above mentioned one of a conditional dayo principle, except in that the conditional dayo principle scenario implies an explicit Divine prediction at Sinai, whereas the no dayo principle scenario assumes no specific Sinaitic transmission on this topic (even if the general authority of the rabbis to judge and maybe innovate may have there been explicitly established). Here again, then, we have a consistent alternative hypothesis that the Gemara did not take into consideration and eliminate, before affirming its own thesis.

The methodology of the Talmud is of course essentially dogmatic. It engages in discussions and arguments, usually genuinely logical; but it does not go all the way with logic, systematically applying its techniques and referring to its results. It accepts some arbitrary ideas. This here seems to be a case in point, where the Gemara seeks to prove some preconceived notion and does everything it can to give the impression that it has. But we must always consider alternatives and evaluate them fairly.

The issue we will explore now is whether the dayo principle is to be regarded as natural, conventional or revealed. By ‘natural’ I mean that it is a law of nature, i.e. more specifically of logic or perhaps of natural ethics. By ‘conventional’ I mean that it is a collective decision of the rabbis, or more generally of human authorities, for whatever motive. And by ‘revealed’ I mean here that it is Divinely-decreed, handed down to us through prophecy or other supernatural means; i.e. more specifically, primarily at the Sinai revelation through Moses, and then written in the Torah or passed on orally through an unbroken tradition.

We have, I believe, definitely established in our above treatment that the dayo principle is not a law of logic. Many people have thought of it – and for a long time, I must confess, I too did so – as signifying that the (predicate of the) conclusion of (purely) a fortiori argument cannot quantitatively surpass the (predicate of the) minor premise. The dayo principle, in that view, corresponds to the principle of deduction, i.e. to a reminder that you cannot get more out of it than you put into it. In that perspective, I used to think the rabbis collectively instituted the dayo principle in order to prevent other people from erroneously drawing a ‘proportional’ conclusion from purely a fortiori premises. I was misled into this belief, perhaps, by the fact that rabbinical a fortiori reasoning is in practice usually correct, and also by the fact that the mentions of qal vachomer in the lists of Hillel and R. Ishmael do not mention the dayo principle as a separate hermeneutic rule, and therefore apparently consider the latter as an integral part of the former’s structure, which though it can be distinguished from it cannot correctly be dissociated from it.[196]

But as we have demonstrated in the present study the dayo principle is something much more complex than that. However, although this principle is not a natural principle in the sense of a law of logic, it might still be considered as a natural principle in the sense of a truth of ethics in a secular perspective. If we were to consider it as such, we would have to say that when the rabbis apply it, they are merely expressing their moral sensibilities as ordinary human beings. In that event, we would have to say that the dayo principle is applicable not only in legal contexts peculiar to the Jewish religion, but in all legal contexts, whether Jewish or non-Jewish, religious or secular. But the latter does not seem true – certainly, if we look at legal rulings in other traditions, the idea of dayo hardly if at all arises. So this idea seems to be a particularly Jewish (indeed, rabbinical) sensibility.

Thus, the dayo principle should rather be viewed as either conventional or revealed. As we have seen, contrary to what the Gemara insists, there is no incontrovertible proof that it is revealed. It may be “of Torah origin” in a broad sense, in the sense of “of Sinaitic origin.” But it is clearly (for any honest observer) not explicitly stated in the Written Torah; so it must be assumed to be part of the Oral Torah. Of course, the Gemara does seem to be claiming this principle to be logically derived from Num. 12:14-15 – but as we have seen, this ‘proof’ is unfortunately circular: it is read into the text rather than out of it. This means that the only way we know that the principle is “of Torah origin” is because the rabbis (led by the Gemara) tell us that it is. Such assertion is considered by the rabbis as sufficient proof that the alleged tradition is indeed Sinaitic. But scientifically it is surely not sufficient, as all sorts of things could have happened in the millennia in between.

Thus, while in the first instance (lehatchila) the rabbis would affirm the principle as derived from the Written Torah, if they are pressed hard enough they would probably as a last resort (bedieved) opt instead for the Oral Torah explanation. But, to my mind at least, this is logically equivalent to saying that the rabbis are the effective source of the principle. That is, it is derabbanan, and not at all deoraita. For we only have their say-so as proof of their assertion. Of course, it is still conceivable that the principle was indeed handed down at Sinai – we have not disproved that, and have no way to do so. But, as there is no way (short of a new revelation) to prove it, either, this conceivable scenario remains a mere speculation. So that the logical status of the principle is pretty much exactly the same as if the rabbis had simply conventionally decided to adopt it. This is the conclusion I adopt as a result of the present study: the dayo principle is of rabbinical origin.

To conclude, it is not clear why the Gemara makes such a big thing about the “Biblical origin” of the dayo principle, even going so far as to construct fictitious inference rules and arguments to prove its point. Did the Gemara have some halakhic purposes in mind, or was it just engaging in idle chatter (pilpul)? As we have seen, the Mishna can well be understood – indeed, in a number of ways – without pressing need to resolve the issue of the origin of the dayo principle. Why then is the Gemara’s commentary so focused on this specific issue, ignoring all other aspects? Perhaps it needs the proposition that the dayo principle is “of Biblical origin” for some other purpose(s), elsewhere. Not being a Talmudic scholar, I cannot answer this question. But in any event, to my mind, whatever the Gemara’s motives may have been, it failed miserably in this particular discourse.

Moreover – let us not forget this fact – when the Gemara refers to the dayo principle, it means just the first expression of that principle, as it is applicable to R. Tarfon’s first argument. The Gemara has not shown any awareness of the existence and significance of R. Tarfon’s second argument, and therefore of the difference in the Sages’ dayo objection to it. Thus, even if it had succeeded to prove somehow that the Sages’ first dayo objection was “of Biblical origin,” it would not have proven that their second objection was of equally elevated origin. This, too, is a disappointment concerning the Gemara: its powers of observation and analytic powers were here also less acute than they ought to have been.

We have thus far considered the issue of the origin of the dayo principle, but now let us look into that of qal vachomer. It is worth noting for a start that qal vachomer and the dayo principle are viewed by the Gemara as two distinct thought processes. The dayo principle is applied ex post facto, to the conclusion of a preexisting qal vachomer. The dayo principle (presumably) cannot be invoked until and unless a qal vachomer is formulated. If the dayo principle is not applied (as is possible in R. Tarfon’s view, according to the Gemara), the qal vachomer stands on its own. Thus, qal vachomer inference is independent of the dayo principle, even if the latter process is not independent of the former. Therefore, claiming that the dayo principle is “of Biblical origin” does not necessarily imply a claim that qal vachomer inference is also so justified. It may thus well be a natural process, if not a rabbinical convention.

In this context it is interesting to note that, in the lists of hermeneutic principles of Hillel and R. Ishmael, the dayo principle is nowhere mentioned, but only qal vachomer is mentioned. Since qal vachomer can occur, according to the Gemara, without the dayo principle, why is the latter not mentioned also as a separate hermeneutic principle? And if the dayo principle is “of Biblical origin,” as the Gemara has it, should it not all the more be mentioned in such lists? Conversely, if qal vachomer is a natural thought process, why does it need to be mentioned is such lists? Perhaps the answer to these questions is simply that the term “qal vachomer” in these lists is intended as an all-inclusive title, meaning “anything to do with qal vachomer, including on occasion application of the dayo principle.” Since, whatever the source of qal vachomer, whenever it is mentioned the question arises as to whether or not the dayo principle is applicable to it, the former always brings to mind the latter. Moreover, the traditional view seems to be that the dayo principle is only applicable to qal vachomer, so this question will not arise in other contexts.

In the Mishna, there is no explicit reference to the issue of the origin of the inference processes used. No explicit claim is made by anyone there that the dayo principle is “of Biblical origin” or any other origin; and nothing of this sort is said of qal vachomer. If we look at R. Tarfon’s wording, we are tempted to say that he regards his reasoning as natural. When he says: “I infer horn from foot” and “does it not stand to reason that we should apply the same strictness to horn?” – he seems to be appealing to logic rather than to some dogmatic given; and furthermore, by saying “I” and “we,” he seems to suggest that the decision process is in human hands. The Sages do not in their replies reprove him for this naturalistic approach; but they merely, it seems, say what they for their part consider to be a wiser ruling.

For the Gemara (i.e. the particular Gemara commentary that concerns us here, and not necessarily the Gemara in general), as we have seen, “qal vachomer” is understood as referring specifically to a crescendo argument, i.e. to a fortiori argument with a ‘proportional’ conclusion. The Gemara bases this understanding on the baraita it quotes. It does not mention purely a fortiori argument, which suggests that it is not aware of such form of argument. This is of course an important error on its part, because without awareness of the difference between purely a fortiori argument and a crescendo argument it cannot realize the logical skill of R. Tarfon’s second argument and the challenge it posed to the Sages’ first formulation of the dayo principle. The Gemara’s blindness to purely a fortiori argument explains its blindness to R. Tarfon’s second argument.

Even so, it is safe to say that the Gemara considers qal vachomer as natural in origin. Certainly, it does not explicitly state it to be “of Biblical origin,” as it does for the dayo principle. Although the Gemara’s assumption that Num. 12:14-15 contains an example of qal vachomer is reasonable, this Torah passage certainly does not use any verbal expression indicative of it, like “qal vachomer” or “all the more;” so, human insight is needed to see the implicit qal vachomer. The Gemara cannot be said to regard qal vachomer as a conventional construct by the rabbis, since the argument is in its view already found in the Torah. Since the Gemara does not even raise the issue (though it could and should have), it may be supposed to regard qal vachomer as ordinary human reasoning.

We might, however, suppose that the Gemara considers that the Miriam example is also given in the Torah to teach us that the correct conclusion of qal vachomer is ‘proportional’ – i.e. that this rule of inference was Divinely-ordained together with the dayo principle. But such a supposition is objectively nonsensical, since a fortiori argument is in fact not universally ‘proportional’. It would suggest that God, well after the Creation, may tell us to disregard logic and judge contrary to its laws. Yet, the laws of logic are not arbitrary dictates that can be discarded at will – even at Divine will – they are inextricably tied to the world as it is and our rational cognition of it. Therefore, to attribute such opinion to the Gemara would be to its discredit.

If we look at the three other a fortiori arguments in the Pentateuch listed in Genesis Rabbah, there is as in the Miriam instance no explicit ‘proportionality’, but we could in two of them at least similarly assume implicit ‘proportionality’, namely Ex. 6:12 and Deut. 31:27. Moreover, there is one passage in the Pentateuch that is explicitly ‘proportional’, namely Gen. 4.24: “If Cain shall be avenged sevenfold, truly Lamekh seventy and seven-fold”[197] –– but the speaker of this statement being Lamekh, someone apparently not regarded as exemplary, it can hardly be considered as halakhically authoritative. There are also many passages in the rest of the Bible that seem either explicitly or implicitly ‘proportional’, and so could be brought to bear in the present context. But the Gemara does not (at least, not here) find it necessary to mention any of them.

Thus, it is reasonable to suppose that the Gemara views qal vachomer (or at least its ‘proportional’ version) as natural argumentation – i.e. as not needing a special Divine dispensation to be credible. In other words, it is purely logical. In Talmudic terminology, this would qualify qal vachomer as a sort of svara, an inference naturally obvious to human reason. This seems to be the way most rabbis throughout history would characterize the argument. Certainly, most of the exceptional rules and dispensations they have enacted in relation to this argument form suggest it; although the fact that some have tried to interdict its free use suggests a doubt in their mind in this regard.

But even though svara refers to natural and universal logical insight, qal vachomer is always counted as one of the “midot,” i.e. of the rabbinical hermeneutic principles. There is a difficulty in this fact, because a hermeneutic principle is thought of as a discursive tool (ordained directly by God or indirectly by rabbinical decision) for use specifically in Torah interpretation. Such principles being essentially non-natural, they may well be not rationally evident or even perhaps contrary to logic. Not so in the case of qal vachomer. So there is a problem with its inclusion in the lists of midot. The solution of this paradox, I would say, is simply that the rabbis themselves did not make such fine distinctions between natural and conventional logic. Or equally well: they could lump qal vachomer with more uncommon forms of reasoning, because in their minds all are “logical.” This is indeed suggested in many rabbinical texts in English, where the word “midot” is translated as “principles of logic.”

2.    Measure for measure

The Gemara perhaps sought to justify the dayo principle by claiming it to be “of Biblical origin” – but there was no pressing need for it to do so, since other explanations were readily available and perhaps less problematic. It seems that the Gemara, not having previously analyzed qal vachomer reasoning in formal terms, was unable to precisely perceive its constituent premises, and under what conditions they resulted in this or that conclusion; and thence, how such an argument could be rebutted. In the Gemara author’s mind, therefore, apparently, the status of a Divine decree (“Biblical origin”) was necessary for the dayo principle to have the power to rebut the qal vachomer argument (as he saw it).

As we have shown, the two arguments proposed by R. Tarfon and the dayo objections to them put forward by the Sages can be interpreted in a number of ways. R. Tarfon’s two arguments could have been (1) intended as two mere arguments by analogy (more precisely, pro rata); or (2) the first one may have been pro rata, while the second was (purely) a fortiori; or (3) they could (as the Gemara did) both be construed as having been a crescendo. The Sages’ dayo statements, could be viewed as (a) particular ad hoc objections, decided by the rabbis collegially; or (b) as general objections, either (i) clearly given in the Written Torah or deduced from it (as the Gemara wrongly claims); or (ii) inductively or rhetorically derived from it (as the Gemara actually attempted); or (iii) known from the Oral Torah (i.e. by unbroken tradition since the Sinai revelation); or again (iv) decided by the rabbis.

If we said that R. Tarfon’s first argument was purely a fortiori, we would thereby imply that he did not know how to reason correctly in the a fortiori mode; nevertheless, if he did so reason incorrectly, the Sages’ dayo objection to his argument would in that event be equivalent to the principle of deduction, interdicting a ‘proportional’ conclusion from the given premises. Many commentators have so interpreted the debate, but in truth they did so without paying attention to R. Tarfon’s second argument, which could also be considered as purely a fortiori and yet be free of the Sages’ same objection. So this hypothesis is farfetched and unconvincing, and best brushed aside.

More probably, R. Tarfon put forward his first argument in pro rata or a crescendo form; and the Sages objected “dayo” to it in particular or in general, as already said. The purpose of this objection was to annul the premise of ‘proportionality’ inherent in R. Tarfon first argument. R. Tarfon, being an intelligent man, got the message and proposed instead a neat second argument, which was not subject to the same rebuttal, for the simple reason that whatever its form (pro rata, a crescendo or purely a fortiori) it yielded one and the same seemingly ‘proportional’ conclusion. Nevertheless, the Sages again objected “dayo” to it, in particular or in general, in exactly the same terms. By so doing, the Sages enlarged the meaning of their dayo objection, since it could here only refer to the generalization process preceding the deduction, since annulling the premise of ‘proportionality’ was useless.

As earlier explained, the principle of deduction is that the putative conclusion of any deductive argument whatsoever must in its entirety follow necessarily from (i.e. be logically implied by) the given premise(s), and therefore cannot contain any information not found explicitly or implicitly in the said premise(s). If a putative conclusion contains additional information and yet seems true, that information must be proved or corroborated from some other deductive or inductive source(s). This principle is true not only of valid a fortiori argument, but of all other valid forms of deductive argument, such as for instances syllogism or dilemma. Inference in accord with this principle is truly deductive. Inference not in accord with this principle may still be inductively valid, but is certainly not deductively valid.

It seems evident that when the Gemara says “a fortiori” (qal vachomer) it means a crescendo. Yet the Gemara does not clearly acknowledge the implications of such an assumption (at least not in the sugya under scrutiny). To be fully credible, the Gemara should have demonstrated its understanding that the arguments it characterized as a fortiori were not purely so, but involved an additional premise, one which establishes a pro rata relationship between the subsidiary and middle items. The issue is not merely verbal, note well, but depends on acknowledging a logical precondition for validity. Unfortunately, (to my knowledge) the Gemara nowhere explicitly acknowledges this crucial precondition. Nevertheless, we can generously suppose that the Gemara unconsciously or tacitly intends it, and move on. Our inquiry must now turn to the question: What is the required additional premise, in more concrete terms?

The tacit premise. It is a principle of justice (perhaps even the essence of it) that: on the positive side, the reward ought to fit the good deed and be commensurate with it; and on the negative side, the punishment ought to fit the wrongdoing and be commensurate with it. If these conditions are not fulfilled, justice has not been entirely served. This principle is in accord with our natural human ‘sense of justice’. It is an insight which cannot be proved, but which expresses (at least in part, if not wholly) what we commonly mean by ‘justice’. It is the basis of many laws legislated by mankind and guides many courts of law (namely, those that are characterized as ‘just’) in their deliberations and their rulings. For examples, a greater penalty is incurred by armed bank robbery than by shoplifting; or by premeditated murder than by murder in a moment of passion. In this negative guise, the principle of justice is known (in Latin) as the lex talionis, or law of retaliation.

Of course, the ‘sense of justice’ is not something literally ‘sensory’, but rather something ‘intuitive’, an insight of sorts. We know from within ourselves what is just and what is not. Of course, such knowledge is mere opinion that has to be confirmed over time using inductive techniques. We individually may see things differently at different times; and different people may see things differently. The sense of justice may be honed by use or blunted by disuse. It may be influenced by surrounding culture, whether incidentally or by deliberate propaganda. All the same, even though this faculty can be put to sleep or smothered, swayed or manipulated, each of us (as a being capable of personally suffering in a similar situation) does have an underlying sense of justice.

Of course, it is not always easy to intuit, much less demonstrate indubitably, what is ‘fitting’ and ‘commensurate’ reward or punishment. Justice is not an exact science. In Judaism, where this principle is known as midah keneged midah (meaning: measure for measure), the right measure is determined either by Divine fiat or by rabbinical decision; in the latter case the wisdom of the rabbis being assumed to be above average. I have not seriously researched the issue as to when this principle began to play an explicit role in rabbinical decision making, but I assume it was very early in view of its implicit presence in many stories and commandments of the Jewish Bible (Torah and Nakh).

The story of Miriam’s punishment for criticizing Moses, which the Gemara focuses on so insistently, is a case in point. In the Mishna debate, it is obvious that R. Tarfon’s two arguments are motivated by the measure for measure principle, even though not in so many words, but in the background, pre-verbally. Some commentators see the statement by God in Gen. 9:6, “Whoso sheddeth man’s blood, by man shall his blood be shed,” as the Biblical precursor of the measure for measure principle, even though it is more specific, in view of its symmetrical format (shed blood justifies blood shedding). The value and importance of justice in Judaism may be seen, for instance, in the Deut. 16:20 injunction: “Justice, justice shalt thou pursue.”

As regards stories, an illustration often appealed to, of God’s practice of ‘measure for measure’, is the correspondence between the crimes of the Egyptians against the Israelites and the punishments that later befell them; for example: they wanted to drown the babies (Ex. 1:22) – their army was drowned in the sea (Ex. 14:28). In Joshua 7:25, “Why hast thou troubled us? Hashem shall trouble thee this day,” a ‘tit for tat’ is clearly implied. The principle is well-nigh explicit in 2 Samuel 22:24-28; for instance in v. 26, David says: “With the merciful Thou dost show Thyself merciful, with the upright man Thou dost show Thyself upright.” Or compare Proverbs 1:11 and 1:18. Many examples of such reciprocity can also be found in the Talmud; see for instance Sotah 8b-11b. The concept is certainly older than the name attached to it.

I have not to date managed to find out when and where the exact Hebrew phrase “midah keneged midah” first appears. But I found a Mishna (Sotah 1:7) with very similar words: “By the measure that a man measures, so is he measured (במדה שאדם מודד בה מודדין לו, bemidah sheadam moded bah, modedin lo)”[198]. The meaning is admittedly not literally identical, since ‘measure for measure’ is understood to mean more broadly that the way a man behaves determines his recompense. However, if we understand “the measure that a man measures” as signifying the thoughts which determine his behavior, and “so is he measured” as referring to the Divine judgment in consequence of his actions, which determines his recompense, the two ideas may be pretty well equated.[199]

On the basis of this equity principle, it appears reasonable to us (for instance) that someone who has offended God deserves more punishment than someone who has merely offended a human being even if the latter be one’s own father. On this basis, then, it appears reasonable to us that, in the episode narrated in Num. 12:14-15, Miriam should indeed, as the Gemara suggests, theoretically deserve a penalty of (say) fourteen days isolation instead of just seven days. The fourteen is perhaps just an illustrative number, because surely offending God deserves more than double the punishment due for offending one’s father. Indeed, even the seven days penalty in the latter case is an arbitrary number – in this case, a Divine decree – so the fourteen days penalty is bound to be so too.[200]

Clearly, the Sages’ dayo principle is not a redundant restatement of the principle of deduction for a fortiori argument, as it might sometimes appear to be; nor does it have any other purely logical purpose. Rather, it serves an important additional, more moral purpose. We could imagine that the Gemara tacitly agrees that, in the Miriam example, the qal vachomer by itself (per se) can only logically yield the conclusion of seven days. But in the present case, even though this is not explicitly said anywhere, the qal vachomer is not ‘by itself’: it happens (per accidens) to be accompanied by an expectation of fourteen days based, not on formal grounds relating to purely a fortiori inference, but on the principle of justice that we have just now enunciated.

The dayo principle then comes to teach us: even in a case like this, where a greater penalty is expected due to implications of the principle of justice, the rabbinical conclusion (i.e. the law, the halakha) should not diverge from the quantity given in the Torah-based premises, whether such premises are used to draw a conclusion by mere analogy or by a fortiori argument or any other inductive or deductive means. The use of inference should not end up concealing and exceeding the penalty amounts mentioned in the premises given by Scripture. Such quantities should be understood as davka (as is), and not used for extrapolations however just those might seem based on human reasoning. The dayo principle is then, as the Gemara suggests, “Biblical,” if only in the sense that it advocates strict adherence to Biblical givens whenever penalties are to be inferred, whether by deduction or by induction.

The motive of the Sages seems obvious enough: the dayo principle is essentially a precautionary measure, enacted to avoid human errors of judgment in processes of inference in legal contexts. When a human court condemns an accused to some penalty, it is taking on a very serious responsibility. If that penalty is Divinely-ordained, i.e. explicitly written in the Torah, the responsibility of the human judges is limited to whether or not they correctly subsumed the case at hand to a given set of laws. Whereas, if the judges add something to the given penalty, on the basis of some ‘proportional’ reasoning, they are taking an additional risk of committing an injustice. So, it is best for them to stick to the Torah-given penalty.

It is interesting to note the comment by R. Obadiah Sforno (Italy, 1475-1550), regarding the principle of “an eye for eye” in Exodus 21:23-25, that “strict justice demanded the principle of measure for measure, but Jewish tradition mitigated it to [monetary] compensation to avoid the possibility of exceeding the exact measure.”[201] This suggests that the idea of compensation was instituted in that context to prevent eventual excess in the application of physical retribution – which, of course, would not be justice, but injustice[202]. We may refer to this idea to perhaps better understand and justify the dayo principle. In instituting this principle, the rabbis were not merely “tempering justice with mercy,” but also making sure that there would not be occasional occurrences of injustice, by mistake or due to excessive zeal. It was, at least in part, a precautionary measure.[203]

Viewed as a restraint on ‘proportional’ inference, the Sages’ dayo principle is not a principle of logic, but a merely hermeneutic principle inclining rabbinical judgment to mercy. It is not intended to regulate the qal vachomer inference as such, but rather to restrict a parallel application of the principle of justice – or perhaps more accurately put, a parallel intuition from our ‘sense of justice’. The Sages are telling us: although our human sense of justice produces in us an expectation that (to take the Gemara’s example) Miriam deserves (say) a fourteen days penalty, nevertheless God mercifully decreed (in the Torah) only seven days penalty for her. On the basis of this exemplary decree in the Biblical story of Miriam, Jewish legislators and law courts must henceforth always judge with the same restraint and limit the concluding penalty to the penalty given in the premise, even when the principle of justice would suggest a more severe punishment.

This is surely the real sense of the Sages’ dayo principle: they were not reiterating any law of logic, but setting a limitation on the principle of justice. And now, having perceived this, we can understand many things in this Talmudic sugya. We can understand why the Gemara would wish to establish that the dayo principle is Divinely-decreed. For it might seem unjust to restrict application of the principle of justice; it might be argued that the conclusion of a strict deduction is as reliable as its premises. Moreover, we can see how it is conceivable that, as the Gemara has it, R. Tarfon can differ from the Sages’ view and ignore the dayo principle in some situations. For no law of logic is being ignored or breached thereby, but only a moral principle; and a moral principle is logically more flexible, i.e. it may apply differently to different situations.[204]

Other angles. The dayo principle as above presented is designed to prevent the rabbis from ruling too severely. What of rulings that are too lenient, we might ask? Surely, ruling too leniently can conceivably be a problem. Justice is not served if criminals are not punished as they deserve (as indeed unfortunately often happens in practice in present day society). Too much leniency can be a bad thing for society, just as too much severity often is. So the dayo principle ought conceivably to forbid excessive mercy, as well as excessive justice.

If we think about it, measure for measure is essentially a principle of justice rather than one of mercy. By definition, mercy is intended to temper strict justice. It is not measure for measure, but beyond measurement. Justice is logical, while mercy is humane. Logically, the judgment should be so and so; mercy mitigates the conclusion. Mercy is surely desirable; but excessive mercy would obviously constitute injustice. Overdoing it would be negation of measure for measure! Thus, the right balance is needed. Arguing thus, we might easily advocate that the dayo principle is applicable to inferences that increase leniency, as well as to those that increase severity.

But my impression from rabbinic discourse generally is that the dayo principle is always intended as a principle of justice, and not occasionally as a principle of mercy. The rabbis are not so worried about irrational bursts of magnanimity; they are worried about inflicting undeserved punishment.

There is another objection that can be raised to our moral interpretation of the dayo principle. It seems reasonable enough in the present negative legal context, where the qal vachomer has as its conclusion a punishment for a wrongdoing. But what of equivalent positive legal contexts, where the qal vachomer has as its conclusion a reward for a good deed? Surely, the rabbis cannot here say that it is merciful to diminish the reward’s proportionality. Also, what of non-legal contexts, when the qal vachomer is constructed in pursuit of a factual conclusion – do the rabbis simply ignore the dayo principle in such cases? The question is, then: how general is the Sages’ dayo principle, or rather: what are the limits of its application?

The answers to these questions are, I think, broadly speaking, as follows. Jewish law, like most law systems, is essentially concerned with sanctions for wrongdoing rather than with rewarding good deeds. For this reason, only the negative side of the measure for measure principle is relevant to the rabbinical legislative process, and applications of the dayo principle occur only in relation to penalties. I doubt that any legalistic a fortiori argument with a conclusion of reward occurs in Jewish law; but if any indeed does, and the principle of measure for measure seems applicable, I very much doubt that the rabbis would block, on the basis of the dayo principle, the inference of increased or decreased rewards.

As regards a fortiori arguments in homiletic and other non-legal contexts, I do believe the dayo principle is indeed ignored in practice. It is admittedly sometimes apparently used – but such use is rhetorical. In other contexts, maintaining the a crescendo conclusion may be preferred. Since the principle has no binding legal impact either way, the decision to use or not-use it depends entirely on what the speaker wishes to communicate.

All the above comments circumscribing use of the dayo principle are of course mere personal impressions and educated guesses; they are open to discussion. They would have to be justified empirically, by thorough systematic research through the whole Talmud and indeed all Jewish law literature. Until such data is gathered by scholars, and fully analyzed by competent logicians, we cannot answer the said questions with much greater precision and certainty than just done. Nevertheless, by asking questions and proposing answers, we have at least raised issues and sketched possible results. It would, of course, be interesting and valuable to find rabbinical statements that clearly justify what has been said.

3.    The dayo principle in formal terms

We shall here review our new interpretation of the dayo principle in more formal terms. This is done with reference to Mishna Baba Qama 2:5, where the principle is traditionally given pride of place, first dealing with the Sages’ objection to R. Tarfon’s first argument, and then with their objection to his second argument. As already seen, these are two distinct expressions of the dayo principle, although they have a common motive. The corresponding Gemara in Baba Qama 25a, as we saw, only seems to have noticed the first version of the dayo principle; but later commentators (notably, it seems, Rashi and Tosafot) did notice the second[205]. We shall show here more precisely why the Gemara’s view is inadequate.

A further reason why we wish to now investigate the dayo principle in more formal terms is because both formulations in the Mishna relate specifically to the positive subjectal form of a crescendo argument. Nothing is there said of eventual applications to the negative subjectal form, or to the positive or negative predicatal forms. Our purpose here is to consider theoretically what such other applications would look like. Whether such other applications actually occur or not in the Talmud (or other rabbinic literature) is not the main issue, here; but it is abstractly conceivable that they might occur. In any case, we are sure to clarify our concept of the dayo principle by this enlarged research.

Let us to begin with deal with the Sages’ dayo objection to the first argument of R. Tarfon. Here, R. Tarfon tried to infer a liability of full payment for damage by horn on private property (conclusion), from a liability of half payment for damage by horn on public property (minor premise). He was thus presumably using a crescendo argument, of positive subjectal form, as follows:

Action P is a more serious breach of a certain law (R) than another action Q is.
Action Q is a breach of that law (R) enough to merit a certain penalty (S).
The magnitude of penalty S is ‘proportional’ to the seriousness of the breach of law R.
Therefore, action P is a breach of that law (R) enough to merit a greater penalty (S+).

The Sages’ dayo objection to this attempt can be stated as: if the minor premise predicates a certain penalty (S) for a certain action (Q), then the conclusion cannot predicate a greater penalty (S+) for a more illegal action (P). This objection can be perceived as neutralizing the additional premise concerning ‘proportionality’. The Sages are saying: although by commonsense such ‘proportionality’ seems just, by Jewish law it is not to be applied, and we can only predicate the same penalty (S) in the conclusion as was previously given (in the minor premise).

What the dayo objection does here is to block, or switch off, as it were, the operation of the additional premise regarding ‘proportionality’: though that moral premise might usually be granted credibility, it is rendered inoperative in the present context, to avoid any possible excess of penalization (as earlier explained). This means that the a crescendo argument is effectively abolished and replaced with a purely a fortiori argument. Evidently, then, the Gemara’s view, according to which the a crescendo argument is allowed to proceed, and then the dayo principle reverses its action[206], is technically incorrect. The action of dayo is preventive, rather than curative; it takes place before the ‘proportional’ conclusion is drawn, and not after.

We can easily, by formal analogy, extend this principle to other forms of a crescendo argument, if only out of theoretical curiosity. The analogous positive predicatal argument would have the following form:

A more serious breach of a certain law (R) is required to merit penalty P than to merit another penalty Q.
Action S is a breach of that law (R) enough to merit penalty P.
The seriousness of the breach of law R is ‘proportional’ to the magnitude of action S.
Therefore, a lesser action (S–) is a breach of that law (R) enough to merit penalty Q.

Notice that the additional premise about ‘proportionality’ is different in subjectal and predicatal arguments. The order is reversed. In the former, the subsidiary term S, being a predicate, is proportional to the middle term R; whereas in the latter, it is the middle term R that is proportional to the subsidiary term S, which is a subject. This is due to the order of things in the minor premise, which the conclusion naturally reflects, where predication is made possible only if the value of R for the subject matches or exceeds the minimum value of R necessary for the predicate.

In this context, the Sages’ dayo objection would be stated as: if the minor premise predicates a certain penalty (P) for a certain action (S), then the conclusion cannot predicate a lesser penalty (Q) for a less illegal action (S–). This objection can be perceived as a denial of the additional premise concerning ‘proportionality’. Here, the Sages might say: although by commonsense such ‘proportionality’ seems just, by Jewish law it is not to be applied, and we can only address the same action S in the conclusion as was given (in the minor premise). This statement, to repeat, is formulated by analogy, merely for theoretical purposes; it is not given in the original Mishna debate.

There is admittedly a difficulty in the latter extension of the dayo principle. For whereas applying dayo to a positive subjectal argument results in preventing potentially excessive justice, by mechanically attributing a greater penalty to a more serious breach of law, the application of dayo to a positive predicatal argument results in the prevention of increasing leniency, which is what attributing a lesser penalty to a less serious breach of law would constitute. We shall return to this issue further on.

As regards the corresponding negative arguments, they can easily be determined using the method of ad absurdum. In each case, the major premise and the additional premise about ‘proportionality’ remain the same, while the negation of the conclusion becomes the new minor premise and the negation of the minor premise becomes the new conclusion. Application of the (first) dayo principle to them would have the effect of inhibiting the deduction of the putative negative a crescendo conclusion from the given negative minor premise, through rejection of the additional premise.

As for implicational arguments, they can be dealt with in comparable ways.

Let us now deal with the Sages’ dayo objection to the second argument of R. Tarfon. Here, R. Tarfon tried to infer a liability of full payment for damage by horn on private property (conclusion), from a liability of full payment for damage by tooth & foot on private property (minor premise). He was thus using an argument, again of positive subjectal form, that yields the same conclusion whether construed as a crescendo argument or as purely a fortiori. This means that the first version of the Sages’ dayo principle would be useless in this second case, for the minor premise and conclusion naturally have the exact same predicate (full payment). Therefore, since the Sages nevertheless declared dayo applicable, they must have been referring to some other feature of the argument.

The only other logical operation they could have been referring to is the inductive formation of the major premise, by generalization from the liability of half payment for damage by horn on public property and the liability of no payment for damage by tooth & foot on public property. That is, the major premise that ‘liability for damage by horn is generally greater than liability for damage by tooth & foot’ was derived from the same given concerning horn as before, namely that ‘liability for damage by horn on public property is half payment’. Here, then, the dayo principle must be stated in such a way as to interdict this preliminary generalization.

The Sages apparently hint at this solution to the problem by restating their second objection in exactly the same terms as the first. There is no other explanation for their using the exact same words. In this context, then, the Sages’ dayo objection would be stated as: if the major premise is inductively based on information about a certain action (P) meriting a certain penalty (S), in one set of circumstances, then the conclusion drawn from it cannot be that the same action (P) in another set of circumstances merits a greater penalty (S+). That is, under the dayo principle, we can only conclude that ‘P is S’, not that ‘P is S+’. Note well how this second version of the dayo principle is very different from the previous.

It is important to realize that, unlike the preceding one, this dayo objection cannot be perceived as neutralizing the additional premise concerning ‘proportionality’. For here, a crescendo and purely a fortiori argument have the exact same conclusion; so that whether or not we ‘switch off’ this third premise makes no difference whatever to the result. This means that, in the present case, the argument is necessarily purely a fortiori, i.e. devoid of an additional premise. No a crescendo argument can usefully be proposed here, since the conclusion is already maximal through purely a fortiori argument. Therefore, in such case, we must prevent the unwanted conclusion further upstream in the reasoning process; that is, at the stage where the major premise is getting formed by means of a generalization.

We can easily, by formal analogy, formulate a similar principle with regard to positive predicatal argument. In this context, the Sages’ dayo objection would be stated as: if the major premise is inductively based on information about a certain action (S) meriting a certain penalty (Q), in certain circumstances, then the conclusion drawn from it cannot be that a lesser action (S–) in whatever other circumstances merits the same penalty (Q). That is, under the dayo principle, we can only conclude that ‘S is Q’, not that ‘S– is Q’. This statement, to repeat, is formulated by analogy, merely for theoretical purposes; it is not given in the original Mishna debate.

Admittedly, our formal extension of the second dayo principle from positive subjectal argument to positive predicatal argument is open to debate. For whereas in the former case dayo serves to prevent increased severity, in the latter case it seems to have the opposite effect of preventing increased leniency. This issue will have to be addressed, further on.

Returning now to the Gemara, we can see from the above formal treatment, that it was wrong in considering the dayo principle as concerned essentially with a crescendo argument. In the first case, which the Gemara did try to analyze, the Sages’ dayo objection effectively advocated a purely a fortiori argument instead of R. Tarfon’s apparent attempt at a crescendo argument. But in the second case, which was unfortunately ignored by the Gemara, the Sages’ dayo objection couldn’t function in a like manner, by blocking the usual velleity of ‘proportionality’, since this would be without effect on the conclusion. It had to apply to a presupposition of R. Tarfon’s argument, however construed – namely the generalization earlier used to construct its major premise.[207]

Let us now return to the issue glimpsed above, as to whether or not the dayo principle is only meaningful in relation to positive subjectal a crescendo argument, which proceeds from a lesser penalty for a lesser infraction to a greater penalty for a greater infraction. We have seen that we can formally enlarge the idea of preventing proportionality implied in dayo application to positive subjectal argument, to negative subjectal, and to positive and negative predicatal arguments – but is such analogy meaningful when more concretely examined? We shall here try to answer this question.

Remember our earlier determination that the dayo principle is not a logical principle, but a “moral” one, i.e. it has to do with ethics or law in the context of the Jewish religion. It is not logically necessitated by the principle of deduction or by the use of a fortiori argument or any other purely logical consideration; no contradiction would arise if we simply ignored it. It is, rather, something Divinely or rabbinically prescribed, to lawmakers and courts of law, for cases where a qal vachomer is being attempted in order to infer a greater penalty for some wrongdoing. It is an artificial injection into the Jewish legislative process apparently motivated by mercy, i.e. to temper justice. There is no reason to apply it in contexts other than the sort just specified, or for that matter in other religions or outside religion.

We could eventually expect the same idea to be extended from penalties to duties. Such conceptual extrapolation might well be found exemplified in the Talmud or other Jewish literature (I have not looked for examples). That is conceivable if we think of penalties and duties as having in common the character of burdens on the individual or community subjected to them. If we look on increased duties (mitzvoth) as positive rewards, in the way that a servant might rejoice at receiving increased responsibilities, the analogy of course fails. But if we look on duties as burdens, an analogy is possible. It that case, the dayo principle could be taken to mean more broadly that burdens in general must not be increased on the basis of a qal vachomer argument from the Torah.

Granting the above clarifications of the dayo principle, the first question to ask is: is its function limited to contexts of positive subjectal qal vachomer – or can this definition be extended to other a fortiori argument formats? The format focused on by the rabbis is, to repeat, positive subjectal, which means that it is minor to major (miqal lechomer), whence the appropriateness of the name qal vachomer. Let us now consider what dayo application to the negative subjectal format would mean. Such argument is, of course, major to minor (michomer leqal) in orientation. It would look as follows:

Action P is a more serious breach of a certain law (R) than another action Q is.
Action P is a breach of that law (R) not enough to merit a certain penalty (S).
The magnitude of penalty S is ‘proportional’ to the seriousness of the breach of law R.
Whence, action Q is a breach of that law (R) not enough to merit a lesser penalty (S–).

The major premise and the additional premise about ‘proportionality’, which (as we saw earlier) is in practice derived from the principle of midah keneged midah (measure for measure), both remain the same, here. What changes is that the minor premise and conclusion are now negative propositions and the major term (P) appears in the former and the minor term (Q) appears in the latter. It remains true that the value of S associated with P is greater than that associated with Q; however, note that here the greater value appears in the minor premise and the lesser in the conclusion.

Our question is: what would be the significance of the dayo principle, in either of its senses, in such negative subjectal context? Note that above argument is formally valid. The question is thus not whether its conclusion follows from its said premises. The question is whether to reject its additional premise (first type of dayo application) or its major premise (second type of dayo application).

At first sight the answer is that the dayo principle would not be called for – because there is no velleity in such a context to use the principle of measure for measure, and dayo is intended as a restraint on such velleities. Since the minor premise and conclusion are negative, we can say that no actual penalty, small or large, is claimed in either of these propositions; in that case, we are not naturally inclined to engage in measure-for-measure reasoning, and therefore no dayo principle is needed to block such reasoning. It would appear, then, that the dayo principle is not useable in such negative context.

However, we could also look upon such negative argument as tacitly positive. Assuming that all law-breaking merits some penalty, we could argue that where an illegal action is not sufficiently illegal to merit a certain penalty we may infer it to positively merit a lesser penalty, though we cannot predict how much less. In that case, the negative subjectal argument would be interpreted as saying that P is illegal enough to positively merit a penalty of magnitude ‘somewhat less than S’, and therefore Q is illegal enough to positively merit a penalty of magnitude even smaller than ‘somewhat less than S’. This thought clearly involves measure-for-measure reasoning; so the dayo principle ought to now be applicable.

But of course it is not in fact applicable, because this new argument infers a decrease in penalty, whereas the dayo principle is essentially aimed at preventing inferences of increase in penalty. It is intended as a principle of mercy, pushing towards leniency rather severity of judgment; therefore, its application here would be inappropriate. In other words, we would not normally try to interdict the conclusion of a negative subjectal argument (even one recast in more positive form), whether by denial of the additional premise or of the major premise, for the simple reason that such reaction would not be in accord with the spirit and intent of the dayo principle.

We can argue in much the same way with respect to positive predicatal a crescendo argument:

A more serious breach of a certain law (R) is required to merit penalty P than to merit another penalty Q.
Action S is a breach of that law (R) enough to merit penalty P.
The seriousness of the breach of law R is ‘proportional’ to the magnitude of action S.
Therefore, a lesser action (S–) is a breach of that law (R) enough to merit penalty Q.

Here again, we have reasoning from major to minor – specifically, from a more illegal action (S) with a greater penalty (P) to a less illegal action (S–) with a smaller penalty (Q) – so, there would be no sense in applying (in either way) the dayo principle to it. Such an argument would, if our analysis of the moral motives of this principle has been correct, be allowed to proceed unhindered.

However, things get more complicated when we turn to negative predicatal argument, since the orientation is again from minor to major, while the minor premise and conclusion are negative in polarity:

A more serious breach of a certain law (R) is required to merit penalty P than to merit another penalty Q.
Action S is a breach of that law (R) not enough to merit penalty Q.
The seriousness of the breach of law R is ‘proportional’ to the magnitude of action S.
Therefore, a greater action (S+) is a breach of that law (R) not enough to merit penalty P.

In view of the negative polarities involved, we are tempted to say that there is no call for the dayo principle since no actual penalties are claimed. However, if we recast the argument in more positive form, following the idea that all law-breaking merits some penalty, we could say that the minor premise concerns some positive penalty of magnitude ‘somewhat less than Q’ (for action S) and likewise the conclusion concerns some positive penalty of magnitude ‘somewhat less than P’ (for action S+). Assuming that ‘somewhat less than P’ is greater than ‘somewhat less than Q’, which seems reasonable granting the additional premise, we can say that this argument is indeed from minor to major in a positive sense. In that case, the dayo principle ought to be applied to it, to prevent justification of the increased penalty advocated by the conclusion. Thus, either the additional premise about ‘proportionality’ or the generalization leading to the major premise will be interdicted.

Thus, to sum up, whereas when we think in bare formalities the four forms of a crescendo argument might seem liable to dayo principle interference, upon reflection it is only the positive subjectal and negative predicatal forms which are concerned, because they go from minor to major. The other two forms, the negative subjectal and the positive predicatal, are not concerned, because they go from major to minor. So the issue is not so much the polarity of the argument as its orientation. All the above can be repeated regarding implicational arguments, of course.

What we have said here, of course, refers to arguments that predicate penalties[208]. Arguments that predicate rewards are not to be treated in an analogous manner, because (as we have seen earlier) the dayo principle is only aimed at preventing increased punishment, not increased reward. But, one might ask, what of decreased rewards? Is not a decrease in reward comparable to an increase in punishment? The answer to that I would suggest is again practical rather than formal: Jewish law is not concerned with rewarding good deeds, but in penalizing bad ones. Furthermore, it does not address all bad deeds, but only some of them – namely, those subject to judgment by rabbinical courts. The purpose of Jewish law, as indeed most law systems, is to ensure at least social peace; it is not to control everything. Accordingly, the dayo principle is not intended to deal with changes in magnitude relating to rewards. It will simply not be invoked in such contexts; and indeed, such contexts are not expected to arise.

This is all assuming, of course, that my understanding of the matter is correct. It is not unthinkable that the empirical truth is a bit different from what I have assumed; and for instance, there are in fact occasional applications of the dayo principle in situations where I have just said it is logically inapplicable. In that event, needless to say, the above account would have to be modified in accord with actual facts. This should not be too difficult, since the formal issues are already transparent. It is not unthinkable that over time the original intent of the Sages’ dayo (given in Mishna Baba Qama 2:5) has been misunderstood, forgotten or intentionally ignored, and the concept of dayo was eventually used more broadly. This is in fact suggested by the broad or vague way that the dayo principle is usually presented in rabbinical literature.

Judging by the study of Mishnaic qal vachomer presented in Appendix 2, we cannot resolve the empirical issue with reference to the Mishna. For, surprisingly, of the 46 arguments found there, only the famous two in Mishna Baba Qama 2:5 involve the dayo principle! This is an important finding. There are nine other arguments which are possibly a crescendo, and therefore could be subject to dayo; but there is no mention of dayo in relation to them – either because they are not really a crescendo or because they do not serve to infer a penalty from the Torah.

Therefore, we must look to the Gemara (and indeed, later rabbinic literature), to find out whether the dayo principle is consistently applied in practice as here postulated. Only after all a fortiori arguments in the whole rabbinic corpus have been identified and properly analyzed will this question be scientifically answered. In Appendix 3, I try to at least partly answer the question, using the Rodkinson English edition of the Talmud. My finding in this pilot study is that there are only six Talmudic contexts where the dayo principle is explicitly appealed to! In five of these cases, the dayo principle may be said to be used as I have predicted, i.e. to prevent increase in legal responsibility through a fortiori argument. In the remaining case, this is partly true (see fuller explanation there).

Considering the prime position given to qal vachomer in the rabbinic lists of middot (hermeneutic principles), and the great attention accorded by rabbinical commentators to the Mishna Baba Qama 2:5 which introduces the dayo principle, one would expect the Tannaim (the rabbis of the Mishnaic period) to resort to dayo objections quite often. That this is statistically not the case is, to repeat, quite surprising. It may well be that more instances of dayo use by Tannaim will be found in some baraitot (statements attributed by Tannaim not included in the Mishna), many (maybe most) of which are quoted by Amoraim (the rabbis of the Gemara period) in different passages of the Talmud. This matter deserves systematic research, if we want to get a realistic idea of the quantity of dayo use by the Tannaim[209].

Besides that, we of course need to further research independent dayo use by the later rabbis, i.e. the Amoraim and their successors, respectively. Its use also in the early and late Midrashic literature deserves close study too. As regards the Amoraim, it is also quite surprising how little they appeal to the principle, at least explicitly, at least in the Rodkinson edition. However, my expectation is that, though some more use of the dayo principle by the Tannaim and the Amoraim may well be found, it will not be significantly much more.

I would like now to deal with a couple of further details, before closing this topic.

To begin with, let us reflect on the fact that rabbinical formulations (apparently of more recent vintage historically) usually describe a fortiori argument as an instrument of legal reasoning that can proceed in both directions, i.e. both from minor to major and from major to minor. For instance, consider the following formulation by R. Feigenbaum:

“Any stringent ruling with regard to the lenient issue must be true of the stringent issue as well; [and] any lenient ruling regarding the stringent issue must be true with regard to the lenient matter as well.”[210]

According to this statement, given that a stringent ruling (S) applies to the lenient issue (Q), it must also apply to the stringent issue (P); and given that a lenient ruling (S) applies to the stringent issue (P), it must also apply to the lenient issue (Q). The first part of that statement matches positive subjectal a fortiori (minor to major). The second part of it presumably refers to the negative subjectal form, since it is major to minor (and obviously not predicatal). Indeed, that is how I interpreted it in my Judaic Logic[211]. My thinking there was that: Given that there has been some breach of law (R), then some penalty is deserved; in that event, “not-deserving a stringent penalty” implies “deserving a lenient penalty”! The terms stringent and lenient being understood as relative to each other, not as absolute.

Thus, a formulation such as R. Feigenbaum’s tacitly assumes that “all law-breaking merits some penalty.” It is only on this basis that we can indeed logically transfer a lenient ruling from a stringent issue to a lenient matter, as he and others postulate. Although his above formula is stated entirely in positive terms, it in fact refers to both positive and negative arguments. Note in passing that the dayo principle is not mentioned in that writer’s formula. That is because he is here thinking in purely a fortiori terms, and not a crescendo like the Gemara. He is not saying that the inferred ruling is to be more stringent or more lenient, but only as much so. The same stringency or leniency is passed on.

Not having R. Feigenbaum’s book in my possession any longer, I do not know what, if anything, he said in it about the dayo principle. I doubt offhand that he distinguished between purely a fortiori and a crescendo argument, and that he related that principle exclusively to the latter form and limited dayo use to increased stringencies. But, using at the language of his above statement, I would say it ought to be amplified as follows. In cases where purely a fortiori inference is appropriate, the same degree of stringency or leniency is concluded, and the dayo principle is irrelevant. But in cases where a crescendo inference is appropriate, the natural conclusion would be more stringency or more leniency. In such cases, if the conclusion is a more stringent penalty than the one proposed in the Torah, dayo should be applied; whereas if it is more lenient it need not be.

Another point I would like to clarify is the idea emitted above that in predicatal a crescendo argument the subsidiary term (the subject of the minor premise and conclusion) is decreased (in the positive mood) or increased (in the negative mood). What does it mean to say, as we did, that an action is lesser or greater? This is best clarified by giving an example. We might, for instance, conceive two kinds of killing: intentional killing and unintentional killing, and argue thus: More badness (middle term, R) is required to merit a more severe penalty (major term, P) than to merit a less severe penalty (minor term, Q); so if, under the law relating to killing, intentional killing (S1) is bad enough to merit a more severe penalty, then unintentional killing (S2) is bad enough to merit a less severe penalty. This is a positive predicatal a crescendo argument.

Formal application of the dayo principle to this reasoning would mean that it is forbidden to here follow the principle of measure for measure and infer a lesser penalty for the less serious crime. Intuitively, such interdiction is obviously contrary to reason: we would rather let the ‘proportional’ conclusion stand since it is more indulgent. Neither justice nor mercy would be well served by applying the dayo principle to such cases. To punish a less serious crime the same way as a more serious one would be contrary to both justice and mercy. To punish a less serious crime less severely than a more serious one is in accord with both our sense of justice and our sense of mercy.

Clearly, then, the dayo principle should remain inoperative in cases of positive predicatal a crescendo argument concerning retribution for crime. Similar reasoning, as we have seen, applies to negative subjectal a crescendo argument. It is only with regard to positive subjectal or negative predicatal a crescendo arguments that the dayo principle makes sense and has relevance, for only in their case may there be an over-enthusiastic upsurge of justice, so that mercy requires a more cautious and temperate approach. In other words, dayo is potentially relevant only to a crescendo arguments that go from minor to major; it plays no role in such arguments that go from major to minor. Dayo is also, of course, irrelevant to purely a fortiori arguments (whether a minori or a majori), since the subsidiary term (whether it is a subject or a predicate) remains unchanged in them.[212]

This is spoken entirely from a theoretical perspective. It does not mean that the rabbis have all always been as conscious as that of the various possibilities. But I suspect they at least subconsciously have indeed reasoned in this way and limited dayo in the ways above described. Exceptions might conceivably be found in the mass of Talmudic and other rabbinic literature. This is an empirical question that must be answered empirically. If examples of upside down application of dayo are found, they would need to be rationalized somehow ad hoc – or, alternatively, they could be viewed as occasional errors of reasoning.

To conclude our formal exposition, we can say that the dayo principle is much leaner than what we may have originally imagined. It is not a formal law of a fortiori logic, but a very specific religiously-inspired rule for Jewish legislators and judges. Moreover, it is not a rule to be applied indiscriminately, but specifically with regard to attempts at increasing penalties on the basis of proportional qal vachomer reasoning. I should add: since a crescendo argument as such, i.e. as distinct from the dayo principle used to freeze its conclusions as just explained, is purely logical – it is inaccurate to call qal vachomer a hermeneutic rule! The first hermeneutic rule in Hillel’s list or in R. Ishmael’s list is, strictly speaking, not the qal vachomer argument, but the dayo principle applied in the context of such argument. We may nevertheless maintain the use of “qal vachomer” as the title of the first rule on the basis that the dayo principle is called for solely in that specific context, because it is only in such context that a quantitative increase (in penalty) might be inferred.

One might unthinkingly assume that the dayo principle might equally well be used in conjunction with other forms of analogical reasoning (e.g. gezerah shavah or binyan av). Indeed, one might argue that if dayo is applicable in such a maximally deductive context as qal vachomer, then it should all the more be applicable in more inductive contexts like gezerah shavah or binyan av. But further reflection should convince that what distinguishes qal vachomer is that it deals with quantities and the dayo principle is a restriction of increase in quantity (of the subsidiary term, to be exact) when inferring a penalty from the Torah. Since gezerah shavah, binyan av and other hermeneutic principles do not prescribe quantitative changes, the dayo principle does not concern them.

It remains conceivable, however, that yet other forms of reasoning could result in quantitative changes that would call for application of dayo. Come to think of it, it does seem like the rabbis “temper justice with mercy” even in situations that do not involve qal vachomer or any other hermeneutic principle. But of course such judgments might not be characterized as based on the dayo principle, since they are made more directly. What I am referring to here is the rabbinical interpretation of the lex talionis (the law of retaliation) found in Exodus 21:23–25 and Leviticus 24:19–21 – the famous “an eye for an eye, a tooth for a tooth” principle. The rabbis do not read this Torah law literally, but as a call for monetary compensation in cases of injury; this is shown using various arguments, including a qal vachomer.[213]

4.    The human element

Looking at rabbinical practices and principles, we can safely say that the rabbis were very careful to acknowledge the human element in reasoning a fortiori, or by means of any other of the listed hermeneutic principles (and by extension, even unlisted thought processes).

This is evident, first of all, in their practice of teshuvah (Heb.) or pirka (Aram.) – usually rendered in English as ‘objection’ or ‘challenge’ – consisting in retorting to or rebutting an argument, and in particular an a fortiori argument, by showing or at least pointing out that one (or more) of its premises is (wholly or partly) open to doubt or false, or that the putative conclusion cannot in fact be drawn from the given premises. This demonstrated their awareness, if only pre-verbally in some instances, of the inductive sources of many of the propositions used in their reasoning. In some cases, as well, such practice on their part demonstrated awareness of the relative artificiality of certain forms of argumentation they used and thence the tenuousness of their conclusions.

Such awareness of the human element in apparently deductive inference is also made evident in their setting a number of explicit restrictions on the use of a fortiori argument. Such argument could only be used for inferring laws by qualified rabbis involved with their peers in the development of Jewish law (meaning in principle members of the Sanhedrin, though in practice some participants were probably not officially members). Inferences made had to be accepted unanimously or by ruling of a majority. Inferences could be made only from written Torah laws, and not from oral Torah traditions, even if they were reputed to go all the way back to Moses, and all the more so if they were considered to be of more recent vintage. One could not infer a new ruling from a previously inferred ruling, i.e. use the conclusion of one a fortiori argument as a premise in the next.

I would additionally suggest, an a fortiori inference from a Torah law would be considered questionable if it was found to conflict with another Torah law. This seems reasonable on the general understanding that written Torah law carries more weight in Judaism than any human inference. An example is apparently given by Louis Jacobs in his The Jewish Religion: A Companion[214] with reference to a responsum of the Radbaz (Spain-Israel, R. David ben Zimra, 1479-1573) to the question why the Torah does not forbid a man’s marriage to his own grandmother, and yet forbids him his wife’s grandmother (who is a more remote relative), although we would expect by a fortiori argument from the prohibition in the latter case that the former case would also be prohibited. Jacobs explains: “Typical of Radbaz’s attitude to the limited role of human reasoning in Judaism is his reply that the a fortiori argument is based on human reasoning, whereas the forbidden degrees of marriage are a divine decree, so that human reasoning is inoperative there. All we can say is that God has so ordained. One degree of relationship is forbidden, the other permitted.”

The a fortiori argument here is: a man’s own grandmother (P) is more closely related (R) to him than his wife’s grandmother (Q); if his wife’s grandmother (Q) is closely related (R) enough to be forbidden in marriage to him (S), then a man’s own grandmother (P) is closely related (R) enough to be forbidden in marriage to him (S). The difficulty is that, although the former is forbidden, the latter is not forbidden. However, I do not see why the rabbis do not accept this a fortiori argument, as they do many others, and simply prohibit marriage to one’s own grandma, since there is no written permission to contend with. The answer given by the Radbaz, and before him by Menahem Meiri (France, 1249-1316), is that there is no need for the inferred prohibition as no one would be likely to do such a thing anyway in view of age differences. That is, more precisely put, while a man might be attracted to his wife’s grandmother (e.g. if his wife is thirteen years old, and her mother twenty-six and her grandmother thirty-nine, and he is forty), he is unlikely to be attracted to his own grandmother (who would be in her mid-sixties at least). But this argument may seem a bit weak, as some men are attracted by much older women, even if rarely.[215]

Another restriction was that a ruling based on a fortiori argument could not take precedence over a Torah law from which it was inferred, if the two happened to come into conflict. For example, it is inferable from the Torah law (Ex. 23:4) that one should return one’s enemy’s lost ox or ass that one should likewise, a fortiori, return one’s friend’s lost ox or ass. One might think that, having thus made a deductive inference, it would follow that when simultaneously encountering two lost animals, one from each of these people, one could legally prefer to return that belonging to one’s friend rather than (or at least before) returning that belonging to one’s enemy. But no: the premise remains more binding than the conclusion, and one must therefore give precedence to the enemy’s animal[216]. Yet another important restriction was that a rabbinical law court could not sentence someone to corporeal punishment on the basis of a legal ruling derived by a fortiori argument. Meaning that, however reliable the justifying deduction might well have been, there was still a drop of doubt in it sufficient to preclude such drastic penalties.

Some of these restrictions were perhaps more theoretical than practical, because if we look at Talmudic discussions (Mishna, Gemara and later commentaries and super-commentaries all included) one is struck by the ease and frequency with which the rabbis engaged in a fortiori argument if only rhetorically. One would have to examine all rabbinic literature in great detail to determine whether these theoretical restrictions have all in fact been consistently adhered to in practice (this is certainly a worthwhile research project for someone). Nevertheless, on the whole, these restrictions show the rabbis’ acute awareness of the natural limits of the human powers of experience and reason.[217]

The dayo principle as I have above described it falls right into this pattern of restricting excessive reliance on logical means. A ruling based on qal vachomer argumentation remains somewhat doubtful, even though the conclusion (if correct) follows the premises with absolute certainty, because there is inevitably some human element in the induction of the premises. These premises may be in part or even largely Torah-based, but still some part(s) of them were inevitably based on human insight or convention, so it is wise to remain a bit open-minded concerning their conclusion[218]. But this is nothing to do with the dayo principle, as we have latterly discovered. This principle is not designed to throw doubt on qal vachomer argumentation as such, but to prevent extrapolation from Torah-based premises by means of the principle of justice.

A question we could ask is: why is the dayo (sufficiency) principle not directly and always applied to the midah keneged midah (measure for measure) principle? In my above treatment of these principles, I have identified the latter as inserting an additional premise of ‘proportionality’ between the minor premise and conclusion, and the former as either blocking the operation of this additional premise or preventing the formation of the major premise through generalization. Thus, we may view the measure for measure principle as tending to turn a purely a fortiori conclusion into an a crescendo one, and the sufficiency principle as on the contrary tending to restrain (in one way or another) such proportionality. The two balance each other out, and the result is that the purely a fortiori conclusion stands unchanged.

The question is: could we not say, more generally: whenever we encounter a midah keneged midah, we must apply dayo? Why does the qal vachomer need to be mentioned at all? Obviously, if such a general rule was promulgated, the two said principles would effectively cancel each other out and cease to exist! Obviously, too, this is not the intent of the dayo principle; i.e. it is not meant to altogether neutralize the midah keneged midah principle. So it is reasonable to suppose the dayo principle to be intended for a specific context; namely, for when a qal vachomer is formulated and we are tempted to extrapolate its conclusion by a thought of measure for measure. And more specifically still, for when the speaker (like R. Tarfon in mBQ 2:5) attempts to infer a larger penalty from a lesser penalty prescribed in the Torah.

If there were no qal vachomer, or other deductive inference, the measure for measure principle might conceivably have been applied without restriction. Why then, we might well ask, was the dayo principle needed in the context of qal vachomer? Perhaps the answer to that important question is that if the measure for measure extrapolation occurs in a non-deductive context, we naturally remain aware of the human element in it and maintain a healthy measure of skepticism. Whereas in a deductive context, especially where the powerful logic of qal vachomer is used, since we have already proved part of the quantity, we are more likely to view its measure for measure extrapolation as also ‘proved’. The dayo principle comes to remind us that the proposed extrapolation does not have the same degree of reliability as the more limited conclusion of the qal vachomer has. Indeed, the dayo principle precludes any temptation to extrapolate rather than let us run the risk wrongful extrapolation.

This may conceivably have been the justification of the dayo principle in the rabbis’ minds. Even if they did not fully realize that it concerned a thought of midah keneged midah accompanying a qal vachomer, rather than the latter argument per se, they would have sensed the danger of unbridled extrapolation. And according to the Gemara, as we have seen, the preemptive measure against such extrapolation (viz. the dayo principle) was not a mere rabbinical ruling (by the Sages), but a Divine decree (through Num. 12:14-15). It perhaps had to be a Torah-based hermeneutic rule, so that it could not in turn be open to doubt as a human construct. Even so, as we have seen, R. Tarfon and others did (according to the Gemara) claim the dayo principle could in some situations be bypassed or even ignored. But, for the most part, the Sages’ posture has prevailed.

It is worth noting lastly that, according to later authorities (at least some of them), qal vachomer argument (or more precisely the dayo principle associated with it) could only be used in the Talmudic law making process. After the closure of this process, it was considered illegal to use this hermeneutic principle, or any other of the thirteen rules of R. Ishmael for that matter, to interpret the written Torah for legislative purposes. The references for this sweeping ruling are given by R. Bergman[219] as: “Maharik Shoresh 139; Ra’ah to Ketubos cited in Yad Malachi 144.” This limitation in time is additional evidence that Judaism does not view the dayo principle as a law of logic but as a revealed ad hoc religious law. Laws of logic cannot be abrogated; decrees can. Similarly for the other hermeneutic principles.

Why this limitation in time? Because, I presume, the hermeneutic rules were a prerogative of the Sanhedrin, the Jewish Supreme Court; when its deliberations were interrupted due to foreign conquest and rule, rabbis were no longer empowered to use these interpretative principles. An implication of this explanation is that if – or when – the Sanhedrin is reinstituted (presumably by the Messiah) the dayo principle and other such guidelines will again be useable by its members. This is a neat answer to the question, except that most of the Babylonian Talmud’s deliberations took place in Babylon, far from the traditional seat of the Sanhedrin in the Land of Israel. Presumably, the Babylonian rabbis involved were considered to be worthy successors to the Sanhedrin. The reason for the time limitation would then simply be that the Talmud was ‘closed’ in about 500 CE (say), and subsequent rabbis were considered as at a lower spiritual level than their teachers.

5.    Qal vachomer without dayo

It should be pointed out that Talmudic use of qal vachomer does not always require application of the dayo principle, for the simple reason that the conclusion sometimes naturally lacks the required quantitative aspect, i.e. there is no propensity to ‘proportionality’ that needs to be interdicted. In other words, the argument is purely a fortiori rather than a crescendo. Consider the following argument:

“All these things they [the rabbis] prescribed [as culpable] on a Festival, how much more [are they culpable] on Sabbath. The Festival differs from the Sabbath only in respect of the preparation of food.” (Mishna Beitzah, 5:2.)[220]

There is, surprisingly, no remark in the corresponding Gemara (Yom Tov, 37a) on this significantly different use of a fortiori reasoning. Here, unlike in the Miriam example and cognate cases, there is no appeal to the dayo principle. Does the Talmud notice and discuss this difference anywhere else? I do not know. In any case, this example is very interesting and worth analyzing further.

The Mishna here clearly teaches that: what is forbidden (assur) on a Festival is, a fortiori, also forbidden on the Sabbath. We can express this in a standard form of a fortiori argument (namely, the positive subjectal, from minor to major) as follows:

The Sabbath (P) is more religiously important (R) than any Festival (Q); whence:
if a certain action on a Festival (Q) is important (R) enough to be forbidden (S),
it follows that the same action on the Sabbath (P) is important (R) enough to be forbidden (S).

This is a passable representation of the argument. However, if we ask what we mean here by more “religiously important,” we might reply that the Sabbath is more “demanding” (or strictly regulated) than any Festival. In that perspective, the argument would seem to be, though still ‘minor to major’, more precisely negative predicatal in form, and we should preferably formulate it as follows:

More holiness (R) is required to observe the Sabbath (P) than to observe any Festival (Q).
If some action[221] (S) is not sufficiently holy (R) to be compatible with observance of a Festival (and thus must[222] be forbidden on it) (Q),
then that action (S) is not sufficiently holy (R) to be compatible with observance of the Sabbath (and thus must be forbidden on it) (P).

Note that I have inserted “holiness” (of an action) as this argument’s operative middle term (R) on the basis of rabbinical explanatory statements in the present context that the holiness of the Sabbath is greater than that of any Festival day. The way I have used this word is a bit awkward, I’ll admit; but it does the job anyway.

More fully expressed the argument has three components: (a) Given that (in the minor premise) S implies not-Q, it follows by contraposition that if Q is prescribed, S must be forbidden. (b) And given that S implies not-Q, it follows by a fortiori that S implies not-P. Finally, (c) since (in the conclusion) S implies not-P, it follows by contraposition that if P is prescribed, S must be forbidden. The two ‘contrapositions’ used are simple ethical logic: anything that interferes with achievement of a set goal is obviously to be prohibited; the means must be compatible with the ends.

We can present the corresponding positive predicatal (major to minor) as follows:

More holiness (R) is required to observe the Sabbath (P) than to observe any Festival (Q).
If some action (S) is sufficiently holy (R) to be compatible with observance of the Sabbath (and thus may be permitted on it) (P).
then that action (S) is sufficiently holy (R) to be compatible with observance of a Festival (and thus may be permitted on it) (Q),

This follows from the negative form by reductio ad absurdum, of course. The meaning of this new argument is: what is permitted (i.e. not forbidden) (mutar) on the Sabbath is, a fortiori, also permitted on a Festival. That is, the argument could as well be put in negative subjectal form, as follows:

The Sabbath (P) is more religiously important (R) than any Festival (Q); whence:
if a certain action on the Sabbath (P) is important (R) not enough to be forbidden (S),
it follows that the same action on a Festival (Q) is important (R) not enough to be forbidden (S).

The expression “not enough to be forbidden” may be taken to imply that the action in in fact “permitted.”

Obviously, we cannot reverse these two statements, viz. that what is forbidden on a Festival must be forbidden on the Sabbath, and what is permitted on the latter must be permitted on the former. Obviously, something forbidden on the Sabbath (e.g. cooking food) is not necessarily also forbidden on a Festival. Something permitted on a Festival (e.g. cooking food) is not necessarily also permitted on the Sabbath. Reasoning of the latter sort would be fallacious by the ordinary rules of a fortiori logic.

Note also: although I have above classified the two arguments as predicatal (i.e. copulative), it might be more accurate to call them consequental (i.e. implicational). For, what the negative form tells us is that a certain action (S) by a Jew causes some deficiency of, let us say, holiness (R) in him and thus causes him to fail to observe a Festival (Q) or the Sabbath (P); similarly for the positive form, mutatis mutandis. In other words, while it is true that P, Q, R, S are terms, there is an unstated underlying subject (a Jewish man, or woman) in relation to which they are all predicates, so that theses (rather than terms) are in fact tacitly intended here.

Furthermore, according to formal logic, if the above two arguments are true, the following two (in which the negative term not-S replaces the positive term S) must also be true.

More holiness (R) is required to observe the Sabbath (P) than to observe any Festival (Q).
If some inaction[223] (not-S) is not sufficiently holy (R) to be compatible with observance of a Festival (and thus must be forbidden on it) (Q),
then that inaction (not-S) is not sufficiently holy (R) to be compatible with observance of the Sabbath (and thus must be forbidden on it) (P).

This is a negative predicatal (minor to major) argument. The meaning of this new argument is, clearly: what is imperative (chayav) on a Festival is, a fortiori, also imperative on the Sabbath. In this form, it is positive subjectal.

More fully expressed the argument has three components: (a) Given that (in the minor premise) not-S implies not-Q, it follows by contraposition that if Q is prescribed, S must be prescribed. (b) And given that not-S implies not-Q, it follows by a fortiori that not-S implies not-P. (c) Finally, since (in the conclusion) not-S implies not-P, it follows by contraposition that if P is prescribed, S must be prescribed. The two ‘contrapositions’ used are simple ethical logic: anything without which a set goal cannot be achieved is obviously to be prescribed; the means necessary for an end are indispensable.

We can present the corresponding positive predicatal (major to minor) as follows:

More holiness (R) is required to observe the Sabbath (P) than to observe any Festival (Q).
If some inaction (not-S) is sufficiently holy (R) to be compatible with observance of the Sabbath (and thus may be permitted on it) (P).
then that inaction (not-S) is sufficiently holy (R) to be compatible with observance of a Festival (and thus may be permitted on it) (Q),

This follows from the negative form by reductio ad absurdum, of course. The meaning of this new argument is: what is exempted (i.e. not prescribed) (patur) on the Sabbath is, a fortiori, also exempted on a Festival. In this form, it is negative subjectal.

Obviously, here again, we cannot reverse these two statements, viz. that what is imperative on a Festival must be imperative on the Sabbath, and what is exempted on the latter must be exempted on the former. Something imperative on the Sabbath (e.g. the additional sacrifices on it) is not necessarily also imperative on a Festival. Something exempted on a Festival (e.g. the said additional sacrifices) is not necessarily also exempted on the Sabbath. Reasoning of the latter sort would be fallacious by the ordinary rules of a fortiori logic.

Clearly, the Sabbath and the Festivals involve some distinctive practices; and Festivals are not all identical. The Festivals are not merely lighter forms of Sabbath, and the Sabbath is not merely a heavier form of Festival; and the various Festivals involve different rituals. We cannot deductively predict all features of one holy day from the other, or vice versa, but must refer to Biblical injunctions or hints for the special features of each. The above a fortiori arguments do not provide a complete set of relationships, which mechanically exclude innovations from the Biblical proof-text.

What can be inferred from the Sabbath to Festivals or vice versa is a product of two forces: (a) the major premise, which relates these two kinds of holy day through a middle term that we took to be ‘holiness’; and (b) the minor premise, which links one of these holy days to a certain subsidiary term through the same middle term. This limits the possibilities of inference, insofar as the middle term does not have unlimited scope. For a start, ‘holiness’ is a vague abstraction, difficult to establish objectively; moreover, it does not provide links to any and all subsidiary terms, but only at best to a specified few.

Thus, much in these arguments depends on traditional understanding of the terms involved. That is to say, the arguments are descriptive propositions as much as deductive processes. They give verbal expression to pre-existing traditions or traditions taking shape, as well as assist in the inference of information. They are formulas designed to enshrine traditional principles and facilitate logical access to them.

It is perhaps historically in this way, by development from the Beitzah 5:2 example of a fortiori argument, that the more general rabbinic definition of qal vachomer emerged (presumably later)[224]. To take a modern statement, R. Chavel defines the argument as follows:

“A form of reasoning by which a certain stricture applying to a minor matter is established as applying all the more to a major matter. Conversely, if a certain leniency applies to a major matter, it must apply all the more to the minor matter.”[225]

This seems to refer primarily to the first two of our above examples, where the “minor matter” is a Festival day and the “major matter” is the Sabbath, and the “stricture” is the proscribing of some action and the “leniency” is its permission. Stricture, of course, suggests restriction, a negative; but it can here be taken to mean more broadly strictness or stringency and thus also refer to a prescription, just as leniency can also refer to an exemption. This is evident in the similar but more accurately worded description of a fortiori reasoning by R. Feigenbaum:

“Any stringent ruling with regard to the lenient issue must be true of the stringent issue as well; [and] any lenient ruling regarding the stringent issue must be true with regard to the lenient matter as well.”[226]

A similar description may also be found in Steinsaltz’s Reference Guide and many other books. What this tells us is that although the examples traditionally drawn from Beitzah 5:2 initially refer to qal vachomer inferences from prohibition to prohibition and from permission to permission, the rabbis also eventually admit the inferences from imperative to imperative and from exemption to exemption that we have just logically demonstrated.

Mielziner, by the way, shows explicit awareness of all four moods, to the extent that where the conclusion is “assur” (forbidden) he adds in brackets the alternative of “chayav” (imperative), and where the conclusion “eino din sheassur” (permitted) he adds in brackets the alternative of “[eino din] shechayav” (exempt). That is, he makes allowance for both the negative and the positive interpretations. He additionally gives us Talmudic examples of an imperative implying an imperative by such qal vachomer: in Baba Metzia 95a, it is inferred that the borrower must restore what was stolen (from him the borrower by some third party) to the lender; or again, in Baba Metzia 94b, that the borrower must restore what he (the borrower) lost to the lender.[227]

However, I am not sure exactly when, in documented history, the transition occurred from the principle specifically concerning Festivals and Sabbaths given in Mishna Beitzah 5:2, and perhaps other passages of the Mishna with a similar thrust, to the general formulations that authors like Mielziner, Chavel, Feigenbaum or Steinsaltz, give nowadays. I suspect the general formulations are not that modern, and may be found in the Talmud or other early literature. It would be very interesting to discover exactly how the progression from material principle to formal principle occurred, i.e. thanks to whom and on what dates.

To conclude this section, what we need to note well is that no application of the dayo principle is needed or even possible in cases of the sort here considered, since obviously an action is either forbidden or permitted, either imperative or exempted, and there are no degrees in between. Admittedly, as regards permitted actions, some may be more ‘desirable’ or ‘to be preferred’ or ‘recommended’ than others, but these are not degrees of permission as such. Observe that we have no inclination, in the above inference from permission on the Sabbath to permission on a Festival, to regard the latter permission as of a lesser (or greater) degree than the former. Similarly with regard to exemption: it has in itself no degrees. Very often, the conclusion of a fortiori argument is like that – without degree. This is clearly purely a fortiori inference, and not to be confused with a crescendo inference.

I do not know if the rabbis explicitly made this distinction, between qal vachomer use with appeal to dayo principle and qal vachomer without relevance of dayo. As I have explained, the dayo principle is needed to block reasoning through the midah keneged midah (measure for measure) principle or similar ‘proportional’ propositions. It is not directly related to a fortiori argument as such; it is only indirectly related, to prevent a common penchant for ‘proportionality’ in special cases. In many cases, if not in most, there is no such propensity, because there is no parallel principle like midah keneged midah pressing us towards ‘proportionality’, and therefore the issue of dayo does not even arise. In truth, a fortiori reasoning is always the same, irrespective of whether there is ‘proportionality’ or not and whether dayo is thereafter used or not.

In view of all this, it is hard to understand why the Gemara commentary in Baba Qama 25a is so categorical in its treatment, giving the impression that a fortiori argument is necessarily a crescendo, and failing to explicitly note that the dayo principle, whether it is applied to all a crescendo arguments (as the Sages apparently hold, in the Gemara’s view) or only to some (as R. Tarfon holds, according to the Gemara), is not applicable to purely a fortiori arguments, i.e. those which do not involve (explicitly or implicitly) an additional premise about ‘proportionality’. Surely, if the author of this Gemara was aware of the full sweep of Talmudic discourse, he would have noticed these distinctions and taken them into consideration in his commentary.[228]

6.    Three additional Gemara arguments

Further on in tractate Baba Qama, on pp. 25b-26a, the Gemara proposes three a fortiori arguments in which the previously used propositions, about damage by horn and by tooth & foot on public and private grounds, are recycled and reshuffled in various ways, and the resulting conclusions are tested. For this reason, I have dubbed them “experimental” arguments. It is not immediately clear what the purpose(s) of these additional arguments might be. At first sight, their insertion here looks like a process of consistency checking. Possibly, the Gemara is using them to settle some legal matter specified in the larger context. Alternatively, it is merely exploring theoretical possibilities, trying different permutations and seeing where they lead. Or again, perhaps the Gemara is simply engaged in intellectual exercise for its own sake. In any case, we shall here try to throw some light on these arguments by means of logical analysis.

Before we do so, however, let us briefly recall here the original Mishna (BQ 2:5) arguments to which they refer, for this will facilitate our work. The first Mishna argument can be presented in several ways. Its premises and conclusion can be laid out as a set of if-then propositions spelling out the legal liability for damage by different causes in different domains, as follows:

If tooth & foot and public, then no liability (by extreme inversion of Ex. 22:4).

If tooth & foot and private, then full liability (Ex. 22:4).

If horn and public, then half liability (Ex. 21:35).

If horn and private, then full liability (R. Tarfon’s putative conclusion).

If horn and private, then half liability (the Sages’ conclusion, after application of dayo type I).

As we saw in our earlier detailed treatment, this basic argument can be recast in analogical, pro rata, a crescendo or purely a fortiori forms, as follows:

Analogy

Just as, in the case of tooth & foot, damage in the private domain implies more legal liability than damage in the public domain (since the former implies full liability and the latter none).
Likewise, in the case of horn, damage in the private domain implies more legal liability than damage in the public domain (i.e. given half liability in the latter, conclude with full in the former).

Pro rata

The degree of legal liability for damage is ‘proportional’ to the status of the property the damage is made on, with damage in the private domain implying more legal liability than damage in the public domain.
This is true of tooth and foot damage, for which liability is known to be nil in the public domain and full in the private domain.
Therefore, with regard to horn damage, for which liability is known to be half in the public domain, liability may be inferred to be full in the private domain.

A crescendo

Private domain damage (P) is more important (R) than public domain damage (Q) [as we infer by extrapolation from tooth & foot damage (where liability is respectively full and half in the two domains) to all causes of damage, including horn].
Horn damage in the public domain (Q) is important (Rq) enough to make the payment half (Sq).
The payment due (S) is ‘proportional’ to the degree of legal liability (R).
Therefore, horn damage in the private domain (P) is important (Rp) enough to make the payment full (Sp = more than Sq).

Pure a fortiori

Private domain damage (P) is more important (R) than public domain damage (Q) [as we infer by extrapolation from tooth & foot damage, to repeat].
Horn damage in the public domain (Q) is important (R) enough to make the payment half (S).
Therefore, horn damage in the private domain (P) is important (R) enough to make the payment half (S).

As we learned previously, the above analogical, pro rata or a crescendo arguments correspond to R. Tarfon’s reasoning. The Mishna Sages reject his reasoning by means of a dayo objection of the first type, i.e. which denies the ‘proportionality’ assumed by their colleague. Effectively, then, the Sages advocate the purely a fortiori argument exclusively. The second Mishna argument can likewise be presented in several ways. As a set of if-then propositions, it looks as follows:

If tooth & foot and public, then no liability (by extreme inversion of Ex. 22:4).

If horn and public, then half liability (Ex. 21:35).

If tooth & foot and private, then full liability (Ex. 22:4).

If horn and private, then full liability (R. Tarfon’s same putative conclusion).

If horn and private, then half liability (the Sages’ conclusion, after application of dayo type II).

And here again, the basic argument can be recast in analogical, pro rata, a crescendo or purely a fortiori forms, as follows:

Analogy

Just as, in the public domain, damage by horn implies more legal liability than damage by tooth & foot (since the former implies half liability and the latter none).
Likewise, in the private domain, damage by horn implies more legal liability than damage by tooth & foot (i.e. given full liability in the latter, conclude with full in the former).

Pro rata

The degree of legal liability for damage is ‘proportional’ to the intentionality of the cause of damage, with damage by horn implying more legal liability than damage by tooth & foot.
This is true of the public domain, for which liability is known to be nil for damage by tooth and foot and half for damage by horn.
Therefore, with regard to the private domain, for which liability is known to be full for damage by tooth and foot, liability may be inferred to be full for damage by horn.

A crescendo

Horn damage (P) is more important (R) than tooth & foot damage (Q) [as we infer by extrapolation from the public domain (where liability is respectively half and nil in the two cases) to all domains, including the private].
Tooth & foot damage in the private domain, (Q) is important (R) enough to make the payment full (S).
The payment due (S) is ‘proportional’ to the degree of legal liability (R).
Therefore, horn damage in the private domain (P) is important (R) enough to make the payment full (S).

Pure a fortiori

Horn damage (P) is more important (R) than tooth & foot damage (Q) [as we infer by extrapolation from the public domain, to repeat].
Tooth & foot damage in the private domain, (Q) is important (R) enough to make the payment full (S).
Therefore, horn damage in the private domain (P) is important (R) enough to make the payment full (S).

As we found out previously, this time all of the above argument forms, including the purely a fortiori one, match R. Tarfon’s reasoning. So, the Mishna Sages cannot reject his reasoning by means of a dayo objection of the first type, since ‘proportionality’ is not essential to its stringent conclusion of full liability. Nevertheless, they maintain their dayo objection, and again advocate a moderate conclusion of only half liability. Therefore, the latter dayo objection must be of a second type. It is indeed, interdicting the inductive process of generalization through which the major premise of such argument is produced. We need not say more than that here, having already dealt with the issues involved at length.

Now, what is interesting is the way the Gemara takes the final conclusion of the Mishna Sages, namely that horn damage in the private domain implies half liability, and uses it as a constant premise in each of its three experimental arguments. This proposition is of course implied by Ex. 21:35, which specifies half liability for horn damage, without specifying a domain; but the Sages have effectively ruled that it is not a minimum but a maximum[229], i.e. it is to be read as davka half. Nevertheless, the Gemara here additionally uses a watered down version of Ex. 21:35 in two of its arguments (the first two).

Another proposition relevant to all three Gemara arguments is Ex. 22:4, which specifies full liability for tooth & foot damage in the private domain[230]. This proposition is repeated in two of the Gemara arguments (the first and last). In the Mishna, the liability for tooth & foot damage in the public domain is taken to be the extreme inverse of Ex. 22:4, i.e. no liability. And this is also assumed in two Gemara arguments (the last two); however, at the end of one Gemara argument (the first one), a moderate inversion is attempted, i.e. “not full” is taken to mean “half” rather than “nil.”

Let us now examine the three new arguments in the Gemara more closely.

First experiment. The Gemara states: “But should we not let Tooth and Foot involve liability for damage done [even] on public ground because of the following a fortiori:

If in the case of Horn, where [even] for damage done on the plaintiff’s premises only half payment is involved, there is yet liability to pay for damage done on public ground,

does it not necessarily follow that in the case of Tooth and Foot, where for damage done on the plaintiff’s premises the payment is in full, there should be liability for damage done on public ground?

— Scripture, however, says: And it shall feed in another man’s field, excluding thus [damage done on] public ground. But have we ever suggested payment in full? It was only half payment that we were arguing for![231]

Note at the outset the sources of the premises in the Gemara’s argument. One is the earlier conclusion of the Mishna Sages (via their dayo objections to R. Tarfon’s claims) that for damage by horn on private property the ox owner’s liability is half. The other two premises are more directly derived from the Torah (Ex. 22:4 and Ex. 21:35). The conclusion concerns damage by tooth & foot on public property.

Expressed as a set of brief if-then statements, this Gemara argument looks as follows. Note that the first two have in common the factor of private property.

If horn and private, then half liability (ruling of the Mishna Sages).

If tooth & foot and private, then full liability (Ex. 22:4).

If horn and public, then some liability (from Ex. 21:35).

If tooth & foot and public, then some liability (putative conclusion).

Or in analogical format, as follows:

Just as, in the private domain, damage by tooth & foot implies more legal liability than damage by horn, since the former implies full and the latter half.
Likewise, in the public domain, damage by tooth & foot implies more legal liability than damage by horn; whence given that the latter implies some liability (note that although Ex. 21:35 implies a specific amount, the Gemara here deliberately avoids mentioning it in its premise), then the former implies some liability.

Or again, in purely a fortiori format, of positive antecedental form (minor to major), as follows[232]:

Tooth & foot damage (P) is more important (R) than horn damage (Q) [as we infer by extrapolation from their liabilities for damage in the private domain, respectively full and half, to all domains, including the public].
Horn damage in the public domain (Q) is important (R) enough to imply some liability (S).
Therefore, tooth & foot damage in the public domain (P) is important (R) enough to imply some liability (S).

The Gemara is thus justified in describing its argument here as qal vachomer (מקל וחומר), although this must be taken to refer to purely a fortiori argument and not a crescendo. We see clearly from the a fortiori formulation that the major premise is produced by a generalization, from the particular case of private property to all property, and its application to the particular case of public property. On this basis, the minor premise about unspecified liability for horn leads to the conclusion about unspecified liability for tooth & foot.

Now, the main question to ask here is: why is the Gemara opting for such vague language? There are actually two separate questions, here: (a) Why is its premise is deliberately vague, saying “there is yet liability” (חייבת), i.e. some liability, without specifying just how much liability even though the amount is already known from Ex. 21:35 to be precisely half? And (b) Why is its conclusion also vague, saying “there should be liability” (חייב), i.e. some liability, although the amount of this liability may be assumed by partial instead of full denial of Ex. 22:4 to be half? We shall now propose our answers.

The way to answer our question about the vagueness of the minor premise is to consider what would happen if more explicit language were to be used. To start with, had the Gemara used half liability as the consequent of the minor premise, and argued a crescendo instead of purely a fortiori, its conclusion would have been full liability for tooth & foot damage in the public domain, and thus contrary to Ex. 22:4, according to which full liability is reserved for tooth & foot damage in the private domain. This is evident in the following lines:

Tooth & foot damage (P) is more important (R) than horn damage (Q) [as we infer by extrapolation, as before].
Horn damage in the public domain (Q) is important (R) enough to imply half liability (S) (as specified in Ex. 21:35).
The payment due (S) is ‘proportional’ to the degree of legal liability (R).
Therefore, tooth & foot damage in the public domain (P) is important (R) enough to imply full liability (S) (contrary to the davka reading of Ex. 22:4).

Thus, the Gemara’s thinking (consciously or otherwise) in this respect was effectively as follows. Since the full liability conclusion is contrary to a Scriptural given (namely Ex. 22:4, which specifies full liability to be applicable only to private property) the argument must be rejected somehow. Since the major and minor premises are already accepted, and the inference process is clearly valid, the only way to reject the argument is by denying the additional premise about ‘proportionality’ – or, in other words, by applying a dayo objection of type I. That is to say, the a crescendo argument is to be discarded, leaving only the underlying purely a fortiori argument. This leftover argument is similar to the Gemara’s (previously mentioned), except that it infers half liability from half liability[233], instead of some liability from some liability.

Another route the Gemara may have tried is the following. As we learned from R. Tarfon, we can by judicious reshuffling of the premises obtain an alternative a fortiori argument. In the present case, this would be done as shown next. In terms of if-then statements, our competing argument would be as follows. Note that the first two statements, which we use to form our major premise, are both about horn damage.

If horn and private, then half liability (ruling of the Mishna Sages).

If horn and public, then half liability (Ex. 21:35).

If tooth & foot and private, then full liability (Ex. 22:4).

If tooth & foot and public, then full liability (putative conclusion, contrary to Ex. 22:4).

This can be recast in analogical form thusly:

Just as, in the case of horn, damage in the public domain implies as much legal liability as in the private domain (since both imply half liability).
Likewise, in the case of tooth & foot, damage in the public domain implies as much legal liability as in the private domain; whence given that the latter implies full liability, then the former implies full liability (contrary to Ex. 22:4, which specifies full liability to be applicable only to private property).

More to the point, we can formulate it in purely a fortiori format as follows. Note that this argument is positive antecedental and a pari (i.e. egalitarian).

Public domain damage (P) is as important (R) as private domain damage (Q) [as we infer by extrapolation from horn damage (where liability is half in both domains) to all causes of damage, including tooth & foot].
Tooth & foot damage in the private domain (Q) is important (R) enough to imply full liability (S).
Therefore, tooth & foot damage in the public domain (P) is important (R) enough to imply full liability (S) (contrary to Ex. 22:4).

Now, observe why this argument seems more secure than the preceding a crescendo. It also goes from minor to major; but since the minor premise predicates what is already the maximum amount allowable (namely, full liability), the conclusion has to predicate the same maximum amount (i.e. full liability). Yet here again the conclusion is contrary to a Scriptural given (Ex. 22:4, which specifies full liability to be applicable only to private property). Therefore, it must be rejected. The only way to do this is through a dayo objection of type II, i.e. by preventing the generalization that gave rise to its major premise from proceeding. The final conclusion will then again be half liability.

What the above suggests, then, is that the Gemara opted for vague language in the minor premise, speaking of liability indefinitely, because it knew or at least sensed that specifying half liability would in any event lead to a conclusion of full liability, contrary to Scripture; which conclusion would have to be prevented by application of dayo objections of both types. In the Judaic frame of reference, a conclusion contrary to what the Torah teaches is a conclusion contrary to ‘fact’, which must be prevented to avoid inconsistency. Apparently, then, rather than get involved in that long discussion, or pilpul, it opted for a vaguer statement of the minor premise, to arrive at its desired conclusion more directly.

As regards its vague conclusion, a minimum of reflection shows that the liability implied, though stated indefinitely, can only be half liability. This is evident already in the above two arguments from the minor premise of half liability, since their conclusion of full liability is unacceptable because contrary to Scripture. However, we could arrive at the same result by working on the vague conclusion of the Gemara’s own purely a fortiori argument (from some to some liability). Given the conclusion that tooth & foot damage on public property implies some liability, i.e. denies no liability, this can only mean half liability, since full liability is excluded by Ex. 22:4. This seemed so obvious to the Gemara that it did not even see any necessity to say it out loud.

As we have seen, according to the rabbis, based on Biblical practice, the variable “liability” allows in the present context for only three possible values; namely, no liability, half liability and full liability. Therefore, an indefinite amount of liability, i.e. some liability, which is the negation of no liability, means “half or full” liability. Therefore, to say “there is liability,” meaning some liability, is not as open a statement as it might seem – it allows for only two possibilities, viz. half or full liability. So, if one of these is known to be false (in this case, with reference to the Torah), the other must be true. The latter argument is a disjunctive apodosis: “either this or that, but not this, therefore that.”

Note well that the Gemara here proposes an alternative judgment on damage by tooth & foot on public property to that previously accepted (in the debate between R. Tarfon and the Sages). Previously, the Mishna and the Gemara interpreted Ex. 22:4 (“If a man… shall let his beast loose, and it feed in another man’s field, etc.”), which imposes full liability for tooth & foot damage on specifically private grounds, as implying that there is no liability for tooth & foot damage on public grounds. Here, the Gemara (logically enough) proposes an alternative reading for the latter case, such that “not full” is taken to mean “half” instead of the more extreme “nil,” and it backs up this moderate reading by reasoning that so concludes.

Thus, the Gemara’s use of vague language in its first argument was not some subterfuge relying on half-truths; it was just intended as a shortcut to a result that was in any case logically inevitable. The Gemara achieved its objective here, which was to establish that Ex. 22:4, which imposes full liability for tooth & foot damage on private grounds, need not be taken to imply (as it was in the Mishna) that there is no liability for tooth & foot damage on public grounds; for the alternative of half liability is logically equally cogent. That the Gemara was consciously doing this is evident from its statement: “It was only half payment that we were arguing for!” At worst, the Gemara can be criticized for being too laconic; but its reasoning is sound.

Second experiment. The Gemara states: “But should we not let Tooth and Foot doing damage on the plaintiff’s premises involve the liability for half damages only because of the following a fortiori:

If in the case of Horn, where there is liability for damage done even on public ground, there is yet no more than half payment for damage done on the plaintiff’s premises,

does it not follow that in the case of Tooth and Foot, where there is exemption for damage done on public ground, the liability regarding damage done on the plaintiff’s premises should be for half compensation [only]?[234]

— Scripture says: He shall make restitution, meaning full compensation.

We should here again at the outset note that the Gemara’s argument uses as a premise the earlier conclusion of the Mishna Sages (via their dayo objections to R. Tarfon’s claims) that for damage by horn on private property the ox owner’s liability is half. The other two premises are derived from the Torah as follows: one directly, from Ex. 21:35; and the other indirectly, by extreme inversion of Ex. 22:4 (by which I mean that “not full” is here taken to mean “nil” as in the Mishna, instead of “half” as proposed in the preceding experimental argument of the Gemara). The conclusion concerns damage by tooth & foot on private property. The Gemara demonstrates that a conclusion of half liability, contrary to the full liability given in Ex. 22:4, would follow from the said premises.

Expressed as a set of brief if-then statements, this Gemara argument looks as follows. Note that the first two have in common the factor of public property.

If horn and public, then some liability (from Ex. 21:35).

If tooth & foot and public, then no liability (by extreme inversion of Ex. 22:4).

If horn and private, then only half liability (ruling of the Mishna Sages).

If tooth & foot and private, then [only] half liability (putative conclusion, contrary to Ex. 22:4).

This can be expressed in analogical form, as follows. Note that I here use the term “exemption” in the sense of “freedom of liability,” allowing for degrees of zero, half and total exemption; the term is thus intended as the reverse of the range of “liability.”

Just as, in the public domain, damage by tooth & foot implies more legal exemption than damage by horn, since the former implies no liability and the latter some liability (note that although we can infer from Ex. 21:35 the amount to be half, the Gemara here deliberately avoids specifying it in its premise).
Likewise, in the private domain, damage by tooth & foot implies more legal exemption than damage by horn; whence given that the latter implies only half liability, then the former implies only half liability (contrary to Ex. 22:4, which imposes full liability for this).

We can represent the same argument in purely a fortiori form, as follows. Note the negative polarity of the middle term (R) used; this is necessary to ensure that tooth & foot damage emerge as the major term (P) and horn damage as the minor term (Q). The resulting argument is thus minor to major, positive antecedental.

Tooth & foot damage (P) is more unimportant (R) than horn damage (Q) [as we infer by extrapolation from their liabilities for damage in the public domain (respectively none and some) to all domains, including the private].
Horn damage in the private domain (Q) is unimportant (R) enough to imply only half liability (S).
Therefore, tooth & foot damage in the private domain (P) is unimportant (R) enough to imply only half liability (S) (contrary to Ex. 22:4, which imposes full liability for this).

The Gemara is thus justified in describing its argument here as qal vachomer (מק”ו), although again this should be understood to refer to purely a fortiori argument rather than a crescendo. We see clearly from the a fortiori formulation that the major premise is produced by a generalization, from the particular case of public property to all property, and its application to the particular case of private property. On this basis, the minor premise about half liability for horn leads to the conclusion about half liability for tooth & foot.

Thus, whether we reason analogically or purely a fortiori, we obtain a conclusion contrary to Scripture. Since the processes used are faultless, what this means is that one or more of the premises must be wrong. In order to try and understand where the problem lies, let us look again at the Gemara’s formulation. The first question to ask (in view of what we learned in the previous case) is why does the Gemara say vaguely “there is liability” (חייבת) for damage by horn in the public domain, when it is known from Ex. 21:35 that the amount of liability is precisely half? Looking at the major premise of the above a fortiori argument, which is generalized from this information, it is clear that it would have made no difference to it if the Gemara had specified half liability. The argument by analogy would similarly be unaffected. So there seems to be no reason for the Gemara not to have said half[235].

Another question is why does the Gemara find it necessary to say “no more than” (אלא) half regarding the liability for damage by horn on private property? Until now, “half” has always meant precisely half, without need to specify that only half is intended. If more than half liability was possibly included in the term half, the meaning of it would have been “half or full,” and this could be stated as before as indefinite “liability.” Perhaps the answer is that if the liability for damage by horn on private property had been full, as R. Tarfon advocated, then the conclusion here would be full liability for damage by tooth and foot on private property. So the Gemara is specifying “no more than half” merely to indicate that it is abiding by the ruling of the Mishna Sages, and not adopting the contrary opinion of R. Tarfon.

In fact, we could represent almost the same argument in a crescendo form, as follows. Note the similarities to the preceding purely a fortiori formulation, but also the totally different conclusion. Instead of half liability, the conclusion here is no liability. But the effect is the same, in that this is contrary to Ex. 22:4.

Tooth & foot damage (P) is more unimportant (R) than horn damage (Q) [as we infer by extrapolation from their liabilities for damage in the public domain (respectively none and some (or more precisely half)) to all domains, including the private].
Horn damage in the private domain (Q) is unimportant (R) enough to imply half liability (S).
The payment due (S) is ‘proportional’ to the degree of legal liability (R).
Therefore, tooth & foot damage in the private domain (P) is unimportant (R) enough to imply no liability (S) (contrary to Ex. 22:4, which imposes full liability for this).

If, in view of the conflict of this conclusion with Ex. 22:4, we interdicted the premise about ‘proportionality’ by means of a dayo objection of type I, we would obtain the same conclusion as the pure a fortiori argument above; namely, half liability. This would of course still leave us with a conclusion contrary to the Scriptural given of Ex. 22:4. Although the Gemara originally does not express this conclusion, however obtained, as “only half,” it is interesting to note that the translator does add on the qualification of exclusion. This is no doubt to exclude “full” liability (rather than to exclude “no” liability), because this is the crux of the issue in this Gemara argument.

Now, this conclusion of half (i.e. not full) liability is especially troubling because the premises that give rise to it were previously regarded as quite acceptable. The major premise is based on Ex. 21:35 (whether we read it as half liability or more vaguely as some) and on the extreme inversion of Ex. 22:4 (i.e. reading not-full as nil, to the exclusion of half) taken for granted by all participants in the Mishna. And the minor premise is the ruling of the Sages in the Mishna, which is in any case implied in Ex. 21:35 (since this verse does not make an explicit distinction between public and private property). How then can these givens result in a conclusion contrary to Scripture, i.e. to Ex. 22:4? This is the difficulty.

Obviously, the problem must lie with the major premise of the a fortiori argument (whether non-proportional or proportional). The extrapolation of “Tooth & foot damage is more unimportant than horn damage” from public property to private property has to be interdicted by a dayo objection of type II, so as to avoid the antinomic conclusion. This could be considered as the intent of the final statement “Scripture says: He shall make restitution, meaning full compensation,” although there is no explicit mention of dayo here. The Gemara is effectively saying: the conclusion cannot be right, therefore block it from happening. This is regular reductio ad absurdum reasoning.

We could also, by the way, obtain the conclusion of no liability by purely a fortiori argument (instead of a crescendo, as just shown), by imitating the Mishna’s R. Tarfon and using another direction of generalization, as shown next. First, let us reshuffle the initial if-then statements, so that the ones we use to form our major premise are both about horn damage, as follows:

If horn and public, then half liability (Ex. 21:35).

If horn and private, then half liability (ruling of the Mishna Sages).

If tooth & foot and public, then no liability (by extreme inversion of Ex. 22:4).

If tooth & foot and private, then no liability (putative conclusion, contrary to Ex. 22:4).

Next, let us formulate the argument in analogical form, keeping to the language of exemption for symmetry with the previous formulation, as follows:

Just as, in the case of horn, damage in the private domain implies as much legal exemption as in the public domain (since both imply half liability):
So, in the case of tooth & foot, damage in the private domain implies as much legal exemption as damage in the public domain; whence given the latter implies no liability, then the former implies no liability (contrary to Ex. 22:4, which imposes full liability for this).

Lastly, we formulate the argument as a purely a fortiori one, of positive antecedental form (minor to major), as follows:

Damage in the private domain (P) is as unimportant (R) as damage in the public domain (Q) [as we infer by extrapolation from horn damage (where liability is half in both domains) to all causes of damage, including tooth & foot].
Tooth & foot damage in the public domain (Q) is unimportant (R) enough to imply no liability (S).
Therefore, tooth & foot damage in the private domain (P) is unimportant (R) enough to imply no liability (S) (contrary to Ex. 22:4, which imposes full liability for this).

This argument seems more solid than the preceding a crescendo argument because it argues from no liability to no liability, rather than from half to none. So it cannot be prevented by means of a dayo objection of type I. And yet its conclusion is the same, viz. no liability. Which poses a problem, since it is inconsistent with the Scriptural imposition of full liability (in Ex. 22:4). Here, then, we must resort to a dayo objection of type II, interdicting the generalization that led to the major premise. We might then be tempted to accept the next amount of half liability as the final result – but no, this is still contrary to Ex. 22:4, and so must be avoided too.

To sum up, the initial premises used in different ways in the various arguments we considered representing the Gemara’s second experiment cannot readily be rejected, yet they lead to a conclusion contrary to Scripture. To prevent such paradoxical result, we had to again resort to dayo objections of both types. This means that the initial premises are together viable provided we do not indulge in proportional thinking or in generalizations in relation to them. Our room for maneuver with them is severely limited; we must proceed with caution.

Third experiment. The Gemara states: “But should we not [on the other hand] let Horn doing damage on public ground involve no liability at all, because of the following a fortiori:

If in the case of Tooth and Foot, where the payment for damage done on the plaintiff’s premises is in full there is exemption for damage done on public ground.

does it not follow that, in the case of Horn, where the payment for damage done on the plaintiff’s premises is [only][236] half, there should be exemption for damage done on public ground?

— Said R. Johanan: Scripture says. [And the dead also] they shall divide, to emphasise that in respect of half payment there is no distinction between public ground and private premises.

We can here again at the outset note that the Gemara’s argument uses as a premise the earlier conclusion of the Mishna Sages (via their dayo objections to R. Tarfon’s claims) that for damage by horn on private property the ox owner’s liability is half. The other two premises are derived from the Torah as follows: one directly, from Ex. 22:4; and the other indirectly, by extreme inversion of Ex. 22:4 (by which I mean that “not full” is here taken to mean “nil” as in the Mishna, instead of “half” as proposed in the first experimental argument of the Gemara). The conclusion concerns damage by horn on public property. The Gemara demonstrates that a conclusion of no liability, contrary to the half liability given in Ex. 21:35, would follow from the said premises.

Expressed as a set of brief if-then statements, this Gemara argument looks as follows. Note that the first two have in common the factor of private property.

If tooth & foot and private, then full liability (Ex. 22:4).

If horn and private, then [only] half liability (ruling of the Mishna Sages).

If tooth & foot and public, then no liability (by extreme inversion of Ex. 22:4).

If horn and public, then no liability (putative conclusion, contrary to Ex. 21:35).

This can be expressed in analogical form, as follows. Note that I here use the term “exemption” in the sense of “freedom of liability,” allowing for degrees of zero, half and total exemption; the term is thus intended as the reverse of the range of “liability.”

Just as, in the private domain, damage by horn implies more legal exemption than damage by tooth & foot, since the former implies [only] half liability and the latter full liability.
Likewise, in the public domain, damage by horn implies more legal exemption than damage by tooth & foot; whence given that the latter implies no liability, then the former implies no liability (contrary to Ex. 21:35, which imposes half liability).

We can represent the same argument in purely a fortiori form, as follows. Note the negative polarity of the middle term (R) used; this is necessary to ensure that horn damage emerge as the major term (P) and tooth & foot damage as the minor term (Q). The resulting argument is thus minor to major, positive antecedental.

Horn damage (P) is more unimportant (R) than tooth & foot damage (Q) [as we infer by extrapolation from private domain damage (for which the liabilities are half and full respectively) to all domains, including the public].
Tooth & foot damage in the public domain (Q) is unimportant (R) enough to imply no liability (S).
Therefore, horn damage in the public domain (P) is unimportant (R) enough to imply no liability (S) (contrary to Ex. 21:35, which imposes half liability).

The Gemara is thus justified in describing its argument here as qal vachomer (מק”ו), although again this should be understood to refer to purely a fortiori argument rather than a crescendo. We see clearly from the a fortiori formulation that the major premise is produced by a generalization, from the particular case of private property to all property, and its application to the particular case of public property. On this basis, the minor premise about no liability for tooth & foot leads to the conclusion about no liability for horn.

No ‘proportionality’ can be presumed here, for the simple reason that the minor premise and conclusion are already an extreme value (namely, no liability). Thus, an a crescendo argument with the same terms would be identical with the above purely a fortiori argument.

Manifestly, whether we reason analogically or purely a fortiori, we obtain a conclusion contrary to Scripture. Since the processes used are faultless, what this means is that one or more of the premises must be wrong. Examining the Gemara’s formulation, we see that in the present case, unlike the preceding two, there is no ambiguous language. The word exemption (פטורה) is clearly intended here, in both its occurrences, in the sense of full exemption, i.e. zero liability.

It is noteworthy that, although the Gemara originally does not express the liability for damage by horn as “only half,” the translator adds on the qualification of exclusion. But this is no doubt simply to exclude the “full” liability here due according to the dissenting opinion of R. Tarfon; and it does not seriously affect the argument, since if full were adopted instead of half, the major premise would become egalitarian, but the minor premise and conclusion would remain the same.

Now, this conclusion of no liability (instead of half) is obviously problematic, since the premises that give rise to it were previously regarded as quite acceptable. The major premise is based on Ex. 22:4 and on the ruling of the Sages in the Mishna, which is in any case implied in Ex. 21:35 (since this verse does not make an explicit distinction between public and private property, as R. Johanan reminds us[237]). And the minor premise is based on the extreme inversion of Ex. 22:4 (i.e. reading not-full as nil, to the exclusion of half) taken for granted by all participants in the Mishna. How then can these givens result in a conclusion contrary to Scripture, i.e. to Ex. 21:35? This is the difficulty.

Obviously, the problem must lie with the major premise of the a fortiori argument. The extrapolation of “Horn damage is more unimportant than tooth & foot damage” from private property to public property has to be interdicted by a dayo objection of type II, so as to avoid the antinomic conclusion. This could be considered as the intent of the final statement concerning damage by horn that “in respect of half payment there is no distinction between public ground and private premises,” although there is no explicit mention of dayo here. The Gemara is effectively saying: the conclusion cannot be right, therefore block it from happening. This is regular reductio ad absurdum reasoning.

Our next obvious move would be to investigate if a conclusion consistent with Scripture would be obtained by imitating the Mishna’s R. Tarfon, and judiciously reshuffling the given information so as to attempt another direction of generalization. This would proceed as follows. First, we reshuffle the initial if-then statements, so that the ones we use to form our major premise are both about tooth & foot damage, as follows:

If tooth & foot and private, then full liability (Ex. 22:4).

If tooth & foot and public, then no liability (by extreme inversion of Ex. 22:4).

If horn and private, then [only] half liability (ruling of the Mishna Sages).

If horn and public, then [only] half liability (conclusion in accord with Ex. 21:35).

Next, we formulate the argument in analogical form, keeping to the language of exemption for symmetry with the previous formulation, as follows:

Just as, in the case of tooth & foot, damage in the public domain implies more legal exemption than in the private domain (since these respectively imply no and full liability):
So, in the case of horn, damage in the public domain implies more legal exemption than damage in the private domain; whence given the latter implies only half liability, then the former implies only half liability (in accord with Ex. 21:35).

Lastly, we formulate the argument as a purely a fortiori one, of positive antecedental form (minor to major), as follows:

Damage in the public domain (P) is more unimportant (R) than damage in the private domain (Q) [as we infer by extrapolation from tooth & foot damage (for which liability is respectively nil and full) to all causes of damage, including horn].
Horn damage in the private domain (Q) is unimportant (R) enough to imply only half liability (S).
Therefore, horn damage in the public domain (P) is unimportant (R) enough to imply only half liability (S) (in accord with Ex. 21:35).

However, before we can adopt this purely a fortiori argument we must look into the corresponding a crescendo argument. The latter is as follows:

Damage in the public domain (P) is more unimportant (R) than damage in the private domain (Q) [as we infer by extrapolation from tooth & foot damage (for which liability is respectively nil and full) to all causes of damage, including horn].
Horn damage in the private domain (Q) is unimportant (R) enough to imply half liability (S).
The payment due (S) is ‘proportional’ to the degree of legal liability (R).
Therefore, horn damage in the public domain (P) is unimportant (R) enough to imply no liability (S) (contrary to Ex. 21:35, which imposes half liability).

Evidently, arguing a crescendo with these premise results in the undesirable conclusion of no liability for horn damage in the public domain, which is contrary to Scripture (Ex. 21:35). This being the case, such a crescendo argument has to be interdicted by means of a dayo objection of type I. So doing, we return to the purely a fortiori argument formulated just before, which yields the conclusion of half liability. Since the latter conclusion is consistent with Scripture (Ex. 31.35), we have no need to interdict it by means of a dayo objection of type II. We can therefore adopt the said a fortiori argument as a viable alternative to the third one proposed by the Gemara, which yielded an unacceptable conclusion.

From this we see that, while the Gemara’s third experiment is in many ways similar to its second, they are ultimately quite different, in that while the second experiment leaves us without a viable a fortiori counter-argument, the third one does have a viable a fortiori counter-argument. It is surprising that the Gemara did not remark on this significant difference, but remained content with simply listing two arguments with conclusions inconsistent with Scriptural givens.

To sum up. The Gemara’s three experimental arguments have in common as a premise the conclusion of the Sages in the Mishna that damage by horn in the private domain implies half liability. The arguments then seek to determine what conclusion can be drawn from that constant premise about the other situations, viz. tooth & foot damage in the public and private domains, and horn damage in the public domain, respectively. The purpose of the exercise is apparently to compare such conclusions to, respectively, an assumption in the Mishna (viz. that tooth & foot damage on public property implies no liability, based on extreme inversion of Ex. 22:4) and to certain Scriptural givens (viz. Ex. 22:4, which imposes full liability for tooth & foot damage on private property, and Ex. 21:35, which imposes half liability for horn damage on public property).

The Gemara’s logical virtuosity in proposing these three arguments is rather impressive, considering its lack of formal tools. Although the above proposed explicit logical analyses of the three arguments are absent in the Gemara, similar analyses may be reasonably be supposed to have consciously or subconsciously colored the Gemara’s thinking, for otherwise it would be difficult to explain its intent in presenting these arguments. Note in particular that though the dayo principle is nowhere here mentioned by the Gemara, both versions of it are very present in the background of its discourse.[238]

7.    Assessment of the Talmud’s logic

We have in the preceding pages examined in great detail, using up to date methods of formal logic, the a fortiori reasoning of both the Mishna and the Gemara, or at least their reasoning in the immediate vicinity of the present sugya[239] (i.e. mBQ 2:5 and bBQ 25a). We judged these texts on their own merits, note well, and not through the prism of later commentaries. Our general conclusion may well be that both the earlier and later Talmudic sages, the Tannaim and the Amoraim, were amazingly powerful logic practitioners, even if they were not great theoreticians. Judging by the Talmudic material we have looked at here, their reasoning seems on the whole sound, even if too often much is left unstated.

What is amazing is precisely that, albeit the brevity of their statements, the people involved were able to reason with such accuracy. I am amazed because, with my pedestrian mind, without reference to formal methods and without full exposition of all implicit discourse, I would be unable to arrive at similar results with equal aplomb. Nevertheless, it must be said and admitted that self-assurance, however esthetically impressive, is not enough. Logic is not just an art; it is first of all a science. To reason correctly is good; but to know just why one’s reasoning is correct is much better. To reason correctly based only on intuition, i.e. on immediate logical insight, is not as convincing as to do so based on broad theoretical understanding, i.e. on abstract study of the exact conditions for correct reasoning (even if, to be sure, such study is also based on the same faculty of logical insight). In the former case, there is some reliance on luck; in the latter, nothing is left to chance.

Comparing now the logic in the Mishna to that in the Gemara, certain trends are evident. The Mishna’s thinking is more straightforward; the Gemara’s thinking is more tortuous. In the Mishna, R. Tarfon puts forward an argument in support of his contention that the legal liability for damage by an ox on private property ought to be full compensation. This argument is not accepted by his colleagues, the Sages, apparently because it relies on proportionality. R. Tarfon then very skillfully proposes an alternative argument, which is not open to such objection. The Sages nevertheless reject the latter argument, apparently by resorting to another kind of objection.

  1. Tarfon’s two arguments are traditionally presumed to be qal vachomer, i.e. a fortiori arguments, although just what that means (besides the descriptive name) is nowhere defined. In fact, looking at these arguments very objectively, they could be interpreted as arguments by analogy or more precisely as arguments pro rata, or as arguments a crescendo (i.e. proportional a fortiori) or as purely a fortiori arguments. Moreover, there is no attempt to theoretically validate these arguments. But in any event, they are intuitively quite reasonable; and it seems from the text that it is on this logical basis that R. Tarfon advocates them.

The Sages’ objections, labeled dayo (from their opening word, which means “it is enough”) are not likewise justified by any theoretical discussion. What is clear after our detailed analysis is that they are not essentially logical objections; they are not indicative of breaches of deductive logic, though they might be postulated to signify some inductive restraint. They should rather be viewed as arbitrary decisions (I here use the term ‘arbitrary’ non-pejoratively, in the sense of ‘resorting to arbitration’) by the Sages themselves, based on certain ethical considerations. It can reasonably be doubted that the Sages are here evoking some ancient tradition, perhaps a teaching dating back from Sinai, because R. Tarfon, their colleague and equal, evidently does not preemptively take it into consideration in his two arguments.

Turning now to the Gemara, i.e. the later Talmudic commentary on this passage of the Mishna, we find a very different frame of mind. One would expect the Gemara to initiate a thorough theoretical reflection on R. Tarfon’s two lines of reasoning and the difference in the Sages’ dayo objections to them. But no; the Gemara ignores these burning issues and goes off on a tangent, focusing on the relatively not very relevant issue of the distance between R. Tarfon’s and the Sages’ positions. Apparently, the Gemara’s only concern here is whether R. Tarfon knew and agreed with the Sages’ dayo considerations. Obviously, he could not have fully agreed with them, since his conclusions differ from theirs; so the question is how far their views on the dayo principle differ.

In pursuit of the answer to that question, the Gemara engages is a very complicated scenario of its own, according to which R. Tarfon advocated a more conditional dayo principle than the Sages did. Briefly put, it proposes a distinction (which it attributes to R. Tarfon ex post facto) between applications of the dayo principle that “would defeat the purpose of” the qal vachomer and those that “would not defeat” it. In the former case, the ‘proportional’ gain made possible by an a fortiori argument (taken by the Gemara, on the authority of a baraita, to mean a crescendo argument) would be wiped out by dayo, so it should not be applied; whereas in the latter case, it would not by wiped out by dayo, so it may be applied.

In defense of this fanciful scenario, the Gemara proposes different readings of a Torah text, viz. Num. 12:14-15, by R. Tarfon and the Sages. However, both these readings are far removed from the plain meaning of the text, in that they do not take all of it into consideration. Most important, the view attributed by the Gemara to R. Tarfon assumes an a fortiori argument to be intended in the text while discarding the verses that would justify such assumption! It thus mendaciously infers an a crescendo conclusion of fourteen days ex nihilo, instead of with reference to the textual given of seven days. This means that the Gemara’s whole idea, of a distinction between applications of the dayo principle that “would defeat the purpose of” the qal vachomer and those that “would not defeat” it, is an outright deception. The bottom line is that the Gemara in fact fails to achieve its stated goal of harmonizing the opinions of the Mishna contestants.

Now, this is a bit of a shock, but not too astonishing. Anyone who has studied the Gemara to any extent can see for himself that its thinking, though based on the Mishna to some extent, is often more convoluted and open to doubt. Of course, more fundamentalist readers would never agree with such an assessment, but instead insist that in such cases the Gemara has intellectual intentions and ways too sublime for us ornery folk to grasp. But we, while making no claim to infallibility or omniscience, do claim to be honest and lucid, and stand by what is evident to the senses and to reason. In the present case, the Gemara’s ideas must obviously not be confused with the discourses found in the Mishna. With regard to this, the following general comment of Louis Jacobs in Rabbinic Thought in the Talmud (pp. 17-18) is apropos:

“A much discussed question is whether the interpretations of the Mishnah found in the Gemara are really a reading of ideas into the Mishnah or whether they are authentic accounts of what the Mishnah itself intended. Now students of the Mishnah in the Middle Ages noted that some, at least, of the Gemara’s interpretations of the Mishnah are so far-fetched and artificial that they cannot possibly be accepted as real interpretations of what the Mishnah intends to say, which is why Maimonides and other early commentators were prepared to disregard the Gemara to interpret the Mishnah on its own terms. To conclude from this that the Gemara has, at times, ‘misunderstood’ the Mishnah is precarious. It is possible that the Gemara, at times, consciously departs from the plain meaning of the Mishnah in order to produce its own original work….”

There is no harm, in our view, in producing original work, provided it is openly acknowledged as such. Unfortunately, the traditionalist’s way of thinking is that what he reads into a text must have been intended in the original; to him, interpretation is a sort of deduction. This is applicable at all levels – from the Mishna reading meanings into the Torah, to the Gemara reading them into the Mishna, to later commentators reading them into the Talmud[240]. And this applies to both halakhic and haggadic material. What is sorely needed to cure this serious intellectual malady is to understand the inductive nature of interpretation. An interpretation is a theory designed to fit the ‘facts’ that the given text constitutes. Its logical status is that of a hypothesis, which may and probably does have competing hypotheses. Rarely is an explanation the only conceivable hypothesis, though this happens occasionally. Therefore, a reading should always be acknowledged to be one possible interpretation, even if it fits the given data.

But in the case under consideration, as we have definitely shown, the Gemara’s proposed interpretation of the Mishna simply does not convince. It is not a credible theory, because it is built on illusion, on make-believe. Furthermore, the Gemara does not demonstrate its having noticed and understood the differences between R. Tarfon’s two arguments and between the Sages’ two dayo objections to them. That later commentators have projected such understanding into the Gemara does not prove that the Gemara in fact had it, only at best that it might have. Such ex post facto attribution of knowledge to the Gemara is only evidence of the faith later commentators had in it. The Gemara itself does not explicitly remark on these crucial issues, nor even implicitly suggest them.

It is interesting to note that, whereas the Mishna participants are involved in a purely legal debate, without stepping aside and reflecting on the methodological issues it implies, the Gemara does, in an attempt to clarify the primary, legal discussion, initiate a secondary, more methodological reflection. The latter discourse is intended as an accessory to the former, in that the legal conclusions that might be drawn depend on the methodological lessons learned. Thus, we can say that there is in the Gemara a sugya within a sugya, or there are two intertwined layers of discussion – the main one being legalistic in content, whereas the accessory one is methodological in content. The problem is that, although the Gemara could have used this opportunity to develop a deep reflection on the methodological issues involved, it disappointingly engaged in a very tangential and artificial discourse, driven by quite ideological considerations.

The present work being a treatise on logic, with an emphasis on a fortiori logic, our concern is naturally with the methodological topic of conversation; we are not really interested in the legal topic except possibly as an example. I personally have no legal axe to grind; I am not out to modify or overturn any halakhah. I certainly have no desire to put down anyone, either. Our interest in this research is relatively abstract, and certainly impartial. Our present study is aimed at logic theory and history; it is not essentially Talmudic in orientation, in contradistinction to the rabbis, whose main interest is always legal rather than logical. Nevertheless, we had to consider the legal debate in some detail, since it houses information we needed for historical purposes and to empirically judge the level of rabbinical understanding of a fortiori reasoning.

The author of the Gemara commentary we have studied is obviously someone with an intelligent, imaginative and logically sharp mind. But it is not an entirely scientific mind, which frankly considers all alternatives, lays out all the pros and cons, and judges the matter fairly in accord with objective standards. It is an authoritarian mind, which therefore functions to some extent manipulatively. The Gemara’s author does not derive a conclusion from given premises in an unbiased manner; he starts with a desired conclusion and proceeds to give the impression of having proven it by intricate argument. He is satisfied with the result, even though he in fact did not prove it, either because he fools himself or because he assumes no one would notice the logical trickery involved in his argument and dare cry foul. In the latter event, he relies on the psychology denounced in H. C. Anderson’s The Emperor’s New Clothes.

What is evident looking at Baba Qama 25a is that the Mishna’s narrative and that of the Gemara are quite distinct. The Gemara presents itself as a mere conduit, authoritatively clarifying and explaining the Mishna – but it speaks for itself alone. There is no evidence that it truly represents the views of R. Tarfon and the Sages. When the Gemara speaks in their names, it is just telling us what it thinks they said or meant. The thesis the Gemara presents must be treated as just a hypothesis, even as a mere speculation, since there is no way to establish its historicity. The dialogues it puts forward are imaginary. Its argument is rhetorical and not logical.

The Gemara’s doctrinal goal seems to be to reconcile the seemingly antagonistic positions of the participants in the Mishna, i.e. R. Tarfon and the Sages. This is in accord, we may remark in passing, with the general rabbinical dogma that everything a Talmudic rabbi (or indeed an important later rabbi) says is essentially right, even if it seems to conflict with what others say. This doctrine that “the Torah has seventy facets” presumably arose ex post facto, first implicitly and then explicitly, perhaps somewhere mid-course in the Talmud, maybe only in the Gemara (when exactly, I do not know[241]). In any case, it clearly plays an active role in the present portion of the Gemara, and this is important to keep in mind.

To repeat, the Gemara’s treatment of the Mishna is quite superficial, failing to spot and take into consideration important details in the proof-text. The Gemara cheerfully refers to the Miriam story as its model for understanding the Mishna, failing to notice that though this passage of the Torah can be used to throw light on the first argument of the Mishna, it is useless with regard to the second. Moreover, the Gemara not having even tried to make a preparatory theoretical analysis of qal vachomer, fails to realize the different possibilities of interpretation inherent in the Mishna. It takes for granted without reflection that the qal vachomer inferences in the Mishna are all ‘proportional’, and does not see the possibility in it of purely a fortiori arguments or even non a fortiori arguments.

The Gemara then embarks on a quite abstruse theory of qal vachomer, which it attributes to R. Tarfon, according to which (in positive subjectal a fortiori argument) a conclusion can only be drawn from a minor premise with the same subject. The Gemara does not notice that this imagined narrative is not in accord with what is explicitly given in the Mishna, let alone realize that it has no basis in formal logic. It does this to justify making a distinction between the argument implied in the Miriam story and the argument (it only perceives one, the first) given in the Mishna, so as to explain the difference of opinion there between R. Tarfon and the Sages in relation to a presumed dayo principle. Furthermore, in attempting to depict this theory, the Gemara has R. Tarfon drawing an alleged qal vachomer conclusion from no premises at all when he applies it to the Miriam story.

The Gemara’s general idea that the Miriam story contains an a crescendo conclusion of fourteen days restricted by a dayo principle to seven days (rather than a straight a fortiori argument with a conclusion of seven days) is still not unthinkable, note well. Even though the Gemara does not admit v. 14a (about offending one’s father) as a premise of this argument, and takes v. 14b (about seven days quarantine) as its final conclusion, the argument can be imagined as occurring in between these verses. It must however be stressed that, contrary to what the Gemara claims, there is no actual concrete hint of this scenario in the Biblical text. Even a purely (i.e. non-proportional) a fortiori reading is open to debate; all the more so an a crescendo one. Consequently, any claim that the passage points to a dayo principle is also open to debate.

The a fortiori reading is not inevitable; but it is a reasonable assumption, provided it is made to explain the connection between the first and second part of v. 14. Note well that the a fortiori argument is not just used to infer a number of days, but especially the punishment of isolation away from the community. The seven days prescribed are only a qualification of this predicate, serving to quantify the penalty; they are not the main issue of the argument. Thus, the final exchange in the Gemara between R. Tarfon and the Sages, regarding where in the Biblical text the alleged two sets of “seven days” come into play focuses on a side issue, diverting attention from the main one. The Gemara gives the impression that the qal vachomer is all about numbers of days; this is misleading.

Our wisest course is to blame the Gemara alone for these various rationalizations. The Gemara is plainly indulging in sophistry, masquerading as rational discourse. Its narrative is an obvious and absurd invention, which has little to do with the Mishna’s. If we accept the scenario the Gemara advocates, we would be unfairly imputing the errors of reasoning it commits to R. Tarfon and the Sages. We cannot justify lumping together the players in the Mishna with those perhaps two or three centuries later (and some five hundred miles away) in the Gemara, just out of some ideological desire to make them appear to all speak with one voice. It is better to blame the author of this Gemara in particular for them than to insist they are true and embarrass everyone else! These very critical remarks of mine are sure to revolt traditional Talmudists, but they are unavoidable.[242]

I am, of course, well aware that such statements undermine rabbinical authority. We can say, having found such casuistry, that the rabbis are not always right, i.e. that their logic is not infallible. But I knew this already, having uncovered much problematic reasoning by them in the course of my earlier research on rabbinic hermeneutics, as detailed in my Judaic Logic[243]. Many more instances are uncovered in the present work; see for examples the fallacies discussed in chapters 3.4, 9.7, and especially 18.2.

The issue is how often do they reason incorrectly? This question cannot be answered offhand but requires systematic and thorough research throughout Talmudic literature – by competent researchers, I might add (for someone who does not know logic much better than them cannot judge theirs). If errors are only occasional, that is surely not too serious, since we are all human beings with limitations; if they are very frequent, that is certainly quite serious, since some inexcusable negligence is involved. It might be possible to lay the blame for all or most errors found on some specific rabbis. This would somewhat improve the logical credibility of the rabbis collectively, although we could still wonder why the errors were not spotted and corrected by other rabbis.

In any case, our approach as logicians must be objective and impartial, and not swayed by any imagined or actual threat of hostility and rejection. From a metaphysical point of view, if God is the ultimate reality of the world we experience, and the meaning of human life is to tend towards Him, then truth is a paramount value and honesty is an indispensable virtue. There is no rational excuse for evading or stonewalling, let alone opposing and denigrating, just criticism. It would be unrealistic to expect utter perfection from any human being, even if he is an important rabbi. When we come across logical faults, we should not deny them, but humbly admit them and try to correct them. While some might consider criticism of rabbinical arguments as cause for condemnation, we should rather view such events as welcome opportunities for improvement.

By this I mean that once we realize and admit that Talmudic and more broadly rabbinic logic is not inerrant – but sometimes debatable, contrived or erroneous – we open a safe door to halakhic review and revision. This of course cannot be taken as a blanket license for general change in Judaism as convenient; but there may be circumscribed opportunities for evolution based on ad hoc logical analysis. For the law must surely be in accord not only with empirical scientific knowledge of nature and history, but also with logic. Just as ignorance of the former is bound to lead to error in law, so is faulty logic also bound to lead to such error.

One of the major rabbinical authorities of modern times, R. Moses Sofer (Germany, 1762-1839) wrote this about logic (higayon): “whoever mixes words of logic with matters of Torah offends against the law of: ‘Thou shalt not plough with an ox and an ass together’ [Deut. 22:10].”[244] But logic is not, as this farfetched statement suggests, something arbitrary that we have a choice about using or not. Mentally, we are of course able disregard it; but intellectually, if we are honest, we cannot do that, because logic is our main means for verifying and certifying the truthfulness and consistency of our judgments. If any verse of the Torah is to be brought to bear in this matter, it is rather this: “Thou shalt not have in thy bag diverse weights” (Deut. 25:13). But there is no need for that; it is obvious.

Apologists for religion reproach secular scientific knowledge of nature and history for varying in time. They suggest that such variation is proof of its unreliability. But this is of course a spurious argument. Scientific knowledge varies because it is essentially inductive, freely and dynamically adapting to new empirical discovery and rational review. This is not a fault or weakness – it is the very virtue of science. The truly scientific view[245] at any point in time is comparatively the best hypothesis human beings as a group have to offer. That it may later change does not make it any the less ‘the best’ at the time concerned. Certainly, it is always better than a static hypothesis based on religious dogma that is out of touch with empirical fact and rational scrutiny.

Browbeating is not a form of proof. Religion must learn to humbly adapt to scientific change. This would certainly not be the end of religion, because religion is a necessary expression and instrument of human spirituality. See how those who lost it suffer, from the lack of direction in their lives. Just as science makes possible the accumulation and transmission of human knowledge of nature and history, so religion makes possible the accumulation and transmission of human knowledge of spirituality. Of course, the latter tends to be more plural than the former, because spirituality allows for many paths. But in any case, whatever the chosen path, empirical science and logic must be taken into consideration to ensure its full truth.

8.    The syllogistic Midot

As regards syllogism, it is also naturally found in rabbinic thinking and even within many of their hermeneutic techniques (midot). This is said to contradict the claim of many commentators that none of the rabbinical hermeneutic techniques are syllogistic.

This position, for instance, is to some extent found in Louis Jacobs’ treatment, insofar as the only rabbinic argument he sees as syllogistic is the one referred to as ha-kol. In his Studies (1961), he says: “There is a form of Talmudic reasoning which has no connection with the qal wa-chomer but bears a remarkable affinity to the Syllogism;” and he goes further in a footnote, saying: “the ‘ha-kol’ formula… is identical with the Syllogism,” and giving as example the following argument implied in Avot 6:3:

“He who learns from his fellow has to pay him honour;

I have learned from my fellow;

Therefore I am obliged to pay him honour.”

Michael Avraham, for his part, asserts categorically (in the English abstract of a 1992 Hebrew paper) that none of the 13 principles of R. Ishmael are syllogistic; as he puts it: “the Kal Vachomer – like the rest of the 13 ‘Middot’ – is not a syllogism” (my italics). This opinion is apparently not new, judging by a statement made by Aviram Ravitsky (my italics):

“Maimonides viewed most of the halakhic world as conventional, and this view enabled him to treat the halakhic arguments as dialectical ones, although he did not think that halakhic arguments could be reduced to syllogistic figures.” [246]

But in my Judaic Logic (1995), I show that many of the thirteen midot of R. Ishmael involve syllogistic thought processes. For a start, a fortiori argument is in part based on hypothetical syllogism. Syllogistic reasoning is implicit in the midot dealing with the scope of terms, collectively called klalim uphratim (rules 4-7), insofar as these have to do with subsumption and exclusion of cases in classes[247]. But more to the point, most of the midot dealing with harmonization (specifically the rules 8-11) are clearly syllogistic, so much so that they can be represented and resolved diagrammatically. While my work on a fortiori argument has attracted some attention, my work on these harmonization midot has apparently not been noticed. For this reason, I think it useful to reiterate some of these findings in the present context, to show how a lot of rabbinic thinking is syllogistic.

The first three (actually, four) of the principles of R. Ishmael concerned with harmonization begin with the phrase kol davar shehayah bikhlal veyatsa…, meaning literally “anything which was in a generality and came out…”. Broadly put, in formal terms, these rules are concerned with the following exegetic situation:

Given the three premises (#s 1, 2, 3), common to the four harmonization rules 8a, 8b, 9, 10:

All S1 are P1                                                                (common major premise, #1),

and All S2 are P2                                                          (common minor premise, #2),

where All S2 are S1, but not all S1 are S2                      (common subjectal premise, #3),

and the fourth premise (#4), as applicable in each of these rules:

P1 and P2 are in some relation f{P1, P2}  (d)                 (distinctive predicatal premise, #4):

  • In rule No. 8a, P2 implies but is not implied by P1; that is:

All P2 are P1, but not all P1 are P2.

  • In rule No. 8b, P1 implies P2 (and P2 may or not imply P1); that is:

All P1 are P2 (whether All P2 are P1 or some P2 are not P1).

  • In rule No. 9, P1 and P2 are otherwise compatible; that is:

Some P1 are P2 and some P1 are not P2; some P2 are P1 and some P2 are not P1.[248]

  • In rule No. 10, P1 and P2 are incompatible; that is:

No P1 is P2 and No P2 is P1.[249]

What, other than the above given, are resulting relations (conclusions)?

Between S1 and P2 (this is the primary issue, #5);

and (secondarily) between S2 and P1, and between S1 and P1, and between S2 and P2.

We can for a start, by means of syllogism, draw the following conclusions, common to all four rules, from the first three premises, without reference to the fourth premise:

  • From the minor and subjectal premises, Some S1 are P2             (mood 3/AAI).
  • From the major and subjectal premises, All S2 are P1 (mood 1/AAA).
  • From the major and subjectal premises, Some P1 are not S2 (mood 3/OAO).

What this means is that, no matter which predicatal premise is used, it cannot logically yield a conclusion incompatible with ‘Some S1 are P2’. The following specifies what can additionally be said in each of the four rules under scrutiny (the sources and discussion of the examples here proposed are given ad loc. in my Judaic Logic):

In rule No. 8a, nothing further about S1 and P2 can be deductively inferred; yet R. Ishmael apparently claims ‘All S1 are P2’ (which is too much). For example: A sorceress (or by extension, a sorcerer) is liable to the death penalty (#1); a male or female medium or necromancer is liable to death by stoning (#2); a male or female medium or necromancer is a sorcerer or sorceress (#3); death by stoning is a species of death penalty (#4); therefore, all sorts of sorcerers or sorceresses are liable to be stoned (#5).

In rule No. 8b, we can syllogistically infer (mood 1/AAA) that ‘All S1 are P2’; yet R. Ishmael apparently claims ‘Some S1 are not P2’ (which is inconsistent). For example: whoever approaches holy offerings while impure is liable to the penalty of excision (#1); anyone who eats peace-offerings while impure is liable to the penalty of excision (#2); peace-offerings are holy offerings (#3); the penalty is the same in both cases, viz. excision (#4); therefore, the consumption of offerings of lesser holiness than peace-offerings is not subject to the penalty of excision (#5).

In rule No. 9, we can syllogistically infer (mood 2/AOO) that ‘Some P2 are not S1’; it is not clear how R. Ishmael’s proposed conclusion here should be presented in formal terms (such lack of clarity being of course a deficiency). I have not found a sufficiently informative example of application of this rule[250].

In rule No. 10, the predicatal premise is logically incompatible with the other three premises, so no syllogistic inference is possible; R. Ishmael apparently resolved the conflict by modifying the major premise to read ‘Some, but not all, S1 are P1’ (which is logically acceptable, though not the only option open to us). For example: the release of a Hebrew slave is subject to a certain set of laws (#1); the release of a daughter sold as maid-servant is subject to another set of laws (#2); a daughter sold as maid-servant is nominally a subcategory of Hebrew slave (#3); yet, the laws for the maid-servant and those for the Hebrew slave in general are very different (#4); therefore, the category of Hebrew slave intended here is in fact not so broad as to subsume such maid-servants (#5).

From these reflections, we learn that at least four of the rules of R. Ishmael (as I have tentatively interpreted them, based on a small number of examples) are syllogistic in form. These four all include at least the syllogism: ‘All S2 are S1 and All S2 are P2, therefore Some S1 are P2’ (3/AAI). Two of the four involve an additional syllogism (of form 1/AAA in rule 8b, and of form 2/AOO in rule 9); one rule involves no additional syllogism (rule 8a); and the fourth rule involves inconsistent premises. It is interesting to note that R. Ishmael’s apparent solutions to these four syllogistic problems are in some way or other deficient. Nevertheless, it does not change the fact that these four rules are essentially of a syllogistic nature.[251]

I have also demonstrated, in an earlier chapter of the present volume, in the section on analogical argument (5.1), the presence of syllogistic thinking in rabbinic analogical arguments, namely in rule 2 (gezerah shavah), rule 3 (binyan av) and rule 12 (meinyano and misofo) in R. Ishmael’s list. These arguments are not solely syllogistic – they involve inductive processes too – but they definitely do include syllogism. These findings are indubitable, and they put to rest once and for all the rather widespread notion in some quarters that the rabbinic hermeneutic principles do not depend on syllogistic reasoning.

Syllogism can, I suspect, be discerned in yet other midot, if we examine them closely enough. I would go much further than that, and assert that these examples drawn from the 13 midot are only the tip of the iceberg. The midot are by far not a full listing of the reasoning processes actually used by the rabbis; it is certain many of their actual reasoning processes are not included in their listings. The listings only bring together certain forms of thought which the rabbis considered worthy of notice and emphasis for some reason. But like all human beings, they used many thought processes unconsciously – including the process of syllogism. It is impossible for anyone to reason without certain basic thought forms; and the syllogism is definitely one of these unavoidable thought forms, since it is required for all mental acts of inclusion or exclusion.

9.    Historical questions

There is, I would say, a significant difference between a fortiori use in Talmudic contexts its use in other ancient literature, such as in Platonic or Aristotelian texts[252]. In the latter, it is probably more accurate to speak of a fortiori discourse rather than a fortiori argument, because it is used more as a rhetorical device than as a form of reasoning. The author in such cases could well have rephrased his text in such a way as to pass the same message without using a fortiori language. Whereas, in Talmudic contexts, the use of a fortiori is definitely argumentative; it is necessary to prove something that has a legal impact and that could not be arrived at by other means. So when we speak of a fortiori use in the Talmud, we are referring to something much more serious.

When we speak of Talmudic and rabbinic logic, we must have in mind and look for both explicit theories and implicit practices by people concerned. Theorizing has different levels: just being aware that one is engaged in an argument is one level; the next higher level is awareness that the argument is of a certain peculiar type, and a name is assigned to it (such as ‘a fortiori argument’); the third level consists in attempting to give form to such argument, using symbols in the place of terms; and the highest level is wondering at the argument’s validity and seeking to establish it once and for all. Study of the history of an argument is also theory, though of a more intellectual-cultural sort. As regards practice, it may be far ahead of theory. Theory can improve further practice, but is generally based on prior ‘intuitive’ practice. Therefore any investigation that aims to understand the logic of some group of people or humanity in general must focus strongly on actual practices.

Even if much conscious research has been carried out on Talmudic and rabbinic logic (including hermeneutics), I wager that there is still a lot to discover in this field. We shall never arrive at an accurate, scientific history – or indeed, theory – of Talmudic logic, and in particular of Talmudic a fortiori logic, without a thorough, systematic listing and competent analysis of all the arguments in the Talmud and related texts. Someone has to do this major work some day; or else we shall always be dealing in rough hypotheses based on limited samples.

Take for instance a fortiori argument, which is our object of study here. What we need, for a start, is a table listing all the apparent cases of a fortiori argument. In each case, we should note its location in the Talmud, and who (named or unnamed) is apparently formulating it, so that the best estimate of its date can later to be put forward. We must distinguish the person(s) formulating it from the person(s) commenting upon it in subsequent developments of the Talmud (by which I mean here, the two Talmuds, that of the Land of Israel and the Babylonian, and related contemporary literature).

Each argument must be analyzed, first by classifying it, i.e. identifying which of the eight standard moods it fits into. Is it copulative or implicational? Label the major term (or thesis) P, the minor term (or thesis) Q, the middle term (or thesis) R and the subsidiary term (or thesis) S. Is the argument subjectal (or antecedental) or predicatal (or consequental)? Is it positive or negative? Moreover, is the argument purely a fortiori or a crescendo? Was dayo applied? Very often, some creative work will be necessary, insofar as the a fortiori argument is not entirely explicit. It may, for instance, be necessary to construct an appropriate major premise, and the operative middle term (or thesis) may have to be suggested by the researcher. Obviously, any such contributions to the argument made by the researcher must be noted as such and not confused with the raw data. That is to say, the fact that the original argument is in some way incompletely formulated is a significant detail of the analysis.

Once we have such an exhaustive database of the a fortiori arguments in the Talmud, we can begin to develop a truly scientific account of this argument form in that document. We can say with certainty what moods (if not all) of the argument were known to the named participants and anonymous redactors, and how well they understood them. We can compare the logical skills in this domain of the different players involved. We can find out more precisely what their theoretical understanding of a fortiori forms were and what terminology they used for them. We must not forget that the Talmud is a document built-up over centuries, by hundreds of people. The Talmud is not a monolith, but has many temporal and geographical layers[253]. Therefore, research must also try to trace the development of skills and understanding of a fortiori and other argumentation across time and place.

We can also more accurately compare Talmudic use and knowledge of the a fortiori argument to use and knowledge in surrounding cultures – notably the Greek and Roman as regards the Mishna and the Jerusalem Talmud, and possibly further afield as regards the Gemara of the Babylonian Talmud, since it was developed in Babylon where perhaps some Indian influences might have occurred. This too, of course, has a long timeline. Take for example the distinction between miqal le chomer (from minor to major) and michomer leqal (from major to minor). This distinction is taken for granted today – but it surely has a rich history. Does it appear anywhere in the Talmud, or is it a later discovery? It is found in later rabbinic literature – but the question is when and where did it first appear? Was this an independent Jewish discovery, or can Greek or Roman or other ancient influence, or later on Christian or Moslem influence, be traced? If the distinction is made in the Talmud, just when and by whom, in what context(s)?

This distinction, note well, signifies some level of conceptual analysis of a fortiori reasoning. But it is still relatively vague or equivocal, insofar as ‘minor to major’ can signify either positive subjectal or negative predicatal argument, while ‘major to minor’ can signify either negative subjectal or positive predicatal argument[254]. A question to ask is, therefore: what was the original intent of this distinction – was it meant as a distinction between positive and negative moods of subjectal a fortiori argument, or was it a distinction between positive subjectal and positive predicatal arguments, or was there awareness of all four possibilities, or did it remain vague? Indeed, granting that positive subjectal argument is the most obvious and widespread form, when were negative subjectal and positive and negative predicatal argument forms first realized in Judaic logic (or elsewhere, for that matter)?

And so on. There are evidently many questions worth asking and the answers cannot be settled till we have a thorough database, as already said. It should be noted that today, with the digitalization of most ancient texts well on the way if not already completed, the job is immensely facilitated, since exhaustive searches of different verbal strings are possible in a jiffy and information can be cut and pasted without difficulty! The historical work and the logical analysis involved may or may not be done by the same person(s). The ideal scholar would be a good wide-ranging historical researcher, knowledgeable and at ease in the Talmud and other significant texts in the original languages, and a good logician to boot. These qualities are not necessarily all found in the same person, but a multi-disciplinary team might be constituted by a university. I do hope some people someday realize the need and value of such research and organize a determined effort in that direction.

9.  Post-Talmudic rabbis

1.    Logic and history issues

In the present chapter, our object shall be to discuss and to some extent trace some of the developments in rabbinic and more broadly Jewish thought concerning the a fortiori argument, and to a lesser extent more broadly the hermeneutic principles. This is of course a massive task that we cannot remotely hope to carry out exhaustively in the present study; we can however hopefully reflect on some of the issues involved and give scattered examples of the kind of research and evaluation that are needed in this context.

The first thing to make clear is the distinction between hermeneutic and logical principles. Although the rabbis to some extent regarded their hermeneutic principles as logical principles, the truth is that logic was not a prime interest for them: their primary interest was in justifying the traditional legal system enshrined in the Mishna and expanded on in the Gemara and subsequently. I will not here even try to roughly trace the development of Jewish law from its Biblical beginnings, through the formative period from Ezra to the Mishna, followed by the Gemara and later rabbinic work. I can only recommend to the reader who has not already done so to read works (preferably critical) on the subject, such as Mielziner’s Introduction to the Talmud. The important thing, in the present context, is to take to heart what Mielziner writes regarding the “circumstances that necessitated artificial interpretation”:

“As long as the validity of this oral law had not been questioned, there was no need of founding it on a Scriptural basis. It stood on its own footing, and was shielded by the authority of tradition. From the time however when the Sadducean ideas began to spread, which tended to undermine the authority of the traditional law and reject everything not founded on the Scriptures, the effort was made by the teachers to place the traditions under the shield of the word to the Thora. To accomplish this task, the plain and natural interpretation did not always suffice. More artificial methods had to be devised by which the sphere of the written law could be extended so as to offer a basis and support for every traditional law, and, at the same time, to enrich the substance of this law with new provisions for cases not yet provided for. This artificial interpretation which originated in the urgent desire to ingraft the traditions on the stem of Scripture or harmonize the oral with the written law, could, of course, in many instances not be effected without strained constructions and the exercise of some violence on the biblical text…” (pp. 120-121).[255]

Two ideas should be emphasized in this context. The first is that the hermeneutic principles have a history. They did not come out of the blue all of a sudden, whether at Sinai or later, but were gradually developed in response to specific needs by specific persons, and often against conflicting opinions by other persons. Changes evidently occurred over time. This development can be precisely traced to some extent, even though traditional commentators make every effort to deny significance to the known history. The second idea is that the hermeneutic principles are not necessarily logical. Mielziner rightly refers to “artificial” as against “natural” methods, using exactly the same terms as I did independently fifteen years ago when I wrote my Judaic Logic. In that work, I showed, clearly and by formal means, to what extent the hermeneutic principles could be regarded as logical and to what extent they could not. Mielziner, in his reference to “strained constructions and the exercise of some violence on the biblical text,” had the honesty and courage to admit the limits of rabbinic logic.

In the present work, following detailed logical analyses mainly of the Mishna Baba Qama 2:5 and the related Gemara Baba Qama 25a, I have developed a more precise assessment of Talmudic logic. It appeared from this exploration that the a fortiori logic found in the Mishna is more natural, less artificial, than that found in the Gemara. Judging from the Talmudic passage we examined, the understanding of a fortiori argument by the earlier rabbis was simpler and more straightforward, while that of the later rabbis was more complicated and tortuous. The two groups should not be lumped together. This is as regards their practice; neither group engages in much theoretical reflection (if any) on the subject. So the artificiality that Mielziner speaks of is more centered in the Gemara than in the Mishna (at least as regards a fortiori argument).

What is clear from our research is that it is misleading and futile to try to interpret and justify the rabbinical hermeneutic principles entirely through logic. They undoubtedly have some logical character, and are often thought of and intended as logic, but they are not purely and entirely logical. They are, as Mielziner well described them, ad hoc responses to the problem of anchoring the oral law so-called, i.e. the Jewish legal tradition existing at a certain period of history, in the more authoritative written Torah. Sometimes that anchoring is possible by quite natural (i.e. purely logical) means; but sometimes some intellectual artifices are necessary to achieve the desired end. With this frank admission in mind, we can more clearly trace the history of commentaries on the hermeneutic principles and practices in general, and on the a fortiori argument in particular, from two points of view.

The first viewpoint is that of the uncritical traditionalists. Their writings or lectures on the a fortiori argument or on hermeneutics are simply designed to pass on as clearly as possible the information received from tradition. This teaching is presumed true and valid without question, and the only role of the teacher is to clarify it and give examples of it. The second viewpoint is that of the critical logicians, among which I count myself. Their written or oral reflections on the subject are aimed at scientific evaluation, and are therefore perforce more formal and not necessarily in agreement with tradition. Truth and validity are not automatically granted, simply because the argument in question is claimed to be, directly or indirectly, of Divine or prophetic origin, or to have the stamp of approval by rabbinical or whatever authorities. These two viewpoints are pretty well bound to be at odds in some cases, though not in all.

Many indices can be used as litmus tests for the classification of a commentator in one camp or the other. We must look and see where each commentator stands in relation to the debate between R. Tarfon and the Sages in the Mishna; how he perceives the argument(s) of the former and the objection(s) of the latter. We must also pay attention to his eventual reactions to the Gemara: to its general equation (on the basis of a baraita) of a crescendo argument with a fortiori; to its readings of the argument about the isolation of Miriam in Num. 12:14-15; to its claims about R. Tarfon’s ideas about when the dayo principle may or may not be applied to an a fortiori argument. In short, we must look out for the depth and breadth of a commentator’s awareness of the issues involved. Certain authors will judge such matters dogmatically: they are the traditionalists. Others will be more circumspect: to the extent they are so, they belong to the critical school.

That is our theoretical stance; but in practice, as we shall presently discover, there is rarely need to get that fancy, because most commentators on the a fortiori argument treat the issues relatively superficially.

2.    Philo of Alexandria

Philo of Alexandria, aka Philo Judaeus or the Jew (20 BCE – 50 CE), has some importance in the history of philosophy, because he embodies an early confluence between Jewish religion (and philosophy) and Greek (religion and) philosophy, and also of course because he later had considerable influence on Christian religion (and philosophy). He interpreted the Jewish Bible (the Septuagint version, in Greek) allegorically, by adaptation of Greek and Hellenistic ideas (derived from works of Plato, Aristotle, and the Stoics) as well as reference to Jewish traditions.

I include him here because he lived, though outside of the Holy Land, in the Mishnaic period. According to Encyclopedia Judaica article on him, there is no mention of Philo is any ancient Jewish source other than Josephus (Antiquities 18:8). This could mean that mentions of him by rabbis of the Mishnaic and/or Talmudic periods have been lost; or that they did not know of his work; or that they knew of his work but disapproved of it. Philo has occasionally anticipated positions that rabbis in the Talmud later adopted, presumably independently, since they did not acknowledge his authorship. He was subsequently (much later) not considered by the rabbis as particularly kosher, despite his evidently intense faith in God and the Torah, and his services to the Jewish community of Alexandria[256], due no doubt partly to the Greek influences evident in his views and partly because Christians eventually adopted many of them.

Philo did not, to my knowledge, discuss a fortiori argument, but he did use it. I found one instance of a fortiori argument in a work by Philo, namely De Fuga et Inventione[257] (§84, p. 163), translated from the Greek original into French by Esther Starobinski-Safran: “No blasphemer has a right to forgiveness. For if those that cursed their mortal parents are chastised, what punishment ought to be meted to those who dare blaspheme the Father and Creator of the universe?” The minor premise of this argument comes from the Biblical passage: “May he who curses his father and his mother perish” (Exodus 21:17).[258]

The reference to blasphemy in the conclusion is derived from the earlier Biblical passage: “And if a man come presumptuously upon his neighbor, to slay him with guile; thou shalt take him from Mine altar, that he may die” (Exodus 21:14). According to Philo, an intentional murderer seeking refuge at the altar is effectively guilty of “blasphemy,” since by this gesture he blames God for his sins. The “blasphemy” perceived by Philo is, note well, not explicit, but implicit in the blame of God, which is in turn implicit in the act of seeking refuge with Him albeit guilty of intentional murder (as explained in §79-80)[259].

The proposed argument is thus: if someone who curses his parents deserves punishment, then someone who blasphemes God deserves punishment – possibly greater punishment[260]. Evidently, the argument is positive subjectal; and possibly a crescendo (note that Philo only asks “what punishment?” – he does not explicitly give an answer, whether equal or greater). Notice its similarity to the argument in the Torah concerning Miriam (Num. 12:14-15), which also proceeds from parents to God. I have not looked for use of a fortiori argument by Philo in other books.

I do not know why Philo finds it necessary to infer from the penalty for cursing parents an equal (or maybe greater) penalty for blaspheming God, when the Torah already states, in Leviticus 24:15-16: “Whoever curseth his God shall bear his sin. And he that blasphemeth the name of the Lord, he shall surely be put to death; all the congregation shall certainly stone him….” Perhaps he differentiated these two verses, since v. 15, which refers to cursing (Heb. yeqalel), does not mention a death penalty, whereas v. 16, which refers to blasphemy (Heb. noqev), does mention a death penalty[261]. But in that case, his purpose would be to prove (by a fortiori argument from parents) that the death penalty for cursing parents also applies to cursing God. But if so, why use the word “blaspheme” instead of “curse” in his stated argument? So there is some confusion or redundancy, here. In any case, note that he does not mention the death penalty at all in his argument, but only chastisement or punishment[262].

I should say that I noticed Philo’s a fortiori argument thanks to Starobinski-Safran, who in her introductory commentary (p. 38) reads the argument as: “If the outrage inflicted on parents is punished by [bodily] death (Ex. 21:15-16[263]), then the outrage to God, which is infinitely more grave, calls for death of the soul, an even heavier penalty” (again, my translation from the French). She does not specify ‘bodily’ death (my addition in square brackets), but we may assume this to be her intent from the explicit contrast to death “of the soul.” Note that the penalty for cursing parents is implicitly specified in Leviticus 20:9 as death by stoning (according to the rabbis, who point out that it there says: “his blood shall be upon him”), the same as the penalty explicitly specified in Lev. 21:16 for blasphemy. So as far as this-worldly punishment goes, the two cases are identical; but nothing is said (or denied) concerning other-worldly punishments.

The distinction here made by Starobinski-Safran between death (of the body) and death of the soul, and the proportionality of punishment she assumes, are not (as far as I can discern) explicitly given in Philo’s wording. The commentator is reading these details into the text. Perhaps she mentions “death of the soul” because no actual verbal blasphemy (meriting bodily death under Jewish law) has occurred in the case under consideration, but only blasphemy by implication (from the attempt to take refuge at the altar illicitly). She does not say what she precisely means by “death of the soul;” maybe she means the penalty of karet (usually translated as ‘excision’)? Certainly, even if Philo did have similar intentions in mind, the matter is open to debate, just as the intent of the argument concerning Miriam is debatable.

The trouble with allegory, as we see in this example, is that while it may provide a neat explanation and integration of the text at hand, there is a risk that it ends up being taken literally by people and thus have untoward legal consequences. The plain meaning of the Torah text referred to here is that an intentional killer who seeks refuge at the altar may be forcibly taken from there and executed anyway, because he committed murder and this crime is liable to the death penalty (Ex. 21:14). On the other hand, Philo allegorically explains the killer’s seeking refuge at the altar as effectively an act of blaming God for his sin, and thence accuses the killer of “blasphemy;” this gives the impression that when the killer is pulled away from the altar to be executed, he is being executed for “blasphemy;” but what he has done is not strictly-speaking blasphemy according to the law (Lev. 24:16).

An additional note. In today’s political context, when fundamentalist Moslems are trying to impose worldwide restrictions on the freedom of speech, by pressuring governments to outlaw what they call “blasphemy,” even by non-Moslems, and when many Western politicians, journalists, academics and sundry institutions, seem about ready to abjectly comply, it is important to add the following disclaimer. Whereas we frequently read in news bulletins about Moslems savagely lynching people they accuse of “blasphemy” (meaning, any doubt or criticism of Islamic beliefs), it must be stressed that Judaism – like Christianity – nowadays no longer applies past laws against blasphemy (meaning, speaking ill of God). Jews today believe in and are happy with the rights of thought, speech and conscience they have in the West, including Israel; and moreover they do not approve of capital punishment for blasphemy or other religious transgressions anywhere in the world. I am not aware of any segment of the Jewish population who would say otherwise.

The outlook of today’s Jews on Torah passages like those imposing the death penalty for blasphemy or for cursing parents is not literal, but symbolic. Their practical purpose is simply to make people aware of the great immorality of such acts. The more severe the penalty theoretically prescribed by the Torah for a certain act, the greater the implied immorality of that act. But the prescribed penalty is merely potential, not actual. That is, retribution for such immoral deeds is today de facto and de jure no longer a rabbinic responsibility; it is left to God[264]. Thus, the aim of such passages is no more than moral education – to raise awareness of the full weight of thoughts, words and deeds. Each person remains free to choose – and, of course, to pay the natural consequences for his or her choice (including Divine retribution, if God so chooses). And that is a good thing. We certainly do not want the development of a Taliban mentality in our community, and most of us would surely combat it as regressive nonsense if it ever arose.

In this context, it is worth citing the Mishna: “A Sanhedrin that effects an execution once in seven years, is branded a destructive tribunal. R. Eliezer b. Azariah says: once in seventy years. R. Tarfon and R. Akiba say: were we members of a Sanhedrin, no person would ever be put to death. [Thereupon] Rabban Simeon b. Gamaliel remarked, [yea] and they would also multiply shedders of blood in Israel!” (Makkoth 1:10)[265]. For most crimes deserving capital punishment, with the possible exception of murder, the penalty was never in fact applied past the Biblical period. Moreover, capital punishment was effectively abandoned altogether during the Mishnaic period, although of course it has remained a topic of theoretical discussions. Jewish law is clearly very different in spirit and practice from Islamic law.

3.    Sifra

The Sifra is a halakhic midrash to Leviticus, which is occasionally called Torat Kohanim like the Torah book (Leviticus) that it is an exegesis of (JE[266]). According to Jacob Neusner (USA, b. 1932), in Rabbinic Literature: An Essential Guide[267], it is considered as dated ca. 300 CE (p. 3). I nevertheless include it in the present chapter – as an extra-Talmudic document, rather than as a post-Talmudic one, for lack of a better place. Indeed, though later than the Mishna, it is often referred to in the Talmud (JE). Neusner describes it as an effort to more thoroughly anchor the ‘oral Torah’ – meaning the Mishna (and the Tosefta) – in the ‘written Torah’ – i.e. essentially the Pentateuch (pp. 56-57). Neusner does not mention the work’s author, but JE discusses the matter[268].

My interest here is in certain features of Sifra’s logic that are mentioned by Neusner. I have not personally read Sifra, but take Neusner’s description of these features for granted. As Neusner puts it, Sifra’s purpose is to show that “the Mishnah is subordinated to Scripture and validated only through Scripture;” and it does so by means of a “critique of the Mishnah” which too often seems to rely on its own logic rather than explicitly refer to the Pentateuch (p. 58ff). This critique, as we shall see, focuses both on syllogistic and a fortiori logic. Sifra reportedly makes (or seems to make) assertions concerning these fields that simply, as I will definitively show in formal terms, cannot be upheld.

Syllogism. First, Sifra disputes “that we can classify things on our own by appeal to the traits or indicative characteristics, that is, utterly without reference to Scripture.” According to Sifra (or according to Neusner’s reading of it), “on our own, we cannot classify species into genera. Everything is different from everything else in some way. But Scripture tells us what thing are like what other things for what purposes, hence Scripture imposes on things the definitive classifications, not traits we discern in the things themselves.” And again:

“The thrust of Sifra’s authorship’s attack on taxonomic logic is easily discerned… things have so many and such diverse and contradictory indicative traits that, comparing one thing to something else, we can always distinguish one species from another. Even though we find something in common, we can also discern some other trait characteristic of one thing but not the other.”

If I understand such statements correctly, what Sifra is saying (or more probably, just implying through its many particular discursive acts, since rabbinic literature is rarely if ever so abstract in its approach) is that antithetical syllogisms can consistently be constructed. This would mean the following in formal terms:

All S1 are G; No S2 are G;
and X is S1. and X is S2.
Therefore, X is G. Therefore, X is not G.

On the surface, such a situation might seem conceivable. The individual or class called X might be classified under species S1 in some respects and under species S2 in other respects; and S1 might fall under genus G, while S2 does not fall under genus G. The two major premises do not seem incompatible, since they concern different subjects, S1 and S2; and the two minor premises do not seem incompatible, since a term may well have different predicates, S1 and S2. Yet the two conclusions are clearly incompatible!

However, logic is quite able to show where in the said premises the contradiction lies, by constructing a 2nd figure syllogism using the two initial major premises:

No S2 are G
All S1 are G
Therefore, No S1 is S2.

Using the latter conclusion as our new major premise, it follows by syllogism that if X is S1, it cannot be S2, and vice versa. That is, despite surface appearances, the two species, S1 and S2, are in fact mutually exclusive, by virtue of being related in contrary ways to the genus G. Thus, in fact, the two minor premises ‘X is S1’ and ‘X is S2’ cannot both be true at once. Therefore, the contradiction between ‘X is G’ and ‘X is not G’ will in fact never arise.

That is to say, the apparent argument of Sifra that contradictions are possible if we rely only on logic, so that appeal to Scripture is necessary to help us choose one side or the other, is not credible. It only seems credible due to superficial appeal to syllogistic reasoning; but in fact such quandaries cannot occur in practice for someone who truly knows logic. It should be said that the supposition that such quandaries are conceivable is not peculiar to Sifra; Greek and Roman sophists have also often imagined them possible.

Of course, Sifra may not be saying what I have here assumed it to say. It may just be saying that without Scripture’s guidance we cannot know whether X is S1 or S2; or perhaps (more likely) we cannot know whether species S1 or S2 falls under genus G or not, where G is some law or legal ruling. Such arguments would be logically acceptable. But what is sure, anyway, is that no one can legitimately argue that the initially listed two 1st figure syllogisms are compatible. This is not open to discussion.

The same of course can be said with regard to rival hypothetical syllogisms:

If B1 then C; If B2 then not C;
and if A then B1. and if A then B2.
Therefore, if A then C. Therefore, if A then not C.

The if–then premises of such arguments may offhand seem compatible, but their conflicting conclusions (assuming thesis A is not a paradoxical proposition) show them to be in fact incompatible. However, it should be obvious that this restriction is only applicable in cases of strict implication; if some of the implications involved are less firm, a situation of rivalry might conceivably occur. If A deductively implies B and B deductively implies C, then the conclusion is that A deductively implies C. But if A merely inductively implies B and/or B merely inductively implies C, then the conclusion is that A merely inductively implies C. Whereas ‘deductive implication’ signifies a 100% certainty, what I call ‘inductive implication’ refers to a looser relationship where the antecedent probably (with less than 100% certainty) implies the consequent. In such cases, the conclusions ‘if A, maybe then C’ and ‘if A, maybe then not C’ may both be justified, even though there is some degree of tension between them.

We can similarly admit that potential (though not actual) conflicts might occur in categorical syllogism. If for instance the rival syllogisms have as major premises that Most S1 are G and Most S2 are not G, and as minor premises that X is S1 and X is S2, then the conclusions will be respectively that X is probably G and X is probably not G. Though these two conclusions are in tension, they are not strictly speaking incompatible, and therefore they might conceivably occur together (especially if their probabilities are expressed so vaguely). It is probably the possibility of such tendencies to conflict that the author of Sifra had in mind. Another possibility is that there are unstated conditions to the premises of the categorical or hypothetical syllogisms, which make the rival arguments compatible although they superficially seem incompatible.

A fortiori argument. Second, concerning the argument a fortiori or qal vachomer, Neusner tells us in the name of Sifra that it “will not serve” – for “if on the basis of one set of traits that yield a given classification we place into hierarchical order two or more items on the basis of a different set of traits, we have either a different classification altogether or, much more commonly, simply a different hierarchy.” This is intended as a critique of “the Mishnah’s… logic of hierarchical classification.” To wit: “Things are not merely like or unlike, therefore following one rule or its opposite, Things are also weightier or less weighty.”

Here, the suggestion is that we can construct compatible a fortiori arguments, with reference to different middle terms (R1, R2), which yield contrary conclusions. This is a very similar suggestion to the previous one, but one specifically centered on a fortiori argument. It should again be stated that Sifra is not alone in this error (if it indeed makes it) – many people seem to think that such a situation is logically possible. Such people do not truly understand the logic involved, as I will now formally show. Consider the following two arguments:

P is more R1 than Q is; P is more R2 than Q is;
and Q is R1 enough to be S. and Q is R2 enough not to be S.
Therefore, P is R1 enough to be S. Therefore, P is R2 enough not to be S.

On the surface, looking at the premises superficially, such a situation may seem possible. After all, such major premises certainly occur in practice. P and Q may be in a certain relation within the hierarchy R1 and in a very different (even opposite) relation in another hierarchy R2. But such differences could not give rise to contrary conclusions, one implying that ‘P is S’ and the other that ‘P is not S’ – for the simple reason that the minor premises are incompatible, one implying that ‘Q is S’ and the other that ‘Q is not S’. Thus, in fact, such a situation is logically inconceivable.

Thus, contrary to what Sifra seems to be (according to Neusner’s analysis) suggesting, we do not need to appeal to Scripture to choose between this hierarchy and that one so as to avoid contradiction. Two hierarchies that lead to contrary conclusions will never be true together. This is logically obvious and demonstrable. Of course, here again, we might defend Sifra by saying that it perhaps does not claim such antithetical a fortiori arguments, but merely says that Scripture is required to establish the major and/or minor premises. This would present no problem. But if the claim is indeed one to viable antitheses, it is untenable.

We could also defend Sifra by pointing out that many rival arguments that seem to adhere to the formal conflict presented above are in fact not intended so strictly. The major premises, which tell us that P is more R than Q, may be tacitly intended to mean that P is usually (though not always) more R than Q; and/or the minor premises may really have the form ‘Q is usually R enough to be S’; in which cases, the conclusions will also be probabilistic at best. Thus, there may be an appearance of conflict, when in fact there is only some logical tension. This, I believe, often occurs in practice, and may well be what the author of Sifra had in mind when he raised this issue. Or again, there may be unstated conditions to the premises of the rival a fortiori arguments, which make them compatible although they superficially seem incompatible.

To sum up and conclude. If, as Neusner seems to be implying, Sifra criticizes the Mishna on the ground that it relies on logic independently of Scripture, and that by doing so it opens itself to irresolvable contradictions, Sifra can and must be opposed on purely formal grounds. Logic does not lead to contradictions, but on the contrary deflects them, or uncovers and resolves them. If, however, Sifra is only saying that the Mishna has to refer to Scripture for its major and/or minor premises, i.e. for the content of its propositions – that is another matter entirely: it is then an issue not of logic, but of fact or even of moral and legal evaluation of fact.

But upon reflection, even in the latter cases we must distinguish between deductive and inductive logic. It is true that deductive logic cannot prescribe facts and even less so their evaluations (though it can be used to ensure that such prescriptions are kept internally consistent). But inductive logic certainly can strongly impinge on issues of fact or even of moral and legal evaluation of facts. Through experience and scientific method we can, for instances, contest that hare are to be classified with ruminants, or that there are no fish that have scales but lack fins. And moreover, from purely factual material, we can put in doubt the credibility of evaluations; for example, how we conceive sunrise and sunset to occur directly affects the times prescribed for beginning and ending the Sabbath.[269]

4.    The Korach arguments

The Midrash called Bemidbar (Numbers) Rabbah, which is closely related to the Midrash called Tanhuma (named after a rabbi), is (or at least its earliest portions are) thought to date from the 5th century CE, apparently before the completion of the Babylonian Talmud[270]. What interests us in it here is its commentary regarding Numbers 16:1, which reads:

“‘Now Korach… took’. What is written in the preceding passage (Num. 15:38)? ‘Bid them that they make them… fringes (Heb. tzitzith)… and that they put with the fringe of each corner a thread of blue (Heb. techeleth)’. Korach jumped up and asked Moses: ‘If a cloak is entirely of blue, what is the law as regards its being exempted from the obligation of fringes?’ Moses answered him: ‘It is subject to the obligation of fringes’. Korach retorted: ‘A cloak that is entirely composed of blue cannot free itself from the obligation, yet the four blue threads do free it?!’

He [Korach] asked again: ‘If a house is full of Scriptural books, what is the law as regards its being exempt from the obligation of mezuzah [a small scroll with a selection of Torah verses, which is affixed to the doorposts of Jewish gates and homes]?’ He [Moses] answered him: ‘It is [still] under the obligation of having a mezuzah.’ He [Korach] argued: ‘The whole Torah, which contains two hundred and seventy-five sections, cannot exempt the house, yet the one section in the mezuzah exempts it?! These are things which you have not been commanded, but you are inventing them out of your own mind!’”

There is, note well, no evidence of this discourse in the Torah itself; it only appears much later in history, in the Midrash. These two arguments attributed to Korach are traditionally regarded as samples of qal vachomer, although (I presume) most commentators view them as qal vachomer of a fallacious sort. For that reason, they are especially interesting, in that they illustrate a possibility of erroneous reasoning in the a fortiori mode. We may paraphrase the two arguments briefly as follows:

  1. If mere threads of blue wool (on each of the four corners) are sufficient to make a garment lawful to wear, then surely if the whole garment is made of such blue wool (even if without the corner threads) it is likewise lawful.
  2. If a few passages of the Torah (in a mezuzah affixed to the doorposts) are sufficient to make a house lawful to live in, then surely if the whole Torah is stored in a house (even if without a mezuzah) it is likewise lawful.

These two arguments have the following form in common: If a small quantity of something (Q) is sufficiently in accord with the norm (R) to make so-and-so be declared lawful (S), then surely a large quantity of that thing (P) is sufficiently in accord with the norm (R) to make so-and-so be declared lawful (S). The preceding hypothetical proposition comprises the minor premise and conclusion of the a fortiori argument. Its tacit major premise is therefore: a large quantity of something (P) is more in accord with the norm (R) than a small quantity of same (Q). The argument is clearly positive subjectal, from minor to major.

What is wrong with this argument? The answer is obvious: its major premise does not have the logical necessity it is implied to have. While on the surface it might seem like a large quantity is preferable to a small one, this is not necessarily the case, because the two quantities may present significant qualitative differences. That is, the terms of the proposed major premise are incompletely specified, and therein lies the fallacy. The minor premise, regarding the sufficiency of a small quantity (Q) to satisfy the norm (R) for a certain result (S), may be true only provided that this quantity fulfills certain qualitative criteria (which may have additional quantitative aspects). If the larger quantity (P) does not fulfill these same qualitative criteria, it may well not be able to satisfy the norm (R) for a certain result (S). Therefore, the major premise should, to be truly universal, more precisely read: a large quantity of something precisely specified (P) is more in accord with the norm (R) than a small quantity of the exact same thing (Q).

Returning now to the two Korach arguments for the purpose of illustration, we can say the following. In both cases, the sophistry consisted in occulting the details given in brackets. In (a), what makes the garment kosher is not merely that it contains blue threads, but that it contains them on the four corners. In (b), what makes the house kosher is not merely that it contains Torah words, but that it contains them on the doorposts. The details do matter – they are not expendable. Therefore, in effect, Korach’s two arguments may be said to commit the fallacy of having more than four terms. The major and minor terms in the major premise are made to appear the same as the subjects in the minor premise and conclusion, but they are in fact different from them.

The two arguments might have been a bit more credible, had they respectively advocated an inference from a garment not made of blue wool yet having kosher tzitzit, to a garment entirely made of blue wool as well as having kosher tzitzit; or from a small mezuzah affixed to the doorposts, to a giant mezuzah affixed to doorposts. But even then, such inference would not be necessarily true, because there is no formal reason why the law might not interdict garments made entirely of blue wool (even with kosher tzitzit) or giant mezuzot (even affixed to doorposts). The major premise in use in any argument must be in fact true, for a true conclusion to be drawn from it. Very rarely is the major premise logically necessary; it is only so if its contradictory is self-contradictory. In most cases, the major premise has to be determined empirically – or, in such a religious context, be given in the proof text.

In my opinion, the two arguments attributed to Korach are not factual reports, but post facto fabrications with an educational purpose. Because: either Korach had some brains and could see for himself the fallacy of his reasoning, but he cynically proposed these arguments anyway, thinking no one else would notice; or he was not intelligent enough to realize his own errors of logic. But in either case, surely Moses had the intelligence needed to see through the fallacy, and would have publicly reproved Korach for his dishonesty or his intellectual deficiency, so as to stop the rebellion in its tracks by discrediting its leader. However, since according to the Torah account Divine intervention was used, we can infer that Moses did not use this logical means. I think the arguments were imagined by the author(s) of the Midrash for three reasons. One was to flesh out the story of Korach with some (most likely anachronistic) Talmudic-style legal debate, showing up the perversity and stupidity of the rebel. Another was perhaps to intimidate eventual readers, saying in effect: if you behave like Korach, expressing doubts in the law of Moses, you will be punished like Korach. The third was perhaps to teach people some a fortiori logic, to make sure they do not make similar errors of reasoning.

However, this is not how some later commentators have understood the purpose of this Midrash. They have taken it to mean, not that Korach was arguing fallaciously, but that Korach was being too logical, so that we ought to learn from this story to suspend our rational judgment now and then. For instance, R. Ephraim Buchwald, in an essay called “The Excesses of Rationality” (2007)[271], explains the matter as follows:

“According to Korach, human logic always prevails. Korach is certain that the rational processes are the ultimate determinant of right and wrong. Since the laws handed down from Moses and Sinai have no internal logic, they must be summarily rejected. It is for that very reason that parashat Chukat follows parashat Korach. The Torah, in Numbers 19:2, declares: ‘Zoat chukat HaTorah’: This is the statute of the Torah! There is no logic to the laws of the Red Heifer. Reason is of little value when it comes to this irrational ritual. The Red Heifer comes to confirm to Korach and all his fellow rationalists, that the ultimate authority is the law of Moses and Sinai, not mortal logic! … While Judaism in general is a most rational and logical faith, true believers must eventually conclude that there are certain aspects of the religion that one can not rationally fathom or master. It is that leap of faith that a believer must make, and this doubt that we all must overcome, and for which we are ultimately rewarded.”

This is obviously, in view of what our analysis above has demonstrated, an erroneous interpretation of the Midrash. The commentator evidently does not have great logical knowhow, since he seems to think that the two Korach arguments are valid. He is therefore not qualified to discuss the limits of human logic. Korach cannot be presented on the basis of the two arguments attributed to him as a “rationalist,” or proponent of reason, since they are in fact not in accord with logic. If he was not an idiot, he was a sophist who cynically faked logical argument. In the Midrashic story, Moses does not answer Korach by sullenly saying: “your arguments are sound, but I will stick dogmatically to my positions,” as our commentator implies. Rather, I’d say, Moses refutes Korach, as often done in Talmudic debate, by denying his conclusion, thereby tacitly implying that at least one of his premises is incorrect; and since the minor premise is in accord with the law of Moses, it must be the major premise that is mistaken. In other words, the correct interpretation is that Moses does not concede Korach’s reasoning powers, but rather challenges them.

  1. Buchwald is, of course, relying on the traditional commentaries regarding the statute of the red heifer (Numbers 19). They find it odd that the ritually clean people involved in preparing the ashes of the red heifer should be made unclean (v. 7, 8, 10), while those ashes are used to ritually clean people who are unclean due to having come in contact with a dead person (v. 12). Rashi comments, citing Yoma 67b: “Because Satan and the nations of the world taunt Israel, saying, ‘What is this commandment, and what purpose does it have?’ Therefore, the Torah uses the term “statute.” I have decreed it; You have no right to challenge it.”

But in truth, what has this to do with logic? It is not logically inconceivable that the same substance (the ashes of the red heifer) might have one effect (ritual uncleanliness) on one set of people (the people producing or handling it) and another, opposite effect (ritual cleaning) on another set of people (the people it is sprinkled on). Such complex relations can readily be found in nature – e.g. a chemical substance might be harmful to one kind of organism and beneficial to another. Or consider, to take an extreme example, the particle-wave duality in quantum mechanics, where the same phenomenon seems different viewed from different perspectives.

The red heifer ritual is no more ‘illogical’ than the ritual of sacrificing animals to purify people of their sins, or the rituals of tzitzit or mezuzah, or that of matza, or those of shofar, lulav and succah, or any other religious ritual. When dealing with the supernatural, everything is equally artificial, i.e. inexplicable by natural means. Rituals are not given in nature, or rationally inferred from it. Such truths (if they are indeed true) can only be known through revelation or similar (alleged) extraordinary means. Belief in them – at least in the case of people without prophetic powers of their own, and maybe even for prophets – depends on faith. Even prayer, the most natural expression of belief in God, depends on faith.[272]

Moreover, the inexplicability of alleged spiritual practices is not a reflection on human logic. Human logic does not promise omniscience. There are many things we do not, and perhaps can never, understand, even in the natural world; all the more so, in the (presumed) spiritual world. The fact that there are limits (whether short or long term) to the power of logic can never be used as an argument against the power of logic within its natural limits. There is no logical argument by which logic might be invalidated, because such argument would be claiming to have some logic, and thus be self-defeating. Even if logic admittedly cannot predict all truth, it can certainly eliminate quite a bit of falsehood. For this reason, we should not hasten to ditch it just because it does not deliver everything we wish for.

  1. Buchwald’s attempt to compare the Korach argumentation to the red heifer statute is, anyway, ingenuous. He regards the Korach arguments as perplexing because though sound (in his view), they lead to conclusions that are contrary-to-fact (i.e. to Biblical fact); and he regards the red heifer ashes as perplexing, because (I presume, though he does not say so) they have contrary behavior patterns in relation to different subjects. But even supposing these two perplexities are justified, they are certainly logically very different and cannot be lumped together. If they are, as he supposes, both ‘illogical’, they are ‘illogical’ in significantly different ways.

In any case, there is one kind of illogic that no amount of faith can ignore or cure – and that is any breach of the laws of thought. Faith is acceptable where there is some gap or uncertainty in knowledge; but if a claim – however ‘authoritative’ – goes against these fundamental laws, we can be absolutely sure it is incorrect. This applies equally well to other-worldly claims as to this-worldly ones. Our reaction in such case should not be blind faith, but to demand a credible resolution of the paradox. This is the adult, mentally-healthy reaction to such conundrums. In this sense, logic has much to say even about spiritual claims. Logic is mankind’s main protection against falsehood of any kind.

5.    Saadia Gaon

When I found out that Saadia Gaon, ben Yosef (Egypt, ca. 882 – Iraq, 942), had written a short book, entitled in Hebrew Perush Shelosh Esre Midot (Explanation of the Thirteen Hermeneutic Principles), and actually found a copy of it on the Internet[273], I was overjoyed, hoping to find in it some interesting original insights into qal vachomer. However, upon reading it (with the help of a friend), I was rather disappointed. Saadia Gaon there in fact says nothing theoretical about qal vachomer, other than to say that it may be used for non-legal as well as legal purposes.

He does not analyze the argument in any way, but is content to present five rabbinical examples of it – without, by the way, explaining why he chose those particular ones. If a man is obligated to take good care of his second wife, all the more so his first wife. Since, if one finds one’s enemy’s strayed animal, one is obligated to return it to him, it follows a fortiori that one must do that for a friend. And so forth. All these examples are in fact legal in content; he does not actually give any with non-legal content, but simply repeats (somewhat lamely, as if he could not think of any offhand) that non-legal content is possible. That’s it. Of course, examples have their importance; but they are certainly not enough.

According to the introduction (in French) to the Œuvres Complètes[274], Saadia does not always thus limit his commentary on the midot to examples, but in some cases gives explanations, even if his explanations are sometimes obscure (e.g. as to what distinguishes the 7th and 8th rules). In any case, he does not go into the details concerning the rules. Moreover, we are told, Saadia considers that anyone has a right to put forward new applications of the thirteen rules, which liberty is far from admitted by other commentators. Nevertheless, I should add, Saadia is known to have defended the rabbinic tradition that the thirteen midot were Divinely revealed to Moses at Mt. Sinai[275]. He no doubt did so in the context of his polemics with the Karaites, who of course rejected rabbinic interpretation[276].

So I was taught, anyhow; but I have not offhand found an explicit statement to that effect. Perhaps he merely implied it. We might, for example, so interpret his citation of Sanhedrin 88b, “With the increase in numbers of the disciples of Shammai and Hillel, who did not advance far enough in their studies, the controversies increased” (The Book of Doctrines and Beliefs, pp. 32-33), to explain the existence of disagreements between rabbis. The implication is that originally, when the Torah was first given, there were no doubts; these developed over time, when levels of learning diminished. This matter could be further pursued, but I will leave it at that for now and move on.

I would like, rather, to take this opportunity to quote Saadia Gaon on the value of empiricism and rationalism:

“Furthermore [authentic tradition] verifies for us the validity of the intuition of reason. It enjoins us, namely, to speak the truth and not to lie. Thus it says: For my mouth shall utter truth…. Besides that it confirms for us the validity of knowledge inferred by logical necessity, [that is to say] that whatever leads to the rejection of the perception of the senses or rational intuition is false…. Next [tradition] informs us that all sciences are [ultimately] based on what we grasp with our aforementioned senses, from which they are deduced and derived.” (The Book of Beliefs and Opinions, Pp. 18-19.)

It is also interesting to note here certain rules for inference set by Saadia Gaon:

“In endeavoring to establish the truth of inferential knowledge, we shall henceforth be on guard against these five possible forms of mistakes, namely: (1) that it does not conflict with knowledge established by sense-perception; (2) that is does not conflict with knowledge established by Reason; (3) that it should not conflict with some other truths; (4) that it should not be self-contradictory; still more, that it should not (5) involve a difficulty more serious than the one intended to avoid.” (The Book of Doctrines and Beliefs, p. 42.)[277]

Taking Saadia at his word, we can predict that were he placed squarely before new facts and shown the validity of certain logical inferences, he would have the intellectual and moral integrity to admit them, and would not dogmatically insist on contrary, more traditional ‘facts’ or ‘inferences’. Unfortunately, there are still today some people who think they do religion a service by refusing to face facts and logic. Just yesterday, I had the hilarious experience of watching an online video showing an Islamic apologist claiming in 2007 on Iraqi TV that the earth is flat and much larger than the sun, which is also flat![278] Fortunately, apologists for Judaism never go so far; but they also sometimes show considerable resistance to change.

I say this here because readers of the present volume must obviously be prepared to adapt to new discoveries and insights, and not cling at all costs to traditional views. I want to emphasize in passing that to be critical does not signify to be hostile and willfully negative. Though critical, I have personally no desire to contradict or denigrate our religious tradition. Not all critical commentators are so moderate in their views or intentions; some are very eager to find fault with the rabbis or the Torah. For my part, I would prefer to always justify the rabbis and the Torah, and confirm their wisdom, and it is only reluctantly that I criticize some of their claims. Nevertheless, I try to be scrupulously fair and honest – i.e. to be scientific – and to admit that there is a problem when there indeed appears to be one. This is the golden mean – neither dishonestly attacking nor dishonestly defending, but sincerely looking for the truth.

6.    Rashi and Tosafot

Concerning the contribution of Rashi, i.e. R. Shlomo ben Yitzhak (France, 1040-1105), to the understanding of the hermeneutic principles, Mielziner tells us that he “occasionally explained, in his lucid way, the single rules where they are applied in the Talmudic discussions.” There is, he adds, “a separate treatise on the hermeneutic rules ascribed to this commentator and published under the title of Perush Rashi al Hamidot,” which however “seems to be spurious.” This is found “in Kobak’s Jeschurun, vi, Hebrew part, pp. 38-44, 201-204; the remaining commentaries on the thirteen rules are enumerated by [Adolf] Jellinek in Ḳonṭres ha-Kelalim, Nos. 163-175.”

I have no access to these various sources, so must make do with a more ad hoc treatment. The question that interests me here is: firstly, what does Rashi say about the qal vachomer in Numbers 12:14-15 (and eventually, the other cases found in the Torah, and maybe also those in the Nakh)? And secondly, what does he say about the discussion concerning the dayo principle in Baba Qama 25a-b? I shall also try and determine the viewpoints on these topics of Rashi successors, the Tosafot. The basic issue to my mind is: do these post-Talmudic commentators accept the idea seemingly advocated in the Gemara (based on a baraita) that qal vachomer is naturally ‘proportional’ and the dayo principle is designed to reign in such velleity in it? The answer to expect is, obviously: yes, they do.

First, let me mention in passing Rashi’s comments on other a fortiori arguments appearing in the Torah. Concerning Genesis 44:8, all Rashi says is: “This is one of the ten a fortiori inferences that are found in Scripture, which are all listed in Bereishit Rabbah (92:7).” For Exodus 6:12: he is likewise content to say: “This is one of the ten a fortiori inferences in the [Tanakh],” although he additionally explains Moses’ speech defect as an “obstruction of the lips.” He has no comment regarding Deuteronomy 31:27. Evidently, Rashi does not question the Midrashic statistic of just ten qal vachomer in the Tanakh.

As regards Numbers 12:14, Rashi’s comment is: “If her [Miriam’s] father were to display, to her, an angry face, would she not be humiliated for seven days? Certainly, then, in the case of the Divine Presence, [she should be humiliated] for fourteen days. However, it is sufficient that the derivative equal the source of its derivation. Therefore, even with My rebuke, let her be confined for seven days.” As can be seen, this is just a repetition of the thesis given in a baraita transmitted in the said Gemara. If we look for Rashi’s comment opposite that baraita in the Gemara, we find that he has none. That means he considers the matter sufficiently clear as it is and sees no point in adding anything to it. There you have it. Rashi does not ask or answer any theoretical questions concerning qal vachomer reasoning or the dayo principle, but takes them for granted.

Rashi comments somewhat more extensively on another qal vachomer and dayo principle application, namely in Tractate Zevachim 69b (Seder Kodashim)[279]. There, the Mishna explicitly refers to both the argument (by R. Meir) and the application of the principle (by R. Jose), and the Gemara expounds almost exactly in the same words as in Baba Qama 25a, saying: “Does not R. Meir accept the principle of dayo [it is sufficient]? Surely the principle of dayo is Biblical, for it was taught: How is a qal vachomer applied? And the Lord said unto Moses: If her father had but spit in her face, should she not hide in shame seven days? How much more should a divine reproof necessitate [shame for] fourteen days; but it is sufficient for that which is inferred by an argument to be like the premise!” But Rashi does not add much, other than to (rightly) point out that the qal vachomer in the Miriam story is implicit rather than explicit.

That Rashi uncritically accepts the common notion that a fortiori argument is ‘proportional’ is evident not only in his acceptance without comment of the “fourteen days” given in the Gemara of Baba Qama 25a, but also in his comment to Genesis 4:24 (which, it should be noted, is not included in the traditional list of ten a fortiori arguments in the Tanakh). There, based on Tanchuma Bereshit 11, Rashi elucidates Lamekh’s statement “If Cain shall be avenged sevenfold, truly Lamekh seventy and seven-fold” as a qal vachomer, as follows: “If Cain killed intentionally, [and yet] his punishment was delayed for seven generations, [then] I, who killed unintentionally, surely will have my punishment deferred for many periods of seven generations.”

Note, however, that though in the case of Miriam Rashi acknowledges the dayo principle, he does not mention its application in the case of Lamekh; nor does he tell us why he doesn’t. I suggest that the reason why it seems reasonable in one case and not the other is the following. In the example of Miriam, the conclusion (14 days penalty) is more stringent than the minor premise (7 days penalty), in accord with the principle of midah keneged midah (measure for measure), so the dayo principle is required to mitigate the punishment; whereas, in the example of Lamekh, the conclusion, though likewise quantitatively superior (77 instead of 7 generations), is more lenient as regards the sanction than the minor premise (i.e. signifies longer deferral of punishment), and so is not subject to the dayo principle (which if applied would speed up the punishment).

Successors of Rashi, known as Tosafot[280], comment on the Mishna and Gemara in more detail. Essentially, they subscribe to the scenario apparently advocated by the Gemara when interpreting the Mishna of Baba Qama 25a. That is to say, they accept uncritically that R. Tarfon’s two arguments are a crescendo. Nevertheless, to their credit, they consider that his first and second try are logically (and not merely rhetorically) different, due to reshuffling. They also agree with R. Tarfon that his second argument is able to avoid the dayo restriction as set by the Sages against his first argument, because while the first argues from half to full damages, the second argues from full to full damages. As a result of which, they intelligently explain the Sages’ continued insistence on dayo application with reference to premises antecedent to the qal vachomer itself.[281]

However, since Tosafot accept the Gemara reading of both arguments as a crescendo, they also accept the “fourteen days” notion proposed by the Gemara (following a baraita), i.e. the claim that qal vachomer naturally yields a ‘proportional’ conclusion. Without questioning this claim, they only focus on trying to explain this number (rather than any other large number[282]). An explanation they give is to refer to seven days as the minimum period of quarantine in the event of leprosy (Leviticus 13:4); a more severe confinement must be at least another seven day period.[283] Tosafot also consequently make efforts to defend the obscure notion, ascribed by the Gemara to R. Tarfon, that the dayo principle can on occasion be ignored, specifically where it would “defeat the purpose of” the qal vachomer.[284]

But such glosses are superficial in their concerns; they gloss over the more serious underlying issues. I have shown in my detailed analysis of this sugya in the two preceding chapters (7-8) that we cannot countenance some of the commonplace interpretations of this Mishna and Gemara without getting ensnared in a multitude of logical errors, which make at least some of the rabbis involved look very foolish. Once the logical errors are understood, it is seen that many of the explanations proposed in the Gemara, and later by others, including Tosafot, are vain attempts to uphold a very wobbly structure. If we want to redeem the rabbis involved, we must approach the whole matter much more lucidly, and consider a moral instead of logical explanation of the dayo principle.

I do not want to seem to be dismissing Tosafot in a debonair manner, being fully aware of their importance, but simply see no point in repeating here what I demonstrated earlier. So I invite the reader to go there.

7.    Kol zeh assim

A thorough study of the logic in Tosafot, and even just of its a fortiori logic, would doubtless result in a thick and interesting book. Not having the necessary language skills, I cannot myself undertake such a study; but I would certainly recommend that someone duly qualified in both logic (especially as taught in the present volume) and the Talmud do the job. But we can here get an idea of the logical resourcefulness of Tosafot through one example, which has to do Baba Qama 25a. This is thanks to Yisrael Ury, who in his book Charting the Sea of Talmud[285] provides an English translation of a commentary by Tosafot and some useful clarifications as to its intents[286]. This passage of Tosafot is only incidentally concerned with Baba Qama 25a, using it to illustrate a certain form of argument; so we shall not here cite all of it, but only quote or paraphrase the parts of it relevant to our narrower purpose.

The Tosafot commentary, whose precise author is not named, proceeds in three stages, we might say. In a first stage, it refers to one of the arguments originally given in the Mishna Baba Qama 2:5, which it paraphrases as follows:

“Whereas tooth and foot, for which damages are not paid for damage done in the public domain, yet are liable for full damages for damages done in the domain of the damaged party, then horn, for which half damages are paid for damage done in the public domain, certainly should pay full damages for damage done in the damaged party’s domain.”

Looking at this argument, we easily recognize the first argument of R. Tarfon, since it proceeds by mentioning first tooth & foot damage in the public and private domains and then horn damage in the same domains. As I have shown previously, this argument can be put in standard a fortiori form as follows:

Private property damage (P) implies more legal liability (R) than public domain damage (Q) [as we know by extrapolation from the case of tooth & foot[287]].
Public domain damage (Q) implies legal liability (Rq) enough to necessitate half payment for damage by horn (Sq) [this is derived from the Torah[288]].
The payment due (S) is ‘proportional’ to the degree of legal liability (R).
Therefore, private property damage (P) implies legal liability (Rp) enough to necessitate full payment for damage by horn (Sp = more than Sq).

We shall here label this argument as argument (1a). Notice that it is positive antecedental. The major premise is obtained by generalization from the givens regarding damage by tooth & foot. The major and minor terms are ‘damage on private property’ (P) and ‘damage on public domain’ (Q). The middle term is ‘legal liability’ (R); and the subsidiary term is ‘to make the payment for damage by horn have a certain magnitude’ (S). In fact, note well, the argument is not purely a fortiori but a crescendo, since the magnitude of S in the conclusion is greater than that in the minor premise. This means there is a tacit premise to take into consideration, about the proportionality of ‘payment due’ (S) to ‘legal liability’ (R).

Although not directly mentioned by Tosafot, the second argument of R. Tarfon is, as we shall see, also (if not more) relevant to the present discussion; so we shall restate it here, in standard form:

Horn damage (P) implies more legal liability (R) than tooth & foot damage (Q) [as we know by extrapolation from the case of public domain].
Tooth & foot damage (Q) implies legal liability (R) enough to necessitate full payment for damage on private property (S).
Therefore, horn damage (P) implies legal liability (R) enough to necessitate full payment for damage on private property (S).

We shall here label this argument as argument (1b)[289]. Notice that it is also positive antecedental. The major premise is, here, obtained by generalization from the givens regarding damage on public grounds. However, the major and minor terms are ‘damage by horn’ (P) and ‘damage by tooth & foot’ (Q). The middle term is again ‘legal liability’ (R); but the subsidiary term is ‘to make the payment for damage on private property full’ (S). Note that this argument is purely a fortiori, and not a crescendo. But it is clear that it could also be stated in a crescendo form, and that if it were would yield the same conclusion (viz. full payment for horn damage on private property), since no payment greater than full is admitted by the Torah or the rabbis. For this reason, it suffices to state it in pure form.

The second stage of our Tosafot commentary concerns an objection, and the reply to it, put forward in the past by a commentator called the Ri (presumably this refers to R. Isaac ben Samuel, a 12th century French Tosafist). The Ri’s objection is described as follows:

“But consider that the damages of tooth and foot are common!”

To which objection the Ri himself replies:

“Paying full damages in the damaged party’s domain is not a severity (chumra) to be used in an objection (pirka), for it does not at all cause tooth and foot to lead to the requirement of half damages for damage done in the public domain as does horn.”

I have to say that I only understood the Ri’s objection thanks to the clarifications given by Ury, which I presume are traditional. He explains it as follows: because damage caused by tooth & foot is “commonplace,” the ox’s owner is obligated to take extra care “that his animal not cause damage when it comes in proximity to the property of others;” so that if such damage does indeed occur, he is more open to blame. As regards damage by horn, since the goring of another animal by an ox is “a rare event,” it is unexpected by the ox’s owner and he is justified in not taking special precautions against it; so that if such damage does indeed occur, he is not as liable.

Thus, the Ri’s objection means that, whereas on public grounds tooth & foot damage implies less liability than horn damage (no liability against half liability), as the Mishna teaches (based on the Torah), it may well be that on private property tooth & foot damage implies more liability than horn damage (full liability against, say, only half). This reasoning thus constitutes an objection to the original Mishna argument – i.e. it is designed to show that the conclusion that seems inevitable in the latter (namely, full liability for damage by horn) is perhaps not so inevitable. Putting this reasoning in standard form, we obtain the following:

Tooth & foot damage (P) implies more legal liability (R) than horn damage (Q) [since the former is common and the latter is uncommon].
Tooth & foot damage (P) implies legal liability (R) enough to necessitate full payment for damage on private property (S).
From which it does not follow that horn damage (Q) implies legal liability (R) enough to necessitate full payment for damage on private property (S).

We shall label this as argument (2a). This argument should be compared to the second argument of R. Tarfon, which we labeled (1b). Notice that they are very similar, except that the major premise has been reversed so that the putative conclusion no longer follows. In (2a), tooth & foot damage is the major term, while the horn damage is the minor term. The middle term is unchanged. The subject of the minor premise is unchanged (still tooth & foot damage), but now this subject is the major term. The subject of the putative conclusion is unchanged (still horn damage), but now this subject is the minor term. Since the format of the attempted a fortiori argument is still positive antecedental, inference from major to minor is illicit. Thus, we can no longer draw the conclusion of (1b) that ‘horn damage on private property necessitates full payment’.

Such conclusion is now a non sequitur – it is not excluded by the new premises (it does not contradict them), but it is not justified by them, either. This argument is not itself an a fortiori argument, note well, but merely serves to put in doubt R. Tarfon’s second a fortiori argument. It obstructs his conclusion, without needing to actually contradict it. It rejects his argument by reversing its major premise[290]. If, as R. Tarfon takes it, the owner of an ox is more responsible for horn damage than for tooth & foot damage, then the inference from full liability in the latter to full liability in the former is perfectly logical. But if, as the Ri contends with reference to ‘frequencies of occurrence’, the owner of an ox is more responsible for tooth & foot damage than for horn damage, then the inference from full liability in the former to full liability in the latter is debatable.

Another way to look at the objection (2a) is to say that the major premise of R. Tarfon’s first a fortiori argument (1a) – which take note is the one that Tosafot mentions – is no longer granted. This premise, viz. “private property damage (P) implies more legal liability (R) than public domain damage (Q),” was obtained by generalization from the given that damage by tooth & foot implies no liability in the public domain and full liability on private property. However, now the objection makes us aware that this generalization is open to question, since the conditions for legal liability are not the same in the case of damage by horn, due to there being different frequencies of occurrence. Thus, analogy is blocked. What applies to tooth & foot does not necessarily apply to horn.

This, then, is the objection conceived of as possible by the Ri, stated in more formal terms. Let us now try to understand the way he himself neutralized the objection. Remember that we are given by the Mishna (based on certain Torah verses) that tooth & foot damage on public grounds does not necessitate any payment for damages, while horn damage on public grounds necessitates payment of half damages. On this basis, the Ri replies to the objection by saying: (i) that “paying full damages in the damaged party’s domain” ought to “cause tooth and foot to lead to the requirement of half damages for damage done in the public domain as does horn;” and (ii) that since this consequence does not in fact occur, “paying full damages in the damaged party’s domain is not a severity to be used in an objection.”

The first part of his remark (i) refers to an a fortiori argument with the same major premise as (2a) combined with the given information about horn damage on public grounds necessitating half payment; these premises would conclude that tooth & foot damage on public grounds necessitates half payment (at least – more than half, i.e. full, if proportionality is applied). We may label this argument (2b), and put it in standard a fortiori form (positive antecedental, from minor to major) as follows:

Tooth & foot damage (P) implies more legal liability (R) than horn damage (Q) [since the former is common and the latter is uncommon].
Horn damage (Q) implies legal liability (R) enough to necessitate half payment for damage on public grounds (S).
Therefore, tooth & foot damage (P) implies legal liability (R) enough to necessitate half payment for damage on public grounds (S).

But, the Ri tells us in the second part of his remark (ii), this conclusion cannot be true, since the Torah tells us that tooth and foot damage on public grounds is exempt from any payment! Therefore, he concludes, this last a fortiori argument must be rejected. This last argument, which is a reductio ad absurdum, can be labeled argument (2c). It says: since the minor premise of argument (2b) is Torah given, and the process is valid, the only way to reject it is by abandoning its major premise[291]. That is to say, tooth & foot damage cannot be taken to imply more legal liability than horn damage on the basis of the former being more common and the latter being less common, as the objection (2a) initially attempts. Thus, the Ri shows that the objection, although reasonable sounding in itself, leads to absurdity and must be dropped.

The third stage of the Tosafot commentary we are analyzing is introduced by the statement in Hebrew: “vekhol zeh assim bakal vachomer,” meaning in English: “and all this I will put into the a fortiori argument”[292]. The unnamed Tosafist then argues as follows[293]:

“Whereas tooth and foot, even though their damages are common, they are exempt from payment for damage done in the public domain, but necessitate a full payment for damage done on the property of the injured party – then horn, even though its damage is not common, and it necessitates payment of half damages for damage done in the public domain, does it not follow (lit. eino din) that it necessitates payment