FUTURE LOGIC:

Categorical and Conditional Deduction and Induction of the Natural, Temporal, Extensional, and Logical Modalities.

Avi Sion, Ph. D.

(C) Copyright Avi Sion, 1990, 1996.

Protected by international and Pan-American copyright conventions.

All rights reserved.

No part of this book may be reproduced in any manner whatsoever,

or stored in a retrieval system or transmitted,

without express permission of the Author-publisher,

except in case of brief quotations with due acknowledgement.

First published 1990, by Avi Sion, Vancouver Island, B.C., Canada.

Revised Edition, published 1996, by Avi Sion, Geneva, Switzerland.

# Abstract

** Future Logic** is an original, and wide-ranging treatise of formal logic. It deals with deduction and induction, of categorical and conditional propositions, involving the natural, temporal, extensional, and logical modalities.

(Simply put, deduction and induction are inferences of more or less certainty; propositions refer to relations between things; modalities are attributes of relations like necessity, actuality or possibility.)

Traditional and Modern logic have covered in detail only formal deduction from actual categoricals, or from logical conditionals (conjunctives, hypotheticals, and disjunctives). Deduction from modal categoricals has also been considered, though very vaguely and roughly; whereas deduction from natural, temporal and extensional forms of conditioning has been all but totally ignored. As for induction, apart from the elucidation of adductive processes (the scientific method), almost no formal work has been done.

This is *the first work ever*** to strictly formalize the inductive processes of generalization and particularization**, through the novel methods of factorial analysis, factor selection and formula revision.

This is *the first work ever*** to develop a formal logic of the natural, temporal and extensional types of conditioning** (as distinct from logical conditioning), including their production from modal categorical premises.

** Future Logic **contains a great many other new discoveries, organized into a unified, consistent and empirical system, with precise definitions of the various categories and types of modality (including logical modality), and full awareness of the epistemological and ontological issues involved. Though strictly formal, it uses ordinary language, wherever symbols can be avoided.

Among its other contributions: a full list of **the valid modal syllogisms **(which is more restrictive than previous lists); the main formalities of **the logic of change **(which introduces a dynamic instead of merely static approach to classification); the first formal definitions of **the modal types of** **causality**; a new **theory of class logic**, free of the Russell Paradox; as well as a critical review of modern metalogic.

But it is impossible to list briefly all the innovations in logical science — and therefore, epistemology and ontology — this book presents; it has to be read for its scope to be appreciated.

# Contents in brief

Part I.** ****ACTUAL CATEGORICALS**. (Chap. 1-10.)

Introduction. Foundations. Logical Relations. Words and Things. Propositions. Oppositions. Eductions. Syllogisms: Definitions. Syllogisms: Applications. Syllogisms: Validations.

Part II. **MODAL CATEGORICALS**. (Chap. 11-19.)

Modality: Categories and Types. Sources of Modality. Modal Propositions. Modal Oppositions and Eductions. Main Modal Syllogisms. Other Modal Syllogisms. Transitive Categoricals. Permutation. More About Quantity.

Part III. **LOGICAL CONDITIONING**. (Chap. 20-32.)

Credibility. Logical Modality. Contextuality. Conjunction. Hypothetical Propositions. Hypotheticals: Oppositions and Eductions. Disjunction. Intricate Logic. Logical Compositions. Hypothetical Syllogism and Production. Logical Apodosis and Dilemma. Paradoxes. Double Paradoxes.

Part IV. ** DE-RE CONDITIONING**. (Chap. 33-42.)

Conditional Propositions. Natural Conditionals: Features. Natural Conditionals: Oppositions, Eductions. Natural Conditional Syllogism and Production. Natural Apodosis and Dilemma. Temporal Conditionals. Extensionals: Features, Oppositions, Eductions. Extensional Conditional Deduction. Modalities of Subsumption. Condensed Propositions.

Part V. **CLASS-LOGIC, AND ADDUCTION**. (Chap. 43-49.)

The Logic of Classes. Hierarchies and Orders. Illicit Processes in Class Logic. Adduction. Theory Formation. Theory Selection. Synthetic Logic.

Part VI. **FACTORIAL INDUCTION**. (Chap. 50-59.)

Actual Induction. Elements and Compounds. Fractions and Integers. Factorial Analysis. Modal Induction. Factor Selection. Applied Factor Selection. Formula Revision. Gross Formula Revision. Factorial Formula Revision.

Part VII. **PERSPECTIVES**. (Chap. 60-68.)

Phenomena. Consciousness and The Mind. Perception and Recognition. Past Logic. Critique of Modern Logic. Developments in Tropology. Metalogic. Inductive Logic. Future Logic.

**APPENDICES**. On Factorial Analysis. On Majority and Minority.

© Copyright 1990, by Avi Sion.

# Table of Contents

PART I. ACTUAL CATEGORICALS. 10

8 . SYLLOGISM: DEFINITIONS. 41

9 . SYLLOGISM: APPLICATIONS. 44

- The Main Moods. 44
- On the Fourth Figure. 46
- Subaltern Moods. 46
- Singular Moods. 47
- Summary. 48
- Common Attributes. 49
- Imperfect Syllogisms. 49

10 . SYLLOGISM: VALIDATIONS. 50

PART II. MODAL CATEGORICALS. 54

11 . MODALITY: CATEGORIES AND TYPES. 55

- Seeds of Growth. 55
- Categories of Modality. 55
- Types of Modality. 56
- Extensional Modality. 57
- Temporal Modality. 58
- Tense and Duration. 59
- Natural Modality. 60
- Other Types. 61

14 . MODAL OPPOSITIONS AND EDUCTIONS. 70

- Quantification of Oppositions. 70
- Basic Intramodal Oppositions. 71
- Quantified Intramodal Oppositions. 73
- Intermodal Oppositions. 76
- Eductions. 77

15 . MAIN MODAL SYLLOGISMS. 79

16 . OTHER MODAL SYLLOGISMS. 85

17 . TRANSITIVE CATEGORICALS. 90

- Substitution. 99
- Comparatives. 100
- Collectives and Collectionals. 100
- Quantification of Predicate. 101

PART III. LOGICAL CONDITIONING. 103

- The Singular Modalities. 109
- The Plural Modalities. 110
- Analogies and Contrasts. 111
- Apodictic Knowledge. 112

24 . HYPOTHETICAL PROPOSITIONS. 124

- Kinds of Conditioning. 124
- Defining Hypotheticals. 124
- Strict or Material Implication. 126
- Full List of Forms. 127

25 . HYPOTHETICALS: OPPOSITIONS AND EDUCTIONS. 130

28 . LOGICAL COMPOSITIONS. 148

29 . HYPOTHETICAL SYLLOGISM AND PRODUCTION. 154

30 . LOGICAL APODOSIS AND DILEMMA. 163

PART IV. DE RE CONDITIONING. 182

33 . CONDITIONAL PROPOSITIONS. 183

34 . NATURAL CONDITIONALS: FEATURES. 186

35 . NATURALS CONDITIONALS: OPPOSITIONS AND EDUCTIONS. 194

36 . NATURAL CONDITIONAL SYLLOGISM AND PRODUCTION. 197

37 . NATURAL APODOSIS AND DILEMMA. 206

38 . TEMPORAL CONDITIONALS. 212

39 . EXTENSIONALS: FEATURES, OPPOSITIONS, EDUCTIONS. 215

40 . EXTENSIONAL CONDITIONAL DEDUCTION. 221

41 . MODALITIES OF SUBSUMPTION. 230

42 . CONDENSED PROPOSITIONS. 236

43 . THE LOGIC OF CLASSES. 240

44 . HIERARCHIES AND ORDERS. 247

45 . ILLICIT PROCESSES IN CLASS LOGIC. 250

- Logical Probability. 255
- Providing Evidence. 256
- Weighting Evidence. 259
- Other Types of Probability. 261

PART VI. FACTORIAL INDUCTION. 276

- The Problem. 277
- Induction of Particulars. 278
- Generalization. 278
- Particularization. 279
- Validation. 280

51 . ELEMENTS AND COMPOUNDS. 282

- Elements and Compounds. 282
- Gross Formulas. 282
- Oppositions. 287
- Double Syllogisms. 288
- Complements. 288

52 . FRACTIONS AND INTEGERS. 290

- Factorization. 297
- Applications. 297
- Overlap Issues. 299
- More Factorial Formulas. 299
- Open System Analysis. 300

- Knowability. 302
- Equality of Status. 302
- Stages of Induction. 303
- Generalization
*vs*. Particularization. 304 - The Paradigm of Induction. 305
- The Pursuit of Integers. 305

56 . APPLIED FACTOR SELECTION. 310

- Closed Systems Results. 310
- Some Overall Comments. 311
- Rules of Generalization. 312
- Review of Valid Moods. 313
- Open System Results. 316

58 . GROSS FORMULA REVISION. 325

- Amplification. 325
- Harmonization. 325
- Unequal Gross Formulas. 326
- Equal Gross Formulas. 327
- Applications. 327

59 . FACTORIAL FORMULA REVISION. 331

- Adding Fractions to Integers. 331
- Reconciliation of Integers. 333
- Indefinite Denial of Integers. 334
- Other Formula Revisions. 336
- Revision of Deficient Formulas. 337

- Empirical or Hypothetical. 340
- Physical or Mental. 340
- Concrete and Abstract. 343
- Presentative or Representative. 344

61 . CONSCIOUSNESS AND THE MIND. 346

62 . PERCEPTION AND RECOGNITION. 353

- The Immediacy of Sense-Perception. 353
- Logical Conditions of Recognition. 355
- Other Applications. 357

- Historical Judgment. 359
- Aristotle, and Hellenic Logic. 360
- Roman, Arab, and Medieval European Logic. 362
- Oriental Logic
*.*364 - Modern Tendencies. 365
- In The 20th Century. 367

64 . CRITIQUE OF MODERN LOGIC. 371

- Formalization and Symbolization. 371
- Systematization and Axiomatization. 374
- Modern Attitudes. 376
- Improvements and Innovations. 379
- The Cutting Edge. 382

65 . DEVELOPMENTS IN TROPOLOGY. 384

- Tropology. 384
- Roots. 385
- Shifts in Emphasis. 387
- Setting the Stage. 389
- Contemporary Currents. 393
- Philosophical Discussions. 397

Appendix 1. On Factorial Analysis. 443

Appendix 2. On Majority & Minority. 456

# PART I. ACTUAL CATEGORICALS.

## 1. INTRODUCTION.

### 1. What is Logic?

*Definition.*

Logic is first of all an instinctive* art*. We all, from an early age, try to ‘sort out’ our experiences and ‘make sense’ of the world around us — and this thought process is to varying degrees ‘logical’. It is logical to the extent that we try to consider the evidence, avoid contradictions, and try to understand. We call this using ‘common sense’.

On a higher level, logic is a *science*, which developed out of the self-awareness of thinkers. They began to wonder why some thoughts were more credible, forceful, and informative than others, and gradually discerned the patterns of logical intelligence, the apparatus of reasoning. A logic theorist is called a logician. Note that we also call ‘a logic’, any specific field of or approach to logical science.

Logic as a field of inquiry has two goals, then. On a *practical* level, we want it to provide us with a guide book and exercise manual, which tells us how to think straight and trains us to do so efficiently. On a *theoretical* level, we seek the assurance that human knowledge does, or can be made to, conform to reality. How these methodological and philosophical tasks are fulfilled, will become apparent as we proceed.

Logic is of value to all individuals, bettering their daily reasoning processes, and thus their efficacy in dealing with their lives, and their work. It teaches you organization, enabling you to arrive at the solution of problems more efficiently. It helps you to formulate more pondered opinions and values.

Be you an artist, a parent, a university professor, a doctor, a psychologist, a civil engineer, an auto-mechanic, a bank manager, an office worker, an investor, a planner, an organizer, a negotiator, a lawmaker, judge or lawyer, a politician or journalist, a systems analyst, a statistician, a computer or robot programmer, whatever your profession or walk of life — you are sure to find the study of logic useful.

It is of value to scientists of all disciplines, helping them to clarify issues and formulate solutions to problems. There is no area of human interest or endeavor where logic does not have a say, and where the study of logic would not be effective in improving our situation.

Logic is worth studying also, for the sheer esthetic joy of it. There is no describing the mind’s response to this beautiful, colorful achievement of the human spirit. I hope the reader will have as much fun reading this book, as I had writing it. It can be hard work, but it is rewarding. My own favorite topic is *de-re *modality; I find it closer to earth than logical modality.

*Method.*

Logic teaches us to pursue and verify knowledge. It is based on an acknowledgment of the possibility of human error, but also implies our ability to correct errors. Where veracity or falsity is hard to establish, it tells us at least how ‘reasonable’ or ‘forced’ our judgments are.

It is essentially a holistic science, teaching us to take everything into consideration when forming judgments. Truth is not to be found in a limited viewpoint, but through a global perspective, an awareness of all aspects of an issue, all proposed answers to a question.

Logical science shows us what to look for in the course of knowledge acquisition, by listing and clarifying the main forms of relation among things and ideas (whence the name ‘formal logic’). It is the ‘systems analysis’ of human thought.

Logic is concerned with the formalities of reasoning, without so much regard to its subject-matter. It allows for objective assessments of inferential processes, precisely because its principles make minimal references to specific contents of thought. It is emotionally detached, it has no double standards, it is open-minded and fair.

Logic is a tool of interpretation, understanding, and prediction. It is a method for drawing the maximum amount of useful information from new experiences, or enveloped in previous knowledge, so as to fully exploit the lessons of the world of matter and mind, appearing all around us all the time.

What logic does is to help us to take all impressions and intuitions in stride, and resolve any disagreements which may arise. What is sure, is that, in reality, things themselves can never be in contradiction. It is ideas which conflict with each other or with primary experiences. Sometimes it is the idea that there is a conflict which turns out to be wrong.

The job of logicians is, not to reword what is already known, but to uncover and enhance the logical capabilities of everyday language. This is achieved by first singling out any concept which seems to infiltrate all fields of human interest. Often, the colloquial expression relating to it has many meanings; in such case, we make an agreement to use those words in only the selected sense, which is usually their most common connotation. Once all risk of ambiguity or equivocation is set aside, we can develop a clear and rigorous understanding of the logical properties of the concept under consideration.

The so-called logical order of development is satisfying to trained logicians (from the general to the particular, as it were), and has also some didactic value. But it is often the opposite of the way an individual or a researcher normally arrives at knowledge (building up from specific discoveries, then formulating a comprehensive theory); sometimes, replicating the natural order is a more effective teaching method.

Sometimes these two kinds of orders coincide. In the last analysis, they are always to some extent both involved, working in tandem; logical practice is an integral part of logical theorizing.

As for the historical order, it follows the natural order pretty closely, though with some redundancies. Some other consciousness must precede self-consciousness. Logic has developed on both the deductive and inductive sides alternately, and not in a systematic fashion.

*Goals.*

The goal of logic is to make the facts and their relations *transparent*; it teaches us to focus the object until its most firm manifestation is captured. Logic cannot immediately solve all problems, but it always brings us closer to the solutions.

For the individual, this self-discipline is the source of realism and understanding. ‘Think for yourself’, do your own thinking, ‘use your head’, be creative, think things through. The goal is not a mind a-buzz with words, a slave to words; but the inner peace and self-respect of efficacy.

In communication with others, transparency means expressing one’s thoughts clearly, so that, as far as possible at the time, there is no doubt or ambiguity as to just what one is trying to say, and on the basis of what processes. ‘Say what you mean, and mean what you say’. Information is freely and helpfully shared; points or areas of ignorance or error are easily admitted.

This is the idea of ‘*glasnost*’, transparency, a mutual respect and openness policy, a cooperative attitude, without unnecessary frictions. Too often, politicians, media, and others, use words to hide or distort, and do not in turn pay attention to input. You may prove something to them incontrovertibly; they remain unfazed, *comme si de rien n’était*.

Clarity of expression, accuracy of observation and thought, passing knowledge on honestly, reasonableness on all sides, are essential to vibrant democracy and social peace. Logic is a civilized way to resolve disputes.

This means self-criticism, the ability to review one’s own proposals, and anticipate possible objections, and try to deal with them as well as one can. We often gloss over possible problems in our own ideas, hoping no one will spot them; but this wastes one’s time, and everybody else’s. Logic is taking the time to double check one’s projects, shifting them this way and that way, to see how well focused they are in the largest context.

On the other hand, when receiving ideas, one’s should not look at them with an overly-critical eye, at least until one has properly understood them. Like rigid bone, hasty and excessive skepticism can inhibit the growth of knowledge. ‘Stop, look, listen’, hear, consider, make the effort to assimilate it. Learn before you try to teach.

While I am not of the opinion that logic is relative and arbitrary, there is more often than not at least some helpful truth to be found in other people’s concerns. One should not reject offhand, though still reserve one’s judgement. One should neither fool nor be fooled. Be humble, but keep your standards high.

### 2. What Logic is Not.

I get some very funny reactions from people at the mention of the word ‘logic’. One should not reject logic offhand, because of a mistaken notion of what it is about.

Logic is not a method of inferring all knowledge from a limited number of abstract premises; it is not a magical tool of omniscience. It depends for its action on moment by moment impressions or intuitions, which in some cases turn out to be unfounded. Nor is logic merely a mechanized manner of pursuing solutions to specific problems.

People often wrongly regard and use logic as a square-headed, narrow-minded activity. But in my opinion, logic is, straight and tough on a level of details, but overall very broad and open minded. Obstinacy and prejudice, are rather attributes of people unwilling to listen to reason, not even to at least consider alternative viewpoints. This is the very antithesis of a logician’s attitude.

People often oppose ‘logic’ to feeling; they believe it discards the emotional side of life. But logic does not mean ignoring feelings, but rather recommends taking the feelings — including their inner meaning, their intuited significance — as one set of data among others in the total picture; rationalistic data must also, however, be given their due weight.

Some people complain that ‘logic’ sometimes leads to evil conclusions. But value-judgments involve inferences from standards. So either the norms are unsound, or they have not been given their due weights in comparison to other norms, or the proposed means are not the exclusive ways to achieve the norms. Thus, the failure involved may precisely be a weakness in logical abilities, rather than any inherent coldness of logic.

Logic is only a tool — it cannot be blamed for errors made in its name, nor can it control the moral choices of individuals who utilize it. Its only possible danger is that the efficacy it endows on thought and action may be used for nefarious ends. But even then, a person who sees things truly clearly, with the broad conception logic gives, is less likely to have twisted values.

Logic is an important component of both mental health and moral responsibility. It requests that we face facts and listen to the voice of reason: this does not exclude having a heart or paying attention to one’s intuition. A person who does not keep in close touch with reality, can easily develop unhealthy emotions and make counter-productive choices. Rationality is a sign of maturity.

Another wrong impression people have of logic is that it is a meaningless manipulation of symbols, or at best a branch of mathematics. One man recently told me the following sad story. He thought of himself as a ‘logical person’, and being inclined to constantly improve his education, he enrolled for a University course on the subject in San Francisco. He was so put off by the lessons he attended, that he now hesitates to call himself ‘logical’!

### 3. Modus Operandi.

*Title.*

This is a book on logic, a formal and detailed study.

I called it ‘Future Logic’ to dedicate it to the future, to suggest its potential for improvement of human thinking and doing. It is also a logic about the future, aimed at knowledge of the possible and necessary. Lastly, it is futuristic, in that it is new, not of the past, unbound by previous limits. Hence, it is a young and optimistic logic, for and of the future, full of strength and energy.

I also called it ‘Future Logic’, because writing it has seemed an endless process. And it is really without end; I have left many things unsaid, only hinting at directions future logicians may take.

I would subtitle the book ‘modal logic’, to stress that all logic is ‘modal’, but not to imply that it concerns a specific sector of logic. The book ranges over virtually the whole of logic, constructing a well-integrated and fruitful *system* of logic, by means of an investigation of modality. A ‘system’ in the grand, traditional sense, not in the narrow sense used by modern logicians with reference to certain manipulations of limited scope.

‘Modality’, simply put, refers to concepts like possibility and necessity, which pervade knowledge in many different senses. Thought without modality is very limited in scope; much of our thinking depends on conceiving what the alternative possibilities are.

Modality is an incredibly creative force, which, like a crystal instantly solidifying a liquid, rushes through every topic and restructures it in new and interesting ways. I want to show how logic is forcefully pushed in a multitude of directions, as soon as modality and its ramifications are taken into account.

*Targets.*

In writing this book my ambition was to invigorate logic — to contribute to the science, and to revive interest in it by all segments of society.

Thus, it is intended equally for laypersons and scientists, for students and educators, and for professional logicians. It is equally a popularizing book, a text-book, and a research report.

My goal is not only to explore new avenues for the science of logic, but especially to make its teachings accessible to a wide public. For this reason, even while attempting to write a scholarly treatise, I do my best to keep it readable by anyone.

The book is full of ground-breaking discoveries, which should impress any logic theorist, and perhaps put him or her back to work. I mean, not just a peppering of incidental insights, but entirely original areas of concern, directions, and techniques, as will be seen. Though well-nigh encyclopedic in scope, it is not a compilation, but presents a unified system.

Although addressed to a wide audience, this is not an elementary work; it is an attempt to transmit advanced logic to everyone. My faith is that we have all reached a level of education high enough to absorb it and use it.

The book moves from the more obvious to the less so, from the simple to the complex, and from the old to the new, so that a layperson or student lacking any previous acquaintance with the subject-matter can grasp it all, granting a little effort. The order of development is thus natural and didactic, rather than strictly ‘logical’ in the sense of geometry. It is easy, at the end, when we know what we are talking about, to review the whole, and suggest a ‘logical’ ordering which consolidates it.

*Strategies.*

My approach is strongly influenced by Aristotle; all I do is push his methods into a much broader field. The primary purpose of logic should be to teach people to think clearly. For this reason, I try to develop the subject in ordinary language, and avoid any excessive symbolization.

Modern logicians have managed to overturn the very spirit of the discipline of logic, and made it cryptic, obscure, and esoteric. This was a disservice to the public, depriving it of an important tool for living, since most people lack the patience to decipher symbols.

Logical science as such has also suffered from this development. Logic has no intrinsic need of symbols other than those provided by ordinary language. An artificial language in principle adds nothing to knowledge, just as renaming things never does. Symbolization as such is just a quaint footnote to logic, not a real advance.

Symbols are to some extent valuable, *to summarize information in a minimum of space, or to discover and highlight patterns in the data*. But taken to an extreme, symbolism can lock us into simplistic mind-sets, and arrest further insight, limiting us to making trivial embellishments. Worse still, it can distance us from empirical inputs, turning logic into a game, a conventional, mechanical manipulation of arbitrary constructs, without referents, divorced from reality.

Also, I try as much as possible in this volume to avoid philosophical issues and metaphysical speculations, anything too controversial or digressive — and to concentrate on the matter at hand, which is formal logic. Some comments on such topics are inserted at the end, for the record.

A logician is of course bound to get involved in some wider issues. Every logical analysis intimates something about ‘thought processes’ and something about ‘external reality’. Logic somehow concerns the interface of these parallel dimensions of epistemology (the study of knowing) and ontology (the study of being), and it is hard to draw the lines.

*Tactics.*

I try to be brief. But I also try and touch on all relevant topics. Every issue is of course many-faceted, and capable of interminable treatment, with every layer uncovered seemingly more crucial than the previous. I still have great quantities of unused manuscript, and therefore know how much more remains to be said. But the reader may find that his questions at any stage, are readily answered in a later stage, in a wider context. One cannot do everything at once.

Often I am obliged to stop the further development of ideas. If I feel that an idea is already drawn clearly enough, and there would be boring repetitions of previously encountered patterns, I merely indicate the expected changes in pattern, and call on the reader to explore further on his or her own. This may be likened to the use of perspective and shading in artwork. Knowledge is infinite anyway, and as the saying goes ‘there is no end to words’.

The informed reader may find that there is too much elementary logic — but I am forced to include some at first, to make the discussion comprehensible to all, and to show the more advanced developments in their proper context. In any case, even in a discussion of traditional logic, an expert may find novel details or viewpoints, as the various aspects of a topic are unraveled.

I apologize to the novice for my failure to give many examples, but this disadvantage seems to me outweighed by the advantage of brevity. I assume the reader capable of searching for appropriate examples, and it is a good exercise. The neophyte reader is warned to beware of our use of many words in selective, specialized senses, which may be based on common connotations or even be neologisms; the context hopefully always makes the intention clear.

I also keep historical notes to a minimum in the course of the text, more intent on being a logician than a historian of logic. However, an effort to attribute authorship of the main lines of thought, is made towards the end, when I seek to place my own contributions in their historical context. My critical evaluations of modern trends in logic are also included at that stage.

My style of writing is no doubt not uniformly good. Repeated editing is bound to sometimes result in obscure discontinuities in the text. Little errors may creep in. I hope the reader will nevertheless be tolerant, because the substance is well worth it.

### 4. Scope.

The book is divided into 7 parts, with a total of 68 chapters; each of the chapters is split into on average 4 sections.

**Part I** starts with the three ‘laws of thought’, then presents the logic of actual categoricals (propositions of the form ‘X is Y’), including their features, their oppositions and immediate inferences, and syllogistic argument. Most of the credit for this seminal work can be attributed to Aristotle, although many later logicians were involved in the further development and systematization of his findings.

**Part II** defines the modalities called ‘*de re*’, and develops the logic of modal categoricals, following the same pattern as was established in the previous part. Although Aristotle wrote a great deal about concepts like potentiality, and described some modal arguments, he did not investigate this area of logic with the same thoroughness as the previous; nor have logicians since done much more, in my opinion. I introduce some new techniques, and arrive at some original results.

This part also, for the sake of completeness, analyses other forms of categorical proposition (among which, those concerning change) and other logical processes (such as ‘substitution’), some of which seem to have been previously ignored or underrated.

**Part III** defines logical modality, and analyses logical conditioning. This concerns ‘if-then’ (and ‘either-or’) propositions, which have been dealt with in great detail by modern logicians. While my own results concur with theirs on the whole, my approach differs in many respects; especially different are the definitions of logical modalities, but there are many significant technical innovations too (such as ‘production’).

**Part IV** introduces ‘*de re*’ conditioning, whose properties are found to be very distinct from those of logical conditioning. This is (to my knowledge) an entirely new class of propositions for logical science to consider, although commonly used in our everyday thought. The research emerged from the insights into modality obtained in part II, and provides us with original and important formal tools for the study of causality (and, incidentally, a better understanding of subsumption).

**Part V** begins with a new logic of classes (including a definitive solution of the Russell Paradox). Then I present the now-traditional discussion of scientific method (confirmation and discrediting of hypotheses), but from the viewpoint of logical modality.

**Part VI** contains an altogether original theory of induction based on ‘factorial analysis’. Consideration of modality in its various senses gives rise to a need for a completely new area of logic: how to induce modal propositions and how to resolve contradictions between them. The problems of generalization and particularization are solved systematically, using very formal techniques. Every aspect of this research — the tasks set, and the ways they are fulfilled — is a major breakthrough for logic.

The practical importance of factorial induction cannot be overstated. How far and in what direction can one generalize any finding? What happens when conflicting data is uncovered — how far and in what direction should one retreat from previous positions? What is the middle ground or compromise position or synthesis between competing views?

**Part VII** considers some of the ontological and epistemological implications of all my previous findings, with a sketch of my theory of cognition. The last few chapters provide a critical, historical and philosophical review of the whole field; this segment is more of interest to academic logicians than to the ordinary reader. Finally, the work is summarized, and I point out some of the opportunities for further research.

The reader is invited to peruse the table of contents for a more precise overview. I recommend that you return to it from time to time, so as to place the topic you are studying in its proper perspective.

## 2. FOUNDATIONS.

Logic is founded on certain ‘laws of thought’, which were first formulated by Aristotle, an ancient Greek philosopher. We shall describe them separately here, and later consider their collective significance.

### 1. The Law of Identity.

The Law of Identity is an imperative that we consider all evidence at its face value, to begin with. Aristotle expressed this first law of thought by saying ‘A is A’, meaning ‘whatever is, is whatever it is’.

There are three ways we look upon phenomena, the things which appear before us, however they happen to do so: at their face value, and as real or illusory.

We can be sure of every appearance, that it is, and is what it is. (i) *Something* has presented itself to us, whether we thereafter judge it real or illusory, and (ii) this something displays *a certain configuration*, whether we thereafter describe and interpret it rightly or wrongly. The present is present, the absent is absent.

Every appearance as such is objectively given and has a certain content or specificity. We can and should and commonly do initially regard it with a simple attitude of receptiveness and attention to detail. Every appearance is in itself neutral; the qualification of an appearance (thus broadly defined) as a ‘reality’ or an ‘illusion’, is a subsequent issue.

That statement is only an admission that any phenomenon minimally exists and has given characteristics, *without making claims about the source and significance* of this existence or these characteristics. The moment we manage to *but think* of something, it is already at least ‘apparent’. No assumption need be made at this stage about the nature of being and knowledge in general, nor any detailed categorizations, descriptions or explanations of them.

Regarded in this way, at their face value, *all* phenomena are evident data, to be at least taken into consideration. The world of appearances thus offers us *something to work with, some reliable data* with which we can build the edifice of knowledge, a starting point of sorts. We need make no distinctions such as those between the physical/material and the mental, or sense-data and hallucinations, or concrete percepts and abstract concepts; these are later developments.

The law of identity is thus merely an acknowledgement of the world of appearances, without prejudice as to its ultimate value. It defines ‘the world’ *so broadly*, that there is no way to counter it with any other ‘world’. When we lay claim to another ‘world’, we *merely expand* this one. All we can ever do is subdivide the world of appearances into two domains, one of ‘reality’ and one of ‘illusion’; but these domains *can never abolish* each other’s existence and content.

What needs to be grasped here is that every judgment implies the acceptance, at some stage, of some sort of appearance as real. There is no escape from that; to claim that nothing is real, is to claim that the appearance that ‘everything is illusory’ is real. We are first of all observers, and only thereafter can we be judges.

Reality and illusion are simply terms more loaded with meaning than appearance or phenomenon — they imply an evaluation to have taken place. This value-judgement is a final characterization of the object, requiring a more complex process, a reflection. It implies we went beyond the immediately apparent. It implies a broader perspective, more empirical research, more rigorous reasoning. But what we finally have is still ‘appearance’, though in a less pejorative sense than initially.

Thus, ‘real’ or ‘illusory’ are themselves always, ultimately, just appearances. They are themselves, like the objects of consciousness which they evaluate, distinct objects of consciousness. We could say that, there is a bit of the real in the illusory and a bit of the illusory in the real; what they have something in common is appearance. However, these terms lose their meaning if we try to equate them too seriously.

*On what basis* an appearance may or should be classified as real or illusory is of course a big issue, which needs to be addressed. That is the overall task of logic, to set precise guidelines for such classification. But the first step is to admit the available evidence, the phenomenal world as such: this gives us a data-base.

### 2. The Law of Contradiction.

The Law of Contradiction is an imperative to reject as illusory and not real, any apparent presence together of contradictories. This second law of thought could be stated as ‘Nothing is both A and not-A’, or ‘whatever is, is not whatever it is not’.

We cannot say of anything that it is *both* present and absent at once: *what is present, is not absent*. If the world of appearance displays some content with an identity, then it has effectively failed to display nothing. Contradictory appearances cannot coexist, concur, overlap: they are ‘incompatible extremes’.

We can say of something that it ‘is’ something else, in the sense of having a certain relation to something distinct from itself, but we cannot say of it that it both has and lacks that relation, in one and the same respect, at one and the same place and time.

It is evident, and therefore incontrovertible (by the previous law), that appearances are variegated, changing, and diverse. Phenomena have a variety of aspects and are usually composed of different elements, they often change, and differ from each other in many ways. However, for any respect, place and time, we pinpoint, the appearance as such is, and is whatever it is — *and not at once otherwise.*

The law of contradiction is not a mere rephrasing of the law of identity, note well, but goes one step further: it sets a standard for relegating some appearances to the status of illusions; in a sense, it begins to define what we mean by ‘illusion’. It does not, however, thereby claim that all what is leftover in the field of appearance is real with finality; nor does it deny that some of the leftovers are real (as is assured us by the law of identity).

By the law of identity, whatever appears is *given some credence*: therefore, one might suggest, the coexistence of opposites has some credence. The law of contradiction interposes itself at this point, and says: no, such events *carry no conviction* for us, once clearly discerned. The first law continues to function as a recognition that there is an apparent contradiction; but the second law imposes on us the need to resolve that contradiction somehow.

The law of contradiction is itself, like anything else, an appearance among others, but it strikes us as an *especially credible* one, capable of overriding the initial credibility of all other considerations. It does not conflict with the message of the law of identity, since the latter is open to any event, including the event that some appearances be more forceful than others. The law of contradiction is precisely one such forceful appearance, an extremely forceful one.

Thus, though the world of appearances presents itself to us with some seeming contradictions, they appear *as* incredible puzzles — their unacceptability is inherent to them, obvious to us. We may verbally speculate about a world with real contradictions, and say that this position is consistent with itself even if inconsistent with itself. But the fact remains that whenever we are face to face with a specific contradiction (including that one) we are unavoidably skeptical — something seems ‘wrong’.

The way we understand the apparent existence of contradictions is by viewing the world of appearances as *layered*, or stratified. Our first impressions of things are often superficial; as our experience grows, our consciousness penetrates *more deeply* into them. Thus, though each level is what it is (law of identity), parallel levels may be in contradiction; when a contradiction occurs, it is because we are superimposing different layers (law of contradiction). In this way, we resolve the ‘general contradiction’ of contradiction as such — we separate the conflicting elements from each other.

(Note in passing, as an alternative to the metaphor of ‘depth’, which likens consciousness to a beam of light, we also sometimes refer to ‘height’. Here, the suggestion is that the essence of things is more elevated, and we have to raise ourselves up to make contact with it.)

That resolution of contradiction refers to the diversity and change in the world of appearance as due to the perspectives of consciousness. Thus, the appearance of the phenomena we classified as ‘illusory’ is due to the limitations of ordinary consciousness, its failure to know everything. This restriction in the power of consciousness may be viewed as a ‘fault’ of our minds, and in that sense ‘illusion’ is a ‘product’ of our minds. For that reason, we regard the illusory as in some sense ‘imaginary’ — this is our explanation of it.

On a more objective plane, we may of course accept diversity and change as real enough, and explain them with reference to the space and time dimensions, or to uniform and unchanging essences. In such cases, we are able to meet the demands of the law of contradiction without using the concept of ‘illusion’; only when space, time, and respect, are clearly specified, does a contradiction signify illusion.

### 3. The Law of the Excluded Middle.

The Law of the Excluded Middle is an imperative to reject as illusory and not real, any apparent absence together of contradictories. This third law of thought could be stated as ‘nothing is neither A nor not-A’, or ‘whatever is, either is some thing or is-not that thing’.

We cannot say of anything that it is at once neither present nor absent: *what is not present, is absent*. If the world of appearance fails to display some content with an identity, then it has effectively displayed nothing. There is no third alternative to these two events (whence the expression ‘excluded middle’): they are exhaustive.

We may well say that some parts or aspects of the world are inaccessible to our limited faculties, but (as pointed out in the discussion of identity) we cannot claim a world beyond that of appearances: the moment we mention it, we include it.

It may be that we neither know that something is so and so, nor know that it is not so and so, but this concerns knowledge only, and in reality that thing either is or is-not so and so. Whatever we consider must either be there or not-there, in the specified respect, place and time, even if we cannot discern things enough to tell at this time or ever. There is an answer to every meaningful question; uncertainty is a ‘state of mind’, without ‘objective’ equivalent.

Moreover, *strictly speaking*, ‘questions’ are artificial attempts to anticipate undisplayed layers of appearance. As things appear now, if nothing is being displayed, *that* is the (current) ‘answer’ of the world of appearances; in the world of appearances there are no ‘questions’. ‘Questions’ merely express our resolve to pursue the matter further, and try to uncover other layers of appearance; they are not statements about reality.

If we choose to, *loosely speaking*, regard doubts as kinds of assertions, the law of the excluded middle enjoins us to class them at the outset as illusory, and admit that in reality things are definite. Problematic statements like ‘it might or might not be thus’ are not intended to affirm that ‘neither thus nor not-thus’ *appeared*, but that what did appear (whether it was ‘thus’ or ‘not-thus’ — one of them did, for sure) was not sufficiently forceful to satisfy our curiosity.

Even if no phenomenon is encountered which confirms or discredits an idea, there must be a phenomenon capable of doing so, in the world somewhere, sometime. We have to focus on the evidence, and try and distinguish the appearance or nonappearance of that imagined phenomenon.

Thus, the law of the excluded middle serves to create a breach of sorts between the ‘objective world’ and the ‘world of ideas’, and establishes the pre-eminence of the former over the latter. The breach is not an unbridgeable gap, but allows us to expand our language, in such a way that we can discuss eventual layers of appearance besides those so far encountered, even while we admit of the evidence at hand.

Such an artifice is made possible by our general awareness from past experience that appearances do change in *some* cases, but should not be taken to mean that any given appearance *will* change. It is only the expression of a (commendable) ‘open-mindedness’ *in principle*, with no specific justification in any given case.

What we have done, effectively, is to expand what we mean by ‘appearance’, so as to include future appearance, in addition to appearances until now in evidence. Thus far, our implicit understanding was that appearance was *actual*, including present realities and present illusions. Now, we reflect further, and decide to embrace our anticipations of *‘possible’ *appearances as a kind of actuality, too.

Such hypothetical projections are also, in a sense, ‘apparent’. But they are clearly imaginary, inventions of the mind. Their status as appearances is therefore immediately that of ‘illusions’; that is their present status, whatever their future outcome. However, they are illusory with less finality than the phenomena so labeled by the law of contradiction; they retain some degree of credibility.

## 3. LOGICAL RELATIONS.

### 1. True or False.

Reality and illusion are attributes of phenomena. When we turn our attention to the implicit ‘consciousness’ of these phenomena, we correspondingly regard the consciousness as realistic or unrealistic. The consciousness, as a sort of peculiar relation between a Subject (us) and an Object (a phenomenon), is essentially the same; only, in one case the appearance falls in the reality class, in the other it falls in the illusion class.

Why some thoughts turn out to be illusory, when considered in a broader context, varies. For example, I may see a shape in the distance, and assume it that of a man, but as I approach it, it turns out to be a tree stump; this latter conclusion is preferred because the appearance withstands inspection, it is firmer, more often confirmed. A phenomenon always exists as such, but it may ‘exist’ in the realms of illusion, rather than in that of reality. The fact that I saw some shape is undeniable: the only question is whether the associations I made in relation to it are valid or not.

‘Propositions’ are statements depicting how things appear to us. Understood as mere *considerations* (or ‘hypothetically’), they contain no judgment as to the reality or illusion of the appearance. Understood as *assertions* (or ‘assertorically’), they contain a judgment of the appearance as real or illusory.

Assertoric propositions must either be ‘true’ or ‘false’. *If we affirm a proposition, we mean that it is true; if we deny a proposition, we mean that it is false.* Our definitions of truth and falsehood must be such that they are mutually exclusive and together exhaustive: what is true, is not false; what is false, is not true; what is not true, is false; what is not false, is true.

Strictly speaking, we call an assertion *true*, if it verbally depicts something which appears to us as real; and *false*, if it verbally depicts something which appears to us as illusory. In this ideal, absolute sense, true and false signify total or zero credibility, respectively, and allow of no degrees.

However, the expressions true and false are also used in *less stringent senses*, with reference to less than extreme degrees of credibility. Here, we call a proposition (relatively, practically) true if the appearance is more credible than any conflicting appearance; and (effectively) false, if the appearance is not the most credible of a set of conflicting appearances. Here, we can speak of more or less true or false.

The ultimate goal of logic is knowledge of reality, and avoidance of illusion. Logic is only incidentally interested in the less than extreme degrees of credibility. The reference to intermediate credibility merely allows us to gauge tendencies: how close we approach toward realism, or how far from it we stray. Note that the second versions of truth and falsehood are simply wider; they include the first versions as special, limiting cases.

Propositions which cannot be classed as true or false right now are said to be *‘problematic’*. Both sets of definitions of truth and falsehood leave us with gaps. The first system fails to address all propositions of intermediate credibility; the second system disregards situations where all the conflicting appearances are equally credible.

If we indeed cannot tip the scales one way or the other, we are in a quandary: if the alternatives are all labeled true, we violate the law of contradiction; if they are all labeled false, we violate the law of the excluded middle. Thus, we must remain with a suspended judgment, and though we have a proposition to consider, we lack an assertion.[1]

### 2. Branches of Logic.

The concepts of truth and falsehood will be clarified more and more as we proceed. In a sense, the whole of the science of logic constitutes a definition of what we mean by them — what they are and how they are arrived at. We shall also learn how to treat problematic propositions, and gradually turn them into assertions.

The task of sorting out truth from falsehood, case by case, is precisely what logic is all about. What is sure, however, is that that is in principle feasible.

If thought was regarded as not intimately bound with the phenomena it is intended to refer to, it would be from the start disqualified. In that case, the skeptical statement in question itself would be meaningless and self-contradictory. The only way to resolve this conflict and paradox is to admit the opposite thesis, viz. that some thoughts are valid; that thesis, being the only internally consistent of the two, therefore stands as proven.

This is a very important first principle, supplied to us by logic, for all discussion of knowledge. *We cannot consistently deny the ultimate realism of (some) knowledge.* We cannot logically accept a theory of knowledge which in effect invalidates knowledge. *That* we know is unquestionable; *how* we know is another question.

Now, logical processes are called *deductive* (or analytic) to the extent that they yield indisputable results of zero or total credibility; and *inductive* (or synthetic) insofar as their results are more qualified, and of intermediate credibility. Deductive logic is conceived as concerned with truth and falsehood in their strict senses; inductive logic is content to deal with truth and falsehood in their not so strict senses.

This distinction is initially of some convenience, but it ultimately blurs. Logical theory begins by considering deductive processes, because they seem easier; but as it develops, its results are found extendible to lesser truths. Likewise, inductive logic begins with humble goals, but is eventually found to embrace deduction as a limiting case.

As we shall see, both these branches of logic require intuition of logical relations, and both presuppose some reliance on other phenomena. Both concern both concrete percepts and abstract concepts. Both involve the three faculties of experience, reason and imagination; only their emphasis differs somewhat. There is, at the end, no clear line of demarcation between them.

### 3. Tools of Logic.

The following are three logical relations which we will often refer to in this study: implication, incompatibility, and exhaustiveness. We symbolize propositions by letters like P or Q for the sake of brevity; their negations are referred to as notP (or nonP) and notQ, respectively.

. One proposition (P) is said to imply another (Q) if it cannot happen that the former is true and the latter false. Thus, if P is true, so must Q be; and if Q is false, so must P be — by definition. It does not follow that P is in turn implied by Q, nor is this possibility excluded. This relationship may be expressed as “If P, then Q”, or equally as “If nonQ, then nonP”. We can deny that Q is implicit in P by the formula “If P, not-then Q”, or “If nonQ, not-then nonP”.*Implication*

When we use expressions like ‘it follows that’, ‘then’, ‘therefore’, ‘hence’, ‘thence’, ‘so that’, ‘consequently’, ‘it presupposes that’ — we are suggesting a relation of implication.

(or inconsistency or mutual exclusion). Two propositions (P, Q) are said to be incompatible if they cannot both be true. This relation is also called ‘exclusive disjunction’, and expressed by the formula ‘P or else Q’. Thus, if either is true, the other is false. The possibility that both be false is not excluded, nor is it affirmed. This relation can be formulated as “If P, then nonQ”, or equally as “If Q, then nonP”. The denial of such a relation would be stated as “If P, not-then nonQ”., or “If Q, not-then nonP”.*Incompatibility*

We can also say of more than two propositions that they are incompatible; meaning, if any one of them is true, all the others must be false (though they might well all be false).

. Two propositions (P, Q) are said to be exhaustive if they cannot both be false. This relation is also called ‘inclusive disjunction’, and expressed by the formula ‘P and/or Q’. Thus, if either is false, the other is true. The possibility that both be true is not excluded, nor is it affirmed. This relation can be formulated as “If nonP, then Q”, or equally as “If nonQ, then P”. The denial of such a relation would be stated as “If nonP, not-then Q”., or “If nonQ, not-then P”.*Exhaustiveness*

We can also say of more than two propositions that they are exhaustive; meaning, if all but one of them is false, the remaining one must be true (though they might well be all true).

We note that whereas implication and its denial are directional relations, incompatibility and exhaustiveness and their denials are symmetrical relations.

Also, underlying them all is the concept of ‘conjunction’, whether or not one can say one thing with or without the other. Consequently, these expressions are interconnected; we could rephrase any one in terms of any other. For example, ‘P implies Q’ could be restated as ‘P is incompatible with notQ’ or as ‘notP and Q are exhaustive’.

The following table summarizes the above through analysis of the possibilities of combination of the affirmations and denials of two propositions, P and Q, which are given as having a certain logical relation, specified in the left column. ‘**No**’ indicates logically impossible combinations, ‘**yes**’ combinations specified as possible, and ‘**?**’ signifies that the status of the combination as it stands, without further specification, is undetermined by the logical relation concerned.

POSSIBILITY OF: | P+Q | P+nonQ | nonP+Q | nonP+nonQ |

Implication | yes | no | ? | yes |

Incompatibility | no | yes | yes | ? |

Exhaustiveness | ? | yes | yes | no |

Unimplication | ? | yes | ? | ? |

Compatibility | yes | ? | ? | ? |

Inexhaustiveness | ? | ? | ? | yes |

We shall have occasion to review these relations in more detail later, and also define what we mean by logical possibility or impossibility. Their study is a big part of logic. For now, it is enough to just point them out, for practical purposes.

### 4. Axioms of Logic.

We can now re-state the laws of thought with regard to the truth or falsehood of (assertoric) propositions as follows. These principles (or the most primary among them) may be viewed as the axioms of logic, while however keeping in mind our later comments (ch. 20) on the issue of their development.

: Every assertion implies itself as ‘true’. However, this self-implication is only a claim, and does not by itself prove the statement.*The law of identity*

More broadly, whatever is implied by a true proposition is also true; and whatever implies a false proposition is also false. (However, a proposition may well be implied by a false one, and still be true; and a proposition may well imply a true one, and still be false.)

: If an affirmation is true, then its denial is false; if the denial is true, then the affirmation is false. They cannot be both true. (It follows that if two assertions are indeed both true, they are consistent.)*The law of contradiction*

A special case is: any assertion which implies itself to be false, is false (this is called self-contradiction, and disproves the assertion; not all false assertions have this property, however).

More broadly, if two propositions are mutually exclusive, the truth of either implies the falsehood of the other, and furthermore implies that any proposition which implies that other is also false

: If an affirmation is false, then its denial is true; if the denial is false, then the affirmation is true. They cannot both be false. (It follows that if two assertions are indeed both false, they are not exhaustive).*The law of the excluded middle*

A special case is: any assertion whose negation implies itself to be false, is true (this is called self-evidence, and proves the assertion; not all true assertions have this property, however).

More broadly, if two propositions are together exhaustive, the falsehood of either implies the truth of the other, and furthermore implies that any proposition which that other implies is also true (though propositions which imply that other may still be false).

Thus, in summary, every statement implies itself true and its negation false; it must be either true or false: it cannot be both and it cannot be neither. In special cases, as we shall see, a statement may additionally be self-contradictory or self-evident.

Some of these principles are obvious, others require more reflection and will be justified later. They are hopefully at least easy enough to understand; that suffices for our immediate needs.

Note in passing that each of the laws exemplifies one of the logical relations earlier introduced. Identity illustrates implication, contradiction illustrates incompatibility, excluded-middle illustrates exhaustiveness.

Although we introduced the logical relations before the laws of thought, here (for the sake of clarity and since we speak the same language), it should be obvious that, conceptually, the reverse order would be more accurate.

First, come the intuitions of identity, contradiction, and excluded-middle, with the underlying notions (visual images, with velleities of movement), of equality (‘to go together’), conflict (‘to keep apart’), and limitation (‘to circumscribe’). Thereafter, with these given instances in mind, we construct the more definite ideas of implication, incompatibility, and exhaustion.

## 4. WORDS AND THINGS.

### 1. Verbalizing.

A major function of the discipline of logic is to teach us to express our thoughts explicitly, clearly and unambiguously. We consider thought as serial, because words are strung together; but the underlying perception or conceptual insight is often more global.

A perception or conceptual insight may be wordless, even a logical process of thought and inference may occur in inner silence. We often feel in ourselves or see in other people a facial or bodily reaction, like a smile of assent or sardonic grin of doubt, and know some thought has taken place on a subconscious or unconscious level, though we cannot say what or why.

We can be aware of a phenomenon without labeling it; but we often label things, to mentally process or socially communicate a thought concerning them. Our thinking is usually expressed by the formation of sounds inside our heads, or we voice or write or even gesture our thoughts.

To label something, we need only *point to it*, physically or mentally, and utter a word; we then understand that henceforth this word is to direct our attention to that thing. When we point to something for our own purposes, we know immediately what we mean; but when we do so in an attempt to communicate our intention to others, we may of course be misunderstood.

If one does not understand the significance of ‘pointing’, one cannot grasp the intention of words. Physical pointing seems to be a sending out of ‘energy’ in the desired direction, enough to draw the respondent’s attention along that line till it meets the object concerned. Animals seem not to comprehend it usually, though sometimes they seem to.

The *‘meaning’* of a word, then, is primarily the phenomenon or group of phenomena we pointed to, one way or another, when we introduced it — and all its eventual manifestations. However, we may later narrow the meaning down, and gradually attach the word to a more distinctive and invariable aspect of it all.

Words are symbols. The mind usually assigns one word (if any) for each thing, though sometimes more than one word may be assigned to one thing (equivocation), or one word may be assigned to more than one thing (ambiguity).[2]

Any object of our consciousness may be distinctively named. But most literally single phenomena are ephemeral, and naming them all would be pointless and confusing. Mostly, we label things by reference to their *similarities and differences*. We look for *repetitive, yet distinct* experiences, and assign names to *groups of phenomena* which have some permanence and relevance to our lives. Even ‘individual’ things are groups of phenomena; ‘kinds’ of things are doubly so.

In the case of *proper names*, of persons or pets, all the manifestations of an individual entity are referred to; for example, ‘Aristotle’ refers to all the accumulated impressions of that person. In the case of *common names*, like ‘man’, a group of similar entities is intended, and all their manifestations as individuals. We do not, of course, have to give proper names to every instance of a kind; we can distinguish them *indicatively*, as in ‘this flower’.

In any case, the existence of *a continuity* is always presumed by our use of words; as is our ability to *recognize* such continuity, in spite of changes of the individual across time or differences from one individual to the next. Many differences are discounted. The labeling is open-ended, confident of our power to apply it as we proceed; if we managed previously, why not also subsequently?

We can limit our vocabulary further, by making statements involving strings of words, instead of inventing new words. Things appear to us not in isolation, but as having various relations. ‘Relationships’ are of course themselves phenomena, which we group and name if found interesting.

When we encounter a relational phenomenon, rather than viewing it as a unity, we distinguish the things related and the relation, and verbally express our perception or conceptual insight as a sentence. Still more complex phenomena may require finer analysis through the use of many sentences.

Thus, words serve first to capture our concrete or abstract experiences. When the phenomenon is relational, we may express it verbally through a sentence or series of sentences. A language is an agreed upon collection of words, a vocabulary, and a convention as to the ways the words may be put together into sentences, a grammar.

The mental or vocal sound, or written symbol, or gesture, acquires the status of a word, only if we once pointed to something (with the index finger, or saying ‘look there!’), giving its ‘coordinates’, or address in space and time. Eventually, we could name something described in terms of other words previously based on such pointing. A ‘word’ without some ultimate points of reference is a meaningless entity.

A word establishes a conventional correspondence between word and thing. We may imagine a ‘line of relation’ joining word and thing, and call it ‘meaning’ or ‘intent’. Once invented in this way, the word may be used as an instrument of thought. Henceforth the word becomes, as it were, our substitute for the thing, representing it like an ambassador. We can focus on it, manipulate it, store it away in or recall it from memory, or pass it on to other people (communicate it).

Simply put, ‘memory’ is any locale where words are laid to rest pending our resumption of attention to them. Words may be externally stored: written in a book or taped on a cassette; or they may be internally stored in our own ‘minds’. However, memory must include not only the word, but also somehow what it refers to.

Recalling the word shape or sound would not constitute full remembering, unless we are also awakened to the meaning of the word as well. On the other hand, remembering may be wordless. Therefore, the essence of memory, however it works, is its ability to cause our awareness to return to the original object or some comparable re-enactment of it.

The words involved are incidental; what counts is the underlying act of consciousness. Still, words are useful instruments, not mere appendages. The words we read or hear act as ‘switches’, which re-trigger and direct our attention to specific experiences or reproductions of them.

### 2. Same and Different.

Note that we may decide together that this sound and that visual symbol will be ‘the same word’, and be used to refer to the same thing. For example, the sound ‘dog’ and the written ‘d-o-g’ are considered equivalent, though they are substantially different.

Furthermore, a ‘word’ is always *a class of symbols*: many individual sounds or visual symbols which resemble each other, or are accepted as one and the same ‘word’. Any word that I utter or hear or write or read today is a different individual manifestation from its previous occurrence, yet their similarity of sound or look, allow me to recognize them as ‘one’ word.

For this reason, it is absurd to contend that ‘the only thing which allegedly similar objects have in common is the name we assign them’. If nothing was similar to anything else, or we could not recognize things, then even words (as themselves objects) would have no resemblances, and be unrepeatable.

Thus, the existence of *some* similarity, and its knowability *in principle*, are inescapable. How we come to know that things are same or different is a big question, but it need not concern us at this stage, since logic assures us that we at least sometimes do manage to know it.

An entity is a unique complex knot of time, place, attributes, motions, relationships of various kinds. The ‘boundaries’ of an appearance are themselves usually given as a component of the total phenomenon, though occasionally we may delimit some arbitrary part of a continuum as a unit for consideration. Nothing seems to exist which appears unrelated in some way or other to other things. Something can always be said about anything.

Especially, the relations of sameness and difference seems to be pervasive; everywhere we look, we get these impressions of resemblance and differentiation. If the world contained absolutely only one uniform thing, there would be no call for concepts of similarity or difference. Such utter inimitable and undifferentiatable Unity perhaps concerns G-d, prior to Creation. But the world we know, the world of appearances, is given as a multiplicity of experiences, with more than one object and at least one subject of consciousness.

A world of many things, but which are entirely *without any similarities* between them, a world where nothing has anything in common with anything else and everything is ‘an island unto itself’, is unimaginable. If such a world contains more than one thing, they have in common at least ‘existence’, ‘singularity’, and ‘dissimilarity’.

A world of many things, but which are entirely *without any differences* between them, a world where everything has everything in common with everything else and is an ‘exact replica’ of each other thing, is also unimaginable. If such a world contains more than one thing, they must differ at least in their space and time coordinates to be apart, to be ‘many’.

In comparing two or more individual appearances, we may find that they seem to have certain distinguishable factors in common, and our response is to look upon these distinct similarities as significant enough to be named and treated as thought-units. In philosophy, the apparent common factors of things are called *‘universals’*.

The simplest way to think of universals is to regard them as substances scattered throughout the world, mingling in different combinations, together constituting entities. Thus, greenness may color objects as distinct as a leaf or a computer screen; a leaf is a meeting point, a sum, of shape, size, color, temperature, and so on. This is the common-sense view, which we will accept as good enough for our purposes here.

When a word is assigned to a new appearance, we do so because the phenomenon seems distinct from any other previously encountered. If further experience shows this initial impression erroneous, because the phenomenon is not novel, then the word becomes an equivocation or falls into disuse.

Likewise, we may wrongly assign a previously created word, or combination of words, to a new phenomenon, which at first seemed to, but on closer inspection ceased to, resemble the old, so that ambiguity arises, or we must reclassify the experience under another word or formulate another sentence.

Thus, naming and verbalizing of our experiences suggests analogies which may later be found inadequate, or which may stand the test of time and further experience. In the former case, we judge the initial assumption illusory; in the latter case, real. But the experience in question remains what it was, however we judge it. Whether real or illusory, it is an ‘appearance’, something presenting itself to us as object of consciousness.

A big issue in philosophy is whether these intuited commonalities, these resemblances (re-appearances, seeming repetitions), are rooted in the mind somehow (subjective), or whether they exist out there in the object somehow (objective), or both somehow. How can something (a universal) be at once one and many? Theorists have suggested a variety of possible scenarios on either side, but never to everyone’s full satisfaction. There may be truth in what they say, but further follow-up is needed.

From the point of view of Logic, no such theory can stand which concludes in the denial that these similarities have some status of reality. For such theory itself, being formulated in conceptual terms, would thereby imply itself untrue. Whatever our theory, the result must be to justify, rather than cause rejection of, the assumption of similarity; for only such result is logically tenable.

As far as concerns Logic, if there is an appearance of resemblance, it is to be considered at its face value. Logically, the appearance of resemblance cannot be declared wrong in principle, even though its exact nature is admittedly yet unclear to us. We may initially assume it to be realistic, without a priori excluding the possibility that *some* such appearances may (as any appearance may) turn out to be illusory.

On the basis of our apparent knowledge of similarity, we tend to group individual phenomena into classes, defined by some selected common factor. In what sense that common factor is itself essentially singular, while being scattered in the many class members, is a mystery. Logic leaves such issues to philosophers and moves on.

### 3. On Definition.

Some comments concerning definition are in order here. One way to define a word is to point to a material object with one’s index finger and say the word; or we may mentally focus on something and think the word in our heads. Alternatively, we may notice that other people repeatedly use a word in the face of a certain experience, and thus we learn that this word refers to that experience.

Yet another way, is to describe something using other words, and assign the new word to this description. Effectively, such definition serves to draw the mind’s attention to the object intended: it is not a mere equation of words. We may later realize that the description we gave was not accurate, and propose a new verbal definition. The word can stay unchanged, we ‘know’ what we were trying to mean by it, only now we have a clearer description of that phenomenon.

Were definition a mere conventional equation of words, a definition would be unchangeable, since the meaning of the word would change when the definition was altered, and we would be talking about a different object than originally intended. But because a definition is an attempt at description, merely designed to direct the mind towards an object of wordless consciousness, it is changeable.

Definition is an attempt to express what appears to be the ‘essential’ character of the object concerned. Nonetheless, it must be stressed that the assumed essence is itself only an appearance: it may at a later stage appear unessential, or even be found to not always be displayed by the object, and other definitions may replace it. Although definition is, like any other aspect of knowledge, flexible, that does not make it any less useful or valid.

In principle, note well, not everything is definable. To suggest that every word must be defined in other words, is to make an impossible demand for an infinite chain of derivations. There has to be some primary meanings, known directly, on which later descriptive meanings may be built. The phenomenon in question may be so fundamental, that we cannot discern any simpler components in it, but can only discern it as a component of more complex phenomena.

Thus, there is no rational basis for forbidding ‘circular’ definition as such. Some definitions are merely formulated to clarify, but make no claim to being much more than tautologies. Even as we make one, we may know that the words we are using are not themselves definable, and may just be other words meaning the same things. But the definition may still be useful in directing the mind more precisely where we want it, by linking together disparate pointings and namings.

## 5. PROPOSITIONS.

### 1. Terms and Copula.

Logic looks upon sentences as attempts to record or predict reality, which may or may not be correct. For this reason, it calls them *propositions*, to stress their fallibility. Logic develops by scrutiny of ordinary thought and language, but also sets especially rigid structural standards in order to be able to develop systematically.

Looking at many propositions, we see that irrespective of their particular contents, they appear to share certain ‘forms’. Our job is to analyze each form, how it is structured, what it means and implies, what are its interrelationships with other propositions, and how it can be known to be true.

Our study begins with one form shared by many propositions, ‘S is P’. Propositions of this sort are characterized as *categorical*, meaning that they are unconditional. We call ‘S’ the *subject*; ‘is’, the *copula*; and ‘P’ the *predicate*. The subject and predicate are both called *terms*. The copula relates the terms together in a certain way. We may view the subject as our center of interest, while the predication (copula and predicate) provides us with additional information concerning it.

Note well how the terms are treated as ‘variables’, while other features such as the copula (so far) are kept ‘constant’, like in algebra. In this way, we can theoretically concentrate on the properties of a kind of proposition, without regard to the specific ‘values’ which might take the place of the symbols S and P. Form is released from content.

We owe this artifice to Aristotle’s genius. In one stroke, it made possible the development of a science of logic, because the study of relations and processes was thereby greatly facilitated, as we shall see.

We will concentrate mainly on categoricals called *classificatory*. Here, the subject and predicate are classes, and their copula informs us that they contain members in common. Typically, in a general proposition, the subject is a species and the predicate a genus; for example, ‘trees are plants’. Other forms will be dealt with eventually.

### 2. Polarity and Quantity.

Propositions may be distinguished by the *polarity* of their copula. Thus, ‘S is P’ is said to have a *positive* copula; ‘S is not P’, a *negative* one. (Polarity is traditionally also known as ‘quality’, note, but since this word has other meanings it will be avoided here.)

We could view ‘is’ and ‘is not’ as two distinct relations (which happen to be contradictory), or as respectively the presence and absence of the same relation of ‘being’ (so that ‘is-not’ means ‘not-is’); logically, these viewpoints are equivalent.

The characterization of propositions as affirmations or denials has accordingly two senses, one absolute and the other relative. Normally, an assertion with a positive copula is called affirmative, and that with a negative copula is called denying; but also, we say of either polarity that it affirms itself and denies the other.

Another relevant distinction between propositions refers to their *quantity*. This primarily concerns the subject, clarifying how much of it we intend by our statement. The quantity is often left tacit in everyday discourse, but for the purposes of science, we have to be more explicit.

If S is a specific, recognizable individual, we use the designation ‘this S’, and the proposition is said to be *singular* (and indicative). Any proposition which is not singular may be called *plural*. If S refers to the whole class, we say ‘all S’, and the proposition is called *general* or universal. If S is a loose reference to some unspecified member(s) of the class, we say ‘some S’, and the proposition is called *particular*.

Other quantifiers define ‘some’ more precisely. Thus, ‘a few’ or ‘many’ mean, a small or large number; ‘few’ or ‘most’ mean, a minority or majority, a small or large proportion. These for most purposes have the same logical properties as particulars, though the latter two sometimes require special treatment.

By combining these different features, the various polarities and quantities, we obtain the following list of classificatory propositions. These are traditionally assigned symbols as shown to facilitate treatment (from the Latin words **A**ff**IR**mo and n**EGO**, which serve as mnemonics).

A | All S are P | E | No S is P |

R | This S is P | G | This S is not P |

I | Some S are P | O | Some S are not P |

The other quantities are also applicable to the two polarities, of course, as in ‘Few or Most S are or are not P’, but have not been traditionally symbolized.

All such propositions are called *actual*, because they suggest the relation they describe as taking place in the present. In that case, they imply that the units which their terms referred to do exist, i.e. that there are S’s and P’s in the world at the time concerned. This claim is open to debate, but will be taken for granted for now — later, we will clarify the issues involved, and look into the implications of not making such an assumption.

### 3. Distribution.

Plural propositions normally refer us to their class members *each one singly*; the plural is simply a shorthand statement of a number of independent singulars. Each individual, subsumed by the subject, and included in the all or some enumeration, is separately and equally related to the predicate. The predication is intended to be ‘dispensively’ applied; meaning severally, not jointly or collectively.

Thus, ‘All S are P’ or ‘Some S are P’, here means ‘S1 is P’, ‘S2 is P’, ‘S3 is P’, … and so on; ‘No S is P’ or ‘Some S are not P’ here means ‘S1 is not P’, ‘S2 is not P’, …etc. — until every S, this one, that one, and the others, which are included by the quantity have been listed.

The doctrine of distribution is that if all the members of a class are covered, the term is called ‘distributive’; otherwise it is not.

This means that the subjects of universals, **A** and **E**, are distributive; whereas those of particulars, **I** and **O**, are not, since the instances involved are not fully enumerated. With regard to singulars, **R** and **G**, they are effectively distributive, insofar as they point to unique subjects.

What of the distribution of predicates? The predicates of negatives, **E**, **G**, and **O**, are distributive, because P is altogether absent from the cases of S concerned ; while in affirmatives, **A**, **R**, and **I**, the predicates are undistributive, since things other than the cases of S concerned might be P.

These properties can be illustrated by means of Euler diagrams, named after the Swiss logician who invented them. In these, S and P are represented by the areas of circles, which overlap or fail to overlap to varying degrees. The reader is invited to explore these analogies. (Very similar are Venn diagrams, named after another logician; the latter differ in that they stress the areas outside the circles, the areas of nonS or nonP.)

In **A** propositions, the S circle is wholly within the P circle, and smaller or equal in size to it. In **E**, the circles are apart, whatever their relative sizes. In **I** propositions, the two circles at least partly intersect, whether each covers only a part of the other’s area, or S is wholly embraced by P, or P by S, or they both cover one and the same area. In **O**, the two circles at least partly do not overlap, whether each only covers only a part of the other’s area, or neither covers any part of the other’s area.

The forms in current use, listed above, are so designed that we can specify alternate quantities for the predicate, if necessary, simply by making an additional statement, in which the original predicate is subject and the original subject is predicate, with the appropriate distributions.

As a result of the distribution doctrine, there have been attempts to invent forms which quantify the predicate, but they have not aroused much interest, being artificial to our normal ways of thinking.

### 4. Permutation.

Classification is a special outlook, but one we can use to develop Logic with efficiently, because it allows us to standardize statements. Classification is more mathematical in nature, and so easier of treatment, than other relations. The process of rewording a proposition, so that its terms are overlapping classes, is called ‘permutation’.

Note that, in formal logic, the word ‘universal’ is used in a quantitative sense, to apply to general propositions, which address the totality of a class. But in philosophy, a ‘universal’ is understood as the common factor, resemblance, similarity, which led us or allowed us to group certain units into a class; in this sense every term is a universal for its members, and even a particular proposition contains universals, except that they happen to be only partially addressed.

Likewise, the word ‘particular’ refers to less than general propositions, in formal logic; whereas, in philosophy, it is understood to mean concrete individuals, as distinct from abstract essences. Normally, the context makes clear what sense of each word we intend.

- The equivocation of the word ‘universal’ is not entirely an historical accident. A proposition may have a ‘quality as such’ as its subject, and only incidentally imply a quantifiable subject-class. Thus, for example, ‘greenness is a (kind of) color’ and ‘all green things are colored’ do not mean quite the same, though their truths are related.

Propositions which have as their subject a quality as such, a universal in the philosophical sense, are virtually singular in format. To be quantified, their subject must be reworded somewhat. This is called permutation of the subject.

- As Logic has developed, it has come to focus especially on the classificatory sense of ‘is’, because attribution, and other relations, can be reduced to it. Colloquially, the ‘is’ copula first suggests that the subject has a certain attribute, viz. the predicate, as in ‘trees are green’. But attribution is a more complex and qualitative relational format than classification, requiring more philosophical analysis.

Many propositions which normally are thought without the classifying ‘is’ copula, can be restructured to fit into it, while more or less retaining the same meaning. Thus, in our example, we would shift from the sense ‘trees have greenness’ to the sense ‘trees are greenness-having-things’. This is called permutation of the predicate.

Most logical processing of categoricals assumes that the statements involved have been permuted into classificatory form. Note well that permutation merely *conceals* the previously intended relationship in a new term, it does not annul or replace it. The difficult relation is once-removed, put out of the way; it is not defined.

## 6. OPPOSITIONS.

### 1. Definitions.

By the ‘opposition’ of two propositions, is meant: the exact logical relation existing between them — whether the truth or falsehood of either affects, or not, the truth or falsehood of the other.

In this context, note, the expression ‘opposition’ is a technical term not necessarily connoting conflict. We commonly say of two statements that they are ‘opposite’, in the sense of incompatible. But here, the meaning is wider; it refers to any mental confrontation, any logical face-off, between distinguishable propositions. In this sense, even forms which imply each other may be viewed as ‘opposed’ by virtue of their contradistinction, though to a much lesser degree than contradictories. Thus, the various relations of opposition make up a continuum.

Now, upon reflection, the logical relations of implication, incompatibility, and exhaustiveness, defined earlier, are found to be incomplete insofar as they leave certain issues open. There is therefore a need to combine them in various ways, to obtain a list of seven fully defining kinds of ‘oppositions’:

**Mutual Implication**(or**implicance**): is defined as the relation between two propositions which are either both true or both false. Each is called an implicant and is said to implicate the other. P implies Q, and Q implies P; and, nonQ implies nonP, and nonP implies nonQ.**Subalternation**: is the relation between two propositions which are either both true or both false, or one — called the subalternant — false and the other — called the subaltern — true; the occurrence of ‘subalternant true and subaltern false’ being excluded by definition. The subalternant and subaltern may be referred to jointly as the subalternatives. This relation is, therefore, one-way implication. P implies Q, but Q does not imply P; and, nonQ implies nonP, but nonP does not imply nonQ.

Subalternation, may be counted as two distinct relations, subalternating, and being subalternated, each of whose direction must be specified. This is in contrast to the other five oppositions, which are symmetrical.

**Contradiction**: exists between two propositions which cannot be both true and cannot be both false. If either is true, the other is false; and if either is false, the other is true. They are said to be contradictories. Their affirmations are incompatible and their denials are incompatible. P implies nonQ, and nonP implies Q; and, Q implies nonP, and nonQ implies P.**Contrariety**: two propositions are contrary if they cannot both be true, but may both be false. If either is true the other is false, but if either is false the truth or falsehood of the other is possible. They are said to be contraries. Their affirmations are incompatible, but not their denials. P implies nonQ, but nonP does not imply Q; and, Q implies nonP, but nonQ does not imply P.**Subcontrariety**: occurs when two propositions cannot be both false, but may be both true. If either is false, the other is true; but the truth of either leaves that of the other indeterminate. They are said to be subcontraries. Their denials are incompatible, but not their affirmations. nonP implies Q, but P does not imply nonQ; and, nonQ implies P, but Q does not imply nonP.**Unconnectedness**(or neutrality): two propositions are ‘opposed’ in this way, if neither formally implies the other, and they are not incompatible, and they are not exhaustive. Note that this definition does not exclude that unconnecteds may, under certain conditions, become connected (or remain unconnected under all conditions).

Note that these seven types of opposition define both directions of the relations concerned, in contrast to the basic logical relations. For this reason they may be called ‘full’ relations: they leave no question marks. They are logically exhaustive, allowing us to classify the relation of any pair of propositions.

There are other kinds of compound logical relations, besides the above mentioned seven. These concern paradoxical propositions, which imply even their own contradictory, or some contradiction. For example, ‘X is not X’ formally implies both that ‘something, called X, exists’ (by the law of identity), and that ‘there is no such thing as X’ (by the law of contradiction).

However, paradoxes are very rare in formal logic; rather they occur, only a bit less rarely, with specific contents. Formal logic is mainly interested in the oppositions between normal propositions, which are in principle consistent in form. More will be said about paradoxes later, when we look into the logic of logic.

The official terminology for the various kinds of opposition, here suggested, may not always accord with common usage. Especially note that in practice the word ‘contradiction’ is very often taken as equivalent to ‘incompatibility’, signifying (in official parlance) ‘either contradiction or contrariety’; thus, for instance, with the expression ‘law of contradiction’; we mean incompatibility. Also, the word ‘opposite’ is sometimes used to mean contradictory.

It is curious to note, too, that the words ‘subaltern’ and ‘subcontrary’, though quite old, are rarely used in practice; I have only seen them used by logicians. Such failures of words or meanings to enter the mainstream of language, are sad testimonies to the popular disinterest in studying logic.

The following table summarizes the above through analysis of the possibilities of combination of the affirmations and denials of two propositions, P and Q, which are given as being related by a certain opposition, specified in the left column. ‘**Yes**’ indicates possible combinations, ‘**no**’ impossible ones.

POSSIBILITY OF: | P+Q | P+nonQ | nonP+Q | nonP+nonQ |

Implicance | yes | no | no | yes |

Subalternating | yes | no | yes | yes |

Being Subalternated | yes | yes | no | yes |

Contradiction | no | yes | yes | no |

Contrariety | no | yes | yes | yes |

Subcontrariety | yes | yes | yes | no |

Unconnectedness | yes | yes | yes | yes |

Note that incompatibles are either contradictory or contrary, while exhaustives are either contradictory or subcontrary. Also worth noting, compatibles may be either implicant, or subalternative (in one or the other direction), or subcontrary, or unconnected. The seven definite oppositional relations are mutually exclusive (i.e. contrary, to be exact), but one of the seven must hold.

### 2. Applications.

The doctrine of opposition arose out of the need to apply the laws of thought to propositions more complex than the initial forms ‘S is P’ and ‘S is not P’. The concepts of equality, conflict, and limitation, had to be expanded upon, to reflect the more qualified relations found to exist between forms once they are quantified.

We know that two singular propositions differing only in polarity (viz. **R**, **G**) are contradictory, for at any given instant This-S cannot both be P and not-be P, and must be one or the other. But what of the plural versions of these forms? The following diagram shows their interrelationships.

We note, to begin with, that for each polarity, the universal (all) subalternates the singular (any specific individual), which in turn subalternates the particular (some is an indefinite quantity, meaning one or more). Next, A universal and particular of opposite polarity (**A** and **O**, or **E** and **I**) are contradictory, just as two singulars (**R** and **G**) concerning one and the same individual are contradictory. Lastly, universals are contrary to universals or singulars of opposite polarity (**A** and **E**, **A** and **G**, **R** and **E**), and particulars are subcontrary to particulars or singulars of opposite polarity (**I** and **O**, **I** and **G**, **R** and **O**).

We may summarize these findings in the form of a ‘truth-table’. This tells us which other propositions must be true (**T**) or false (**F**), or may be either (**.**), in the context of each form given on the left under heading **T** being true, or each form given on the right under **F** being false.

T | A | R | I | E | G | O | F |

A | T | T | T | F | F | F | O |

R | . | T | T | F | F | . | G |

I | . | . | T | F | . | . | E |

E | F | F | F | T | T | T | I |

G | F | F | . | . | T | T | R |

O | F | . | . | . | . | T | A |

The conjunction of **I** and **O** may be viewed as a form of proposition in its own right, though composite. If we oppose this to the above standard forms, we obtain the following. Since ‘**I** + **O**’ subalternates **I** and **O** (considered separately), it is contrary to **A** and **E**. It is unconnected to **R** and **G**, since either may be true or false without affecting it.

Also note in passing the position of forms quantified by ‘most’ or ‘few’, which we mentioned earlier. See **Appendix 2** for remarks on this topic.

Note that two propositions with the same subject, but with different predicates, may be considered opposites, if the predicates are well known to be antithetical. Thus, ‘S is P’ and ‘S is Q’ may implicitly intend ‘S is P (but not Q)’ and ‘S is Q (but not P)’, respectively. In such case, the forms may of course be treated as effective contradictories.

### 3. Validations.

These oppositions are proved as follows. Remember that each of the plural propositions can be defined by a series of singular propositions of the same polarity. Thus, **A** and **I** are reducible to a series S1 is P, S2 is P, S3 is P, etc., differing in that All-S covers the whole class of S, whereas Some-S covers only part of the same class. Likewise in the case of negatives, **E** and **O**. Thus the subalternation of singular or particular, to a generality of like polarity, is simply the inclusion by the whole of the class of any part thereof. This relation is unidirectional in that if the whole is affirmed or denied so is every part of it, whereas if some part is affirmed or denied it does not follow that other parts are.

Similarly, the contradictions of **A** and **O**, or **E** and **I**, are proven by consideration of their subsumptions. If all the members of a class are included in a predication, then any which is declared excluded would be found to be both P and nonP, an impossibility. The same can be argued in the negative case: if all are excluded, then none can be included without inconsistency.

With regard to **I** and **O** (or **I** and **G**, or **R** and **O**), they are subcontrary insofar as conflicting predicates can consistently be applied to different parts of the same subject-class, although it is impossible to evade either affirming or denying any predicate of a subject, i.e. one must be true. The contrariety of **A** and **E** (or **A** and **G**, or **R** and **E**) is due to the observation that, while they cannot be both true without implying some singular case(s) of inconsistency, they could be both false without antinomy, as occurs in the case of **I** and **O** being both true.

The concepts of inclusion and exclusion are geometrically evident. They were implicit in the original formulation of the laws of thought, when we referred to the whole or part of a singular phenomenon. In this logical discipline, we broaden the laws of thought, by treating individual instances as parts of a larger phenomenon we call a class or universal, and then applying our laws to this new whole. Essentially, no information has been added, we have merely in fact elucidated inherent data.

To conclude, let us point out that ‘opposition’ can be viewed as a kind of immediate inference, like eduction. This is especially obvious when we draw out an implicant or subaltern, but can also be said about affirming a proposition on the basis of another’s falsehood, or denying one on the basis of another’s truth or falsehood. Opposition is not a mere theoretical construct for logicians, but of practical value to the layman.

## 7. EDUCTION.

### 1. Definitions.

Immediate inference is the process of discovering another proposition implicit in a given proposition, without use of additional information. It differs from syllogistic reasoning, in that the latter draws a new proposition from two or more previous ones. We have come across one sample of such inference in the foregoing text, namely opposition. Here we will deal systematically with another, which may be called eduction.

What eduction does is to change the position and/or polarity of the terms; this often results in a proposition of different polarity or quantity. The original proposition is the premise, the educed proposition an implication of it.

Often, to fully understand a proposition, we have to restate it in another way, some hidden character of it is thereby revealed, facilitating further thought. The structural change we effect in the given form yields new information, although a simple process.

Starting from an S-P format, we may be able to obtain propositions through transposition and/or negation of terms, in the following ways.

Process: | From S-P to: |

Obversion | S-nonP |

Conversion | P-S |

Obverted Conversion | P-nonS |

Conversion by Negation | nonP-S |

Contraposition | nonP-nonS |

Inversion | nonS-nonP |

Obverted Inversion | nonS-P |

The source proposition is then called obvertend, convertend, contraponent, invertend, and so on, while the target proposition is called obverse, converse, contraposite, inverse, as the case may be.

Whereas such processes are generally possible with one or both of the universals, they are not always feasible in the case of singulars or particulars, as we shall see. Note also that some processes are *reversible*, and some are not: only in some cases may the source proposition be educed again from its implication (by the same or any other eductive process).

### 2. Applications.

We shall now list the implications of the various plural forms, and then validate the processes involved. Although these may tedious details, they do constitute an important training for the mind.

- Obversions (S-P to S-nonP).

A | All S are P | implies | E | No S is nonP |

E | No S is P | implies | A | All S are nonP |

R | This S is P | implies | G | This S is not nonP |

G | This S is not P | implies | R | This S is nonP |

I | Some S are P | implies | O | Some S are not nonP |

O | Some S are not P | implies | I | Some S are nonP |

Thus, all forms are obvertible, and so reversibly.

- Conversions (S-P to P-S).

A | All S are P | implies | I | Some P are S |

E | No S is P | implies | E | No P is S |

R | This S is P | implies | I | Some P are S |

I | Some S are P | implies | I | Some P are S |

Thus, affirmatives yield a particular. Only **I**’s and **E**’s conversions are reversible. **G** and **O** propositions are not convertible.

- Obverted Conversions (S-P to P-nonS).

A | All S are P | implies | O | Some P are not nonS |

E | No S is P | implies | A | All P are nonS |

R | This S is P | implies | O | Some P are not nonS |

I | Some S are P | implies | O | Some P are not nonS |

Thus, affirmatives yield a particular. Only **I**’s and **E**’s obverted conversions are reversible. **G** and **O** propositions lack an obverted converse.

- Conversions by Negation (S-P to nonP-S).

A | All S are P | implies | E | No nonP is S |

E | No S is P | implies | I | Some nonP are S |

G | This S is not P | implies | I | Some nonP are S |

O | Some S are not P | implies | I | Some nonP are S |

Thus, negatives yield a particular. Only **A**’s and **O**’s conversions by negation are reversible. **R** and **I** propositions have no converse by negation.

- Contrapositions (S-P to nonP-nonS).

A | All S are P | implies | A | All nonP are nonS |

E | No S is P | implies | O | Some nonP are not nonS |

G | This S is not P | implies | O | Some nonP are not nonS |

O | Some S are not P | implies | O | Some nonP are not nonS |

Thus, negatives yield a particular. Only **A**’s and **O**’s contrapositions are reversible. **R** and **I** propositions are not contraposable.

- Inversions (S-P to nonS-nonP).

A | All S are P | implies | I | Some nonS are nonP |

E | No S is P | implies | O | Some nonS are not nonP |

Only universals are invertible, and that irreversibly, to particular form. **R**, **G**, **I**, and **O** propositions are not invertible.

- Obverted Inversions (S-P to nonS-P).

A | All S are P | implies | O | Some nonS are not P |

E | No S is P | implies | I | Some nonS are P |

Only universals may be subjected to obverted inversion, and that irreversibly, to particular form. Process not applicable to **R**, **G**, **I**, and **O** propositions.

We note at the outset that while quantity may be lost, it cannot be gained. A universal or singular proposition may yield a particular, but a singular or particular cannot produce a universal. It is also clear that, with the exception of obversion, the processes applicable to singulars are so only by virtue of the corresponding particulars implicit in them by opposition. This is true also of **A** in conversion and obverted conversion, **E** in conversion by negation and contraposition. Universality plays an active role only in conversion and obverted conversion of **E**, in conversion by negation and contraposition of **A**, and in inversion and obverted inversion.

### 3. Validations.

We can validate all these processes by working on two: obversion and conversion, for the others follow.

**Obversion**.

To say S is P posits that P is found in S; to say S is-not nonP means that the absence of P is absent from S. Which is not to imply, in either case, that S is not simultaneously other things than P or nonP — Q, R, etc. So, S can be P and something other than P, although it cannot both exhibit and not-exhibit P. These arguments thus define the copula is-not and the term nonP more precisely.

What is true here in the case of singular propositions, can be argued equally for plural propositions, since the latter subsume the former. That is, they collect them together as a unit while at the same time dealing with them each one singly; so that the statement does not concern them as either a count of individuals or as a collective unity, but is merely an abbreviated statement being distributed out to its instances equivalently.

Thus S-P merely means S1-P1, S2-P2, S3-P3, etc. Here again, this doctrine provides an opportunity to more precisely define formal concepts.

*Conversion**.*Here each quantitative is considered separately.

- For
**I**: ‘Some S are P’ and ‘Some P are S’ each means ‘Some things are both S and P’; we are seeing S and P together, we may attribute either to the other; this defines the generality of our copula is, and proceeds from the law of Identity. - For
**E**: likewise, ‘No S is P’ and ‘No P is S’ each means ‘Nothing is both S and P’; S and P never appear together, have no instances in common. This clarifies our copula is-not, telling us that S is-not P is the same as P is-not S; and also reminding us that ‘No X is Y’ means ‘All X are-not Y’. - For
**A**: ‘All S are P’ by subsumption implies that ‘Some S are P’, and therefore also that ‘Some P are S’ as shown above. However, it could-not imply ‘All P are S’, although in some cases such mutual inclusion occurs, because there are cases where it does not. Here again, we are better defining our form, in accord with its common usage. - For
**O**: from ‘Some S are not P’ we cannot infer ‘Some P are not S’, for it happens that ‘All P are S’; that is, it happens that only S are P, i.e. that P does not occur elsewhere; our form is intended as that broad and inclusive of possible circumstance.

Other approaches to these validations are possible. But the intent here was to show that these need not be viewed as ‘proofs’, so much as focusing more precisely on the forms our consciousness naturally uses, and inspecting every aspect of their selected meanings to delimit the extent of their application to phenomena as they appear to us.

With regard to the *other types* of eduction, they can be reduced to combinations of the above two processes, and thus validated. Thus:

**Obverted Conversion**. Convert, then obvert.**Conversion by Negation**. Obvert, then convert.**Contraposition**. Obvert, then convert, then obvert.**Inversion**. For**A**: contrapose, then convert. For**E**: convert, then contrapose.**Obverted Inversion**. Invert, then obvert.

Note lastly, one can say ‘some nonS are nonP’ (or the converse) for just about any S and P chosen at random, with the exception of certain very broad terms, like ‘existence’, which have no real negatives. So processes which yield such conclusions are not very informative.

Addendum: “No S is P” means that S and P are incompatible – if one of them is present, the other one cannot also be present. “No nonS is nonP” means that S and P are exhaustive – if one of them is absent the other cannot also be absent. To affirm both these propositions is to say the two terms S and P (or nonS and nonP) are contradictory. To affirm the first and deny the second is to say S and P are contrary. To deny the first and affirm the second is to say S and P are subcontrary. To deny both is to say some S are P and some nonS are nonP – i.e. they are compatible and inexhaustive.

## 8. SYLLOGISM: DEFINITIONS.

### 1. Generalities.

We call inference the mental process of becoming aware of information implicit in given information, be it concrete or abstract. When we draw ideas from experience or generalities from particulars, we are involved in induction; otherwise, it is deduction. In any case, the original data is called the premises, and the logically derived proposition, the conclusion.

When the conclusion is already known to us, and we are considering its validity in the context of other knowledge, we are said to argue. Furthermore, if the motive of our argument was to arrive at the conclusion for its own sake, we are said to be proving it; if on the other hand our motive was to show the contradictory or a contrary of our conclusion to be false, we are said to be engaged in a process of refutation.

The difference in connotation between inference and argument is merely one of sequence: what was posited first, premise(s) or conclusion? The distinction between proof and refutation lies in our motive. But the logical form of all these processes is the same, so their names are used interchangeably here.

The term deduction is sometimes used in a restricted sense which excludes eduction. Eduction has already been defined as eliciting information from one proposition (granting that the logical principles involved in this are not regarded as premises too). The deductive process which concerns us here, in contrast, is drawing information implicit in two or more propositions together, and not separately. P and Q are true, ergo R is true.

This is called mediate inference, because it is found that the premises must have some factor in common, which serves as the medium of inference, making possible the eliciting of a conclusion. This might be thought of as ‘conduction’. The technical name for it is syllogism, from Greek, the language of Aristotle.

It can be shown that arguments involving more than two categorical propositions are reducible to a series of syllogisms and eductions. In this analysis, we will concentrate on categorical syllogism, that involving only categorical propositions. Argument involving noncategorical propositions will be dealt with later.

### 2. Valid/Invalid.

Now, an argument may be valid or invalid. The science of Logic shows that the validity of the method is independent of the truth or falsehood of the premises or conclusion. A formal argument only claims that if the premises are true, the conclusion must be true; if the conclusion is found false, then one or more of the premises must be false. It may happen that the premises are false, yet the conclusion is independently true; rejection of the premises does not necessarily put the conclusion in doubt. The validity or invalidity of an argument is a formal issue, irrespective of the content of the propositions involved.

Logic analyses the variety of forms possible, and distinguishes the valid from the invalid, by reference to the Laws of Thought. The results are analyzed, in the search for general rules. Strictly speaking, only valid syllogisms are ultimately so called; invalid syllogisms are mere fallacies. But at the outset, Logic lists all possible combinations of propositions on an equal footing, to ensure the exhaustiveness of its treatment; then it finds out which are good and which bad.

Its ultimate aim is of course to draw the maximum consequent information from any data. This allows us to correlate the different aspects of our experience, and improve our knowledge of the world. By comparing and connecting together all our beliefs, we can through logic discover inconsistencies, which cause us to reassess our assumptions at some level, and correct our data banks. In this way our beliefs are ‘proved’; at least until there is good reason to think otherwise.

Scientific proof always depends on the context of knowledge. It is always conceivable that some aspect of knowledge turns out to be open to doubt, even after seeming fundamental and unassailable for ages. For instance, certain axioms of Euclidean geometry. So proof never entirely frees a conclusion from review, given some new motive. Finding an inconsistency does not in itself guarantee that we will succeed in finding the source of the error, i.e. some false premise. In such cases, we register that there is some doubt yet to resolve, and either wait for new experience or search for an answer imaginatively.

### 3. Figures.

A syllogism, then, involves three propositions, two premises and a conclusion. These together involve three, and only three, terms. They are: the middle term, one common to both premises, but absent in the conclusion; the minor term, which is the subject of the conclusion, and present in one of the premises; and the major term, which is the predicate of the conclusion, and present in the other premise. The minor and major term are also called the extremes; the middle term acts as intermediary between them, to yield the conclusion. The premise involving the minor term is called the minor premise, that with the major term the major premise.

The position of the middle term in the premises, that is, whether it is subject or predicate in each, determines what is called the ‘figure’ of the syllogism. (The colloquial expression for thought, ‘to figure’ or ‘to figure out’ may derive from this usage.) There are four possible figures of the syllogism. They are shown in the following table, with S, M, P symbolizing the minor, middle and major terms, respectively:

Figure | First | Second | Third | Fourth |

Major premise: | M-P | P-M | M-P | P-M |

Minor premise: | S-M | S-M | M-S | M-S |

Conclusion: | S-P | S-P | S-P | S-P |

Note well the variety in the position of the terms. The order of the propositions in Logic is conventionally set as major-minor-conclusion, so that symbolic references can always be understood. But of course in actual thought any order of appearance may occur. Thus it is seen that syllogism is mediate inference; from their respective relationships to a middle term, a relationship may be found to follow between the extremes.

Each figure of the syllogism reflects a structure of our thinking. In practice, the Fourth figure is not regarded by many logicians as very significant. Aristotle, though aware of its existence, had this viewpoint. Galen, however, introduced it as a formal alternative for the sake of completeness.

### 4. Moods.

We previously identified six categorical forms, **A**, **E**. **I**, **O**, **R** and **G**, which can be involved in such syllogism. Each of the propositions in each figure might at first glance have any of these six forms. So there are 6X6X6 = 216 possibilities per group of proposition in each figure. Each of these combinations is called a mood of the syllogism. Altogether, in the four figures, there are 216X4 = 864 imaginable syllogistic forms. Each such form can be designated clearly by mentioning its figure and mood; for example, ‘mood **EAA** in the first figure’, or more briefly, ‘**1/EAA**’.

Our task is differentiate the valid from the invalid, in this multiplicity of theoretical constructs. It will be seen that very few actually pass the test. The valid moods per figure should be justified, and the invalid ones shown wrong. This will enable us to know when a conclusion can be drawn from given premises, and when not.

Note that each of the propositions may be positive (**+**) or negative (**–**), so that there are 2X2X2 = 8 possible combinations of polarity in each figure; they are: **+++**, **++-**, **+-+**,**+–**, **-++**, **-+-**, **–+**, **—**. Likewise, as three quantities exist, viz. universal (**u**), particular (**p**), and singular (**s**), there are 3X3X3 = 27 possible combinations of quantity in each figure; which are: **uuu**, **uup**, **uus**, **upu**, **upp**, **ups**, **puu**, **pup**, **pus**, **ppu**, **ppp**, **pps**, and so on. It will be seen that many of these combinations are nonsensical, and rules concerning polarity and quantity can be formulated. Some rules are general to all figures, some are specific to each. In any case, the conclusion sought is always the maximal one; if a universal can be concluded, the subaltern conclusion is not of interest, though it follows *a-fortiori*.

A more traditional way to express the task of logic with respect to syllogism is as follows. In each figure, which of the 6X6 = 36 combination(s) of premises yield a conclusion? Or which of the 2X2 = 4 combination(s) of polarity:** ++**,** +-**,** -+**,** —**? And which 3X3 = 9 combination(s) of quantity: **uu**, **up**, **pu**, **pp**, **us**, **su**, **sp**, **ps**, **ss**?

### 5. Psychology.

Some critics of Logic have accused it of puerility, arguing that the syllogism is too simple in form, and yields no new information, whereas actual thinking is somehow a more creative and complex process. But the ‘event’ of syllogistic reasoning is not as mechanical and automatic as it is made to appear on paper. Logic presents a static picture of what is psychologically a very dynamic and often difficult process.

There is a mental effort in bringing together the concepts which form the separate propositions involved; this requires complex differential perceptions and insights. We also have to think of bringing together the propositions which constitute our premises; they are not always joined and compared automatically, sometimes a veritable inspiration is required to achieve this. And even then, actual drawing of the conclusion is not mechanically inevitable; honesty, will, and intelligence are needed.

Thus, Logic merely establishes standards of proper reasoning, identifying common aspects of thought and justifying its sequences. But mentally, in practice, the processes are complexes of differentiation and integration. Sometimes such events are easy to produce, but often years of study and even genius are necessary to produce even a single result. Virtues such as open-mindedness, reality-orientation, perceptiveness, intuition, will-power are involved.

## 9. SYLLOGISM: APPLICATIONS.

In this chapter we will list the valid moods of the syllogism, and make some generalizations and comments, so as to acquaint the reader with the central subject of our discussion. Thereafter, validation will be dealt with in a separate chapter. Please remember that we are dealing here specifically with one type of proposition, the actual, classificatory, categorical. Other types of proposition require eventual treatment, of course.

Our main concern here is classical logic in all its beauty, the showpiece of the science, which we owe to Aristotle and subsequent masters. There are related topics of lesser importance, these will be mentioned in the course of development.

### 1. The Main Moods

Syllogism is inference from two propositions of a third whose truth follows from the given two. In categorical syllogism, we deduce a relation between two terms by virtue of their being each related to a third term. According to the direction of their relationship to the third term, the syllogism is said to form different figures, or “movements of thought” (Joseph). The polarities and quantities of the premises, because of their diverse ways of distributing their terms, generally affect the character and validity of the conclusion. These differences are used to distinguish moods of the syllogism in each figure, which may reflect a variety of approaches through which our minds analyze a subject to attain understanding of it.

In this section, we will list the principal valid moods of plural syllogism, that is, of syllogism both of whose premises are plural. They are the most important in this doctrine. Valid moods involving one or two singular premises will be listed in the next section. Derivatively valid syllogisms, of an artificial or subaltern nature, or involving atypical conclusions, will be discussed separately. Moods not included in these listings of valid moods are to be regarded as paralogisms, they are either *non-sequitur*s (‘it does not follow’ in Latin) or self-contradictory.

- First Figure.

Form:

Major premise | M-P |

Minor premise | S-M |

Conclusion | S-P. |

AAA | AII |

All M are P | All M are P |

All S are M | Some S are M |

\ All S are P | \ Some S are P |

EAE | EIO |

No M is P | No M is P |

All S are M | Some S are M |

\ No S is P | \ Some S are not P |

We may observe that the major premise is always universal, and the minor premise always affirmative, here. The principle of such reasoning, called the first canon of logic, could be expressed as ‘Whatever satisfies fully the condition of a rule, falls under the rule’. The condition here means ‘being M’, and the rule means ‘being P’ or ‘not being P’.

- Second Figure.

Form:

Major premise | P-M |

Minor premise | S-M |

Conclusion | S-P. |

AEE | AOO |

All P are M | All P are M |

No S is M | Some S are not M |

\ No S is P | \ Some S are not P |

EAE | EIO |

No P is M | No P is M |

All S are M | Some S are M |

\ No S is P | \ Some S are not P |

We observe that the major premise is always universal, and the conclusion always negative. The second canon of logic, implicit in these moods, can be stated as ‘Whatever does not fall under a rule, does not satisfy any full condition to the rule’. The condition here meaning ‘being P’ and the rule ‘being, or not-being, M’.

- Third Figure.

Form:

Major premise | M-P |

Minor premise | M-S |

Conclusion | S-P. |

AII | EIO |

All M are P | No M is P |

Some M are S | Some M are S |

\ Some S are P | \ Some S are not P |

IAI | OAO |

Some M are P | Some M are not P |

All M are S | All M are S |

\ Some S are P | \ Some S are not P |

We observe that the minor premise is always affirmative, and the conclusion is always particular. Two more moods, **AAI** and **EAO**, are normally included by logicians with the above; but these are true only by virtue of the truth of **AII** and **EIO**, respectively, whose minor premises theirs imply; I have therefore chosen to exclude them. The principle here, our third canon, is expressed as ‘Rules following from the same condition are in that instance at least compatible’. The common condition being instances of subsumed M in both premises, and the rules being their relations to S and P.

- The Fourth Figure.

Form:

Major premise | P-M |

Minor premise | M-S |

Conclusion | S-P. |

EIO | |

No P is M | |

Some M are S | |

\ Some S are not P |

We note that the major premise is a negative universal, the minor is affirmative, and the conclusion a negative particular one. (The mood **EAO** might also have been included here, but its validity is only due to its minor premise implying that of **EIO**.) This figure is rather controversial. It formally has three more valid moods, **AEE**, **IAI** and **AAI**, but these are left out as too insignificant for such central exposure. This topic will be further discussed. No canon is normally formulated for this figure.

There are therefore a total of 4+4+4+1 = 13 moods of the plural syllogism which are valid, nonderivative, and significant.

### 2. On the Fourth Figure.

If we consider the second and third figures, we see that transposition of the premises does not change the figure, although the conclusion if any will have transposed terms; the middle term remains common subject or predicate, as the case may be, of the premises. But in the first figure, if the major and minor premises are transposed, not only are the major and minor terms transposed in the conclusion, but a new figure emerges, the fourth. The reverse is also true, shifting from fourth to first. Yet, the order of appearances of the premises is essentially conventional, and should not matter.

It is doubtful whether anyone ever thinks in fourth figure terms, probably because of the double complication it involves. The minor term shifts from being a predicate in its premise to being a subject in the conclusion, and the major term switches from subject in its premise to predicate in the conclusion. While each of these changes does occur in the third and second figures respectively, in the fourth figure both of these mental acrobatics are required. We have difficulty in reasoning thus, whereas the process should be obvious enough for the mind to concentrate on content.

Some logicians have opted for ignoring the fourth figure altogether, on such grounds. Others have insisted on including it as a formal possibility, arguing that the science of logic should be exhaustive and systematic, and show us all the information we can draw from any given data.

My own position is a compromise one. The valid moods **AEE**, **IAI**, and **AAI** (which is implicit in **IAI**, incidentally), clearly do not present us with information not available in the first figure (after transposition of premises). Given the two premises, we are sure to process them mentally in the first figure, and then, if we need to, convert their conclusions as a separate act of thought. In the case of valid mood **EIO** (and likewise **EAO**, which is implicit in it), however, the conclusion ‘Some S are not P’ would not be inferable in the first figure, since **O**-propositions have no converse. It follows that it must be retained to achieve a complete analysis of possibilities, even if rarely used in practice.

This position can be further justified by observing the lack of uniformity in these five moods. They do not have clear common attributes like the valid moods of other figures; they rather seem to form three distinct groups when we consider their polarities and quantities. **EIO** (and **EAO**) make up one group; **AEE**, another; **IAI** (and **AAI**), yet another.

### 3. Subaltern Moods.

Under this heading we may firstly include the two third figure moods, **AAI** and **EAO**, and the fourth figure mood, **EAO**, which were mentioned earlier as mere derivatives. The reason why logicians have traditionally counted them among the principal moods, was that they inform us that in the cases concerned, only a particular conclusion is obtainable from universal premises; but I have chosen to stress rather their implicitness in the corresponding moods with a particular minor premise, so that from this perspective they give us no added information. They do not constitute an independent process, but are reducible to an eduction followed by a deduction, or vice versa. Note in passing that the insignificant mood **AAI** in the fourth figure is such a derivative of **IAI**, also insignificant.

We can also call subaltern, moods which simply contain the subaltern conclusion to any higher conclusion found valid. Thus, though valid, they are regarded as products of eduction after the main deduction. They are: in the first figure, **AAI** and **EAO**; in the second figure, **AEO** and **EAO**; the third figure has none; in the fourth figure, **AEO**.

Thus, there are altogether of 2+2+2+3 = 9 plural moods which, though valid, are subaltern, in the four figures.

### 4. Singular Moods.

These contain one or more singular propositions. The valid ones are as follows.

In the first figure, **ARR** and **ERG**; in the second figure, **AGG** and **ERG**. In these figures, we have singular conclusions, higher than in the corresponding valid particular moods (since singulars are not implied by particulars), and so novel syllogisms. They are worth listing.

First Figure:

ARR | ERG |

All M are P | No M is P |

This S is M | This S is M |

\ This S is P | \ This S is not P |

Second Figure:

AGG | ERG |

All P are M | No P is M |

This S is not M | This S is M |

\ This S is not P | \ This S is not P |

I would not regard the moods **AAR** and **EAG** in the first figure as valid, in spite of their apparent subalternation by **ARR** and **ERG**, respectively, because they introduce a ‘this’ in the conclusion which was not in the premises (so that there is an implicit third premise ‘this is S’). Likewise in the second figure for **AEG** and **EAG**, they are not true derivatives of **AGG** and **ERG**. This issue will be confronted more deeply later.

The subalterns of these valid moods, viz. in first figure, **ARI** and **ERO**, and in the second figure, **AGO** and **ERO**, are of course also valid, but not of interest.

In the third figure, the two moods **RRI** and **GRI** are worthy of attention. Each exceptionally draws a conclusion from two singular premises, without involvement of a universal premise; this is of course due to the position of the middle term as individual subject of both premises. This reflects the fact that one instance often suffices to make a particular point (and is sometimes enough to disprove a general postulate). Note that the conclusion is particular, and not singular, because the ‘this’ cannot be passed on from a subject to a predicate.

Third Figure.

RRI | GRO |

This M is P | This M is not P |

This M is S | This M is S |

\ Some S are P | \ Some S are not P |

Also valid in the third figure, are **ARI**, **ERO**, **RAI**, and **GAO**. But in these cases the conclusions from singular premise moods are no more powerful than those from their particular premise equivalents, so that we have mere subaltern forms.

In the fourth figure, **ERO** and **RAI** are valid, but as they offer no new conclusion, they may be ignored as subaltern. Because in this figure validation occurs through the first figure, after conversion of premises or conclusion, and a singular proposition converts only to a particular, there cannot be any special valid singular syllogisms.

The total number of valid singular moods, which are not subaltern, is thus 2+2+2+0 = 6. Additionally, we mentioned 2+2+4+2 = 10 subalterns.

Regarding syllogisms involving propositions which concern a majority or minority of a class, we get results similar to those obtained with singular moods.

Thus, in the first figure, there are four main valid moods, their form being: ‘If All M are (or are-not) P, and Most (or Few) S are M, then Most (or Few) S are (or are-not) P’. In the second figure, there are four main valid moods, too, with the form: ‘If All P are (or are-not) M, and Most (or Few) S are-not (or are) M, then Most (or Few) S are not P’.

In the third figure, we have only two main valid moods. They are especially noteworthy in that they manage without a universal premise. Their form is: ‘Most M are (or are-not) P, Most M are S, therefore Some S are (or are-not) P’. Note that the two premises are majoritive, and the conclusion is only particular. The validity of these is due to the assumption that ‘most’ includes more than half of the middle term class, so that there is overlap in some instances.

There are no nonsubaltern valid moods in the fourth figure. Subaltern versions of the above listed syllogisms, involving majoritive or minoritive premises, exist, but will not be listed here.

### 5. Summary.

The following table lists the 19 moods of the syllogism in the four figures, which were found valid, nonsubaltern, and sufficiently significant. These may be called the primary valid moods, because of their relative independence and originality. Another 25 moods are valid, but are either subaltern to the primary syllogisms or insignificant fourth figure moods. These may be grouped together under the name of secondary valid moods.

Figure | First | Second | Third | Fourth |

Primary | AAA | AEE | AII | EIO |

Moods | EAE | EAE | EIO | |

AII | AOO | IAI | ||

EIO | EIO | OAO | ||

ARR | AGG | RRI | ||

ERG | ERG | GRO | ||

Secondary | AAI | AEO | AAI | EAO |

Moods | EAO | EAO | EAO | ERO |

ARI | AGO | ARI | AEE | |

ERO | ERO | ERO | AEO | |

RAI | AAI | |||

GAO | IAI | |||

RAI |

The count of primary valid moods is thus (secondaries in brackets): 6 (+4) in figure one, 6 (+4) in figure two, 6 (+6) in figure three, 1 (+7) in figure four. Thus out of 864 imaginable moods, barely 2.2% are valid and significant. A further 2.9% are logically possible, but of comparatively little interest, for reasons already given. These calculations show the need for a science of Logic. If there is a 95% chance of our thought-processes being in error, it is very wise to study the matter, and not leave it to instinct.

### 6. Common Attributes.

We may observe some characteristics the valid moods have in common, relating to polarity or quantity.

- Polarity.

- One premise is always affirmative. Two negative premises are inconclusive.
- If both premises are affirmative, so is the conclusion.
- If either premise is negative, so is the conclusion.

- Quantity.

- Only when both premises are universal, may the conclusion be so; though in some cases two universals only yield a particular.
- One premise is always universal. Two particular premises are inconclusive. (Exceptions occur in Figure Three, if both premises are singular or majoritive; the conclusion is in such cases particular.)
- If either premise is particular, so is the conclusion. (To note, additionally, a singular conclusion may sometimes be drawn from a singular premise, in Figures One and Two. Likewise for majoritives and minoritives.)

Comparing these, it is interesting to note how polarity relations are almost similar to quantity relations. Positive is a connection superior in force to negative, much like as universal is to stronger than particular.

These half-dozen ‘general rules of the syllogism’ (as they are called), together with the couple of specific rules mentioned above within each individual figure, are intended to be sufficient, if memorized, to allow us to reject moods which do not fit into any one of them. They apply to the main forms under discussion, though some exceptions occur in a wider context, as will be seen.

*Distribution**.*Additional rules have been formulated, which focus on the distribution of terms. These rules help explain the generalities encountered in the previous approach. They are:

- The middle term must be distributive once at least. That is, there must be common instances between the members of the middle term class subsumed in the two premises; this explains the general need of a universal, as well as the mentioned exceptions.
- A minor or major term which was not distributive in its premise, cannot become distributive in the conclusion. That is, we cannot elicit more information concerning a class than was implicit in the given data.

Euler diagrams are very helpful in this context. Through drawing the extensions of the three classes, we can observe on paper the transition from minor to major via the middle.

### 7. Imperfect Syllogisms.

If logic is viewed as having the task of drawing the most information from given data, then certain additional formal possibilities of deduction from some pairs of categorical premises should be mentioned. We have seen that normal syllogisms always yield a conclusion with the minor term as subject and the major as predicate, ‘S-P’. We may ask if there are cases where such a typical conclusion may not be drawn, but the deduction of some other form of conclusion, at least, is still possible.

It is found that indeed this occurs in certain cases. The conclusion involved always has the form ‘Some nonS are nonP’, a particular proposition connecting as subject and predicate the negations of the minor and major terms, instead of the terms themselves. The list of such imperfect moods is as follows. In the first figure, **EE**, **OE** (and **GE**); in the second figure, **AA**, **EE**; in the third figure, **EE**, **EO**, **OE** (and **EG**, **GE**); in the fourth figure, **EE**.

These syllogisms are of course very artificial, and will not be discussed further.

## 10. SYLLOGISM: VALIDATIONS.

### 1. Function.

Validation of a syllogism consists in showing its consistency with the axioms of logic. If it is shown that the conclusion follows from the premises, the form of thought is justified. When we encounter a syllogism which results in some antinomy, we obviously reject it; when we reject a sequence of premises and conclusion, we automatically validate that sequence of premises with a contradictory conclusion. Only thus is the balance of consistency restored. This defines validation.

Note well that the conclusion must follow from the premises; mere compatibility between the propositions is not sufficient to imply a connection between them. Thus, invalid syllogisms will display either a conclusion incompatible with the premises somehow; or a conclusion which, though compatible with them, is not more compatible than its contradictory is. Thus, validation could be viewed as the discovery of those forms of thought which satisfy a precondition set by Logic, namely that the premises be shown to imply the conclusion. Invalidity signifies failure of the syllogism to fall under the class so defined.

The validation process itself uses logic; but this circularity does not logically put it in doubt. This apparent paradox can easily be explained as follows. The science of Logic is merely a verbalization of observed fundamental phenomena (identity, inclusion, the need for consistency); these phenomena are out-there and in accordance with themselves; our science’s task is to apprehend the exact extent and limit of their manifestations. If the use by logic, for the validation of its processes, resulted in an inconsistency with any of the apparent controlling principles of our world, we would be justified in questioning it. But so long as no inconsistency is found, it must be trusted. For to say that the validation processes depend on their own conclusions to work, merely confirms how basic this science is. Whereas, the attempt to cast doubt on logic itself appeals to our logical instincts for its credibility, and therefore constitutes an inconsistency, that between the primary denial and hidden dependence on logic. Of the two theses only the former, then, is self-consistent. We conclude that validation is meaningful as a process of clarifying the consistency of valid logical processes with the axiomatic basics.

### 2. Methods.

Many approaches to validation have been developed by logicians. From the start, Aristotle was aware that each figure had its own character, and was able to identify the method of validation most appropriate to each. However, other methods are always worth exploring, to obtain further confirmation, to be exhaustive, and to train the mind.

- The First Figure.

This is the most basic figure, and essentially defines the nature of subsumption and inclusion. It is validated by ‘exposition’. Aristotle formulated the Law of Identity as “What is, is what it is”. If a thing exists, it has certain attributes. If according to our perceptions and insights it appears that anything which is X is Y, then anything which appears to be X must appear Y. Our suppositions are justified, until otherwise proven, and we must submit to the reality of our experiences and to the meaning of our words. This reflects the self-evidence of the world around us, and attaches our words to their intention.

Thus, if one says ‘All M are P’, then indeed anything which is M, is likewise P, so that the (all or some) S which are M must also be P. If any S were not P, this would signify that some M are not P, and contradict our original assumption that all M are P. Therefore, granting the two premises, the conclusion follows, and **AAA**, **AII**, and **ARR**, are valid. The same can be argued in the case of a negative major, or we can reduce the forms **EAE**, **EIO**, and **ERG**, to the affirmative form, by obverting the major.

Subaltern forms of course follow by eduction. Syllogism with the contradictory conclusions to the above, are proved invalid, by opposition. No conclusion logically follows from all the remaining pairs of premises, so they have no validity as syllogistic processes.

- The Second Figure.

Reduction consists in demonstrating that the validity of one inference proceeds from that of another, already established. Reductio ad absurdum shows that a major (A) and minor (B) premise together imply a conclusion (C), because if A were asserted together with the negation of C, they would together imply the negation of B, through an already validated process. This method, with the major premise kept constant while the rest is tested, is used to validate the second figure by reference to the first.

Thus, ‘All P are M and No S is M’ imply ‘No S is P’, for granting that all P are M, if some S were P, then some S would be M, which contradicts our original minor premise that no S is M. In this way, **AEE** in the second figure is reduced to **AII** in the first figure, through a syllogism involving the original major term as middle term. We can proceed likewise to validate the other valid moods of the second figure.

Although so provable, the second figure should also be viewed as reasonable on its own merit, by exposition. Because essentially it defines for us the mechanics of exclusion, just as the first figure reflected more those of inclusion. If we consider two things one of which is excluded and another is included in a third, they cannot reasonably be visualized as contiguous.

- The Third Figure.

Validation by exposition seems to be the method most suited for the third figure, although again other approaches are possible, because the conclusion is always particular. We proceed by showing that in certain instances under scrutiny, two events are contiguous (because the whole includes the part), so that the conclusion holds. This is a positive approach, which can be buttressed by reductions.

Thus, we could take **AII** in the third figure, and reduce it ad absurdum through **EIO** in the first figure. We test the effect of contradicting the conclusion while holding on to the minor premise, this time. The resulting syllogism has the original subject as its middle term, and its conclusion contradicts the original major premise. We can likewise validate the other valid moods of the third figure.

Of course, we could use similar methods to reduce the third figure to the second, or vice versa by changing our constant.

- The Fourth Figure.

Here, direct reduction is the most natural treatment. The two premises of **EIO** in the fourth figure are each converted, to yield **EIO** in the first figure, which results in the same conclusion. Alternatively, convert the minor premise only, and reduce to the second figure; or convert the major only, to obtain the third figure.

Reductio ad absurdum is an indirect form of reduction, which we use quite often in everyday thinking. Another approach, just sampled, is direct reduction. This is more formal minded, in that one or both premises are subjected to an eductive process to reduce the syllogism to a first figure mood with the same conclusion or one implying it. This method is not restricted to the fourth figure, but can equally be practiced in the second and third. For example, for **AEE** second figure, the major is converted by negation, and the minor obverted, to obtain **EAE** in the first figure. Again, for **EIO** in the third figure, the minor is converted, to yield **EIO** in the first. The full list of such processes is easy to develop and well established, and available in most logic text books, so we will not here belabor the reader with excessive detail.

- Secondary Syllogisms.

Though subalterns could be analyzed independently, once a subalternant syllogism is established, its subalterns are easily seen to follow by eduction.

With regard to the imperfect syllogism, combinations of premise which do not yield a normal S-P conclusion, but nevertheless can be wrung-dry to yield a nonS-nonP conclusion, they are dealt with by direct reduction through valid third figure arguments. For the first figure, use obversion of the major and obverted conversion of the minor; for the second, contrapose the two premises if positive, or use obverted conversion if negative; for the third figure, obvert both premises; for the fourth figure, draw the obverted converse of the major and obvert the minor.

- Rejection.

As already indicated, this is the process of invalidation, and of course should be applied to each and every invalid mood systematically. The method is similar, since a mood which concludes something contradictory or contrary to our valid forms must be rejected. More broadly, forms which are not established as valid somehow, are automatically kept apart: the onus of proof can be left to them, as it were.

### 3. In Practice.

The science of Logic has, as above, analyzed validation and invalidation processes used to establish the general truth of the reasoning processes described in the previous chapter. Whereas it works in formal terms, we normally do not refer to formal logic in practice to verify our thinking or spot fallacies in it. We repeat the expository or reductive processes, every time we need to understand or convince ourselves of an argument, with the specific contents of our propositions. Going through such a process serves to integrate our knowledge, comparing its elements and checking their consistency.

Once, however, one is trained in logic, one may well refer to the science’s findings to unravel some argument. In this context, the rules and the canons of Logic may be appealed to intellectually. Analysis of the quantities and polarities involved, consideration of the distribution of terms, are then valuable tools, if one has them well in mind.

A popular way to verify that arguments are kept in accord with logical rigor, is through application of the fallacy tests developed by Aristotle and logicians since. These warn of common pitfalls which one may encounter. They reveal how one may, through hidden equivocation (the Four Terms), confusing suggestions (as in the Many Questions), self-contradiction (Begging the Question), or other such devices, befuddle ourselves or others. Study of these, found in most text books, is of course valuable training.

### 4. Derivative Arguments.

We have stated that syllogism involves three, and only three, propositions; and likewise three and, only three, terms. In practice, it may seem that other possibilities exist. But logic shows that such atypical argument is actually either abridged or compound syllogism, which can be reduced to the standard formats.

are syllogism a premise or the conclusion of which is left unstated, but which is clearly taken to be understood or implied. This artifice is common in normal discourse, as when we rely on context, and can only be formally validated by bringing the suppressed proposition out in the open, and checking that the argument obeys the rules of logic.*Enthymemes*

An ** epicheirema** is an argument in which one or both of the premises is supported by a reason. This simply means that the explained premise is itself the result of a prior syllogism.

- We often have trains of thought: these may be reduced to chains of two or more syllogisms, of any kind. Such an entangling of argumentation is called a
. The name is more traditionally applied specifically to certain regular chains of argument in the first figure, which suppress intermediate conclusions. These are as follows:*sorites*

All (or Some) A are B,

All B are C,

All C are D,

All D are E,

All (or No) E are F,

therefore All (or Some) A are (or are not) F’.

We move from a universal or particular, but always affirmative, minor premise, through one or more intermediate universal affirmative premises, to a final affirmative or negative, but always universal, major premise, to obtain a conclusion with the quantity and subject of the minor premise and the polarity and predicate of the major.

There are thus four valid moods. **AAAA**, **AAEE**, **IAAI**, **IAEO**, for each set of three or more premises. The validation of these is achieved by listing a series of syllogism with the same result. For instance:

A is B and B is C, therefore A is C;

A is C and C is D, therefore A is D;

A is D and D is E, therefore A is E;

A is E and E is F, therefore A is F.

The conclusion of each syllogism is used as premise in the next, if any. Clearly, the middle terms must all be distributive.

The name ‘sorites’ could equally be applied to any complex of arguments, in any combination of figures, instead of just to such a regular series of first figure syllogism. Irregular sorites takes the conclusion of any unit of argument, and transfers it to another argument where it serves as a premise.

Thus, sorites in the widest sense is simply the multiple branching of thought in all directions. Each unit argument within this network may be indicated by only a highlight — a premise or two, or a conclusion — the most significant or controversial part. A sorites is a collection of such highlights, an abridged argument.

- Certain arguments called immediate inference by
**added determinants**or by complex conception, seem like immediate inference, but are really mediate inference. This refers to arguments like ‘since X is Y, then ZX is ZY’. If the qualifying Z is an adjective, the argument is valid, since if some X are Y, and all X are Z, we may infer, in a third figure syllogism, that some Y (those which are X) are indeed Z. But if the Z clause does not fit in such a valid syllogism, it in some cases cannot be passed on.

In practice, such argument can easily be fallacious, as a result of double meanings (as in ‘science is fun, so scientists are funny’), or the use of terms in inappropriate ways (as in ‘horses are fast, so the head of a horse is the head of a fast’).

Such rough logic is not very reliable, and should not be considered a part of formal logic. It is better to insist on strict conformity to formal processes. If a specific kind of content allows for special logical rules, then these may be clarified explicitly in a small field of logic all of their own.

# PART II. MODAL CATEGORICALS.

## 11. MODALITY: CATEGORIES AND TYPES.

### 1. Seeds of Growth.

Aristotelean Logic, we have seen, deals with categorical propositions of the form ‘S is P’. The copula ‘is’ is often conceived as having an absolute or timeless quality; it is viewed as the essential relationship between things in a scientific body of knowledge. Although this knowledge may involve particular statements, their role is merely that of either stepping stones towards eventual general statements or tools for denying general statements. Science’s goal is mainly to discover universals. For this reason, time, change, and causality were not given formal attention in the traditional approach.

But if Logic as a science is to be universal in scope, it must go into deeper detail, and analyze the full range of existing phenomena reflected in language and everyday thought processes. This is painstaking, perhaps never-ending, work. Logic is not to be confused with grammar; it is not primarily concerned with the structure of sentences, which may vary from one language to another, and indeed sometimes seem illogical. But Logic can observe commonplace statements to identify possible areas of interest for treatment in its peculiar way. In any case, its ultimate goal is to say some general things about reality, and about how we may properly think about it.

In this perspective, then, classical Logic is but a beginning, a specialized investigation which needs to be pushed further gradually. In this chapter, we will indicate some of the possible areas of expansion for our discipline.

The concept of modality is extremely interesting, because its detailed development has a powerful systematizing effect on logical science. From the seeds of thought provided by a few insights, like postulates, a large chunk of knowledge can be organized into a formal whole. A relentless progression of problem solutions and predictions is put in motion, providing us with exciting tools for the growth of knowledge.

A theory is ultimately judged not only by its consistency and truth, but also by its fruitfulness. The distinctions and classifications, the understandings and guidance, which modality generates, show its importance to advanced logic, and thereby to the broader concerns of philosophy.

Modalities are certain qualifications of relations, expressing the frequency of events, within some framework. In the deepest sense, modality is concerned with the differing and varying *levels of being*; hence its central place in both ontology and epistemology. The study of modality could be called ‘Tropology’: it is a broad field.

The term modality may be used in the sense of a ‘category of modality’, or in the sense of a ‘type of modality’. Category refers to the frequency aspect, type defines the framework. When referring to the modality of a relation, we may mean either of these senses, or their intersection. For we find, within each type of modality, the same categories of modality, only with a somewhat different meaning.

The types and categories define the multiplicity of ways in which anything may said to ‘exist’.

### 2. Categories of Modality.

We call categories of modality the concepts of possibility or necessity, impossibility or unnecessity, contingency or incontingency, probability or improbability and their degrees — as well as presence or absence.

These terms will all be more fully defined further on. Meanwhile, let us note that they are interrelated in various ways. The following tree illustrates some aspects of their interrelationships.

Presence signifies the occurrence of an ostensible individual phenomenon, a unit clearly defined in time and place; and absence is the negation of this. Presence is a class standing under possibility and above necessity; absence, between unnecessity and impossibility. Presence or absence occur either because of incontingency, or through the realization of contingency.

Possibility may be viewed as a generic concept which embraces either contingency or necessity. Likewise, contingency and impossibility may be viewed as mutually exclusive species of unnecessity. Contingency signifies possibility and unnecessity taken together. Incontingency a genus for necessity or impossibility. The various degrees of probability are subcategories of possibility or unnecessity.

In practice, these concepts are expressed in sentences by words like ‘in some cases’, ‘sometimes’, ‘can’, ‘may’, ‘might’, ‘possibly’, ‘potentially’, ‘permissibly’, ‘perhaps’, and all their related terms. The differences between these modal expressions are not merely verbal.

Indeed, in normal discourse, we tend to interchange terminology indiscriminately. For instances, in some cases we say ‘always’ to mean ‘all’; in some cases, ‘can always’ means ‘all can’. This is not our concern as logicians: we identify the connotations closest to what we are trying to discuss, and henceforth adopt restrictions which serve our purposes.

### 3. Types of Modality.

I have identified *five* main types of modality, five senses in which the various categories of modality may be understood. Within each type, all the categories occur, but with other meanings than in the other types. The categories have similar interrelationships and properties within each type. These uniformities allow us to abstract them, but ultimately each type needs to be considered separately. The interactions between types must also be analyzed.

Quantity, or extensional modality, is the primary type of modality, and is the one which was thoroughly dealt with by Aristotle. Two more, temporal modality and natural modality, will presently be analyzed in detail; they interact intimately with quantity. The last two types, logical modality and ethical modality, are each sui generis, and require independent treatment.

It will be soon be evident that the temporal and natural modalities have characteristics in common with quantity. They represent different ways the subject and predicate might be related. They can be combined in certain ways with quantity, to form complex propositions. They are mutually related, in fact form a continuum, although they cannot be compounded together as they can be with quantity. They are subject to rules resembling those found for quantity, because they derive from the same geometric fundamentals.

Each type of modality has its own character. Quantity refers to the proportion of a whole class that is subject to a certain relation to a predicate. Temporal modality refers to the proportion of its whole existence in time that any individual subject happens to have a certain relation to a predicate. Natural modality expresses the degree of causal conditionality concerning such relation.

Extensional modality recognizes the variations which can be found to exist between instances of similar phenomena, be they static or dynamic. Temporal modality proceeds from the occurrence of change in individual things during their existence. Natural modality stems from the belief that ‘laws’ guide events. Our world is diverse in all these senses. There is thus an ontological basis for such distinctions.

Furthermore, Logic must investigate the differences and similarities in behavior of such phenomena, and the results of their interplay. Here, then, is a possible area of new activity for Logic, clarifying the meanings of forms involving modality, and analyzing their oppositions, the eductions possible from them, and the syllogistic arguments involving them. This topic will be dealt with in considerable detail in this treatise.

The two types of modality we are introducing here are effectively qualifications of terms similar to distribution, although strictly speaking they apply to the relationships of terms. Such propositions are complex variations of the standard forms researched by Aristotle, involving an additional factor, modality, which can be subjected to whole-and-part, inclusion-exclusion type analyses, as was done with quantity.

### 4. Extensional Modality.

Consider the ways in which we use expressions of possibility or necessity. As stated previously, we are in everyday discourse not consistent in our use of terms like ‘sometimes’, ‘can’, ‘may’, ‘might’, ‘must’, and so on. Ultimately these are semantic issues, not important to us, though they need pointing out. Logic simply establishes conventions for terminology, and focuses on the material issues.

Now, it happens, for instance, that when we say ‘S may be P’ we mean ‘some S are P’. This use of a modal-looking qualification to express quantity is not accidental. When considering a specimen of S, we may want to note that the fact that other S have been found P suggests that this one also could fall in that group for all we know.

But is there an ontological basis for considering quantity a type of modality, or are we just dealing with a mode of thinking, a useful artifice? Quantity is essentially a qualification of universals, which we suppose to have some kind of reality, although we cannot yet understand their nature adequately. When we say that some S are P, we are not merely intending to express a quantitative fact, but to affirm the compatibility between ‘S-ness’ and ‘P-ness’.

This is traditionally known as the distinction between viewing a concept in its extension (the units it applies to) and its intension (its meaning). A universal may be viewed as a ‘substance’ (or stuff) which is scattered in the world. When two universals, S and P, coincide in some entities, we learn more than simply the fact of contiguity; we learn that the natures of the two universals do not intrinsically prevent such occurrences, and this is for us significant information.

Similar argument is possible for the other quantities. They tell us of the compatibility or incompatibility, necessity or contingency, high or low probability of coincidence between universals. The numerical aspect of quantity is incidental, though Logic develops by concentrating on it because of its manageability.

Social statistics, for example, are mostly based on this approach. The information we obtain concerning a social group is applied to each individual in the group, with the corresponding degree of probability. The mere fact that most individuals in a sample behave in a certain way, should not imply that there is any possibility that individuals who did not behave in that way at all could have. And yet we do feel justified in so reasoning, because we believe that reality functions through the forces inherent in universals.

As stated before, although many skeptical philosophers have denied validity to such modes of thinking, my position is pragmatic optimism. This is the position of science: that even if an appearance is not fully understood, it is received with an open mind, provided or so long as no inconsistency arises from the belief.

Humans inevitably conceive the world in terms of universals; therefore it appears that they exist. That they are difficult to fully grasp does not mean they are untrue. Only if they were logically contradictory to evidence, would doubt be reasonable. But no credible cause for doubt has arisen. Indeed, most importantly, to deny universals through some speech, is using universals to deny them: that position is the inconsistent one of the two, and therefore absolutely false.

So when we say that ‘some S are P and some are not P’, we still believe that there was a ‘possibility’ even for S which are not P, to have been P (or vice versa), although they did not happen to concretize in this way. We think this, because the universals S and P (or nonP) have displayed compatibility in some cases of their existence.

Thus, we mentally distribute not only generals, but even particulars to each of all the individuals involved, via the universals, while remaining aware that the factual concretization of the universals in that contingent way is final. In this sense, quantity can legitimately be viewed as a type of modality.

It must be stressed, however, that extensional modality differs radically from temporal and natural modalities, in that it can be combined with either of them, whereas they cannot be superimposed with each other. That is because they are really part of the same modal continuum of individual capabilities, whereas quantity remains essentially a factor concerning groups of phenomena.

Temporal and natural modality may be called ‘intrinsic’ modalities, because they concern the properties of concrete individuals; extensional modality is comparatively ‘extrinsic’, in that it focuses on abstract universals.

### 5. Temporal Modality.

While it is true that often the copula ‘is’ is intended in a timeless sense, we sometimes use the word with a more restrictive connotation involving temporal limits.

The temporal equivalent of what is a singular instance in extension, is a momentary occurrence; this is the unit under consideration here. When we say ‘S is P’ we may mean either that S is always P, or that S is now P, or even that S is sometimes P. This ambiguity must be taken into consideration by Logic explicitly. A possible modification of standard propositions is therefore through the factor of temporal frequency.

We can say of an individual S that it is now or not-at-this-time P, or sometimes or always, or sometimes-not or never P, or usually or rarely P. We recognize that a thing can vary in attributes during time, and often use such forms to express such experiences. Such propositions can in turn be quantified, so that complex combinations emerge.

According to the traditional approach, we are supposed to deal with these forms simply by attaching the frequency qualification to the predicate, to obtain a new predicate. This process is called permutation; we encountered it previously, in the context of changing propositions into the “is” form, and obversion is a sample of it, too. Tradition has assumed that once permuted, such propositions can be processed in the normal way, through Aristotelean syllogism.

But this first impression was wrong; the device is misleading where modality is concerned, for three reasons. Firstly, it fails to account for a large number of practical inference, whose validity can only be established through analysis of the propositions in their original forms. With such propositions in their permuted forms, syllogisms would contain a middle term which is not identical in the two premises (for example, ‘S is sometimes-M, M are always-P’), or a minor or major term not identical in premises and conclusion (for example, the conclusion ‘S is sometimes-P’ from the said premises).

Secondly, and even more importantly, permutation can result in erroneous inference. For, in fact, as analysis shows, we cannot always transmit a frequency unchanged from premises to conclusion (for example, as in ‘S is M, M are always-P, therefore S is always-P’), and sometimes not at all (for example as in ‘S is M, M are sometimes-P, therefore S is sometimes P’).

Thirdly, in some cases, we can deduce from a given frequency, not capable of being itself simply transmitted, another, lower frequency; if we merely relied on permutation, the conclusion would not be formally valid. (For example, in ‘S is M, M are always-P, therefore S is P’). So we have no choice but to demand special treatment; the issues are more complex than we are led to believe by the permutation theory.

All this will become clearer by and by. It will be seen, as the analysis of modal forms proceeds in full detail, that, although our method of analysis is similar to Aristotle’s, we cannot mechanically reduce temporal modality arguments to traditional forms. Such situations must be investigated systematically, and special principles must be formulated to guide our reasoning in relation to them. The results obtained are often unexpected and instructive, and justify our research effort.

Temporal modality is especially useful, when reporting the behavior patterns of organisms; this is especially true for animals, who have powers of volition, and even more so for humans, who we consider as having free will. For, with regard to certain actions or states of such subjects, we cannot say that they ‘must’ or ‘cannot’ do or have them, in the sense of natural determinism, but only that they always or occasionally or never do so.

Thus, for instance, we can study the psychology of people, and predict their reactions to some extent, without having to postulate a more rigid degree of necessity than mere constancy, and before being able to explain volition or free will.

### 6. Tense and Duration.

We have indicated that the unit considered by temporal modality is a moment of existence. But ‘now’ is not the only individual moment we can refer to. The individual moment involved may be located anywhere in time, past, present, or future; and that location may be expressed precisely, by date and time o’clock, or roughly. This issue is known to grammar as tense, and we may adopt the same name for it in logic.

Also, the individual moments we speak of vary in size. The segment of time involved may be a fleeting moment, or an extended period of time; it may be expressed vaguely, or precisely, as a year, week, hour, or microsecond. This is an issue of duration.

These different units in the continuum of time, defined by the tense and duration of existence, of the subject and predicate relation under scrutiny, are the instances of the ‘class’ under consideration in the context of temporal modality, in analogy to the cases of a universal in the context of quantity.

We can in principle thus develop an infinite list of possible tense/duration characterizations for propositions, according to where in the time continuum the event is projected, and for how long. Thus ‘things S and P’ could mean: things now S and P, or which were or had earlier been S and P, or which will be or are later going to be S and P; and the time locations and periods tacitly intended could be specified explicitly.

Here again, following the permutation idea, we would suppose it possible to merge the tense into a term so related, to form a new term capable of timeless treatment; for example, ‘S was P’ would become “S is a ‘was-P’”. This presupposes that, provided no equivocation was involved, a proposition so altered could then enter into a syllogism without causing problems.

However, in fact, this artifice does not work; it conceals the validity of certain arguments which it assumes false, and it causes us to assume certain arguments correct, which closer inspection reveals false. So a specific analysis is required. These claims will be seen evident as formal treatment proceeds.

Tense is not in itself a distinct type of modality qualification; but an integral part of the doctrine of frequency. It simply defines the possible variety of locations in time, besides the elementary ‘now’; without awareness of them, we might make logical mistakes.

However, apart from these general guidelines, the topic of tense will not be developed in full detail in this paper. It is enough for our present purposes to make the reader aware that our use of the expression ‘now’ is intended to include past and future nows, and nows of any size. So long as the now involved in any argument is one and the same, the rules we will establish for such arguments will work. The possible interactions of different nows will not be covered, however.

### 7. Natural Modality.

The most significant type of modality is what I call natural modality. This refers to propositions such as ‘S can be P’, ‘S cannot be P’, ‘S can not-be P’, and ‘S must be P’, with the sense of real, out-there potential or necessity. These relations were effectively recognized by Aristotle in his philosophical discussions, but were not systematically dealt with in the framework of his logic works.

Note in passing that often, when people write ‘S can not be P’, they mean ‘cannot be’ rather than ‘can not-be’; in the former case, the ‘not’ negates ‘can be’ (it means ‘not-can be’ in spite of its position in the phrase), whereas in the latter, the ‘not’ only negates the ‘be’.

Such modality differs radically from temporal modality. We do not here merely recognize that something may be sometimes one thing and sometimes another, or always or never so and so. We tend to go a step further, and regard that there is a character intrinsic to the object which makes it able to behave in this way or that, or incapable of doing so or forced to do so. Thus, temporal and natural modalities represent distinct outlooks, which cannot be freely interchanged.

We can infer from S being sometimes P, the implication that it can be P, arguing that otherwise it would never be P; likewise that S is sometimes not P, implies that it can not-be P, or else it would always be P. But when we say that S can be P or nonP, we mean something deeper than merely an observed conjunction. We often claim, through indirect discovery, to know that S can be P (or nonP), even though this potentiality is never actualized. Whereas, with S is sometimes P (or nonP), we are making a statement that requires the relation of S and P to be actualized at least once.

Similarly, we may induce, in the way of a generalization from experience, from S always being P (or never being P), that it must (or cannot) be P. But when we say that S must (or cannot) be P, we intend a more profound relationship than mere constant recurrence (or nonoccurrence, as the case may be). We claim knowledge of the inner nature of the object (whence my choice of the term ‘natural modality’, by the way); we claim to be explaining why the observed constancy took place. We may thereafter discover indirectly that S can be and can not-be P; we would then conclude that, although S is always (or never) P, this is not a case constancy due to necessity, but just the way a contingency was actualized.

The indications here given should be enough to clarify ostensibly what phenomenon we are trying to refer to. Before discussing the concept of natural modality further, on a more philosophical plane, a pragmatic definition, sufficient for the needs of logical science, will be proposed.

An event is said to be potential if it occurs *in some circumstances*; it is said to be naturally necessary if it occurs *in all circumstances*. Unnecessity is, then, nonoccurrence under some circumstances, and impossibility occurrence under no circumstances.

This concept of circumstance refers us, then, not to time as did temporal modality, but to the assumption that, scattered in the environment of an event, are certain causative factors, be they known or unknown, specified or unspecified.

‘S can be P’ thus means ‘When certain causes occur, S is P’, ‘S can not-be P’ means ‘Under certain conditions, S is nonP’, ‘S must be P’ means ‘In all situations, S remains P’, ‘S cannot be P’ means ‘Whatever the surrounding circumstances, S remains nonP’.

That definition justifies our calling this phenomenon a type of modality, because, like the previous types of modality (temporal and extensional), it is reducible to an issue of enumeration: we use the same ideas of whole and part, inclusion and exclusion, all/this/some, frequency.

In the case of extensional modality, we are dealing with instances of a universal; in that of temporal modality, with moments of an existence; in natural modality, with causal conditions. All these implicit concepts are admittedly inscrutable in their essences, but their applications are numerical and so capable of systematic treatment by logical science.

We can argue, as we did for temporal modality, that natural modality is not permutable. I will not repeat the arguments here, especially since this truth becomes so obvious once we start dealing with formal issues.

### 8. Other Types.

Two other main types of modality, the logical and the ethical, need to be indicated to complete our introductory synopsis of the topic. As previously stated, these types are each sui generis, and worthy of thorough treatment on their own. Logical modality will be dealt with later in this work, but ethical modality is left to some future volume.

What distinguishes these types from those previously considered, is their object of attention. Extensional, temporal and natural modalities tell us something concerning the subject and predicate related themselves. Logical and ethical modalities, in contrast, either report about the state of our knowledge, or make recommendations for action, in connection to those objects.

*Logical Modality.*

To the extent that such an evaluation is scientific, based on rigorous process, thorough, common public knowledge, and so on, it is objective information. To the extent that thought is deficient in its methodology, such modality is subjective.

Whereas the extensional, temporal and natural types of modality may be called ‘materialistic’, in that they refer directly to the world out there, which is mainly material or in any case substantial, logical modality may be called ‘formalistic’, because it operates on a more abstract plane.

*Ethical Modality.*

Ethical modality is of course relative to standards of value. The complex issue of how to establish absolute standards, or whether we are able to, will not be discussed here. Suffices to say that, within a given framework, an ethical statement can in principle be judged true or false like any other.

Subjectivity comes into play here, not only in the matter of selecting basic values, but also to the extent that, in this field more than any other, factual knowledge is often very private.

Logic must, of course, eventually analyze such modality types in detail. But for our present purposes, let us note only that, in either case, the resemblance to the other types of modality is the aspect of conditionality. They are defined through the conditions for their realization.

Their distinction is that they do not concern the object in itself (i.e. the S-P relationship as such) like the others, but involve an additional relation to man the knower of that object, or man the eventual agent of such object. The latter relation is thus a new object, which includes the former, but is not identical with it. Such modalities, then, are not essentially subjective, though they can degenerate into subjectivity, but rather concern another object.

The reader should beware of the various ways the words ‘modality’ or ‘modal’ will be used in this volume. In its broadest sense, ‘modality’ applies to any type and category of modality, which details should be specified, and every proposition is ‘modal’.

In practice, we sometimes use the word ‘modality’ to refer specifically to the natural, temporal or extensional types of modality, to the exclusion of the logical. Sometimes, the sense is restricted to only natural and temporal modality, as distinct from quantity. Likewise, we may in some cases call a proposition ‘modal’, to signify that it is other than actual or singular or factual.

The context should always make the intent clear.

## 12. SOURCES OF MODALITY.

### 1. Diversity.

Underlying the existence and concept of modality, are the phenomena of difference and change. At any given time, the world appears as a multiplicity of distinguishable phenomena, distributed in space; and across time, the world reappears before us, comparatively differently constituted and deployed.

- The concept of difference implies that of similarity. If everything was absolutely different from everything else, things would not coexist: they would have nothing in common, neither existence nor space nor time nor any character, each would have to be a ‘world’ by itself; and only one such ‘world’ could exist, which would have but one point of space and time. Thus, paradoxically, to deny similarity is to deny difference; to posit a world consisting only of diversity logically implies that we believe the world to contain no diversity at all, not even dimensions.

While it seems obvious that the phenomenon of motion requires that we postulate a time dimension, a universe devoid of motion but extended in time seems conceivable. While a universe of only two, or even only one, space dimensions (with or without motion) seems conceivable — a universe devoid of space or time (an unextended point and instant, rather than a minuscule and short-lived one) seems unthinkable, impossible not to measure up against infinity and eternity.

Diversity is of various kinds. A thing may for instance be both green and flat, at the same time and in the same place; this is diversity of character in the purest sense, a coincidence of ‘incomparables’. A thing may be both green and red, but only in different places of it or times in its existence; or it may partly or wholly move, and occupy two different places, though only at different times; these are diversity of comparable characters or of location, made possible by the existence of space and time dimensions.

- With regard to diversity in time, there is no escape from the fact that change appears to be happening; this phenomenon is alone sufficient to demand from us a recognition and concept of change, independently of whether we evaluate given cases as specifically real or illusory.

Some change is indeed illusory, meaning that the newly perceived difference was already there, but was previously unperceived. The ‘change’ is due to the movement of the spotlight of our consciousness, rather than to an event in the object itself. The concept of illusion is built on such ‘changes of mind’; we are not omniscient, our knowledge evolves; every appearance is assumed real, unless or until it fails to be consistent with the mass of other appearances, in which case it is reclassified as illusory.

But some of the change has to be real: to deny this is logically untenable, because even an ‘illusion’ is in itself a specific kind of object. What we call ‘illusory change’ is more precisely a real change from one illusory appearance to another illusion or a reality. The form of change is real enough, it is its particular content which is illusory.

Even if, in an attempt to explain away time and change, all apparent mobility in the material world were attributed to the travels of consciousness, we would still be left with the need to understand the latter movements as themselves changes dependent on time. Therefore, nothing radical is to be gained from such an attempt.

- It may well be that, at some higher plane of being and consciousness, the world merges into undifferentiated and immobile oneness, which is somehow more ‘real’ than our ordinary, sublunary experiences, because it allegedly unifies and explains them. However, this idea does not logically deny the side by side existence, in some respect, of the variegated and dynamic, illusory lower world.

The appearances of difference and change may be illusory, may be inferior on some spiritual scale, but even so, they have to have a sort of existence. It is a phenomenon presented to our consciousness, which we cannot avoid admitting to exist as such, even if we believe it to be a warped image of an otherwise uniform and static reality. It is conceivable that at some past or future time the world was or will become One; but this in no way excludes the current existence of some form of difference and change, as appears, if only as appearance.

- In conclusion, the world must stand somewhere between the extremes of absolute diversity and absolute unity, which are incidentally one and the same idea. The mere experience of difference and change, whether real or illusory, is enough to guarantee this fence-sitting position. We cannot logically evade or wipe out this given phenomenon; we can only at most delimit it to some narrower domain and relegate it to some lower status.

Note that mystics of many traditions claim that the conflict of dualism and monism is itself illusory, and that at some higher level the contradiction disappears convincingly. While keeping an open mind toward such a special experience, we may plod on with a logic designed for our commonplace world.

- Once difference and change are admitted, concepts such as polarity, similarity and quantity, time, modality and causality, are inevitable and needed. If the world was, against experience, without diversity or change, there would be but one polarity, one entity, one character, no space or time, no contingency, only necessity, and no need for causal explanations.

### 2. Time and Change.

Time and change appear to be extremely fundamental phenomena in our world and experience, and simultaneously very mysterious and difficult to analyze conceptually. At first sight, they seem evident and obvious, but once we try to understand them in a deep way we uncover a mass of difficulties and complexities.

As far as I am concerned, no satisfactory solutions to the crucial problems involved have been found. Some of these ontological issues will be touched upon in later discussion, but for the most part we will bypass them and concern ourselves with formal logical issues.

With regard to Time, let us pragmatically accept, as appears intuitively, the existence of past, present and future, and that events somehow occur in measurable relative locations in this continuum, which we imagine to be a dimension similar to the three of space.

As for Change, it may be pragmatically defined as occurring when something has one property at one time, and not at another time; or lacks it, and later has it; or, compositely, exchanges one attribute for another. Things change across time, losing properties, acquiring new properties, or changing in degree of some otherwise enduring property, or replacing properties. They may change place (that is motion), or qualitative attributes (alteration), or even change pattern of movement (acceleration) or vary in uniformity of qualitative change. Some changes are irregular, some cyclical, and so on.

### 3. Causality.

An assumption that man regularly makes in his cognition of the world, is that objects behave in the way they do, not merely by happenstance, but because this is somehow programmed into what they are, as part of their identity, their nature. This reference to the inner nature of things is a reference to causality, in its widest sense. We believe not merely in the coincidence of the thing and its attributes, but that the particular identity of that thing has caused it to display this particular behavior rather than any other.

Some philosophers deny this assumption, and claim that all we can say is that things just occur, not that they somehow had to. As with all insights, it is a function of the rules of logic to resolve the debate. I accept the common sense viewpoint, because I have found it consistent and useful.

It seems to us to be so, that there is such a thing as causality, it is one of the appearances in the world, we instinctively think in such terms. If there were some solid reason to deny the concept, we would have to, but no convincing argument has been presented by the skeptics so far. Doubt on the mere basis of difficulty of precise definition or explanation, is not logically sufficient. Logically, some things are bound to be irreducible; why not causality? We can know that it is there somehow, while admitting our inability to adduce its essence, in view of its fundamental nature. There is no self-contradiction in this position.

Fundamental phenomena, like universals and difference, time and change, causality and necessity, are inevitably difficult to fully describe and understand, and perhaps even ultimately, in principle, undefinable and incomprehensible. They are so radical to our world that they cannot be reduced to something else. But this in itself is not a reason to altogether reject them. And indeed, even if some philosophers choose to reject them gratuitously, it changes nothing. People will rightly continue to think in these terms, trusting appearance, unintimidated. It works.

The concept of causality is indeed extremely difficult, if not impossible, to define. It is, like attribution, something we intuitively understand, but which is so fundamental that, although we can discuss it to some extent, we can never pin it down. What we can do with relative ease, however, is identify its varieties.

In the widest sense, any event signifies causality; the nature of an object is viewed as the underlying cause of its ‘behavior’. In this sense, any attribute or change, be it permanent or inevitable, or transient or accidental, is caused by the thing being what it innately is.

But more specifically, causality is limited to the suggestion of necessity. It is most often related by philosophers to time and change, or the explanation of movements. But in fact, in practice, even in the empirical sciences, we conceive it as a force explaining static, as well as dynamic, connections.

Indeed, it will be shown further on that there is one type of causality corresponding to each type of modality. ‘Extensional’ causality concerns uniformities or differences relating to universals. ‘Temporal’ causality concerns constancies or changes across time. ‘Natural’ causality relates to necessity or contingency on a deeper level. ‘Logical’ causality concerns the relationships of ideas.

The pragmatic definition of causality by David Hume as merely “constant conjunction”, simply does not adequately capture what we intend by this concept. J. S. Mill equated natural to temporal modality, in an attempt to bypass philosophical problems relating to the former’s definition. He defined the way we induce causality by generalization, as a substitute for telling us what it is.

## 13. MODAL PROPOSITIONS.

### 1. Categories and Types.

Let us review some of the modal concepts introduced thus far, before examining them in more detail.

Modality in its widest sense is an attribute of relationships. The paradigm of modality is the quantity attribute of (the terms of) propositions. When phenomena are observed to be alike in some way, they may be grouped into a class, and be regarded as instances of that class.

We may refer to such units in various ways. The units intended by a reference are said to be included in it; those not so, excluded. When a unit is focused on individually and specifically (if only through a pointing to it), the reference is singular; otherwise, our focus is plural.

When we refer to a fraction of the class, it is particular; when to its totality, it is general. The greater division of a class is a majority; the smaller, a minority. Singular and particular frequencies concern mere incidence; the other plurals — generality, or majority or minority — are relative frequencies, and describe prevalence.

Quantity is one type of modality, namely the extensional. Other types of concern to us here are temporal modality and natural Modality. These have in common with quantity the mode of analysis defined above. However, the classes under consideration are not the terms of propositions, but respectively the temporal existence or the causal conditions of the connection between the terms.

Just as quantity concerns the application of a term to one, some, all, most or few of its instances; so temporal modality analyses the application of the predicate to one, some, all, most or few of the moments of its given subject’s existence; and natural modality concerns the application of the subject and predicate relation to one, some, all, most or few, of the circumstances surrounding such happening.

These common factors may be called the categories of modality. They are: presence (unitary event), possibility (partial reference to the events-class), necessity (complete reference to it), high or low probability (inclusive of more or less than half the units). Derivative concepts are: absence (presence of negation, or negation of presence), possibility-not (possibility of negation, or negation of necessity), contingency (sum of possibility and possibility-not), impossibility (negation of possibility, or necessity of negation), and incontingency (either necessity or impossibility). These general categories may be given specialized names when applied to each type of modality.

In extensional modality, the main ones are, as we have seen, singularity, particularity, generality (or universality). In temporal modality, we will use the words momentariness, temporariness, constancy, for the corresponding concepts. In natural modality, actuality, potentiality, necessity.

Note that the sub-categories of possibility should not be taken to imply contingency, as often the case in everyday discourse; they are compatible with necessity. Also note our double use of words such as necessity for both abstract categories and especially natural modality sub-categories.

Additionally, let us point out that presence may be usefully viewed either as stemming from necessity or as an occasion of contingency. This way of viewing presence, as the realization of a deeper phenomenon of necessity or contingency, follows from the oppositional relations between these concepts, which will be analyzed below. Accordingly a singular instance may be viewed as the concretization, of either a generality or a distinction. A momentary event may be viewed as the eventualization, of either a constancy or a variability. An actual occurrence may be viewed as the actualization of either a (natural) necessity or a (natural) contingency. Similarly, on the negative side.

We reserve the following terminologies in formal treatment of these three types. This, some, all, most, few, will express quantity. Now, sometimes, always, usually, rarely, will be used to express temporal modality. Is, can be, must be, is likely to be, is unlikely to be, will express natural modality. In ordinary discourse, these various expressions of frequency, quantifiers and modifiers, are of course often interchanged.

It is stressed that all plural such expressions are intended to include the units they subsume on a one by one basis. That is, ‘in some or all cases’ means ‘in each and every one of the cases in the part or whole of the group under consideration’. It is not a collective reference to the units considered together. This quality applies equally to all three types of modality, each in its own domain (extension, time, circumstances).

Every proposition has quantity (implicitly if not explicitly); and every proposition has either temporal or natural modality. The unitary forms of these latter two modalities coincide; but their plural forms cannot be combined, being factors in one and the same continuum. That is, when we colloquially say ‘X can always be Y’, for instance, we may mean formally-speaking ‘All X can be Y’, but it is not possible to combine ‘can’ or ‘must’ with ‘sometimes’ or ‘always’ in the reserved senses of words, because, strictly, must implies always implies sometimes implies can, i.e. these concepts are related in specific ways, as will be seen.

### 2 List and Notation.

Aristotelean logic recognized six main propositional forms, as we have seen, labeled **A**, **E**, **I**, **O** and **R**, **G**. Actually, classical logic is usually developed in terms of the first four of these, i.e. the universal and particular. I added on the last two, i.e. the singular, to complete the picture systematically; they were not unknown to Aristotle, anyway. The labeling above mentioned is of course mere convention. Another notation could have been devised, using the letters **u**, **p**, **s** for quantity specification, and **+**, **–** for polarity. In that case, **A**=**u+**, **E**=**u-**, **I**=**p+**, **O**=**p-**, **R**=**s+**, **G**=**s-**. Generally, I have found it practical to continue using the letters **A**, **E**, **R**, **G**, **I**, **O**, in most work, though the separate labeling of quantity and polarity are sometimes valuable.

The value of this alternative notation becomes more evident once modality is introduced, because the laws of inference in Aristotelean logic can thereby be brought out more clearly. (Note how I often use the term modality in a restrictive sense excluding quantity.) By analogy to **u**, **p**, **s**, we may introduce the symbols **c**, **t**, **m**, for constant, temporary and momentary propositions, respectively; and **n**, **p**, **a** for naturally necessary, potential and actual propositions, respectively. (The equivocal use of ‘**p**’ for particularity and potentiality is perhaps unfortunate, but context will always make clear which of the two is meant, so it is not serious). The modality symbols may be used as subscripts to the standard six letters. The following is a list of all the categorical forms under consideration in this study.

- Propositions involving natural modality. These, for the purposes of definition, could equally be expressed in the form ‘In all/this/some circumstance(s), all/this/some S is/is-not P’ (Or, ‘Under any/the given/certain conditions, all/this/some S is/is-not P’.) Note well the difference between ‘cannot be’ (which should have been written ‘not-can be’, to signify negation of potentiality) and ‘can not-be’ (signifying potentiality of negation).

An | All S must be P | En | No S can be P |

Rn | This S must be P | Gn | This S cannot be P |

In | Some S must be P | On | Some S cannot be P |

A | All S are P | E | No S is P |

R | This S is P | G | This S is not P |

I | Some S are P | O | Some S are not P |

Ap | All S can be P | Ep | All S can not-be P |

Rp | This S can be P | Gp | This S can not-be P |

Ip | Some S can be P | Op | Some S can not-be P |

- Propositions characterized by temporal modality. These can be defined by the overall form ‘At all/this/some time(s), all/this/some S is/is-not P’. Note that we here use the word ‘now’ equivalently to ‘at this time’, to avoid getting involved with issues of tense in this context.

Ac | All S are always P | Ec | No S is ever P |

Rc | This S is always P | Gc | This S is never P |

Ic | Some S are always P | Oc | Some S are never P |

A | All S are now P | E | No S is now P |

R | This S is now P | G | This S is not now P |

I | Some S are now P | O | Some S are not now P |

At | All S are sometimes P | Et | All S are sometimes not P |

Rt | This S is sometimes P | Gt | This S is sometimes not P |

It | Some S are sometimes P | Ot | Some S are sometimes not P |

It will be observed that, in the above listing, we left out subscription of actual propositions with an ‘**a**’, and momentary propositions with an ‘**m**’. This was an intentional ambiguity, which will now be explained. If we analyze common usage of the form ‘S is P’, we find that it is really very vague and capable of many interpretations. This is not said as a criticism of Aristotle’s logic; in a way it has been one of its strengths, the reason why he seemed to have succeeded in describing human thought processes fully. But logic requires that ambiguities be brought out in the open, to ensure that nothing is left to chance. That is precisely why I have taken the trouble to develop a theory of modal logic, and researched it in such detail.

In its broadest sense, ‘S is P’ could be understood to mean any of the following: ‘S must be P’ (an absolute sense, often though not exclusively encountered in theoretical sciences), or ‘S is always P’ (a timeless sense, often found in empirical sciences), or ‘S is in the present circumstances P’ or ‘S is at the present time P’ (such meanings are usually intended in everyday descriptions of social events), or even no more than ‘S can be P’ or ‘S is sometimes P’ (with the qualification left tacit for purposes of stress). We are sometimes not aware of just how high or low on this scale our thoughts or statements fall; sometimes, though aware, we allow our meaning to be suggested by the context, or regard the distinction as not important enough to call for explicit expression. Sometimes, of course, our intention is not left tacit, and we say exactly what we mean.

To further complicate matters, the ‘S is P’ form is sometimes used in a likewise indefinite, but more restricted sense; that is, one not including natural necessity or potentiality, but broad enough to include any temporal modality. In this sense, ‘S is P’ signifies a generic actuality, capable of embracing either constancy or momentariness or temporariness.

As far as formal logic is concerned, the ‘broadest sense’ described above, means no more than ‘S can be P’, which is its least assuming interpretation. Likewise, the ‘more restricted sense’ next described, must be taken by formal logic at its minimal power, meaning ‘S is sometimes P’. Thus, paradoxically, the broader the possible meaning, the lower is its logical value; that is, given a more or less indefinite ‘S is P’ statement, without further specification, we are forced to adopt its most all-inclusive interpretation. Logical science therefore ignores such vague references, and prefers to deal in fully specified forms.

This leaves us with one more ambiguity. If an ‘S is P’ statement is not intended in the above vague senses, is it intended in the sense of actuality (in this circumstance) or in that of momentariness (at the present time)? Are these parallel but different, or are they essentially one and the same? I suggest that the latter answer is ultimately to be preferred. The concepts of ‘present circumstance’ and ‘present time’ indeed have somewhat different conceptual roots, namely causality and time; but they represent the point of intersection of these two frameworks.

Just as a singular proposition points to ‘this’ instance and not merely ‘an’ unspecified instance of the subject-concept, so in natural and temporal modality, there is an mentally understood environment to the event under scrutiny (i.e. S being P). In a natural modality perspective, we view this vague environment as the surrounding disposition or layout of other objects, constituting an undefined set of causal conditions, which may have given rise to our event. In a temporal modality perspective, we merely locate the event in time, but it is taken for granted that the underlying circumstances, however unclear precisely which, may be involved somehow in our event.

Thus, the difference between a-forms and m-forms, in their most definite senses, is merely one of perspective, but they both point to the same factual material. We may therefore regard them as identical, when the interactions of natural and temporal modal propositions are analyzed.

We thus have 18 natural modality forms and 18 frequency forms, or a total of only 30 forms, according to our perspective. We may deal with the two modalities as separate phenomena, or as part of the same continuum of modality. The interrelationships between these various forms will be much clarified by oppositional analysis.

### 3. Distributions.

The concept of distribution of terms, which was developed in the context of Aristotelean logic, can be broadened to apply to modality. It has been found a useful doctrine, often aided by pictorial representations, for understanding the workings of arguments, and its utility would be increased. We defined a term as being distributive if, as a result of the structure of the proposition, it was found to be referring to all the instances of the class concerned; otherwise, the term was being used undistributively. Now, this concerns quantity, the extensional type of modality, and could be called extensional distribution.

We could then by analogy consider a term as naturally distributive if it was being referred to under all conditions, and naturally undistributive if the reference was dependent on circumstance. Likewise, temporal distribution would indicate reference to all or some of the times concerning a term. The following properties can then be formulated.

- Whatever the polarity, concerning the subject: universals are extensionally distributive; but particulars are not; necessaries are naturally distributive, but not potentials; constants are temporally distributive, but not so temporaries.
- The predicates of negatives are distributive in all three senses, whereas those of affirmatives are in all senses undistributive.

Thus, a given proposition may be distributive of this or that term in one sense, but not in another. In this way, we can explain why a certain inference is possible, or why another is not. This is not a very important doctrine, but, as already stated, a useful tool.

## 14. MODAL OPPOSITIONS AND EDUCTIONS.

We have already encountered the oppositions of actuals or momentaries in classical logic. There is subalternation from **A** to **R**, to **I**; and from **E** to **G**, to **O**. **A** and **E**, **A** and **G**, **R** and **E**, are pairs of contraries; **A** and **O**, **I** and **E**, **R** and **G**, are pairs of contradictories, **R** and **O**, **I** and **G**, **I** and **O**, are pairs of subcontraries. These relationships were shown to proceed from analysis of the forms’ meanings and application of the laws of thought. In the wider context of modal logic, we are concerned with the oppositions of, not only these six forms, but 24 more.

Remember that subalternation is one-way implication, contradictories can neither be both true nor both false, contraries cannot be both true but may be both false, subcontraries may be both true but cannot be both false, and unconnecteds do not affect each others’ truth or falsehood.

### 1. Quantification of Oppositions.

At this point, I would like show how, given a certain oppositional relation to exist between two singular propositions (**s1**, **s2**), referring to the same instance of the same subject-concept, we can systematically predict the oppositions involving one or two of the corresponding universal (**u1**, **u2**) and particular (**p1**, **p2**) forms. This doctrine may be called quantification of oppositions, meaning more precisely opposition of quantified forms. It allows us to introduce quantity into basic figures of opposition, such as that between the categories or types of modality which will presented in the next sections. Consider the following general-model figure of opposition.

Grant that we already know the subalternations, labeled (1), to be true, since universality includes singularity, which includes particularity. For any given opposition between singulars, labeled (2) horizontal, we need to discover the remaining lines of oppositions, namely (2) diagonal, (3), and (4). The following results are obtained.

If the singulars are *implicants*, then all horizontal lines signify implicance, and all diagonals signify subalternation, downward. Proof for the horizontals: since it is given any pair of singular forms **s1**, **s2** mutually imply each other, then any full or partial enumeration of such pairs, as in **u1**, **u2**, or **p1**, **p2**, will likewise mutually imply each other, provided the extensions involved are the same. For the diagonals: since **u1** implies **s1**, and **s1** implies **s2**, then **u1** implies **s2**. Since **u1**, **s1**, imply **s2**, and **s2** implies **p2**, then they also imply **p2**. Likewise, **u2**, **s2** can be shown to imply **s1**, **p1**.

If the singulars are *subalternative*, left implying right, then all horizontal or left down to right diagonals signify subalternation in that direction, and all right down to left diagonals signify unconnectedness. Proof: similar to previous case, though the relations involved here are unidirectional. Unconnectedness, of course, applies when no more finite opposition can be established.

If the singulars are *contradictory*, then all lines labeled (2) signify contradiction, all lines labeled (3) contrariety, all lines labeled (4) subcontrariety. Proof for the upper square: given that **s1** and **s2** cannot both be true, then any enumerations which include them both, such as **u1** + **s2**, **s1** + **u2**, or **u1** + **u2**, cannot be both true (so, for instance, if **u1**=T, then **u2**=F; i.e. if **u1**, then not-**u2**). Proof for the lower square: given that **s1** and **s2 **cannot both be false, then any enumerations which exclude them both such as not-**p1** + not-**s2**, not-**s1** + not-**p2**, or not-**p1** + not-**p2**, cannot both be true (so, for instance, if not-**p1** = true, then not-**p2** = false; i.e. if not-**p1**, then **p2**). So far, we have proven the claimed contrarieties and subcontrarieties. But what of the contradictions of **u1** + **p2**, or **p1** + **u2**? If we affirm such a pair, we do not necessarily thereby affirm a specific **s1** + **s2** pair true, but we do imply that some unspecified pair(s) of **s1** and **s2**, referring to one and the same individual, would be posited together; this shows the incompatibility of **u1** + **p2**, or **p1** + **u2**. Likewise, for the incompatibility of not-**u1** + not-**p2**, or not-**p1** + not-**u2**, there is bound to be some unspecified case(s) of not-**s1** + not-**s2** subsumed, against our given information.

If the singulars are *contrary*, then all lines labeled (2) or (3) signify contrariety, and all lines labeled (4) unconnectedness. Proof: see the relevant (‘not both true’) parts of the arguments above for contradiction.

If the singulars are *subcontrary*, then all lines labeled (2) or (4) signify subcontrariety, and all lines labeled (3) unconnectedness. Proof: see the relevant (‘not both false’) parts of the arguments above for contradiction.

These general rules of opposition can now be used in any context, saving us from having to deal with each case of quantification anew.

### 2. Basic Intramodal Oppositions.

The following diagram concerns singular propositions only, and is designed to illustrate the relationships of the different categories of modality, whether of the natural type or of the temporal type (each type separately).

The above is equivalent to the figure of oppositions of the six quantities of Aristotelean propositions, and may be established by similar argument. The vertical, downward subalternations proceed from the definitions of the concepts involved; ‘all’ the circumstances or times includes any ‘this one’ we pick, and any specific ‘this one’ implies ‘some’ unspecified number.

The horizontal contradiction is simply the axiomatic presence and absence incompatibility. The diagonal contradictions between necessity and unnecessity, or impossibility and possibility, follow, on the basis that there would otherwise be individual circumstance(s) or time(s) which contained both presence and absence, or neither.

For the rest, the proofs are very mechanical consequences of the above. For example, using the symbols **n**, **a**, **p**, with subscripts **+** and **–**, we can say: **n+** implies **a+** implies not{**a-**}, whereas not{**n+**} does not imply not{**a+**}, nor therefore **a-**, so that **n+** and **a-** are contrary; or again, not{**p+**} implies not{**a+**} implies **a-**, whereas **p+** does not imply **a+**, nor therefore not{**a-**}, so that **p+** and **a-** are subcontraries.

With regard to contingency; being defined as the sum of possibility and unnecessity, it subalternates **p+** and **p-**, and is contrary to **n+** and **n-**. Incontingency, its negation, therefore means either necessity or impossibility, and is subalternated by **n+** and **n-**, and subcontrary to **p+** and **p-**. Contingency and incontingency are both oppositionally unconnected to presence and absence. These relationships could be represented in a wedge-shaped diagram.

As for the oppositions of probability forms See **Appendix 2** for remarks on this topic.

Furthermore, necessity implies probability, and impossibility implies probability-not. Improbability implies unnecessity, and improbability-not implies possibility. It follows that high or low probability are contrary to necessity of opposite polarity, and subcontrary to possibility of opposite polarity. These wider relations are easily established.

We can view necessity as the highest form of probability. Also, probability, whether high or low, is merely a more defined form of possibility. If we express a more specific proportion of cases (e.g. 75% or 33%), we obtain sub-categories of probability. Lastly, of course, none of the probability forms are connected oppositionally to the presence/absence forms. Nevertheless, the whole idea of probability thinking is to try and predict the chances of realization of presence or absence.

### 3. Quantified Intramodal Oppositions.

If we take each of the oppositional relations between singulars of natural modality and quantify them with the general rules, we obtain the following table of opposition for all the forms of natural modality.

Key to symbols: | Unconnected | [ | |

Implicant | z | Contradictory | M |

Subalternating | È | Contrary | Ê |

Subalternated | Ã | Subcontrary | Ç |

An | A | Ap | Rn | R | Rp | In | I | Ip | En | E | Ep | Gn | G | Gp | On | O | Op | |

An | z | È | È | È | È | È | È | È | È | Ê | Ê | Ê | Ê | Ê | Ê | Ê | Ê | M |

A | Ã | z | È | [ | È | È | [ | È | È | Ê | Ê | [ | Ê | Ê | [ | Ê | M | Ç |

Ap | Ã | Ã | z | [ | [ | È | [ | [ | È | Ê | [ | [ | Ê | [ | [ | M | Ç | Ç |

Rn | Ã | [ | [ | z | È | È | È | È | È | Ê | Ê | Ê | Ê | Ê | M | [ | [ | Ç |

R | Ã | Ã | [ | Ã | z | È | [ | È | È | Ê | Ê | [ | Ê | M | Ç | [ | Ç | Ç |

Rp | Ã | Ã | Ã | Ã | Ã | z | [ | [ | È | Ê | [ | [ | M | Ç | Ç | Ç | Ç | Ç |

In | Ã | [ | [ | Ã | [ | [ | z | È | È | Ê | Ê | M | [ | [ | Ç | [ | [ | Ç |

I | Ã | Ã | [ | Ã | Ã | [ | Ã | z | È | Ê | M | Ç | [ | Ç | Ç | [ | Ç | Ç |

Ip | Ã | Ã | Ã | Ã | Ã | Ã | Ã | Ã | z | M | Ç | Ç | Ç | Ç | Ç | Ç | Ç | Ç |

En | Ê | Ê | Ê | Ê | Ê | Ê | Ê | Ê | M | z | È | È | È | È | È | È | È | È |

E | Ê | Ê | [ | Ê | Ê | [ | Ê | M | Ç | Ã | z | È | [ | È | È | [ | È | È |

Ep | Ê | [ | [ | Ê | [ | [ | M | Ç | Ç | Ã | Ã | z | [ | [ | È | [ | [ | È |

Gn | Ê | Ê | Ê | Ê | Ê | M | [ | [ | Ç | Ã | [ | [ | z | È | È | È | È | È |

G | Ê | Ê | [ | Ê | M | Ç | [ | Ç | Ç | Ã | Ã | [ | Ã | z | È | [ | È | È |

Gp | Ê | [ | [ | M | Ç | Ç | Ç | Ç | Ç | Ã | Ã | Ã | Ã | Ã | z | [ | [ | È |

On | Ê | Ê | M | [ | [ | Ç | [ | [ | Ç | Ã | [ | [ | Ã | [ | [ | z | È | È |

O | Ê | M | Ç | [ | Ç | Ç | [ | Ç | Ç | Ã | Ã | [ | Ã | Ã | [ | Ã | z | È |

Op | M | Ç | Ç | Ç | Ç | Ç | Ç | Ç | Ç | Ã | Ã | Ã | Ã | Ã | Ã | Ã | Ã | z |

These relationships may be clarified by use of a truth-table. In the table below, given the truth of a proposition listed under column heading **T**, or the falsehood of one under **F**, we see the reactions, along the same row, of all other propositions, listed as column headings. This data follows from the preceding table.

(key: **T** = true, **F = **false, **. = **undetermined.)

T | An | A | Ap | Rn | R | Rp | In | I | Ip | En | E | Ep | Gn | G | Gp | On | O | Op | F |

An | T | T | T | T | T | T | T | T | T | F | F | F | F | F | F | F | F | F | Op |

A | . | T | T | . | T | T | . | T | T | F | F | . | F | F | . | F | F | . | O |

Ap | . | . | T | . | . | T | . | . | T | F | . | . | F | . | . | F | . | . | On |

Rn | . | . | . | T | T | T | T | T | T | F | F | F | F | F | F | . | . | . | Gp |

R | . | . | . | . | T | T | . | T | T | F | F | . | F | F | . | . | . | . | G |

Rp | . | . | . | . | . | T | . | . | T | F | . | . | F | . | . | . | . | . | Gn |

In | . | . | . | . | . | . | T | T | T | F | F | F | . | . | . | . | . | . | Ep |

I | . | . | . | . | . | . | . | T | T | F | F | . | . | . | . | . | . | . | E |

Ip | . | . | . | . | . | . | . | . | T | F | . | . | . | . | . | . | . | . | En |

En | F | F | F | F | F | F | F | F | F | T | T | T | T | T | T | T | T | T | Ip |

E | F | F | . | F | F | . | F | F | . | . | T | T | . | T | T | . | T | T | I |

Ep | F | . | . | F | . | . | F | . | . | . | . | T | . | . | T | . | . | T | In |

Gn | F | F | F | F | F | F | . | . | . | . | . | . | T | T | T | T | T | T | Rp |

G | F | F | . | F | F | . | . | . | . | . | . | . | . | T | T | . | T | T | R |

Gp | F | . | . | F | . | . | . | . | . | . | . | . | . | . | T | . | . | T | Rn |

On | F | F | F | . | . | . | . | . | . | . | . | . | . | . | . | T | T | T | Ap |

O | F | F | . | . | . | . | . | . | . | . | . | . | . | . | . | . | T | T | A |

Op | F | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | T | An |

Needless to say, the easiest way to visualize and transmit all the above information is by means of a figure of opposition. However, since in this context the required diagram is three-dimensional, it is rather difficult to present on paper. Below is a sketch of it, but without the various lines of opposition. Note that the shaded planes have already been presented earlier, with all their lines of opposition shown. The reader can work out the remaining planes, with reference to the above two tables, or more radically to the principles of ‘quantification of oppositions’ developed earlier.

Identical results are obtainable for temporal modality, substituting **c** for **n**, and **t **for **p**, throughout.

### 4. Intermodal Oppositions.

Lastly, needing elucidation, is the inter-opposition of natural and temporal modalities. The following diagram shows the continuum of modality, including both natural and temporal types together. Parts of this diagram have already been presented, when we dealt with each modal type separately. But it is interesting to have an overview, anyway.

This diagram concerns singulars. We know from our analysis of modality that **n** implies **c**, which implies **a** or **m**, which implies **t**, which implies **p**, for either polarity; that is, the illustrated subalternations proceed from the meanings of the concepts involved. From these, and the already established intramodal oppositions, it is easy to infer the contrariety between **n** and **c**, or **n** and **t**, forms of opposite polarity (upper diagonals), and the subcontrariety between **c **and **p**, or **t **and **p**, forms of opposite polarity (lower diagonals).

These relationships between singulars can now be quantified by reference to the general rules of opposition, and the results tabulated as follows.

Key to symbols: | Unconnected | [ | |

Subalternating | È | Contrary | Ê |

Subalternated | Ã | Subcontrary | Ç |

Ac | At | Rc | Rt | Ic | It | Ec | Et | Gc | Gt | Oc | Ot | |

An | È | È | È | È | È | È | Ê | Ê | Ê | Ê | Ê | Ê |

Rn | [ | [ | È | È | È | È | Ê | Ê | Ê | Ê | [ | [ |

In | [ | [ | [ | [ | È | È | Ê | Ê | [ | [ | [ | [ |

Ap | Ã | Ã | [ | [ | [ | [ | [ | [ | [ | [ | Ç | Ç |

Rp | Ã | Ã | Ã | Ã | [ | [ | [ | [ | Ç | Ç | Ç | Ç |

Ip | Ã | Ã | Ã | Ã | Ã | Ã | Ç | Ç | Ç | Ç | Ç | Ç |

En | Ê | Ê | Ê | Ê | Ê | Ê | È | È | È | È | È | È |

Gn | Ê | Ê | Ê | Ê | [ | [ | [ | [ | È | È | È | È |

On | Ê | Ê | [ | [ | [ | [ | [ | [ | [ | [ | È | È |

Ep | [ | [ | [ | [ | Ç | Ç | Ã | Ã | [ | [ | [ | [ |

Gp | [ | [ | Ç | Ç | Ç | Ç | Ã | Ã | Ã | Ã | [ | [ |

Op | Ç | Ç | Ç | Ç | Ç | Ç | Ã | Ã | Ã | Ã | Ã | Ã |

These relationships may be clarified by use of a truth-table. This data follows from the preceding table.

(key: **T** = true, **F = **false, **. = **undetermined.)

T | Ac | At | Rc | Rt | Ic | It | Ec | Et | Gc | Gt | Oc | Ot | F |

An | T | T | T | T | T | T | F | F | F | F | F | F | Op |

Rn | . | . | T | T | T | T | F | F | F | F | . | . | Gp |

In | . | . | . | . | T | T | F | F | . | . | . | . | Ep |

Ap | . | . | . | . | . | . | . | . | . | . | . | . | On |

Rp | . | . | . | . | . | . | . | . | . | . | . | . | Gn |

Ip | . | . | . | . | . | . | . | . | . | . | . | . | En |

En | F | F | F | F | F | F | T | T | T | T | T | T | Ip |

Gn | F | F | F | F | . | . | . | . | T | T | T | T | Rp |

On | F | F | . | . | . | . | . | . | . | . | T | T | Ap |

Ep | . | . | . | . | . | . | . | . | . | . | . | . | In |

Gp | . | . | . | . | . | . | . | . | . | . | . | . | Rn |

Op | . | . | . | . | . | . | . | . | . | . | . | . | An |

In the table above, given the truth of a proposition listed under column heading T, or the falsehood of one under F, we see the reactions, along the same row, of all other propositions, listed as column headings.

### 5. Eductions.

The following is a list of the eductions possible from propositions with natural modality. The methods of validation used for these are similar to those developed for Aristotelean forms. That is, conceptual analyses and appeal to the laws of thought, in the cases of obversion and conversion; and reduction to these first two process, in the other cases.

(S-P to S-nonP).*Obversion*

S must be P implies S cannot be nonP; S cannot be P implies S must be nonP; S can be P implies S can not-be nonP; S can not-be P implies S can be nonP. These are true irrespective of the quantity (all/this/ some) involved; and the obverse has in all cases the same quantity as the obvertend. These results follow from the definitions of the concepts involved and the law of contradiction.

(S-P to P-S).*Conversion*

Affirmatives, be they necessary or potential, general or particular, all convert to a particular potential, Some P can be S, but no better. In the case of negatives, only No S can be P (**En**) is convertible, and that fully to No P can be S; **Ep**, **Gn**, **Gp**, **On**, **Op** are not convertible. These results can be established by consideration of the subsumptions of circumstance involved.

(S-P to P-nonS).*Obverted Conversion*

This process is applicable only to convertibles, which are then all obvertible. Thus, affirmatives all yield Some P can not-be nonS; and **En** yields All P must be nonS.

- Conversion by Negation (S-P to nonP-S).

This is obversion, followed by conversion. Thus, all originally negative propositions can be converted by negation, to yield Some nonP can be S. But of originally affirmative propositions, only All S must be P (**An**) can be so processed, to yield No nonP can be S; **Ap**, **Rn**, **Rp**, **In**, **Ip** are not convertible by negation.

(S-P to nonP-nonS).*Contraposition*

This requires conversion by negation, followed by obversion. Therefore, all negatives are contraposable, and that to Some nonP can not-be nonS. Whereas, in the case of affirmatives, only **An** can be so processed, yielding All nonP must be nonS.

(S-P to nonS-nonP).*Inversion*

Of affirmatives, only **An** can be so treated, by contraposing then converting it, to obtain Some nonS can be nonP. Of negatives, only **En** is invertible, by converting then contraposing it, with the result Some nonS can not-be nonP.

(S-P to nonS-P).*Obverted Inversion*

This being inversion followed by obversion is applicable only to universal necessaries, **An** yielding Some nonS can not-be P, and **En** yielding Some nonS can be P.

We note, in conclusion, that only **An** and **En** (as well as **A** and **E**) can be subjected to all six of these processes.

Similar results can easily be established regarding propositions with temporal modality.

## 15. MAIN MODAL SYLLOGISMS.

### 1. Valid Modes.

We called a mood of syllogism, a combination of formally fully specified premises and conclusion in a given figure (e.g. **1/AAA**). We will call mode, any combination of symbols which does not by itself fully specify a syllogistic form, but which abstracts a specific aspect of such, in a given figure (e.g. **1/uuu**). It was shown, in Aristotelean logic, that the primary valid modes of polarity and quantity are as in the following table.

Figure | First | Second | Third | Fourth |

Polarities | +++ | +– | +++ | -+- |

-+- | -+- | -+- | ||

Quantities | uuu | uuu | upp | upp |

upp | upp | pup | ||

uss | uss | ssp |

We can at the outset, prior to systematic validation, predict that the valid modes for natural and temporal modality will be the following, by analogy to the results obtained for extensional modality.

Figure | First | Second | Third | Fourth |

Natural Modality | aaa | aaa | aaa | aaa |

nnn | nnn | npp | npp | |

npp | npp | pnp | ||

Temporal Modality | mmm | mmm | mmm | mmm |

ccc | ccc | ctt | ctt | |

ctt | ctt | tct |

Note the slight difference between quantity modes and modality modes. The modes **aaa** and **mmm** are valid in all figures, whereas **sss** is not (**3/ssp** is exceptional, and anyway does not yield an **s** conclusion). This is due to modality standing outside the relationship between the terms, whereas quantity concerns the subject more directly.

Natural and temporal modality being essentially analogous, we can concentrate on developing the theory of syllogism for the former, and then generalize the results to the latter. Apart from the above we will need to investigate the valid modes of mixed, natural and temporal syllogism.

In the broadest sense, of course, all syllogism is modal. But for the sake of convenience we will often find it useful to call nonmodal, syllogism both of whose premises are actual or momentary (**aaa** or **mmm**); so that syllogism with one or both premises necessary or possible, can be called modal. Aristotelean logic can then be said to have concerned nonmodal syllogism, while this thesis concerns modal syllogism.

### 2. Valid Moods.

If we combine together the valid modes of polarity and quantity for a given valid mode of modality, in each of the figures, we should obtain the valid moods of syllogism. Let us now do so, using the valid natural modality modes, to develop a full list of natural syllogism, including both the nonmodal (Aristotle’s achievement) and the modal (the new contribution). This is the principal goal of our whole formal research. The notation system used for this, consists in applying modality subscripts (**n**, **p**, **a**) to the six standard symbols, **A**, **E**, **I**, **O**, **R**, **G**.

We see in the list below that only 56 primary moods emerge as logically valid, not counting derivative syllogism. There are 18 valid moods in each of the first three figures, and 2 in the fourth. Since 19 of the above moods are actual, only 37 are original forms.

Mode/Figure | First | Second | Third | Fourth |

aaa | AAA | AEE | AII | EIO |

EAE | EAE | EIO | ||

AII | AOO | IAI | ||

EIO | EIO | OAO | ||

ARR | AGG | RRI | ||

ERG | ERG | GRO | ||

nnn | AnAnAn | AnEnEn | ||

EnAnEn | EnAnEn | |||

AnInIn | AnOnOn | |||

EnInOn | EnInOn | |||

AnRnRn | AnGnGn | |||

EnRnGn | EnRnGn | |||

npp | AnApAp | AnEpEp | AnIpIp | EnIpOp |

EnApEp | EnApEp | EnIpOp | ||

AnIpIp | AnOpOp | InApIp | ||

EnIpOp | EnIpOp | OnApOp | ||

AnRpRp | AnGpGp | RnRpIp | ||

EnRpGp | EnRpGp | GnRpOp | ||

pnp | ApInIp | |||

EpInOp | ||||

IpAnIp | ||||

OpAnOp | ||||

RpRnIp | ||||

GpRnOp |

We will now present these 37 valuable new forms in full, for the record.

*First Figure.*Form: M-P, S-M, S-P.

AnAnAn | EnAnEn |

All M must be P | No M can be P |

All S must be M | All S must be M |

All S must be P | No S can be P |

AnInIn | EnInOn |

All M must be P | No M can be P |

Some S must be M | Some S must be M |

Some S must be P | Some S cannot be P |

AnRnRn | EnRnGn |

All M must be P | No M can be P |

This S must be M | This S must be M |

This S must be P | This S cannot be P |

AnApAp | EnApEp |

All M must be P | No M can be P |

All S can be M | All S can be M |

All S can be P | All S can not-be P |

AnIpIp | EnIpOp |

All M must be P | No M can be P |

Some S can be M | Some S can be M |

Some S can be P | Some S can not-be P |

AnRpRp | EnRpGp |

All M must be P | No M can be P |

This S can be M | This S can be M |

This S can be P | This S can not-be P |

*Second Figure.*Form: P-M, S-M, S-P.

AnEnEn | EnAnEn |

All P must be M | No P can be M |

No S can be M | All S must be M |

No S can be P | No S can be P |

AnOnOn | EnInOn |

All P must be M | No P can be M |

Some S cannot be M | Some S must be M |

Some S cannot be P | Some S cannot be P |

AnGnGn | EnRnGn |

All P must be M | No P can be M |

This S cannot be M | This S must be M |

This S cannot be P | This S cannot be P |

AnEpEp | EnApEp |

All P must be M | No P can be M |

All S can not-be M | All S can be M |

All S can not-be P | All S can not-be P |

AnOpOp | EnIpOp |

All P must be M | No P can be M |

Some S can not-be M | Some S can be M |

Some S can not-be P | Some S can not-be P |

AnGpGp | EnRpGp |

All P must be M | No P can be M |

This S can not-be M | This S can be M |

This S can not-be P | This S can not-be P |

*Third Figure.*Form: M-P, M-S, S-P.

AnIpIp | EnIpOp |

All M must be P | No M can be P |

Some M can be S | Some M can be S |

Some S can be P | Some S can not-be P |

InApIp | OnApOp |

Some M must be P | Some M cannot be P |

All M can be S | All M can be S |

Some S can be P | Some S can not-be P |

RnRpIp | GnRpOp |

This M must be P | This M cannot be P |

This M can be S | This M can be S |

Some S can be P | Some S can not-be P |

ApInIp | EpInOp |

All M can be P | All M can not-be P |

Some M must be S | Some M must be S |

Some S can be P | Some S can not-be P |

IpAnIp | OpAnOp |

Some M can be P | Some M can not-be P |

All M must be S | All M must be S |

Some S can be P | Some S can not-be P |

RpRnIp | GpRnOp |

This M can be P | This M can not-be P |

This M must be S | This M must be S |

Some S can be P | Some S can not-be P |

*Fourth Figure.*Form: P-M, M-S, S-P.

EnIpOp | |

No P can be M | |

Some M can be S | |

Some S can not-be P |

A similar listing would be obtained for temporal syllogism. Secondary modes, valid derivatively and of lesser significance, will discussed later. Mixed syllogism will also be dealt with separately.

### 3. Validations.

We have seen that each figure has a method of validation most appropriate to it. Aristotelean syllogism being identical with our nonmodal (actual or momentary) forms, the task of validation of modal syllogisms is much facilitated. Similar approaches can be used with regard to modal syllogism; and moreover we can appeal, if we need to, to correct nonmodal argument, in the process. The following description of validation and rejection processes for natural modal syllogism, can all be repeated for temporal modes.

- First figure.

We previously defended Aristotle’s valid moods in the first figure, on the basis of the principle that one must mean what one says. Some phenomena have been observed, perceptually and/or conceptually; within a complex of appearances, certain aspects have been distinguished; names have been assigned to their various components; thereby, a framework is established which we are logically required to adhere to; such recognition guarantees the accord between thought and reality (that is, long-term, overall, appearance.)

Now, granting the six valid actual moods of this figure, the corresponding moods in the modes nnn and npp, are to be demonstrated valid.

A necessary proposition ‘X must be Y’ may be viewed as merely a collection of actual propositions ‘In circumstance 1, X is Y’, ‘in circumstance 2, X is Y’, ‘in circumstance 3, X is Y’, and so on; it says ‘Whatever the surrounding circumstances, X is Y’. Likewise, a potential proposition may be viewed as a partial enumeration of circumstances in which the stated relationship of X and Y is actualized. An actual proposition indicates a specific single circumstance in which the event occurs.

Now, let us consider a group of three propositions which, in the actual mode **aaa**, constitute a valid syllogism, e.g. **AAA**. In the case of **AnAnAn**, the **nnn** equivalent, we can predict that in each and any circumstance we may select, we will find the two premises **AA** actual, and yielding the conclusion **A**. It follows that, given the premises **AnAn**, we can say, ‘Whatever the circumstances, the conclusion **A **occurs’; which means that the **An** conclusion is valid. Thus, any mood valid in **aaa** mode is equally valid in **nnn** mode. With similar reasoning, we can demonstrate the validity of **npp**, since the ‘all circumstances’ in the major premise includes the ‘some circumstances’ in the minor premise, which are in turn posited as framing the conclusion, too.

With regard to invalidation of invalid modal modes. Although the onus of proof is on anyone who wants to defend them, as it were, it is important to give special attention to the mode **pnp**, which might at first sight seem reasonable. We might think that the ‘in all circumstances’ of the minor premise, includes the ‘some circumstances’ of the major premise, so that a potential conclusion can be drawn. However, in any modal proposition, the circumstances under consideration apply primarily to the subject of the proposition. When we refer to all the circumstances surrounding the subject’s existence, we do not claim these to be the only circumstances which can coincide with the predicate’s existence, or any other subject’s existence.

In the case of **npp**, the ‘some circumstances’ under consideration, are implied for the middle term since the minor premise is affirmative, and concern the same subject in minor premise and conclusion. But in the case of **pnp**, the specific conditions under which the middle term is addressed in the major premise do not necessarily coincide with any condition concerning the minor term in the minor premise, and so cannot be transferred to it in the conclusion. The change of subject being qualified makes the process illicit. The invalidity of **ppp** is all the more obvious, since it has no misleading unconditional premise. Thus the analogy of valid modal modes to valid quantitative modes is complete.

What of **aaa**, which is posited as valid, although we reject **sss**? Here too, one could argue that the unitary circumstance referred to by each of the two premises may not coincide, since their subjects differ. In truth, this argument against **aaa** is justified, and serves to warn us that the **aaa** mode is valid only on the condition that we know the unitary circumstance involved in the two premises to be one and the same. However, actual propositions by definition concern an ostensible circumstance (which though left tacit is understood). So **aaa** is a valid mode, when we know the *sous-entendu* circumstance to be common.

Although we might attach a similar proviso for the validity of **sss**, we in fact cannot, because of a structural difference between actuality and singularity. The ‘this’ in a singular proposition is more firmly attached to the subject; it identifies the subject itself, and not a circumstance surrounding it. Comparing one subject’s ‘this’ to another’s is nonsensical, as is the idea of moving our mental finger from one to the other; because all they have in common is ‘this-ness’, not this one ‘this-ness’. In actuals, on the other hand, the focus of the ‘this’ is a circumstance standing outside the subject of the proposition, though bounded by its existence; it is not the subject as such which is focused on by that ‘this’. It follows that, here, the two ‘this’ occurrences in the premises may be compared, and the specification transferred to their conclusion.

- Other figures.

The valid moods of the second figure are established by reduction ad absurdum through the first figure. We attach the denial of the conclusion to the major premise, and see that the result would be denial of our original minor premise. Thus, **2/nnn** is reduced through **1/npp**, and **2/npp** follows from **1/nnn**; always of course provided the underlying actual mood has valid polarity and quantity properties. Invalid modes in the second figure are dealt with similarly, by showing that the combination of the major premise with the suggested conclusion results in a contradiction or a non-sequitur, through the first figure.

The third figure modes could be reduced ad absurdum to the first figure for systematic validation; the denial of the conclusion would be combined with the minor premise and result in denial of the original major premise. Rejection of invalid modes could be achieved similarly. However, exposition reflects more accurately the way we deal with this figure in practice. We can reproduce our arguments for the first figure, showing that the circumstances in the premises intersect, and are passed on to the conclusion. This is facilitated by the fact that, in this figure, the two premises have the same subject (the middle term).

For the fourth figure, direct reduction is the appropriate approach. There is, furthermore, only one primary valid mood to consider. The premises **EnIp** are both converted, allowing us to process them in the first figure, and obtain the desired **Op** conclusion. Other validations, and invalidations, are likewise easy to deal with.

## 16. OTHER MODAL SYLLOGISMS.

### 1. Secondary Modes.

Concerning subaltern valid syllogism. Any combination of premises not included in the above list of primary valid moods, but implying one which is included, can obviously be listed as a derivatively valid mood. Likewise, propositions implied by one of the conclusions to the valid moods form subaltern moods with the same premises. There are also fourth figure moods to take into account, which though valid are insignificant.

We have two stages to consider. To begin with, applying the primary valid modal modes to the secondary valid actual (or momentary) moods. And then, listing the secondary valid modal modes, which can be applied to both primary and secondary valid actual (or momentary) moods. These two lists together make up the full list of secondary valid modal moods. We shall for a start deal with the first three figures, before turning our attention to the less regular fourth figure.

Figure | First | Second | Third |

Quantity | uup | uup | uup |

usp | usp | usp | |

sup | |||

Natural Modality | nna | nna | nna |

naa | naa | naa | |

ana | ana | ana | |

nnp | nnp | nnp | |

nap | nap | nap | |

anp | anp | anp | |

aap | aap | aap |

Similarly for Temporal Modality.

The subaltern quantity modes were implicit in the list of secondary moods established previously, when considering Aristotelean syllogism. Note that the subaltern modality modes are the same in the three figures. They could be mostly predicted by analogy; but let us derive them quickly from the primary modes, at least in the case of natural modality. In these three figures, **nna**, **naa** or **ana** are derived from **aaa**, whose premises theirs imply; and **nnp**, **nap**, **anp**, and **aap** follow from these by virtue of their subaltern conclusions. Temporal modality modes can similarly be dealt with.

Now let us count the number of subaltern moods which we can expect to encounter in these three figures. The full list will not be drawn up, being too large and relatively unimportant; the numbers are interesting, however, as will be seen. Each of these figures has 2 valid polarity modes. These are combinable with 3 valid primary quantity modes in each figure; plus 2 valid subaltern quantity modes each, in the first and second figure, and 3 of them, in the third. These are in turn combinable, in any of these figures, with 3 valid primary natural modality modes, plus 7 valid subaltern natural modality modes.

We thus obtain, in the first figure, a total of 2X(3+2)X(3+7) = 100 valid moods. In the second figure, we have the same results. In the third figure, our total is 2X(3+3)X(3+7) = 120. In each of the three figures, there are 2X3X3 = 18 primary moods, and the rest are secondary. Identical results are of course obtainable for temporal modality.

Now, whereas in the first three figures any valid polarity mode can be correctly combined with any valid quantity or modality modes, in the fourth figure only specific combinations are permissible. The fourth figure, as earlier indicated, lacks uniformity, and seems to in effect contain three different sets of valid moods, which we may call **4a**, **4b**, **4c**, for the sake of convenience.

Figure **4a**, whose polarity mode is **-+-**, is the significant one; and we saw that it contained two valid primary moods, **EIO** and **EnIpOp**. Figures **4b** and **4c** are insignificant, being mere derivatives of the first figure by transposition of premises and conversion of conclusion. The prototype of **4b** is **AEE**, with polarity **+–**; and that of **4c** is **IAI **with polarity **+++**. The corresponding modal forms are **AnEnEn** and **IpAnIp**. We shall now list the implicit quantity and modality modes of these forms and their subalterns.

Sub-figure | 4a. | 4b. | 4c. |

Polarity | (-+-) | +– | +++ |

Quantity | (upp) | uuu | pup |

uup | uup | uup | |

usp | sup | ||

Natural Modality | (npp) | nnn | pnp |

(aaa) | aaa | aaa | |

nna | nna | nna | |

naa | naa | naa | |

ana | ana | ana | |

nnp | nnp | nnp | |

nap | nap | nap | |

anp | anp | anp | |

aap | aap | aap |

Similarly for Temporal Modality.

The primary valid modes of the fourth figure are included in the above table in brackets to facilitate reading of the sources of their subaltern modes; these are in **4a**. With regard to **4b** and **4c**, the first quantity and the first two modalities listed for each are the sources of the others, but all are viewed as secondary, as well as the corresponding polarities.

Let us now count the number of moods implied valid. For **4a**, 1X(1+2)X(2+7) = 27, of which only two are primary. For **4b**, 1X2X9 = 18, and for **4c**, 1X3X9 = 27; all these being secondary in one way or another. Thus, figure four consists of two primary valid moods, and another 70 secondary valid moods. This is said for natural modality, and can be repeated for temporal modality, as usual.

Thus, to conclude this section, there are a total of 100+100+120+72 = 392 valid moods in each type of modality, of which 56 are primary, and 336 are secondary. The full list of secondary moods is easily developed given the above lists of modes.

### 2. Mixed Modes.

To systematically cover all possibilities of combination, we now need to investigate mixed syllogism, that is, syllogism involving a mixture of natural modalities (**n**, **p**, **a**) and temporal modalities (**c**, **t**, **m**), in their premises and/or conclusions. It will be seen that valid such combinations are entirely derivable from syllogisms previously encountered under this or that type of modality separately, so that we can say that mixed modes are in fact all secondary.

Analysis shows that the mixed modes we seek are all derivable from the primary modes of temporal modality of each figure. We have seen that these are: in figures **1** and **2**, **mmm**, **ccc**, **ctt**; in figure **3**, **mmm**, **ctt**, **tct**; in figure **4a**, **mmm**, **ctt**; in figure **4b**, **mmm**, **ccc**; in figure **4c**, **mmm**, **tct**. If we analyze the subalternations (through premises and/or conclusion) possible for each of these four modes, we get the following results. From **mmm**: **nnt**, **ncm**, **nct**, **ncp**, **cnm**, **cnt**, **cnp**, **nmt**, **mnt**, **ccp**, **cmp**, **mcp** (12 modes, in common to all figures). From **ccc**: **nnc**, **ncc**, **cnc** (3 modes, applicable to figures **1**, **2**, and **4**b). From **ctt**: **ntt**, **ntp**, **ctp** (3 modes, applicable to figures **1**, **2**, **3**, and **4a**). From **tct**: **tnt**, **tnp**, **tcp** (3 modes, applicable to figures **3** and **4c**).

We thus obtain, for the first three figures, 18 valid mixed modes each; for each subset of figure four, 15 valid modes. Other mixed modes are found not to follow from any valid nonmixed modes, or to be only apparently mixed because of the different symbolization of actual and momentary propositions.

The valid mixed modes may each be combined with the polarities and quantities existing in their respective figures. It follows that the number of valid mixed moods are as follows: 2X5X18 = 180, for each of figures **1** and **2**; 2X6X18 = 216 for figure **3**; and 3X15 = 45 for figure **4a**, 2X15 = 30 for figure **4b**, 3X15 = 45 for figure **4c**. The total number of valid mixed moods is therefore 696. These valid moods, to repeat, count as secondary. Most may be practically useless, but they had indicated for completeness.

The above listed mixed modes, you will observe, do not include combinations involving the a-form and certain combinations involving the **m**-form. The reason for this is simply that the actual and momentary forms are essentially identical, although they appear different. Their distinction is one of perspective, and a verbal one sometimes, but their logical value is the same. It follows, not only that **aaa** and **mmm** are equivalent, but also that other combinations of **a** and **m**, namely **amm**, **ama**, **mam**, **maa**, are all identical. Furthermore, combinations involving modal propositions together with one or both these, are also redundant; this includes groups such as **nnm**, **cca**, **ncm**, **nca**, **nmm**, **nma**, **nam**, and so on.

We must of course avoid the duplicate listing as mixed modes, of syllogism which have already been presented as nonmixed modes, merely because they superficially appear different through the use of different notation, so combinations such as those just mentioned must be left out of our accounts. On the other hand, some combinations involving nonmodal proposition(s) mixed with modal(s), are noteworthy, even though subaltern, because they provide additional logical information. We need only to select either of the symbols **a** or **m**, to represent nonmodal forms, and work with that exclusively. I selected m as more appropriate, after finding that all valid mixed modes could be derived from solely temporal syllogism. But this is strictly-speaking mere convention; modes such as **ncm** or **nmt** could equally have been written **nca** or **nat**. The underlying meaning is the same.

### 3. Summation.

If we examine the principal 37 new syllogism introduced in this paper, it is clear that they are not at first sight obviously valid. An effort of thought is needed to see their truth. This shows that our enterprise, the development of a modal logic, was a worthwhile endeavor, a valuable addition to human knowledge. The justification is still greater, if we analyze our work in this chapter statistically, and sum-up the number of new syllogistic forms introduced.

We saw earlier that there are 2X3X5 = 30 possible categorical forms of the kind under study, a proposition may have one of two polarities (**+** or **–**), one of three quantities (**s**, **u**, or **p**), and one of five modalities (**a** or **m**, or **n**, **c**, **t**, or **p**). A syllogism contains three propositions, in any of four possible figures; therefore the total number of imaginable combinations is (30 cubed)X4 = 108,000 moods, whether valid or invalid. Of these, ((2X3)cubed)X4 = 864 would be wholly nonmodal moods; (((2X3X3)cubed)X4)-864 = 22,464 would be natural modal moods; and again 22,464 would be temporal modal moods; the remainder 62,208 moods would be of mixed modal type.

Now, let us calculate how many out of this theoretical total of possibilities, are in fact valid. We saw that in nonmodal logic, there are 44 valid moods, of which 19 are primary and 25 are secondary. Next, in modal logic, we established 37 primary moods for natural modality and 37 for temporal modality; and we found these to have 336 and 336 secondary moods, respectively. Lastly, we identified 696 mixed moods as valid, and pronounced them all secondary. Thus, the total number of valid moods obtained is 1486, of which only 93 are primary, and the remaining 1393 are secondary.

Thus, only 1486 out of 108,000 = 1.4% of possible combinations are logically valid; versus 98.6% chances of erroneous reasoning. This shows the importance of our thesis, that modality needed to be considered and systematically analyzed by logical science. The number 1486 is of course quite large in itself; this shows the value of the notation system I invented, which made it possible for me to analyze so many combinations with a certainty of exhaustiveness and in a minimum of space.

Of the valid moods, only 6.3% are primary and 93.7% are secondary. Primary moods are the most significant and independent forms of reasoning; secondary moods are relatively less significant and more derivative. This does not mean, however, that secondary moods are necessarily less commonly used in practice; although many of them occur rather rarely, many may nonetheless be as important as primary moods. For example, **naa** moods in figure **1** or **2** are quite valuable, although technically subaltern to **aaa**.

It must be stressed, also, that the recognition of invalid modes of thought is as important as the knowledge of valid modes. We indicated, for example, how at first sight one might suppose moods such as **1/ApAnAp** valid; an analytical effort is required to understand the error involved. Some invalid moods are of course instantly seen to be wrong; but some contain pitfalls for the logically untrained mind.

Further research, which I will not develop in detail in this paper, shows that some invalid moods may be made to yield imperfect conclusions, of the types ‘Some nonS can be nonP’ or ‘Some nonS are sometimes nonP’. Similar cases arose in nonmodal syllogism, with ‘Some nonS are nonP’ conclusions, the reader will recall.

Another kind of atypical conclusion is drawable in some cases; for example, **1/ApAp** (All M can be P and All S can be M) does not conclude **Ap** (All S can be P), but does allow us to infer that ‘All S either can be or can become P’. Indeed, this may be viewed as an explanation why the simple ‘All S can be P’ cannot be inferred.

Syllogism which mix copula in this way might be characterized as mixed-form. They involve another kind of copula (becoming, instead of being). The logic of becoming is a large field on its own, which will be touched upon further in the next chapter.

### 4. General Principles.

To conclude, general rules of the syllogism, which summarize, or explain, the valid moods, may be presented. They reveal the underlying principles of such reasoning, or the common attributes.

. If the extreme terms are positively connected to the middle term in the premises, they will be connected in the conclusion, if any. But if either extreme term is unconnected to the middle term in its premise, it will be unconnected to the other extreme term in the conclusion, if any. If neither extreme term is positively connected to the middle term in the premises, no conclusion can be drawn.*Rules of Polarity*. At least some instance(s) of the middle term must be found in common to both premises, for a conclusion to be conceivable. And no instance of either extreme term may be found referred to in a suggested conclusion which was not covered in the premise. However, instances found in a premise may not-reappear in the conclusion.*Rules of Quantity*. There must be found some circumstance(s) and time(s) in common to both premises, for a conclusion to be capable of being drawn from them. And no circumstance or time may make its appearance in the conclusion, which was not already mentioned in a premise. Though, of course, the conclusion may be circumstantially or temporally more narrow than the premises.*Rules of Modality*

These are common sense conditions; we can see at a glance that they are reasonable. They tell us that the information in our proposed conclusion must have been implicit in our premises, if we are to claim that it was drawn from them. These rules can be expressed graphically or in the language of ‘distribution’.

- The middle term must occur in both premises, and be distributive once at least, extensionally, naturally and temporally, for any conclusion to be possible. Breach of this rule is labeled fallacy of the undistributive middle term.
- One of the terms of each premise must be found in the conclusion, and if such an extreme term was undistributive in its premise, extensionally, naturally, or temporally, it must remain likewise undistributive in the conclusion, if any. Breach of this rule is labeled fallacy of illicit process of the major or minor term, as the case may be.

Such breaches of logic are essentially commissions of the fallacy of four terms. A syllogism has a valid structure only if these rules are obeyed; otherwise it is a paralogism. If the middle term is really different somehow in the two premises, it is as if there were no middle term, and therefore no basis for deduction. Likewise, if either term in the conclusion has a meaning or scope different from that given in the premises, it is as if a new term has been introduced in the equation, so that it cannot be called deduction.

Lastly, note the possibility of **sorites** with modal syllogisms, as with actual ones. A regular sorites, consisting entirely of first figure arguments, would look as follows:

All (or Some) A must (or can) be B,

All B must be C, All C must be D, All D must be E,

All E must be F (or No E can be F)

therefore, All (or Some) A must (or can) be F (or the negative equivalent).

The rules of quantity or polarity are the same as before: only the first premise (the most minor) may be particular, only the last one (the most major) may be negative, and the conclusion follows accordingly. Here, we may add the rule of modality, that only the first premise may be potential, and the conclusion follows accordingly.

## 17. TRANSITIVE CATEGORICALS.

### 1. Being and Becoming.

Consider the wide form of actual proposition ‘This thing, which was/is/will be S at some time, was/is/will be P at some (same or other) time’.

Note that the primary subject is ‘this thing’, an existent which is being mentally or physically being pointed to; however, that designation is further specified by the thing’s characterization as S at some (explicitly specified or tacitly intended) time, say t1. Next, note that the time at which P applies may be the same or a different time, call it t2. This form is uncommitted as to whether, at time t1, this thing is P or nonP, as well as S; nor is it specific as to whether, at time t2, this thing is S or nonS, as well as P.

- We normally understand the form ‘this S is P’ to mean ‘this thing, which is now S, is now P’, implying S and P to be simultaneous. It is a static event, not implying process. The past and future tenses of this likewise only give a still snapshot of the situation at the given time. Such propositions may be called
, since their original function is to record the attributes of things.*attributive*

The copula of ‘being’ is the most fundamental copula, but it is not the only kind of relation Logic needs to consider. The way categorical propositions of this kind are structured limits their applicability to certain phenomena (the majority, no doubt); certain other phenomena seem best dealt with using categorical propositions with other relational features, defined using attributives.

One such alternative copula is, let us say, ‘S changes to P’ (intended in the most generic sense of ‘something, which initially is S, later is P’). It is close to fundamental, though derived from ‘is’ (and less used). Such propositions might be called ** transitive**, since they suggest a process across time. We have to be careful to distinguish between the subject-definition time, and the predicate-activation time.

- However, this verb ‘to change to’ is colloquially used in many senses, and we should first isolate the ones which concern us most here. When we say ‘S has or will, change to, or is changing to, P’, what do we mean? A convention is needed, so we can develop two somewhat more specific forms describing change. Let us agree that:

‘This S will get to be P’ signifies a process starting at S (with or without P) at some earlier time, and ending at P with S at some later time.

‘This S will become P’ signifies a process starting at S (with or without P) at some earlier time, and ending at P without S at some later time.

We have dichotomized change into two kinds, each covering a segment of the phenomena we come across. These forms represent two species of transitive propositions. Sub-classes of these, or also classifications otherwise defined, can be developed as needed, of course. For example, ‘S ceases to be P’, for change from SP to nonP.

Note that in both cases, we start as S, without specifying whether P or nonP applies at the same initial time; and we end at P. The copula ‘will get to be’ is reserved for ‘S and P’ finales, and the copula ‘will become’ is reserved for ‘nonS and P’ finales. If we are not sure whether the finale contains S or nonS, we simply say ‘S will get to be or become P’, to specify our knowledge of P at least; this is equivalent to the generic ‘change to’ copula.

These two copula are enough to cover all processes. With nonP as predicate, we say ‘will get to be nonP’ or ‘will become nonP’; with nonS as subject, we say ‘this nonS will get to be’ or ‘this nonS will become’. To deny processes, we may negate the copula, using ‘won’t’ (or will-not) get to be or become. If we wish to specify whether the initial state includes or excludes P, we may do so separately, by saying ‘this S is at first P (or not P)’.

This distinction allows us to express whether the change in question is: an ** alteration**, with the starting point S remaining or reappearing (= ‘eventual being’) at the end, when P arises; or a

**, with the original subject S disappearing (= ‘becoming’) at the end, when P arises. Thus, we may speak of alterative and mutative propositions.**

*mutation*- Note in passing that there are other ‘types of change’. The above types are based on natural or temporal modality; they concern real changes in objects, transitions through which individual things go, over time. However, there is also ‘extensional change’, as when we say that a genus ‘changes’ differentiae from one species to another; this refers to relatively static differences, and treats a universal as if it was an individual going through changes (because the mind works serially). More broadly, we have ‘logical change’, which traces the mental realization of a previously unknown yet existing truth, as if its to us novel appearance was equivalent to its coming into being.

### 2. Various Features.

- In the interim, between the beginning at S and the result at P, any state or combination of states may take place (S and P, S and nonP, nonS and P, nonS and nonP), or the change may be abrupt and without intervening state. The forms adopted are left open in that respect. If indeed complications arise in the interim, they can be specified separately, in additional statements.

The transition itself may be gradual or sudden; also, there may be a slow build up of surrounding forces before the process in question is put in motion, or it may take off immediately.

- In the past tense, ‘got to be’ and ‘became’ may be used. They report the culmination of processes. In the present continuous tense, ‘is getting to be’ (implying ‘will get to be’) and ‘is becoming’ (implying ‘will become’) may be used. Note that these ‘presents’ do not in fact describe the present stage (i.e. whether S or nonS, P or nonP, are at this time active), but merely define the beginning and end of a process and tell us that it is in progress.

In the cases of ‘will change’ or ‘is changing’, if ‘must change’ is not meant, there is not a real actuality, like for ‘has changed’, but a certain probability of success, assuming no extraordinary interference, without totally excluding failure. A specific example of such future is of course the expression of intention, of resolve of will-power, including free-will.

The negatives of those actual forms are interpreted accordingly. ‘Hasn’t changed’ denies completion of process so far; ‘won’t change’ denies it for the future, or some part thereof; and ‘is not changing’ denies a process is taking place.

However, remember that ‘will’ and ‘won’t’ may strictly-speaking be both wrong as predictions, even though it is, by the law of contradiction, *ex-post-facto* true in general that the event in question must ultimately either happen or not-happen. Modalities are also useable here, to specify more exact statistics of past culminations of process, giving the degree of ‘will’ or ‘won’t’.

- These forms may, of course, all be quantified and naturally or temporally modalized. Thus, quantity and modality, like polarity, are characteristics of relations which transcend the specific copula involved. The plurals ‘all’ and ‘some’ used for these purposes are, as usual, intended subsumptively, addressing each unit singly; the meaning is more rarely collectional or collective.

Note that in plural forms, the times at which the implied individual events happen, and the relative times of start and finish of each event, fade in importance, in comparison to singular propositions; the various events span across time, indefinitely, ‘whenever they happen’.

Potentiality, as in ‘S can get to be or become P’, means that in some circumstances, S does get to be or become P’. Natural necessity here signifies inevitability of the alteration or mutation, in contrast to the suggestion of invariability in the static ‘S must be P’. If ‘can’ is combined with ‘can not-’ (get to be or become), we have contingency. ‘Cannot’ of course means ‘in no circumstances’ does the event occur.

Note that potential alteration and potential mutation are compatible, although they cannot actually happen together, since the one finally keeps its subject while the other loses it. It is also conceivable that ‘S cannot get to be P’ and ‘S cannot become P’ are both true: together they mean that P is entirely closed to S, by whatever process.

Likewise, with temporal modalities.

- The oppositions between alterative or mutative propositions, and also those of these two groups with each other, and with attributives, need to all be worked out in detail, of course. But this job will not be carried out here, to avoid expanding this paper more than necessary.

Simply, beginning at the singular level, examine each form’s definition for implications, and compare pairs of forms for points of agreement or disagreement; once singular oppositions are established, they can be quantified following the rules developed for attributives.

Similarly for eductive arguments.

### 3. Various Contrasts.

- Consider the following examples:

*Attributive*: This egg is soft (or hard).*Alterative*: This egg has hardened (gotten to be hard).*Mutative*: This soft egg has become hard (or a hard egg).

A soft egg can’t be or get to be a hard egg, but only become one; however, soft and hard eggs are both still eggs. The transition from soft to hard is gradual, depending on heat supply; and the terms soft and hard may be viewed as extremes separated by unnamed in-between states, or we could say that all states before hard are degrees of soft, depending on whether we define soft as raw or uncooked.

In terms of class-thinking, each of these copulae plays a distinct role. The first merely classifies. The second indicates a process leading to a classification. The third declassifies on one side and reclassifies on the other. The logic of classification would be incomplete if we did not consider the more dynamic relations of change.

Note that, whereas ‘This S is or gets to be nonS’ are self-contradictory, ‘This S becomes nonS’ is not so, but rather implied whenever an S becomes anything (P or nonP); almost every S eventually becomes a nonS in this world. On the other hand, whereas ‘This S is or gets to be S’ are formally acceptable, ‘This S becomes S’ is indeed self-contradictory, unless we add the word ‘again’, to mean: ‘This S becomes nonS, which in turn becomes S’.

- The similarities and differences between attributives and alteratives should be noticed. ‘This S will be P’ and ‘This S will get to be P’ are obviously very close in meaning; in fact, if you look at the definitions, the former is a special case of the latter, and is implied by it. The difference is merely in the time-definition of the subject; when quantity or modality are introduced, the difference blurs, because that time is less definite.

Still, the one is static and the other dynamic. This is more noticeable in the past tense; compare ‘This S was P’ to ‘This S got to be P’. Similarly, in the present continuous sense, ‘is’ and ‘is getting to be’ are obviously different. For this reason, two distinct copulae are needed.

- We should henceforth not confuse ‘getting to be’ and ‘becoming’, with each other or with the more general ‘changing to’. We make this convention, although these expressions, and others like them (end up as, turn out to be), are colloquially interchangeable, because logic needs fixed terminologies to proceed.

In alteration, the initial term, defining the subject, effectively remains in force at the end, underlying the predication.

Note that, normally, ‘S gets to be P’ implies ‘S is not initially P’, suggesting a switch of predicates. But this issue is best left open, since the form is then more widely useable for cases starting and ending at SP, yet having intermediate stages, of nonS and/or nonP, or even for cases involving no change. Often, the terms S and P are two extremes in a range, and the motion from the one to the other is a gradual transition whose intermediate stages are irrelevant, and often unnamed. If the process starts and ends at SP, yet passes through other states, we would say ‘S gets to be again P ‘.

In mutation, there is a radical exchange; both terms are effectively subjects, and we swap one for the other; the initial subject is gone at the end, replaced by another. There may indeed be an implied substratum to the mutation, an underlying constancy; the changing thing remains always at least a ‘thing’, but often some narrower genus is understood. The paradigm of mutation is of course biological metamorphosis; but such change is found also in physics or other areas.

Note that, ‘This S can or must become P’ do not imply that ‘it cannot become nonP’, because the subject is not defined by P or nonP. There is no obversion here; the polarity of the copula (become) and of the predicate (P) are not interchangeable.

### 4. Some Syllogisms.

We will only concern ourselves here with the some of the more significant moods of syllogism. Our purpose is only to show how introducing transitive categoricals allows us to draw conclusions from arguments which would otherwise be sterile.

The various copula discussed above are in practice often used in tandem, to specify a situation with more precision. We might for instance say the conjunction ‘This S cannot be P, but can become it’. In syllogism, a conclusion in the form of a logical disjunction like ‘This S can (get to) be or become P’ is often possible where otherwise no other conclusion can be drawn.

Systematic treatment of syllogism involving transitives would proceed as follows. We are dealing with four families of proposition: attributives, transitives, alteratives and mutatives; the second of these being a genus for the last two. Each of these classes covers a number of forms varying in polarity, quantity, modality. We must consider, for each figure of the syllogism, every combination of premise, however mixed, and find out what conclusions, of whatever family, can be drawn from them.

However, for now, let us concentrate on a more limited task. We will look at the first figure only, and review only moods whose premises are attributive and/or mutative, and modal. The conclusions obtained have the generic transitive form ‘S can get to be or become P (or nonP)’.

- When both premises are attributive, an attributive conclusion can only be drawn (if at all) from a necessary major premise; if the major is potential, we cannot draw an attributive conclusion. However, a transitive conclusion can be drawn, showing that the modality may have an impact on the copula. The following moods are valid:

All M can be P (or nonP),

All/This/Some S can (or must) be M,

ergo, All/This/Some S can get to be or become P (or nonP).

We know from the premises that what started as S, will in some circumstances be S and M; and that whatever is M, will in some further circumstances be M and P (or nonP); but we cannot predict whether the end result of this process includes or excludes S. It is conceivable that S stays on with P (or nonP), but it is also conceivable that S disappears prior to the arrival at P (or nonP). For this reason, our conclusion cannot be merely ‘S can be P (or nonP), but must be open to the ‘S can become P (or nonP)’ outcome.

- The same can be argued with a mutative major premise, whatever the modalities involved. Thus, the following are valid:

All M can (or must) become P (or nonP),

All/This/Some S can (or must) be M,

ergo, All/This/Some S can get to be or become P (or nonP).

If one or both premises are potential, so is the conclusion, as above; but if both premises are necessary, a necessary conclusion can be drawn, as below:

All M must become P (or nonP),

All/This/Some S must be M,

ergo, All/This/Some S must get to be or become P (or nonP).

- In cases where the minor premise is mutative, whether the major is attributive or mutative, similarly disjunctive conclusions may be drawn. We know from the minor premise that S will disappear to become M, but we cannot be sure whether, in the circumstances when M is or becomes P, S reappears or stays away.

Note that in all the cases so far considered, we could view the conclusion as a logical disjunction as we did, or we could say that a more specific conclusion can be drawn if we know one or the other alternative to be excluded. This would be equivalent to having a third premise, viz. ‘S cannot get to be P’ or ‘S cannot become P’. But formally speaking, this constitutes an additional argument (apodosis) after the syllogism as such.

- All the above only concerns cases with both premises affirmative (whether the predicate be P or nonP). Now, the minor cannot be negative in the first figure, but what if the major premise is negative (in the sense of negating the copula, not merely the predicate)? In such case, we cannot draw a likewise negative conclusion, because we can construct a syllogism with compatible affirmative premises yielding a conflicting affirmative conclusion. Thus, for example, the following mood is invalid:

No M can become P,

This S must be M,

ergo, This S cannot get to be or become P.

This is invalid, because it is conceivable that, though no M can become P, all M can nonetheless be P, in which case the following syllogism could be constructed, as earlier established:

all M can be P,

This S must be M,

ergo, This S can get to be or become P.

We could interpret this to mean that, a (compound) negative conclusion is possible, if the negative major is compound, as in the following mood. Note that major premise and conclusion are conjunctions, not disjunctions, of negatives. The result is due to the attachment of S to M.

No M can be or become P,

This S must be M,

ergo, This S cannot get to be or become P.

If either or both of these premises were potential instead of necessary, a potential conclusion would be drawn.

I shall not, however, develop the matter further, but remain content with having shown some of the impact of transitives on syllogistic reasoning. A full theory should of course list all the valid moods, and do so in all figures. Also, all the above concerns natural modality, but similar arguments can be presented for temporal modality.

## 18. PERMUTATION.

We need to determine the limits of applicability of classificatory forms by looking more closely at the matter of everyday reasoning. We also need to observe common sense practices, and find out if any forms of argument, other than those dealt with previously, are instinctively used by us.

The ultimate goal of Logic is, of course, to bring out into the open for scrutiny all the details of everyday reasoning. We want to encompass into the sphere of Logic, any forms or processes capable or worthy of formal treatment.

### 1. Two Senses of ‘Is’.

As previously pointed out, the form ‘S is P’, as commonly understood in Logic, serves the function of classification. Such use of the copula ‘to be’ is rather abstract and specialized. In this sense, ‘S is P’ tells us that S is an individual, or some or all members of a class, which also count(s) as among the units of some other class. S and P have to some extent the roles of species and genus.

Most often, in practice, ‘S is P’ means that P is an attribute of S; we are describing an object (most typically, an entity) in terms of its qualities. We mean that P somehow is in S, something which S has, part of the being of S, one of the many phenomena which all together add up to the thing we call S, and which are distinguishable within it.

This sense of ‘is’ as attribution is strictly-speaking quite distinct from the use of ‘is’ for classifying; there is an ambiguity, the same word is used for two different relations. The common ground of the two is the information that the ‘universals’ S and P, in some degree intersect. Two domains of reality are compared for overlap, in their instances, in space and time, in causal contexts.

Thus, ‘S is P’, may mean ‘S has P-ness’ or ‘the unit(s) of S is/are unit(s) of P’. The attributive sense may be ‘permuted’ to the classificatory, by saying ‘S is P-ness having’. This puts the specific copula ‘has’ into the predicate, in a proposition with more numerical intent.

But the possessive relation remains; it has value and meaning quite apart from such permutation; otherwise, we would have no need for the concept. Classification cannot occur without prior attribution. Before one can put a unit under a class, the class-defining quality has to be attributed to the unit. Permutation merely conceals or bypasses the attribution, it does not erase it.

The classificatory ‘Ss are Ps’ contains attributive propositions within its terms, since it means ‘things which are S are things which are P’. The ‘are’ in the two terms are attributive, while the ‘are’ of their relation to each other is classificatory.

Attribution concentrates on the substantive (i.e. in terms of universals), qualitative, so-called *‘connotative’* aspect, of this relation of coincidence. Classification focuses on the enumerative, quantitative, so-called *‘denotative’* aspect. Both aspects exist out there (at whatever level the phenomenon might be); we mentally isolate the one from the other somewhat, to stress each in turn.

Although we regard these as two aspects of the same term — we say that in one case, it is taken ‘in its denotation’, and in the other case, it is taken ‘in its connotation’ — it is more accurate to say that these are two kinds of term, which have a close relationship, both objective and verbal. For example, dogs may be viewed as things which have dog-ness (or in better English, caninity), and dog-ness may be viewed as that which dogs have in common and distinctively.

In truth, dog-ness cannot be called the meaning of dogs, as some have suggested; nor is the reverse correct. Each of these terms subsumes slightly different referents: the former refers to every corporeal dog, and the latter (though we regard it as effectively singular) refers to every occurrence of the qualities which make up dog-ness. These two are indeed equal in number, and have no existence apart from each other, but the intention is not identical.

### 2. Other Permutations.

Other propositions are permutable, besides ‘S is (or has) P’. Some propositions are geometrical: they locate the subject in space, by reference to the location of the predicate; thus, ‘S is at or in, next-to, above or below, near or far-from P’. Some involve placement in time, using relatives like ‘earlier’, ‘later’, ‘at the same time’. One can say that ‘S is at P’ implies ‘S is an at-P thing’, but one cannot say that they are equivalent and identical.

Some propositions describe actions, implying change and/or causality. To do, is to change or move in a certain way, or to cause something else to be or to change somehow. The verb may be simple or continuous; ‘S does or is doing P’. Here again, permutation is possible, to ‘S is a P-doer or one of the P-doing things’. We permute when it is useful, but the original form does not lose its utility and disappear.

The logical properties of the classificatory forms are only generic properties, applicable to all permutables. We may well expect that each of the original, more specific, forms, has its own special logical properties. It is part of the long-term task of logical science to gradually intercept all such forms, and confront them for analysis of their peculiar properties. To establish the implications, oppositions, and arguments, which are peculiar to each form.

While modality can be permuted, e.g. ‘S can be P’ to ‘S is one of the things which can be P’, this easily leads to error. The reason is that, normally, when we say of a unit that it belongs to predicate P, we mean this to apply to times and circumstances when it is actually P. This is not equivalent to reference to the class of things only potentially P. As a result, as we have seen, if this process is used inattentively, one may draw wrong conclusions in syllogism.

Similarly with transitive copulae, like becoming. They are too fundamental, too much part of the structure of things, to be permuted with any more than superficial interest. To process ‘S becomes P’ to ‘S is one of the P-becoming things’ merely puts the need to investigate the logic of becoming one step removed, since it still there, but now hidden in a more detailed predicate. Likewise, with regard to causality. To find the specific properties of such a copula, we have to keep it intact, as a relation in its own right.

(Note that any goals pursued by such permutations can, as we shall later see, be fulfilled by using ‘extensional conditional’ propositions.)

Copulae represent relations. Each relation has its own nature. ‘Is’ in the sense of class-thinking, is a broad relation, with a number of rules; ‘is’ (in the attributive), ‘becomes’, ‘causes’, are all perhaps narrower relational concepts, but still worthy of special attention, in search of the rules applicable to them specifically.

Relations like the verbs ‘sings’ or ‘digs’ also no doubt have their own characteristics, and are legitimate topics of inquiry; but their limited scope, assigns them to a secondary position in logical science, while they may well be of primary importance to other sciences. We might make a distinction between formal logic and material logic, on this basis.

### 3. Verbs.

Let us look briefly at propositions with a verb of possession or action. Their general form is ‘X / does-Y / Z’. There is a logical subject (X), a verb (here written ‘does-Y’), and some or no appendages (the ‘Z’ part).

‘Does’ is here meant to include agency and passion, doing and being done to. But not every action implies a patient, note. The agency, by the subject, of the act (verb/relation), may range from an act of free-will to a fixed absolute. An act may be static or dynamic, it may signify a posture, a movement, a forcing. In this context, even ‘has’ is an act.

Note that quantity and modality are applicable to any such proposition, as usual. The simple and continuous present tenses are not interchangeable, of course. The modality involved is often left implicit, and should always be clarified. For instance, ‘animals sleep’ implies ‘animals are sleeping some of the time’ without implying ‘all the time’ or ‘at this time’.

The appendages concern parameters such as: who, what, where, when, why, whence, how, what for, how much. They serve to further specify the relation, and delimit it. Prepositions like ‘by’, ‘at’, ‘to’, ‘for’, are definable in this context.

We may mention any combination of the following: the patient of the action, or agent of the passion (e.g. electrons repel each other); the locale of the incident in time and/or space (e.g. he went west at sunrise); the conditions or causes (e.g. water boils at 100 deg.C, at s.t.p.); the effects or consequences (e.g. it drove him to work harder); the means or ends (e.g. the water was boiled for tea).

Also, generally, any measurement, qualitative or quantitative, of the above, may be mentioned (e.g. she sang beautifully and softly). The ‘measure’ may concern the relationship itself (e.g. he gave liberally), or an appendage of it (e.g. he gave lots of money). Attaching expressions of number, magnitude or degree to a statement (e.g. everyone has some virtue), should not be confused with the attempt of some logicians to ‘quantify’ the predicates of classificatory propositions (as in ‘all X are some Y’).

Often, the proposition can be restructured from categorical form to subjunctive form, and so the logic of subjunction comes into play (e.g. ‘She sings when happy’ is more subjunctive than categorical).

Causality is often concealed in statements which do not mention it. For instance, ‘her beauty attracted him’ implies that the agent caused the patient to act or move in a certain way. Causality of course is of various degrees, ranging from mere influence on a voluntary act or probabilistic tendency, to compulsion or mechanical force.

Thus, we see that many material statements can be analyzed for more formal components, and thereby be subjected to certain rules of formal logic. They may nonetheless yet contain further logical properties, not found in the formal components.

### 4. ‘As Such’ Subjects

We have looked at permutation which encloses a nonclassificatory copula and its appendages into the predicate of a classificatory proposition. But we also need to consider permutation of the subject.

Many propositions have a ‘universal’ as their subject. This may be a quality as such (e.g. turquoise is a bright color), or any act as such (e.g. ‘love is a nice feeling’, ‘running is good for you’). ‘As such’ subjects like these are being focused on, not so much in their capacity as classes embracing the various appearances of the ‘universal’ in the world, but as if the ‘universal’ is an individual thing (a whole, whose various manifestations in the world are its parts).

A class concept regards the individual manifestations of a universal, or of a complex intersection of universals, as separate entities, which happen to display that simple or complex distinct character in common. An ‘as such’ concept, instead, views the particular manifestations as merely the segments of a single, continuous thread, and refers to that more or less uniform whole as a distinct entity.

Before a proposition can be processed according to the logic of classificatories, its subject may need permuting. (Thus, for our examples above, we would say ‘turquoise things are bright-colored’, ‘people in love feel nice’, or ‘runners are healthier’).

### 5. Commutation.

Concerning verbs, the process of commutation should be mentioned.

Relations are generally directional, so that when something is true in one direction, then something else is true in the other direction. The ‘total’ relation between the terms includes both directions, though only one of the components might be named, leaving its correlative implicit. The correlative copulae may be identical (e.g. A=B and B=A) or quite different (e.g. owns and belongs-to, or shot and was-shot).

We may call ‘commutation’ the inference from one direction of a relation to the other. The correlative of ‘is’ is ‘is’, as conversion shows. The proposition ‘X causes Y’ is commutable to ‘Y is caused by X’. ‘He bought IBM shares’ implies and is implied by ‘IBM shares were sold to him’. A lot of our reasoning consists in rewording statements like that, changing the perspective to improve our understanding.

Commutation may apply not only to the main copula of a proposition, but to copulae implicit in its terms. For example: ‘the bodies attract with a force of 10 dynes: the force, with which the bodies attract, is 10 dynes’.

## 19. MORE ABOUT QUANTITY.

In this chapter, we will look into various topics which involve quantitative considerations.

### 1. Substitution.

Substitution is a widely used, yet little noticed logical process, which is open to formal treatment of sorts. It consists in replacing a term with another which has the same units, but views them in a somewhat different perspective. The entity referred to remains the same, only its label changes (*qua* what it is referred to); the substitution is thus justifiable.

We may substitute a generic term for a species, if we keep the same quantity, or a species for an individual. For instance: ‘X has (some number of) Y, All Y are Z, so X has (that many) Z’, or ‘X has this Y, this Y is Z, so X has a Z’. Example: ‘Man has a mind, a mind is an organ, so man has an organ (at least one)’.

Note that this is not a normal, first-figure, classificatory syllogism. Here, the major premise must be or be made affirmative and classificatory; but the minor premise and conclusion are possessive (in this case). Needless to say, verbs other than ‘to have’ are open to substitution, too. Example: ‘Bill hit Joan: Bill hit a woman’.

If the major premise is negative, it should be obverted before the substitution. Thus, ‘X has Y, No Y is Z’ conclude ‘X has nonZ’, rather than ‘X doesn’t have Z’. For example, ‘Tom has a dog, a dog is not a cat’. Likewise, if the major premise is not classificatory, it should first be permuted. Thus, in ‘X has Y, all Y have Z’, the term to substitute would be ‘something which has Z’, rather than just ‘Z’. Example: ‘Tom has a dog, dogs have fleas’. We are said to commit the ‘fallacy of accident’ when we make errors of this kind.

If the minor premise is negative, the conclusion must be formulated very carefully, if at all. One is forced to keep the middle term in the conclusion, only qualifying it with the major term, to ensure we do not change the implicit quantity of units referred to. Examples: ‘Tom doesn’t have a dog: Tom doesn’t have an animal of the dog kind (though he may have some other animal)’, ‘Bill did not hit Joan: Bill did not hit this woman (though perhaps another)’.

We often profit by substituting a subject. This process takes the third figure form: ‘Y is X, Y has Z, so an X has Z’. Here, the major premise may have any copula or polarity (in this case we used ‘has’), while the minor should be affirmative and classificatory. Example: ‘Joe is a man, Joe runs 40 miles a day, so (there’s) one man (which) runs 40 miles a day’. Substitution of a pronoun would take this form.

Even the verb may be substituted, using an exact description of some necessary aspect of it. Examples: ‘the magnet was repelled 3 feet: the magnet was caused to move 3 feet’ or ‘he sprinted to the finish line: he ran to the finish line’.

What logicians call immediate inference by added determinants (e.g. ‘horses are animals: therefore, the heads of horses are heads of animals’) or complex conception (e.g. ‘Physics is a science: therefore, physical treatises are scientific treatises’), involve substitutive syllogism, with a tacit minor premise (e.g. ‘horses have heads’ or ‘some treatises are about physics’), which enables the conclusion to be drawn. These processes are illicit when the rules of substitution are not properly obeyed (e.g. ‘horses are animals: the majority of horses are the majority of animals’ or ‘physics is fun, physical treatises are funny’).

### 2. Comparatives.

Logic interfaces with mathematics, whenever we compare the number, position, magnitude or degree of something, relative to something similar; or of two things measured by reference to a third. We may call such propositions ‘comparative’.

This concerns forms like ‘X is more Y than Z’, ‘X is less Y than Z’, ‘X is as Y as Z’. They affirm X to be greater, smaller or equal to Z, in some respect Y (e.g. this metal is as strong as steel). It is implied that X and Z are each Y, though to different extents.

Sub-categories of these measures may be defined by inserting more precise quantities, like ‘much more’ or ‘30% more’, say. Further complications are often introduced, through the concepts of ‘enough’ and ‘too much’, which evaluate the measurements in relationship to some goals.

Copulae other than ‘is’ may be involved, and the comparison may concern the verb or an appendage (e.g. ‘he ran faster than her’ or ‘they ordered more food’). Often the comparative aspect is verbally concealed (e.g. ‘she was happier today’ or ‘that ball is the closest’).

The corresponding negative forms are defined as follows. ‘X is not more Y than Z’ means ‘either X is not Y and/or Z is not Y or X is less or equally Y compared to Z’. Similarly for ‘not less’ (= not at all, or more or as much), and similarly for ‘not as much’.

Since only one of the three affirmative measures may be true, and, granting that Y is applicable to X and Z, one of them must be true, they are contrary to each other. It follows that the negatives which contradict them are subcontrary to each other.

Comparative propositions can be commuted. If X is (or is not) more/as much/less Y in comparison to Z, then Z is (or is not) less/as much/more Y in comparison to X, respectively. Example: ‘he left just before sunrise’ to ‘the sun rose soon after he left’.

Syllogistic style arguments can be constructed. For instances, using the symbols of mathematics (>, =, <; and / for their negations), to signify the possession or lack, of some common character, we can predict that ‘If A>B and B>C, then A>C’ or that ‘If A¹B and B=C, then A¹C’. In some cases, no deduction is possible; as for instances in ‘A>B and B<C’ or ‘A¹B and B¹C’. These relations are generally well known, and need not be pursued further here.

Comparative propositions are significant in so-called *a-fortiori* arguments. These arguments are quite important and very commonly used in everyday reasoning, but apart from a brief mention of them in a later chapter, they will not be analyzed in detail in this volume. I hope to deal with them in a later work.

### 3. Collectives and Collectionals.

A proposition is ‘dispensive’, or ‘collective’, or ‘collectional’, according to the way its subject subsumes its units for the predication.

- Most propositions are
(many authors prefer the name ‘*dispensive**distributive*’ instead), and this is the way of subsumption we have dealt with so far, in detail. A plural dispensive refers us to its class members severally, each one singly; it is simply a conjunction of a number of mutually independent singulars. Thus, ‘all or some S are (or are not) P’, means ‘this S is P; that S is P;… and so on’, until all the S intended to be included under the all or some quantifier have been enumerated. - In contrast, some propositions are
, applying a predicate to the units included by their subject only if they are taken together, and not separately. Such propositions are effectively singular; they conjoin the units into a group, rather than a class. A collective has the form ‘these S together are P’, meaning ‘this S and that S and…so on, taken as one, are P’. Note that, unlike with dispensives, ‘all S together are P’ does not imply that ‘some S together are P’.*collective*

The group may have a summational property, which sums up the lesser measures or degrees of the same property displayed by the parts (e.g. we each have ten dollars, but both of us together have twenty). Or the group may have a composite property, due to the causal interaction of the parts, which is not found in any measure or degree in the parts themselves (e.g. as individual cells together make up a human being, the whole having various powers the parts lack). In the latter case, a certain arrangement of the parts may be tacitly required for the predication to work, so that a statement more descriptive than mere conjunction may be needed for accuracy.

The logical subject here is ‘these S together’; we may, if we so wish, form a new collective term from it (like ‘crowd’ or ‘society’). Note that, in some propositions, the intention is a dispensive summary of collectives (e.g. ‘fifty books form a big pile’ means any set of fifty books). This may be formalized as follows (where ‘*n*’ signifies some number): ‘Any *n*S together are P’ or ‘Certain *n*S together are P’; here, ‘*n*S together’ refers to a class of collectives.

- Some propositions are
. These differ from dispensives and collectives, in that, although they refer to events each one singly, they also tell us whether these can or cannot, are bound to or may not, be actual jointly — simultaneously, at the same definite or indefinite instant or period of time. This is usually signified by stressing the quantity (by the tone of voice or italics).*collectional*

Thus, here, ‘*All* S can be P’ means ‘the conjunction of this S as P and that S as P and… so on — is potential’: this does imply the dispensive ‘all S can be P’, but further reveals that the actualization of these potentials can take place all at once. We would use ‘*All* S can be nonP’, if we want to say ‘it can happen that all S are simultaneously nonP’.

Accordingly, ‘*All* S cannot be P’ denies the potential for simultaneous actualization, the ‘not’ being directed at the ‘all’ (rather than at the ‘can be’): it is formally compatible with ‘all S can be P’ in a dispensive sense; though usually used in such context, it does not imply it. We would use ‘*All* S cannot be nonP’, if we want to say ‘it cannot happen that all S are nonP at once’.

(In contrast, the form ‘*All* S must be P’ would be interpreted as ‘it cannot happen that some S are not P: if any S are P, then all are P’; similarly with ‘*All* S must not-be P’; to deny these statements, we would say ‘it can happen that some S… etc’.)

The particular versions of such statements, ‘*Some* S… etc’. may be similarly analyzed. There are also singular versions, like ‘this S can be P, alone’, which tells us about the potential for actualization of ‘this S is P’ when all *other* S are nonP. More broadly, any quantity ‘*n*’ may be specified: thus, for instance, ‘*n*S can be P’ informs us that this number of S can be P at the same time; in some cases, we additionally specify ‘at least’ or ‘no more than’ to open or limit the statement.

The above concerns natural modality, but equivalent statements involving temporal modality are conceivable: ‘*All* S are sometimes P’, ‘*All* S are never P’, and so forth. Note that collectionality is used in a modal context; the actual proposition ‘*All* S are not P’ (meaning that not-all S are P, meaning that some are not, though some are), is not really collectional.

Collectional intent is often encountered in the antecedent or consequent of conditional propositions (for examples, ‘when *all* the cog-wheels are aligned, the key is able to turn’ or ‘when the button is pressed, *all* the lights come on’).

I will not here work out the logics of collective and collectional forms in detail. Each form needs to be analyzed for its exact implications, then the interactions of all the forms with each other and with dispensives (including all immediate and mediate inferences) must be looked into.

### 4. Quantification of Predicate.

The forms people currently use, and accordingly adopted by the science of Logic, are so designed that we can specify alternate quantities for the predicate, if necessary, simply by making another, additional statement in which the original predicate is subject and the original subject is predicate, with the appropriate distributions.

However, as an offshoot of the distribution doctrine, there have been attempts to invent forms which explicitly ‘quantify the predicate’ of classificatory propositions. Let us look into them briefly.

- On a singular level, the basic form would be ‘this S is this P’. The contradictory ‘this S is not this P’ would be compatible with ‘this S is that (meaning, some other) P’.

Normally, we need to know, say, ‘whether the girl is or is not (at all) pretty’, rather than ‘whether she is or not that pretty thing’. We may of course say ‘her dress was this shade of brown’; but here the indicative only specifies a kind of color, not an individual qualitative phenomenon. Someone may tell me ‘the girl I mean is the one we met last week’; but here the predicate is intrinsically a one-member class.

Normally, we use indicatives in the subject, rather than the predicate. The indicative is used to ‘hold down’ a first appearance, as our initial designation of the object: once, that is settled, we are only interested in discovering its further attributes as such.

Suppose I see a green and blue object, I may say ‘this green thing is blue’ (or vice versa), but I would have no need to establish class correspondence, since the object is already one and the same right before my eyes. It is not inconceivable that I perceive a green object and later a blue object, and then equate (or distinguish) them, saying ‘this green thing is (or is not) the same as that blue thing’; but this is a rare exception, and is it really classification?

When we say ‘this S is P’ we first intend to qualify the subject by the predicate (e.g. that baby was rather cute). We cannot transfer the designation ‘this’ from the subject to the predicate without missing the point, which is attribution. Also, we normally use ‘this’ to refer to entities, rather than qualities (though we can say ‘this green is rather dark’).

Still, theoretically, ‘this thing’ under discussion is indeed theoretically an instance of P as well as S, so that permutation to classificatory form is feasible. We have to remain formally open in this issue, since we do regard ‘all S’ as implying ‘this S’.

- With regard to plurals, ‘quantification of the predicate’ would give rise to the following forms: ‘all (or some) S are all P’ (both implying that all P are S); ‘No S are certain P’ (implying some P are not S) and ‘some S are not certain P’ (the latter two not excluding that all S be P — i.e. other instances of P).

The forms: ‘all S are some P’, ‘some S are some P’, ‘no S are any P’ and ‘some S are not any P’, would be equivalent to the established **A**, **I**, **E**, **O**. The rest would be relatively new.

Only the form ‘some S are not certain P’ contains information we cannot express in natural language: but that may be simply because we never need to make such a statement of partial exclusion in practice.

These forms have not aroused much interest, because they are artificial to our normal ways of thinking. If we have so far managed very well without them, why complicate things and try to introduce something no one will ever use?

However, to be fair, such statements are indeed used by logicians, if not by laymen, to clarify the distributions of terms. We would speak in that way to explain Euler diagrams, mentioning the one-one correspondence of individual members of distinct classes, or the overlap or separation of segments of classes. Thus, we may view them as specialized, rarely used — but still legitimate.

Quantification of the predicate could also be viewed as a special case of substitution.

# PART III. LOGICAL CONDITIONING.

## 20. CREDIBILITY.

### 1. Laws of Thought.

We began our study by presenting the laws of thought — the Laws of Identity, of Contradiction, and of the Excluded Middle — as the foundations of logic. We can see, as we proceed, that these first principles are repeatedly appealed to in reasoning and validation processes. But in what sense are they ‘laws’?

- Many logicians have been tempted to compare these laws to the
*axioms*of geometry, or the top postulates of natural sciences. According to this view, they are self-consistent hypotheses, which however are incapable of ultimate proof, from which all other propositions of logic are derived.

There is some truth to this view, but it is inaccurate on many counts. The whole concept of ‘systematization’ of knowledge, ordering it into axioms and derivatives, is itself a device developed and validated by the science of logic. It is only *ex post facto* that we can order the information provided by logic in this way; we cannot appeal to it without circularity. If logic was based on so tenuous a foundation, we could design alternative logics (and some indeed have tried), just as Euclidean geometry or Newtonian mechanics were replaced by others.

Logic is prior to methodology. The idea that something may be ‘derived’ from something else, depends for its credibility on the insights provided by the ‘laws of thought’. The ‘laws of thought’ ought not to be viewed as general principles which are *applied* to particular cases, because the process of application itself depends on them.

Rather, we must view *every particular occurrence* of identity, contradiction, and excluded-middle, as *by itself compelling*, an irreducible primary independently of any appeal to large principles. The principles are then merely statements to remind us *that* this compulsion occurs; they are not its source. This means that the ‘laws of thought’ are not general principles in the normal sense, but recognitions that ‘there are such events’. The science of logic is, then, not a systematic application of certain axioms, but *a record of the kind of events which have this compelling character* for us.

Note this well. *Each* occurrence of such events is self-sufficiently evident; it is only thereafter that we can formulate statements about ‘all’ these events. We do not know what to include under the ‘all’ beforehand, so how could we ‘apply’ the laws to anything? These laws cannot be strictly-speaking ‘generalizations’, since generalization presupposes that you have some prior data *to* generalize.

Thus, we must admit that *first* comes specific events of identity, contradiction and excluded-middle, with a force of their own, then we can say ‘these and those are *the kinds of* situations’ where we experience that utter certainty, and only lastly can we *loosely-speaking* format the information in the way of axioms and derivatives.

Nevertheless, it remains true that the laws of thought have a compelling character on their own. There is no way to put these laws in doubt, without implicitly arousing doubt in one’s own claim. Sophisms always conceal their own implications, and tacitly appeal to the laws of thought for support, to gain our credulity. We could, therefore, equally say that the principles as units in themselves are entirely convincing, with utter finality — provided we *also* say that every act of their ‘application’ is likewise indubitable. It comes to the same.

However, the previous position is more accurate, because it explains how people unversed in the laws of thought, can nonetheless think quite logically — and also how we can understand the arguments here made about the laws of thought. The inconsistency of denials of the laws of thought is one instance of those laws, and not their whole basis.

- What, then, is this ‘compulsion’ that we have mentioned? It is evident that people are not forced to think logically, say like physical bodies are forced to behave in certain ways. This is given: we do make errors, and these sometimes seem ‘voluntary’, and sometimes accidental. In any case, if thought was a mechanistic phenomenon, we would have no need of logical guidelines. We may only at best claim that we
*can and should, and sometimes do*, think in perfect accord with these laws.

The answer to this question was implicit in the above discussion. It is or seems evident that things do present themselves and that they do have certain contents (identity), and that these presentations are distinct from their absences (contradiction), and that there is nothing else to refer to (excluded-middle). Because these statements concern appearances as such, it is irrelevant whether we say ‘it *is* evident’ or ‘it *seems* evident’.

The concepts of reality and appearance are identical, with regard to the phenomenal; the concept of illusion is only meaningful as a subdivision of the phenomenal. These laws are therefore *always* evident, whether we are dealing with realities or illusions. We can wrongly interpret or deliberately lie about what we ‘see’ (if anything), but *that* we ‘saw’ and* just what* we ‘saw’ is pure data. Thus, the ‘compulsion’ is presented to us an intrinsic component of the phenomenal world we face.

The practical significance of this can be brought out with reference to the law of contradiction. We are saying, in effect, that whatever seems contradictory, is so. This statement may surprise, since we sometimes ‘change our minds’ about contradictions.

To understand it, consider two phenomena, say P1 and P2, in apparent contradiction, call this C1. One way to resolve C1, is to say that one or both of P1 and P2 are illusory. But we might find, upon closer inspection, that the two phenomena are not in contradiction; call this noncontradiction C2. So we now have two new phenomena, C1 and C2, in apparent contradiction; call this new contradiction C3.

The question is, does C3 imply that one or both of C1 and C2 are illusory? The answer is, no — what happened ‘upon closer inspection’ was not a revision of C1, but a revision of P1 and/or P2. So that in fact C2 does not concern exactly the same phenomena P1 and P2, but a slightly different pair of phenomena with the same names.

Thus, C1 and C2 could never be called illusory (except loosely speaking), because they were never in conflict, because they do not relate the same pair of phenomena. Nor for that matter may C3 be viewed as now erroneous, because the pair of phenomena it, in turn, related have changed.

Which means that our ‘intuition’ of contradiction is invariably correct, *for exactly the data provided* to it. A similar argument can be made with regard to other logical relations. The phenomena related may be unclear and we may confuse phenomena (thinking them the same when they are different) — but, at any level of appearance, the logical relation between phenomena is ‘compulsively evident’, inflexibly fixed, *given*.

In other words, *among phenomena, logical relations are one kind which are always real*; in their case, appearance and reality are one and the same, and there are no illusions. The laws of thought are presented as imperatives, to urge us to focus on and carefully scrutinize *the phenomena related*, and not to suggest that the *logical* intuitions of thought are fallible, once the effort is made to discern the relation.

This is not a claim to any prior omniscience, but a case by case accuracy. As each situation arises, its logical aspects are manifest to the degree that we inspect things clearly. Note well, we do not need to know *how* the intuition functions, to be able to know and prove *that* it functions well. We have called it ‘intuition’ to suggest that it is a direct kind of consciousness, which may well be conceptual rather than perceptual, but these descriptive issues are secondary.

Thus, with regard to the laws of thought, we have no ground for wondering whether they are animal instincts imposed by the structure of the mind, or for wondering whether they control the events external to it as well. In either case, we would be suggesting that there is a chance that they might be illusory and not real. If we claim that the mind is distortive, one way or the other, we put that very claim in doubt.

The mind is doubtless limited. It is common knowledge that mental conditions, structural or psychological or voluntary, can *inhibit* us from comparing phenomena with a view to their logical relation — but that does not mean that when the elements *are* brought together, the comparison may fail.

Nervous system malfunctions, personality disorders, drunkenness, fatigue — such things can only arrest, never alter these intuitions. As for evasions and lies, we may delude ourselves or others, to justify some behavior or through attachment to a dogma — but these are after the fact interventions.

### 2. Functions.

The laws of thought relate to the credibility, or trustworthiness, of phenomena. They clarify things in three stages. At the identity level, appearances are acknowledged and taken as a data base. At the contradiction level, we learn to discriminate clearly between real and illusory appearances. At the excluded-middle level, we introduce a more tempered outlook, without however ignoring the previous lessons. More specifically, their functions are as follows:

The first law assigns a minimal credibility to any thought whatsoever, if only momentarily; the evidence, such as it is, is considered. If, however, the ‘thought’ is found to consist of meaningless words, or is overly vague or obscure — it is as if nothing has appeared, and credibility disappears (until and unless some improvement is made). To the extent that a thought has some meaning, precision, and clarity, it retains some credibility.

The second law puts in doubt any thoughts which somehow give rise to contradictions, and thereby somewhat enhances the credibility of all thoughts which pass this test. In the case of a thought which is self-inconsistent (whether as a whole or through the conflicts of its parts), its credibility falls to zero, and the credibility of denial becomes extreme. In the case of two or more thoughts, each of which is self-consistent, but which are incompatible with each other, the loss of credibility is collective, and so individually less final.

The third law sets bounds for any leftover thoughts (those with more than zero and less than total credibility, according to the previous two laws): if special ways be found to increase or decrease their credibilities, the overall results cannot in any case be such as to transgress the excluded-middle requirement (as well as the no-contradiction requirement, of course). As we shall see, the processes of confirmation and discrediting of hypotheses are ways logic uses to further specify credibilities.

We see that, essentially, the law of identity gives credence to *experience*, in the widest sense, including concrete perceptions and abstract conceptual insights. The law of contradiction essentially justifies the logical intuitions of *reason*. The law of the excluded-middle is essentially directed at the projections of the *imagination*. This division of labor is not exclusive — all three laws come into play at every stage — but it has some pertinence.

The credibility of a phenomenon is, then, a measure of how well it fits into the total picture presented by the world of appearances; it is a component of phenomena, like bodies have weight. This property is in some cases fixed; but in most cases, variable — an outcome of the interactions of phenomena as such.

The laws of thought are, however, only the first steps in a study of credibility. The enterprise called logic is a continual search for additional or subsidiary norms. Logic theory develops, as we shall see, by considering various kinds of situations, and predicting the sorts of inferences which are feasible in each setting.

More broadly the whole of philosophy and science may be viewed as providing us with more or less rough and ready, practical yardsticks for determining the relative credibility of phenomena. However, such norms are not of direct interest to the logician, and are for him (relatively speaking) specific world views. Logic has to make do with the two broadest categories of reality and illusion — at least, to begin with.

### 3. More on Credibility.

Every phenomenon appears to us with some degree of *‘credibility’*, as an inherent component of its appearance; this is an expression of the law of identity. That initially intuitive credibility may be annulled or made extreme, through the law of contradiction; or it may be incrementally increased or decreased, by various techniques (yet to be shown), within the confines of the laws of contradiction and of the excluded middle.

Thus, credibility is primarily an aspect of the phenomenal world, and a specific phenomenon’s degree of credibility is a function of what other phenomena are present in the world of appearances at that stage in its development. Because phenomena interact in this way, and affect each other’s credibilities, credibility may be viewed as a measure of how well or badly any phenomenon ‘fits in’ with the rest.

‘Reality’ and ‘illusion’ are just the extremes of credibility and incredibility, respectively; they are phenomena with that special character of total or zero force of conviction. We cannot refer to a domain beyond that of appearances, for good or bad, without thereby including it within the world of appearances.

How do we know that all appearances must ultimately be real or illusory? How do we know that *median* credibility cannot be a permanent state of affairs in some cases, on a par with the extremes of credibility and incredibility? We answered this question, in broad terms, in our discussions on the laws of thought, as follows. More will be said about it as we proceed.

Reality and illusion are a dichotomy of actual appearances: for them, whatever is inconsistent is illusory, and everything else is real enough. Median credibility only comes into play when we try to anticipate future appearances, but has no equivalent in the given world. In the actual field of concrete and abstract experiences, things have either no credibility or effectively total credibility; it is only through the artificial dimension of mental projections that intermediate credibility arises.

Knowledge is merely consciousness of appearances; the flip-side, as it were, of the event of appearance. Viewed in this perspective, without making claims to anything but the phenomenal, knowledge is always a faithful rendering of the way things appear. We may speak of knowledge itself as being realistic or as unrealistic or as hypothetical, only insofar as we understand that this refers to *the kind of* appearance it reflects. These characterizations refer primarily, not to knowledge, but to its objects.

The difference between knowledge (in its narrower sense of, knowledge of reality) and opinion (in the sense of, the practically known), is thus merely one of degree of credibility *manifested by their objects* (at that time); we cannot point to any essential, structural difference between them. However, this distinction is still significant: it matters a lot that the objects carry different weights of conviction.

Changes or differences in appearances and opinion are to some extent *explained* by reference to variations in our perspective, and breadth and depth of consciousness. But this explanation does not annul the primacy of phenomena, in all their aspects.

In practice, median credibility is often not patiently accepted, but we use our ‘wisdom’ to lean one way or the other a bit, according to which idea seems to ‘hang together’ the best. But a contrary function of wisdom is the ability to see alternatives, or the remote possibility of suggested alternatives, and thus keep an open mind. The intelligent man is able to take positions where others dither, and also to see problems where others see certainties.

### 4. Opinion and Knowledge.

I would like to here mention in passing, without going into details, that work has been done by some logicians, in clarifying the logical properties of belief (or opinion) and knowing.

The logic of belief is concerned with the implications of propositions such as ‘S believes that P’, where S is a subject of consciousness and P is any proposition. Belief, disbelief, and uncertainty are subjectively given: they are facts immediately accessible to the subject, though they may be wrongly remembered or dishonestly reported. There are also iterative forms to consider, like ‘X believes that Y believes that Z’.

The following are some of the formal issues in this field. Mutual oppositions: believing something does not imply disbelieving its contradictory, since people sometimes do (however ‘illogically’) believe both a thesis and its contradictory; therefore, disbelieving and not-believing are not identical. Also, relationships to ‘alethic’ propositions: believing something does not imply that it is true, and the true may be disbelieved.

The logic of knowing may similarly be investigated, for forms like ‘S knows that P’. These topics are not unrelated, since knowing is taken to imply believing (ordinarily, though sometimes we resist), even if we believe some things without the degree of certainty which we qualify as knowing (or perhaps with reference to other standards of judgment). The distinction between conscious awareness and ‘tacit knowledge’ has to be considered.

Knowing is often regarded as based on more rigorous methodology than belief, and hence effectively implying (at least contextual) truth; whereas belief may be groundless or even contrary to reason. Knowing implies some effort of review and control of belief, with reference to logical standards of some sort. If one has trained oneself in logic, avoided all laziness, and tried to be honest, then as far as that subject is concerned his or her belief has become knowledge. For that subject now, though not necessarily at some other time or for other subjects, this is equivalent to the ideal of truth.

Belief, knowing and alethic truth are three parts of the same curve: truth is the ‘vanishing point’ toward which belief and knowing tend. Belief is more inertial, more affected by emotional forces, like peer group pressure or psychological factors. Knowing involves willfully freeing one’s mind of such prejudices and influences, but is still a function of one’s intelligence, logical skills, research efforts, the limitations of one’s cognitive faculties.

## 21. LOGICAL MODALITY.

### 1. The Singular Modalities.

I do not claim that my theory of logical modality as it stands solves all issues, but I think you will find it very productive, an impressive integrative force.

The concepts of ‘logical modality’ enable us to predict systematically all the ways credibility may arise in knowledge over the long-term. Credibility itself is not a type of modality, but the ground and outcome of logical modality. We shall immediately define the primary categories of logical modality, and thereafter discuss their development, their significance, and their justification:

** Truth** is the character of a proposition which seems more convincing than its negation, in a given context of knowledge. In the case of any proposition implied by its own negation, its credibility is extreme.

** Falsehood** is the character of a proposition which seems less convincing than its negation, in a given context of knowledge. In the case of any proposition implying its own negation, its incredibility is extreme.

A proposition is *‘ problematic’*, with regard to its truth or falsehood, if it seems to carry neither more nor less conviction than its negation, in the given context of knowledge. This is indicated by such expressions as ‘might or not be’ or ‘perhaps is and perhaps is not’.

In practical terms, the ** degree of credibility**, whether high, low, or median, of a proposition is a measure of the amount of evidence or counterevidence put forward on its behalf or against it. This refers to the weighting of information by confirmation or undermining, which topic will be dealt with more fully under the heading of adduction.

By (logical) ** context** is meant, the accumulated experiences and conceptual insights of the knower (a person or society) at the time concerned.

The context-specific concepts of logical modality are built on the awareness that: at every stage of knowledge, some things somehow seem ‘true’, other things somehow seem ‘false’, yet others seem ‘problematic’; and that these attributes often vary with the growth of experience and reasoning.

These observations suggest that, although every appearance is accompanied by some such characterization, the characterization is not in all cases firmly attached to the object, but is often a function of the experience and reasoning which have preceded them.

The concepts are thus formed, to begin with, only in recognition that such events occur, and that they are distinguishable by our consciousness, and that they each display such and such properties. Then we say: ‘Let us call this truth or falsehood or problemacy, as the case may be….’

It must be stressed that underlying the foregoing definitions of truth, falsehood, and problemacy, is the assumption that a sincere effort of awareness took place. It is difficult to insert such technical specifications in our definitions explicitly, without engaging in circularity, but there is no doubt that the definitions would lose all their value and significance without this tacit understanding.

A true or false proposition is called *‘assertoric’*, because it makes a definite claim. A problematic proposition is not assertoric: it presents an appearance with equal tendency in both directions, and therefore devoid of tendency; it calls upon us to consider a hypothesis.

Problemacy signifies a suspension of judgment. It does not signify the existence of ‘real’ indeterminacy, but only recognizes the appearance of indeterminacy in contexts less than complete. In reality, we believe, every issue is settled, once the event takes place; in omniscience, there would accordingly be no problemacy — it only arises in more limited viewpoints.

Problemacy has no equivalent outside logical modality; being freely open to change as knowledge evolves, there is no error in saying that any proposition we choose to formulate is at first encounter problematic.

Note that meaningful, precise, and clear, propositions may be true, false or problematic. Meaningless propositions are classified as false. Vague or obscure propositions, as at best problematic, if not false.

Factual assertorics of less than extreme credibility and problematics, give a semblance of co-presence or co-absence of opposites. The laws of contradiction and of the excluded middle are our reminders that that impression is transient; ultimately, everything is either totally credible or completely incredible. In other words, so long as we make no attempt to at once apply both truth and falsehood, or both untruth and unfalsehood, no law is broken; but as soon as we lay claim to more than the propositions suggest, we err.

For this reason, we can effectively discard nonextreme assertions and problems, and say of any proposition: it cannot be both true and false, and cannot be neither true nor false. There is ultimately no mixing or in-between of these attributes; our goal is to arrive to the extremes, not to linger on intermediate stages. There would be no point in constructing a logical system with reference to the finer gradations of credibility: it would be immobile.

### 2. The Plural Modalities.

Truth and falsehood are the categories of logical modality *with a single, given context* as their frame of reference.

Truth is a category of logical modality lying between logical necessity and possibility. Falsehood is the exact contradictory of truth, lying between logical impossibility and unnecessity. Truth is fact and falsehood is fiction, ideally. So we may call them the *‘factual’* level of logical modality; in analogy to the actual level of natural or temporal modality, or the singular level of extensional modality; but this is only an analogy, not an equation.

The categories of logical modality referring to *a plurality of unspecified contexts*:

Logical ** necessity** characterizes a proposition which is true in every context, and in that sense is true irrespective of any given context.

Logical ** impossibility** characterizes a proposition which is false in every context, and in that sense is false irrespective of any given context.

Logical ** contingency** characterizes a proposition which has neither the attribute of necessity nor that of impossibility, as they are above defined, so that it is true in some contexts and false in others.

Logical ** incontingency** is the negation of contingency, the common attribute of necessary and impossible propositions. Logical

*possibility*is the negation of impossibility, the common attribute of necessary and contingent propositions: truth in some contexts. Logical

*unnecessity*is the negation of necessity, the common attribute of impossible and contingent propositions: falsehood in some contexts.

With regard to corresponding concepts of logical ** probability** or

**.**

*improbability*We can say that, in this system, truth or falsehood correspond to mere incidence or nonincidence; necessity or impossibility signify the extremes (100%) of probability or improbability, and contingency concerns intermediate degrees (less than 100%) of these. Thus, to be consistent, we must define the logically probable as what would be true in most contexts (or false in a minority of contexts), and the logically improbable as what would be true in few contexts (or false in a majority of contexts).

These concepts would then enable us to specify our breadth of vision — effectively, how many eventual changes of context we have taken into consideration in making a prediction. The practical feasibility of this, with some precision, and the relation of logical probability and credibility, will be explored when we deal with adduction.

Thus, in summary, ** logical modality** may be defined as a qualification of propositions as such, informing us as to whether each is true or false, in this (i.e. a given) context, only some (unspecified) contexts, or all contexts, or somewhere in between these main categories.

Here again, it must be emphasized that ‘is true’ (meaning, seems more convincing than not) and ‘is false’ (seems less convincing than its contradictory), depend for their plausibility on our having sought out and scrutinized the available information with integrity. This issue is discussed in more detail in the next chapter.

I want to emphasize here that the concepts of logical modality, as here defined, are prior to concepts of logical relation, like implication, which (as we shall see) they are used to define.

The former are built on the vague, notion of a proposition being variously credible *‘in’* some context(s). Although this ‘in’ suggests that a kind of causality is taking place, it is not yet at the stage where specific relations like implication may be discussed. There is only a mental image of items ‘pushing’ others into existence; a very sensory notion.

Likewise, our first encounter with ‘credibility’ is very intuitive, something intrinsic to our every consciousness. The later systematic understanding of credibility, with reference to adduction, is merely a report on when it occurs, not a substitute for that primitive, inner notion.

It is interesting that, in Hebrew, the word for ‘with’ is *‘im’* (spelt ayin-mem), and that for ‘if’ is *‘im’* (spelt aleph-mem). In that language, if I am not mistaken, when verbal roots are that close, it signifies that the thoughts underlying them are also close. I wonder if the English words ‘in’ and ‘if’ have similar origins, rather than those most philologists assume.

Incidentally, also similar in Hebrew, are the words *‘az’* (spelt alef-zayin), meaning ‘then’ in time or logic, and *‘oz’* (spelt ayin-zayin), meaning ‘strength’. This confirms what I said above, that the notion of logical causality is rooted in an intuitive analogy to physical force.

### 3. Analogies and Contrasts.

Various analogies and contrasts between the singular and plural modalities are worthy of note. The former measure credibilities in any one context. The latter take a broader perspective, and compare credibilities in a variety of contexts. Thus, true, false, and problematic are comparable to necessary, impossible, and contingent — but they are not identical.

Contingent truth and falsehood are contextual, whereas necessity and impossibility (incontingent truth and falsehood) effectively transcend context. What holds in every context, holds no matter what the context, whereas the contextual is tied to context and in principle liable to revision (though that may never happen).

Note that it is the *realization* of contingency as truth or falsehood, which is relative to context, but the contingency in itself is no less absolute (with respect to context) than necessity or impossibility.

A careful distinction must be made between the truth, falsehood, or problemacy, of a proposition whose logical necessity, contingency, or impossibility is unspecified — and the truth, falsehood, or problemacy, of any proposed modal specification for that proposition. Failure to distinguish between these perspectives can be very confusing.

A proposition may be problematic to the extent that, not only do we not know whether it is true or false, but we do not even know whether it is logically necessary, contingent, or impossible.

Less extremely, we may know the proposition to be true or false (and thus, possible or unnecessary), yet not know whether it is logically necessary, contingent (possible *and* unnecessary), or impossible. In such case, the singular modality (the proposition per se) is assertoric, but the plural modality is still to some extent problematic.

If a proposition is known to be logically necessary or impossible, then it is assertoric with regard to both its plural modality (the incontingency) and to its singular modality (accordingly, true or false).

If a proposition is known to be logically contingent, it is assertoric with respect to its plural modality (the contingency). We may additionally know that the proposition per se is true or false, in which case it is also assertoric with respect to its singular modality. Or we may still be at a loss as to whether it is true or false, so that it is problematic with respect to its singular modality.

In any case, here again, problemacy does not signify real indeterminacy, but merely absence of sufficient knowledge, remember.

Our definitions make clear that problemacy should not be confused with logical contingency. A proposition may be definitely true or false, and so unproblematic, and still contingent; and a problematic proposition may after serious consideration be found to be necessary or impossible, whereas a properly contingent proposition should not thus change status.

Yet problemacy and contingency have marked technical analogies, which allow us to treat any problematic proposition (and therefore any proposition whatever, at first encounter) as *effectively* contingent in logical properties. Logic repeatedly makes use of this valuable principle. As will be seen, if the proposition is not indeed contingent, it will be automatically revealed so eventually through dilemmatic argument, so that no permanent damage ensues from our assumption.

Note that the definitions of the logical modalities are very similar to those of extensional, natural and temporal modalities. There is a marked quantitative analogy (this, some, all), so that we can refer to them as ‘categories of modality’; and there is a broad qualitative analogy (inclusion or exclusion in a wider perspective), yet with enough difference that we can refer to them as distinct ‘types of modality’.

Logical modality puts more emphasis on epistemology than ontology, in comparison to the other types. It primarily qualifies knowledge, rather than the objects of knowledge. Whereas natural modality refers to the objective circumstantial environment of events, temporal modality to surrounding times, and extensional modality to cognate instances — logical modality looks at the informational setting.

With regard to technical properties, logical modality is often similar to the other types, but some notable differences also occur, as we shall see as we go along.

### 4. Apodictic Knowledge.

The many-contexts concepts of logical modality are formed by reference to the awareness that there are items of knowledge which somehow would seem to be true or false no matter what developments in knowledge may conceivably take shape, while others seem somehow more dependent on empirical evidence for their acceptance or rejection. The former are often called ‘a priori’ or *‘apodictic’*, and the latter ‘a posteriori’.

At first sight, apodictic statements present a difficulty. They seem inaccessible to anyone with less than total knowledge. Only the fully omniscient could know what is necessary or impossible in the widest context. A normally limited mind like ours cannot have foreknowledge of any final verities. Indeed, even if we ever reached omniscience, how could we be sure we have reached it?

However, these skeptical arguments can be rebutted on several grounds. To begin with, they are self-defeating in that they themselves claim knowledge about the capabilities of omniscience, and they do so in no uncertain terms: therefore, they are intrinsically conceptually flawed. Logically, then, it is conceivable for a limited mind to acquire apodictic knowledge, somehow.

Secondly, it is noteworthy that our minds, though admittedly less than omniscient, are not rigidly limited in their powers of imagination. We are able to construct innumerable hypotheses even with a limited amount of factual data to play with. Thus, we are never limited to one context, the present one, but can manipulate ideas which go beyond it. Of course, this does not mean that our imagination is able to foresee all contexts. The more factual data we have to feed on, the more our imagination can stretch out — but we never have all the seeds.

Thirdly, the skeptical arguments misconstrue the issues. We defined the necessary as true, and the impossible as false — ‘in every context’. We did not say, the necessary is what is true, and the impossible is what is false — ‘to the omniscient’. Our definition does not exclude that the quality of necessity or impossibility be *given as such within any single context*, as an inherent component of the appearance. It does not logically mean that we have to foretell what goes on in other contexts besides our own.

And indeed, we find within common knowledge many instances of manifest necessity or impossibility, without need of further investigations. Such events constitute the experiential basis for these concepts.

The primary examples of this are Aristotle’s laws of thought. They strike us as intrinsically overwhelming, as in themselves capable of overriding any other consideration of knowledge. We can only ever deny them reflectively, by obscuring their impact; but the moment we encounter them plainly, their practical force is felt. When we are face to face with a specific contradiction, we see that it is nonsense and that something, somewhere must be amiss. That is why the laws of identity, of contradiction, and the excluded middle are naturally adopted as the axioms of logical science.

But other examples abound. More generally, as we shall see, a proposition is self-evident, if it is implied by its own negation, or implied by any contradictories; and a proposition is self-contradictory, if its affirmation implies its own negation, or implies any contradictories. It will be shown that a self-evident proposition displays the consequent property of being implied by any conceivable proposition, and a self-contradictory proposition that of implying any conceivable proposition. ‘Any’ here means ‘every’ — so that these are cases of logical necessity or impossibility.

This may occur *formally*, for all propositions of a certain kind whatever values be assigned to their variables. Indeed, the science of logic itself may be viewed as a record of all such occurrences. Or it may occur *contentually* (or ‘materially’), in the sense: not for all propositions of a certain kind, but only with certain specific contents. Note that this distinction is somewhat relative, depending on what we hold fixed and what we allow to vary.

Another way apodictic knowledge (or, for that matter, any knowledge) might conceivably be made available to a limited mind is through *revelation*, a communication from an omniscient mind. This is the logical premise of religion. *Faith* might be defined as the conviction that the information does indeed come from an infallible source, G-d. This topic is too vast to be discussed in this treatise, but I merely wanted to indicate the entry point.

Now, if logical necessity or impossibility are somehow given as components of the appearance of things in any context of knowledge, what is their difference from (contingent) truth or falsehood, which are also given?

Theoretically, once a proposition has been seriously scrutinized and found not to be necessary or impossible, it henceforth remains permanently contingent — just as once a proposition is seen to be necessary or impossible, its status is thenceforth established. In practice a mistake might conceivably be made, but this does not affect the principle.

The essence of necessity or impossibility is their property of self-evidence or self-contradiction; it is not their permanence, which is only incidental. Contingent truths or falsehoods may also be permanent; a proposition may happen to remain true or false without change as knowledge evolves, and yet never lose its contingent status. That some contingent truths or falsehoods do change over time, is irrelevant. Even in a total knowledge context, truths or falsehoods may be characterized as contingent.

Thus, we do not regard an obvious empirical truth like ‘it is now raining’, or a well-established law of nature like ‘the amount of matter and energy in the universe are constant’, as logically necessary, even though we believe them to happen to be fixed truths (each in its own way), because they do not seem self-evident; they are both therefore intrinsically logically contingent. The raw, factual finality of the former or the natural necessity of the latter do not affect their common logical status.

On this basis, we can also say that logical contingency is conceptually distinct from problemacy. In omniscience, problemacy disappears, but not logical contingency. The latter remains as a further qualification of certain truths and falsehoods, distinguishing them from logical necessities and impossibilities, respectively. It follows that contingency as such is not a lower status than necessity or impossibility.

Lastly, note, a necessity or impossibility may be immediately apparent to anyone, or we may need to go though a long or complicated reasoning process to make it apparent. But in either case, the sense of obviousness is given within the appearance itself, so that the ease or difficulty with which we were brought to the insight are irrelevant to its finality.

It is hard to distinguish a priori and a posteriori knowledge by reference to the concepts of reason and experience. The former is indeed more purely analytical, but it cannot occur without the minimum of experience on the basis of which the concepts involved are meaningful and clear. Likewise, the latter is indeed more likely to be affected by changes in experience, but its conceptualization and logical evaluation involve a great deal of rational activity.

## 22. CONTEXTUALITY.

### 1. Statics.

We defined logical modalities with reference to the relative credibilities of appearances ‘within contexts’. We will here try to clarify what constitutes a context, and its role.

In a very narrow, ‘logical’ sense, one might refer to the context of a proposition as any arbitrary set of propositions. In this sense, a proposition could be taken in isolation and constitute its own context. It might still appear to us as true (if in itself reasonable looking) or false (if obviously internally inconsistent) or even problematic (if of uncertain meaning). Likewise for any larger set of propositions we choose to focus on exclusively. But this leads to a very restricted sense of truth or falsehood.

In practice, there is no such animal. A more ‘epistemological’ understanding of context is called for. The effective context of any proposition is not arbitrarily delimitable, but is a very wide body of information, which, whether we are conscious of it or not, impinges on our judgement concerning the proposition. It is the ‘status quo’ of knowledge at a given time, for a given individual or group.

A proposition is not just a string of words or symbols written on a piece of paper; it has to mean something to become an object of logical discussion. We cannot consider it in isolation, because our consciousness is, like it or not, always determined by a mass of present or remembered perceptual and conceptual data. This periphery is bound to affect our reaction to the proposition at hand.

It is in acknowledgement of this dependency that our definitions of logical modality must be constructed. The context of a proposition is thus all the things we are experiencing or thinking, or remember or forgot having experienced and thought — which happen to color the proposition at hand as credible or not, to whatever degree.

This is not intended as a psychological observation, suggesting that our judgment is being warped by structural or emotional factors; in some cases it indeed is, in others not. Nor is the issue what we consciously take into consideration; that may have no effect, and there may be unconscious influences anyway.

It is merely a recognition that the appearance of realism or unrealism of any proposition is always a function of a great amount of data, besides it and any artificially selected framework. The contextual data generating such a result include: perceptions, direct conceptual insights, and indirect inductions and deductions. Hence the concept of a context, as here used. It refers to the actual surrounding conditions of our knowledge.

It is hard to pinpoint precisely and with unfailing accuracy just which of the peripheral information impinges on a given proposition’s evaluation. Innumerable wordless sensations, mental images, and intuitions, are involved, and merely having had logically relevant experiences or thoughts, does not entail that they played any effective role in the present result. All we can say with certitude is that a lot of data is involved in the final display of some quality of credibility by a proposition.

*The whole of logical science may be viewed as an ongoing attempt to investigate this aetiology.* Its job is to find just what causes propositions to carry conviction or fail to do so, and how the totality of knowledge can be gradually perfected. We have seen its work in the domain of deduction with certain categorical propositions; now other forms are about to be analyzed. The solution to the problem of knowledge is not found in simplistic and vague pontifications, nor in a step-by-step linear guidebook, but in a vast tapestry of interlocking considerations.

### 2. Dynamics.

The concepts of truth, falsehood, and problemacy, refer to the deployments of credibility in a static context, the ‘state of affairs’ in knowledge at a given stage. The concepts of necessity, impossibility, and contingency, refer to the changes of credibility: they consider knowledge more dynamically.

Knowledge is an evolving thing. We, human beings, are none of us ever omniscient or infallible. If our consciousness was unlimited by space, time, and structural resources, like Gd’s, there would be no problematic knowledge: every proposition would be true or false with finality. Just as reality is one, knowledge would be one and complete.

But reality is opened to our consciousness piecemeal, over time. We are obliged to repeatedly adapt to new factual input. Indeed, we have to actively dig into reality, if we want to approach that ultimate goal of total consciousness of everything.

We know we cannot reach that goal, since we have already missed out on enormous tracts of reality in the distant past, and the whole future is ahead of us, unexplored. We know that innumerable phenomena are happening all around us and within us, all the time, at every level (from the sub-atomic to the astronomical, from the material and physiological to the mental and spiritual); and we cannot keep track of all that. Thus, the data available to us is inevitably restricted.

Furthermore, our faculties of knowledge can play tricks on us, and draw us away from the goal. Our eyes may be myopic, our memory may fail, our reasoning may be muddled, we may be too imaginative, our mind may be moved by very subjective, emotional, considerations. We have to somehow make-do, in spite of all such imperfections in our make-up.

Our response to these limitations, if we are intent on knowing reality, is staying aware of our mental processes, and unflagging reevaluation of what and how much we know or ignore. This is where logical modality comes into play. It provides us with labels we can attach to each and every proposition, which assign it a rank, as we proceed.

Theoretically, we take the full body of everything we have experienced or thought thus far, and order the present information in a hierarchy. Tools may be invented to increase our certainties: eyeglasses, the written word, a science of logic. The sources of information are considered: we distinguish between the fictions of our imagination and the facts of sense data, between vague and clear concepts, between fallacious and rigorous argumentation.

In practice, things are more dynamic than that. We may take some part of our data base, and hold it still long enough to evaluate it with the proper amount of reflection. But, on the whole, the process is on-going, an ad hoc response to the flux of information. Logical modalities allows us to register our value-judgments of this kind as we proceed, like a running commentary.

### 3. Time-Frames.

Now, there are three ways for knowledge to evolve, and credibility to change. We may associate the word ‘context’ to the sum total of knowledge, the whole environment — or, more restrictively, to a given body of fundamental axioms and raw data, a *framework*. Here, let us use it in the latter sense.

We may not have drawn all the possible lessons from these primary givens; the process is not automatic, but has a time dimension. A proposition may be logically implicit in knowledge I already have, but it may take me time and effort to discover it.

There is always a great deal of undigested, unexploited information in our memory banks, and accessing it and assessing it demand time and skill. I mean, Philosophy, for example, requires relatively little raw data to develop considerably, because it pursues facts implicit in every existent. This is internal development, or context *intensifying*.

Or we may receive new input of rational axioms and empirical data to consider. Here, two alternatives exist: either the new facts already existed out there, but unbeknown to us; or some change occurred in these external objects themselves, which we accordingly now absorb as new existents. These are developments fed from the outside, or context *extending*.

Thus, we may distinguish between three time-frames for modality change: the external time in which objects change into new objects; the interfacial time of turning our attention and sensors towards pre-existing objects — to extend context; and the internal time of mental assimilation of memory (analyzing, comparing, checking consistency) — the work of intensifying context.

The first of these essentially pertains to natural and temporal modality; the second, extensional modality; the third, is the time-frame of logical modality. But all of them, if only incidentally, concern logical modality.

### 4. Context Comparisons.

That our definitions of truth and falsehood do not specify the context taken as being final and ideal, is not a relativistic position. It is merely intended as a statement that every proposition’s credibility is conditioned by a totality.

The given context is pragmatically accepted as a starting point for further inquiry, without thereby being regarded as ‘the best of all possible contexts’. It is subject to change, to improvement. Some contexts are to be favored over others — the exact grounds just need to be elucidated.

We might refer to the overall credibility of a context. We could perhaps consider any given context as a whole, and (of course, very roughly) sum-up and average the credibilities of its constituents, and thus get an estimate of its finality or staying-power. But, quite apart from the issue of practical feasibility, I do not think this would be of any use. The relative credibilities given within each context pertain to that context alone, and have no bearing on the relative credibilities in other contexts.

The general principle for comparing contexts seems obvious enough. Contexts are of varying scope and intensity, and it is clear that *the deeper and wider the context, the closer to final* will the impressions of truth or falsehood concerning any proposition in it be; and the less numerous will doubtful cases be. Thus, the bigger and more cohesive the context, the better.

The ideal context of omniscience is beyond man’s power, we can only gradually approach it. But we can say that in that ultimate, limiting case, the impressions of truth or falsehood would be final, subject to no further change or appeal; and furthermore, there would be no in-between impressions of a doubtful kind, since reality once established is determinate. Here, knowledge and reality would correspond entirely.

When we apply the above principle to one person over time, it is relatively easy to say which context is to be preferred. The more information at his or her disposal, the more this information has been carefully sifted for hidden messages, the more certain may that person be. For the individual, improvement is almost inevitable over time, because his or her context is a widening circle.

We always refer to appearance, though we can distinguish between prima-facie impressions and well-tested impressions. The two kinds of impression are essentially the same in nature, but they have different positions in a continuum stretching from subjectivity and mere belief (which still however contain seeds of objectivity and knowledge) to ultimate realism and certainty.

When, however, we compare the contexts of two (or more) people, it is not so easy to say which is better or worse. Each may have data the other lacks, and each may have thought about any item of data they have in common more thoroughly than the other. Thus, they may disagree in their conclusions, and yet both be ‘right’ for their respective contexts. And since their contexts overlap in only some respects, so that neither embraces the other as a whole, the contexts cannot be rated better or worse.

All we can do is focus on specific areas of knowledge, and consider the relative expertise of each individual in that area. If someone is a specialist in some field, we may well assign greater credibility to his or her pronouncements on the subject. On this basis, we may even trust a person we know to be generally very wise, without committing the fallacy of ‘ad hominem’.

### 5. Personal and Social.

We must distinguish, here, between personal and social knowledge.

At the lowest level, is ‘personal knowledge’. Some people are better at knowing than others, because of their healthier faculties, or because they are endowed with more intelligence and insight, or because they are more interested, more careful, and make more of an effort, in this domain. Also, individuals inevitably have different quantities of information at their disposal, both inner and outer.

‘Social knowledge’ is an ideal. We collectively, across cultural boundaries and the generations, gradually compile a record of common knowledge, agreed upon methods, information and conclusions. It is the human heritage, our shared data bank.

An individual may admittedly have more knowledge of some field than everyone else at a given time; he may get to share it, or it may disappear with him. There may be specific disagreements at any time between groups of individuals. It may even happen that the majority of the peer group wrongly rejects an individual’s valuable contribution.

Yet, over time, the collective enterprise we call Science develops, a pool of knowledge greater and truer than any which individuals can fully match, based on a methodological consensus.

Since credibilities depend on context, individuals may assign different credibilities to the same proposition. To that extent, truth and falsehood are often ‘subjective’, since they reflects the mental abilities and dispositions of people.

Still, I may take all the premises of another person and demonstrate that his evaluations are logically incorrect even for his context. In a sense, I start off with the same context as him, and end up with a slightly different version; but in another sense, I have merely clarified the given context, brought out its full potential, without significantly altering it. If he is intelligent and honest enough, he normally bows to the evidence.

Thus, the contextuality of credibility need not imply its utter subjectivity. The evaluation can only ultimately be viewed as subjective in the pejorative sense, if it is contextually wrong.

And even then, such accusation can only be leveled fairly if the individual allowed psychological forces to sway his judgment. He may be intellectually negligent through laziness, or dishonestly evade unpleasant or frightening data or thoughts, or insincerely report his conclusions. If the error was honest, merely due to a failure to notice a connection, we can hardly criticize him, only correct him.

We get around these problems of personal weakness through the institution of social knowledge, science. This allows us to collectively ‘average-out’ the subjective vector. We mutually scrutinize and criticize each other’s contributions, until we are of one mind. There may still be collective delusion, but that at least eliminates personal deviations from logical norms.

We presume that the influence of our collective mind-sets will gradually wither away as knowledge develops further. This assumption is justified by previous developments: we have seen historical examples of liberation from ideas which seemed immovable. The notion that science is inevitably subjective, is derived from such liberations, and cannot be used to denigrate them.

## 23. CONJUNCTION.

### 1. Factual Forms.

In this chapter, we begin to analyze the various ways two or more propositions, or sets of propositions, of any kind, may be correlated. A proposition so considered, in relation to other propositions, is called a ** thesis**; we symbolize theses by using letters such as P, Q, R,…. The negation of any thesis is called its

**; that is its exact logical contradictory: the antithesis of thesis ‘P’ is ‘nonP’, and vice versa.**

*antithesis*The primary form of correlation is conjunction; this is expressed by means of the operator ‘and’, or its negation. On a factual level, the conjunction (or positive conjunction) of two theses typically takes the form *‘***P and Q***’*. The contradictory of ‘P and Q’ would be *‘ Not-*

**{P and Q}**

*’*, where the ‘not’ negates the ‘and’; this may be called nonconjunction (or negative conjunction). In the context of conjunction, a thesis may be called, more specifically, a conjunct.

Most simply, the theses are categorical propositions of any form, so that their conjunction may be viewed as a compound categorical. But, by extension, a thesis may itself be a conjunction of two or more categorical propositions; or it may consist of any other, more complex, kind of proposition, or any mix of various kinds of propositions conjoined together. Thus, a thesis may ultimately be a whole, intricate theory.

Logical conjunction of two theses simply affirms both of them as true, implying that they are true separately as well as together. Thus, ‘P and Q’ (or ‘P with Q’) may be read as ‘{P and Q} is true’, implying ‘{P is true} and {Q is true}’. The ‘is true’ segment may be left tacit or made explicit, as with categorical affirmations.

The contradictory form, ‘not-{P and Q}’ simply denies that the two theses are *both* true, without asserting that they are *both* false. All it tells us is that at least one of the two theses is false, without excluding that an unspecified one of them be true, nor excluding that both be false. Thus, ‘not-{P and Q}’ may be read as ‘{P and Q} is false’, which does not imply that ‘{P is false} and {Q is false}’.

Thus, whereas the ‘and’ relation is fully assertoric, with regard to the parts as well as the whole, the ‘not-and’ relation is much more indefinite. It gives us limited information: it is assertoric with regard to the whole, but leaves the parts problematic. This problemacy should not even be interpreted as a logical contingency: not only do we not know of each thesis in isolation whether it is true or false, we do not even know whether it is contingent or incontingent. Keep that well in mind.

By definition, ‘P and Q’ and ‘Q and P’ are equivalent: the relation is reversible; also, ‘P and P’ is equivalent to ‘P’ alone: repetition of a thesis does not affect it. Likewise, by definition, ‘not-{P and Q}’ and ‘not-{Q and P}’ are equivalent: the relation is reversible; note however that ‘not-{P and P}’ is equivalent to ‘not-{P}’ alone, since ‘P and P’ means ‘P’.

The three forms ‘P and nonQ’ (or, ‘P without Q’), ‘nonP and Q’ (or ‘Q without P’), ‘nonP and nonQ’ (or ‘neither P nor Q’), are derivative forms of positive conjunction, obtained by substituting antitheses for theses in the original formula. Likewise, the three forms ‘not-{P and nonQ}’, ‘not-{nonP and Q}’, ‘not-{nonP and nonQ}’, are derivative forms of negative conjunction, obtained by substituting antitheses for theses in the original formula. We thus have a grand total of eight forms.

Note the we have used the word ‘conjunction’ in two senses. In a wider sense, it includes both the positive and negative forms. In a narrower sense, it includes only the former, the latter being called ‘nonconjunction’. Note that a positive conjunction is denied by negating any one, or any set, or all, of its parts, which means that one of the remaining alternative positive conjunctions must be true; thus, nonconjunction may be viewed as an abridged reference to the outstanding conjunctions.

Conjunction may of course involve more than two theses, as in ‘P and Q and R and..’., signifying that they are all true individually as well as collectively. Conjoining an additional thesis to a conjunction of two or more other theses, just results in a conjunction of all the theses, in a normal string: ‘{P and Q} and {R}’ simply means ‘P and Q and R’. Knowledge as a whole may be viewed as a conjunction of all the propositions in our minds.

Nonconjunction of more than two theses, as in ‘not-{P and Q and R and…}’ accordingly signifies that the theses are not *all* true, without implying any further information concerning each thesis alone. Any combination of theses and antitheses other than the one denied, would be acceptable. We shall develop the theory of conjunction with reference to two-theses forms, but the results can be extended with appropriate carefulness to forms with more than two theses.

### 2. Oppositions of Factuals.

The following table lists the various forms of conjunction (or positive conjunction), and shows the truths (**T**) and falsehoods (**F**) of theses and antitheses they imply. We see that, in contrast, nonconjunctions (or negative conjunctions) leave the individual theses and antitheses problematic (**?**): their information is purely collective. I have labeled these forms **K1**–**K4** and **H1**–**H4**, as shown, for convenience.

Symb. | Conjunction | P | Q | nonP | nonQ |

K1 | P and Q | T | T | F | F |

H1 | not-{P and Q} | ? | ? | ? | ? |

K2 | P and nonQ | T | F | F | T |

H2 | not-{P and nonQ} | ? | ? | ? | ? |

K3 | nonP and Q | F | T | T | F |

H3 | not-{nonP and Q} | ? | ? | ? | ? |

K4 | nonP and nonQ | F | F | T | T |

H4 | not-{nonP and nonQ} | ? | ? | ? | ? |

The four positive conjunctions exhaust the possible ways two theses and their antitheses may be positively conjoined, and are mutually exclusive. That is, *one of them must be true, and three of them must be false*. If any one is true, the other three must be false; but if one of them is false, the status of each the others is undetermined. Thus, the oppositional relation of any pair of positive conjunctions is contrariety.

The oppositions of the four negative versions relative to each other is: *three of them must be true, and one of them must be false*. If one of them false, the other three must be true; but if one of them is true, it is uncertain what the status of each of the others is. This follows from the interrelations of the positive versions. Thus, the oppositional relation of any pair of negative conjunctions is subcontrariety.

The opposition of any pair of positive and negative conjunctions, other than a pair of formal contradictories, is therefore subalternation. Proof: consider any positive conjunction, its truth implies the three others to be false, and therefore implies their contradictories to be true; on the other hand, its falsehood does not have further implications.

Thus, we could present the eight forms of conjunction in a cube of opposition, with the four positive forms in the upper corners and the four negative forms in the lower corners. The top plane involves contrariety, the bottom plane involves subcontrariety, the diagonals through the cube involve contradiction, and the four remaining faces involve subalternation in a downward direction.

### 3. Modal Forms.

The eight factual forms of conjunction are the singular level of logical modality. Let us now investigate the corresponding plural levels of logical modality. Each of the factual conjunctions has a possible equivalent below it and a necessary equivalent above it. Thus, we have to consider 2X8 = 16 modal conjunctions, in addition to the 8 factual ones. They are (always referring to logical modality, needless to repeat):

Positives | Negatives |

{P and Q} is necessary | {P and Q} is impossible |

{nonP and Q} is necessary | {nonP and Q} is impossible |

{P and nonQ} is necessary | {P and nonQ} is impossible |

{nonP and nonQ} is necessary | {nonP and nonQ} is impossible |

{P and Q} is possible | {P and Q} is unnecessary |

{nonP and Q} is possible | {nonP and Q} is unnecessary |

{P and nonQ} is possible | {P and nonQ} is unnecessary |

{nonP and nonQ} is possible | {nonP and nonQ} is unnecessary |

The factuals fit in between these two levels of modality, of course; they are less than necessary, but more than possible.

Now, just as the factual positives implied that their respective theses are not only collectively true, but individually true — so the necessary positives imply that their theses are each (as well as all) necessary, and the possible positives imply that their theses are each (as well as all) possible. However, in the latter case, it does not follow that the antitheses are equally possible, note well.

In contrast, none of the negatives tell us anything about the logical modalities of their respective theses. In all cases, the statuses of the individual theses are left entirely problematic; all we have is collective information. Not only are we left in the dark as to whether any thesis is true or false, but there is no specification as to whether it is necessary or possible or unnecessary or impossible.

Thus, for examples. ‘P and Q are necessary’ implies that P is necessary (and nonP is impossible); and likewise for Q. ‘P and Q are possible’ implies that P is possible, not impossible (and nonP is unnecessary, not necessary) — but without excluding that P be necessary or contingent: both are acceptable; and likewise for Q.

‘P and Q are impossible’ (meaning: ‘not-{P and Q} is necessary’) does not imply that P and Q are each impossible, but is equally compatible with each of them being contingent or necessary — except that in the latter case, if one theses is necessary, the other would needs be impossible, to satisfy the overall requirement of the form. ‘P and Q are unnecessary’ (meaning: ‘not-{P and Q} is possible’) allows for each of the theses to be necessary, contingent or impossible — provided they are not both necessary at once.

Similarly, for the remaining forms. Thus, we see that each form delimits some collective property of the theses, in some cases implying some individual properties; but in most cases, the form leaves some open questions, some areas of doubt, which would require additional statement(s) to specify in full.

Only the necessary positives fully define the factual and modal status of the theses (they are equally necessary). The factual positives establish the factuality and possibility of the theses, but leave their exact modal status (necessary or contingent) undetermined. The possible positives establish the possibility of the theses, but leave their factual and exact modal status untold.

The negatives are even less committed with regard to their theses. It is very significant to note that although a negative conjunction makes mention of a proposition as one of its theses, it does not thereby imply it as even logically possible. One might think that the mere mention of a proposition is always an admission of its possible truth; but here we learn that such assumption is unjustified.

The value of such indeterminacy is that it allows us to verbally capture just precisely those relational details which are of interest to us, without being forced to know more than we do at that point in time. If we could only make statements where every issue is already resolved, we would be left wordless until we had all the requisite details.

Be careful not to confuse problemacy and logical contingency. A proposition may be so problematic, that we do not even know whether it is logically contingent or incontingent, let alone whether it is true or false; or it may be only problematic to the extent that, though we know it to be contingent, we do not know whether this contingency is realized as truth or falsehood on the factual level.

### 4. Oppositions of Modals.

The following table lists the various forms of modal conjunction, and shows the necessity (**N**), impossibility (**M**), possibility (**P**), unnecessity (**U**), or problemacy (**?**), of individual theses and antitheses, implied by each modality (cum polarity) of conjunction, in accordance with our previous comments. Note the labels assigned, namely **K1**–**K4**, **H1**–**H4**, with suffix **n** or **p**, as the case may be, for convenience.

Symb. | Conjunction | Modality | P | Q | nonP | nonQ |

K1n | P and Q | necessary | N | N | M | M |

K1p | P and Q | possible | P | P | U | U |

H1n | P and Q | impossible | ? | ? | ? | ? |

H1p | P and Q | unnecessary | ? | ? | ? | ? |

K2n | P and nonQ | necessary | N | M | M | N |

K2p | P and nonQ | possible | P | U | U | P |

H2n | P and nonQ | impossible | ? | ? | ? | ? |

H2p | P and nonQ | unnecessary | ? | ? | ? | ? |

K3n | nonP and Q | necessary | M | N | N | M |

K3p | nonP and Q | possible | U | P | P | U |

H3n | nonP and Q | impossible | ? | ? | ? | ? |

H3p | nonP and Q | unnecessary | ? | ? | ? | ? |

K4n | nonP and nonQ | necessary | M | M | N | N |

K4p | nonP and nonQ | possible | U | U | P | P |

H4n | nonP and nonQ | impossible | ? | ? | ? | ? |

H4p | nonP and nonQ | unnecessary | ? | ? | ? | ? |

Since the categories of logical modality are by definition distinguished with reference to a quantity of contexts, the oppositions of the various modalities of conjunction among themselves, can be deduced from the oppositions between the corresponding factual conjunctions, given in an earlier section of this chapter, and the general doctrine of ‘quantification of oppositions’, which we worked out in an earlier chapter (14.1) with reference to categoricals.

Thus, since the forms **K1** and **H1** are contradictory, the oppositions between **K1n**, **K1**, **K1p**, **H1n**, **H1**, **H1p**, are the same of those between the categoricals **A**, **R**, **I**, **E**, **G**, **O**. Likewise for similar sets.

Since the forms **K1**, **K2**, **K3**, **K4**, are contrary to each other, it follows that: the forms which subalternate these factuals, **K1n**, **K2n**, **K3n**, **K4n**, are contrary to each other, and to them; and the forms which these factuals in turn subalternate, **K1p**, **K2p**, **K3p**, **K4p**, are neutral to each other, and to them.

Since the forms **H1**, **H2**, **H3**, **H4**, are subcontrary to each other, it follows that: the forms which subalternate these factuals, **H1n**, **H2n**, **H3n**, **H4n**, are neutral to each other, and to them; and the forms which these factuals in turn subalternate, **H1p**, **H2p**, **H3p**, **H4p**, are subcontrary to each other, and to them.

Since the form **K1** subalternates the forms **H2**, **H3**, **H4**, it follows that: **K1n** subalternates **H2n**, **H3n**, **H4n**, and therefore **H2**, **H3**, **H4**, and **H2p**, **H3p**, **H4p**; but **K1** is neutral to **H2n**, **H3n**, **H4n**, though it subalternates **H2p**, **H3p**, **H4p**; and **K1p** is neutral to **H2**, **H3**, **H4**, though it subalternates **H2p**, **H3p**, **H4p**. Likewise, for similar sets.

## 24. HYPOTHETICAL PROPOSITIONS.

### 1. Kinds of Conditioning.

We saw in the previous chapter that two or more propositions may be correlated in various ways, with reference to conjunctions (involving the operator ‘and’) of various polarities and logical modalities.

Implicit in certain conjunctive forms are relationships of ‘conditioning’; they signify a certain amount of interdependence between the truths and/or falsehoods of the theses involved. These relationships are definable entirely with reference to modal conjunction, so that we may fairly view all forms of conjunction, and all forms which may be derived from them, as one large family of propositions called ‘conditionals’.

However, in a narrower sense, and usually, we restrict the name conditional to the derivative forms which employ operators like ‘if’. The remaining derivative forms, which employ operators like ‘or’, are called disjunctive.

These issues of terminology are of course of minor import. What counts is that conjunctive, conditional, and disjunctive propositions are ultimately all different ways of saying the same things, as far as logic is concerned. Nevertheless, because each of these formats reflects a quite distinct turn of thought, they are worthy of separate analyses.

We are in this part of our study concerned with conditioning in the framework of logical modality. But as we shall see eventually, each other type of modality also gives rise to a distinct type of conditioning.

Logical conditionals are more commonly known as ‘hypothetical’ propositions — this more easily distinguishes them from non-logical (not meaning illogical) conditionals, meaning natural, temporal or extensional conditionals, which may therefore simply be called ‘conditionals’, in a narrower sense.

Hypothetical propositions are essentially concerned with the logical relations between propositions, or sets of propositions. This area of Logic is therefore quite important, as it constitutes a self-analysis of the science, to a great extent — the ‘logic of logic’. But it is also a specific investigation, like any other area of Logic, for the purposes of everyday reasoning.

The sequence in hypotheticals, the ordering of their theses, is what we call ‘logical’. It is not essentially temporal, though the mental sequence is of course temporal, one thought preceding the next — we can be aware of only so much at a time, beyond that we function linearly, in trains of thoughts. Some thoughts are linked into chains by precise relational expressions, but their sequence should not be viewed as to do with natural causation between mental phenomena per se. Thought processes are sometimes apparently involuntary, but for the most part there plainly seems to be a volitional element involved; indeed, if thought was automatic, there would be no call for logic.

Logical sequence has rather to do with conceptual breadth. The wider proposition is viewed as including, or implying, its consequences, in a timeless manner. The exclusive proposition ‘Only if P, then Q’, though formally identical to the reciprocal relation ‘If P, then Q, and if Q, then P’, suggests that P and Q are not logically quite interchangeable, but that P has a certain conceptual primacy over Q, that their order matters. The suggested order is not merely in the time of arrival of the thoughts about P and Q, but more deeply concerns the hierarchy of their factual contents.

### 2. Defining Hypotheticals.

- The paradigmatic form of hypothetical proposition is
*‘If P, then Q’*, where P and Q are any theses. The former, P, is known as the antecedent, and the latter, Q, as the consequent. The relation between them is minimally defined by saying that*the conjunction of P and nonQ is impossible*. This means that the affirmation of P and the denial of Q are incompatible; given that P is true, Q cannot be false, and it follows that Q must also be true. We can also say: P implies Q.

Note the correspondence of this proposition to the negative modal conjunction labeled **H2n** in the previous chapter; as we saw, this leaves the individual theses P and Q entirely problematic at the outset: they need not even be logically possible. Note well also that the unmentioned conjunctions ‘P and Q’, ‘nonP and nonQ’, and ‘nonP and Q’ are all left equally problematic; one should not surmise, from the allusion to P being followed by Q, that the conjunction of P and Q is given as logically possible.

The expression ‘if’ normally suggests that the truth of the antecedent ‘P’, and thereby of its consequent ‘Q’, are not established yet; they are still in doubt. Note that the ‘if’ effectively colors both the theses.

The expression ‘then’ (which in practice is often left out, but tacitly understood) informs us that, in the event that the truth of the antecedent is established, the truth of the consequent will logically follow. The form ‘if P, then Q’ does not specify whether P is likewise implied by Q, or not; it takes an additional statement to express a reverse relation.

A note on terminology: officially, in logical science, the whole relation ‘if P, then Q’ is called a hypothetical proposition, in the sense that it includes one thesis in another. The proposition as a whole is assertoric, not problematic (unless we specify uncertainty about it, of course); it is the two theses in it which are normally problematic. But colloquially, we understand the expression ‘hypothetical’ as signifying problemacy, so confusion is possible.

Etymologically, the word ‘hypothesis’ could suggest a thesis which is placed under another, and so might be applied to the consequent; here, the sense is that it is ‘conditioned’ upon the truth of the antecedent (which, however, is normally in turn conditioned by other theses). But, again in practice, we often look upon the antecedent as the ‘hypothesis’, because it is qualified by an ‘if’ and underlies the other thesis; here the sense is that our thesis is placed before the consequent (which, however, is normally more or less equally ‘iffy’).

Be all that as it may, logical science has frozen the various expressions in the special senses described.

- The contradictory of the ‘if P, then Q’ form is
*‘If P, not-then Q’*. This merely informs that*the conjunction of P and nonQ is not impossible*. It tells us that: if P is true, it does not follow that Q is true; Q may or not be true for all we know, given only that P is true. We can also say, P does not imply Q.

Note the correspondence to the positive modal conjunction labeled **K2p** in the previous chapter; as we saw, this implies that P is logically possible and Q is logically unnecessary, though both individual theses are of course left problematic with regard to their factual status. One should not surmise, from the allusion to Q rather than nonQ, that Q is given as logically possible. Note well also that the unmentioned conjunctions ‘P and Q’, ‘nonP and nonQ’, and ‘nonP and Q’ are all left equally problematic.

It is not excluded that P and Q have some other positive relation; for instances, that P together with some additional conditions imply Q, or that Q implies P. It is also conceivable that P is not only compatible with the negation of Q, but implies it; or at the other extreme, that P and Q are totally unrelated to each other. In any case, here again, the theses P, Q are normally problematic, though the proposition as a whole is assertoric.

The name ‘hypothetical’ may be retained for such negative forms insofar as the prefix ‘if’ is equally involved; likewise, the name ‘antecedent’ for P remains correct; but for Q, the name ‘inconsequent’ would be more accurate here. For, whereas the positive form ‘If P, then Q’ suggests that Q is a logical consequence of hypothesizing P, the negative form ‘If P, not-then Q’ denies such connection (for this reason it is called the ‘nonsequitur’ form, the Latin for ‘it does not follow’). We may use the word ‘subsequent’ (without chronological connotations) to mean ‘consequent or inconsequent’; or we may simply use the word ‘consequent’ in an expanded sense.

The form ‘if P, not-then Q’ should not be confused with the form ‘If P, then nonQ’, which means that the conjunction of P and Q are impossible; sometimes we say the latter with the intent to mean the former. There is a world of difference between ‘P does not imply Q’ and ‘P implies nonQ’. To make matters worse, we sometimes leave out the ‘then’, and just say ‘if P, not Q’, which can be interpreted either way.

It is important to note that we commonly assume that ‘if P, not-then Q’ is true, whenever we have searched and found no reason to think that ‘if P, then Q’ is true. This is effectively an inductive principle for negative hypotheticals: strong relations like ‘if P, then Q’ require specific proof, whereas weak relations like ‘if P, not-then Q’ may usually be taken for granted, so long as their contradictory has eluded us.

- The following table clarifies the relations between the antecedent and consequent and their antitheses, in positive and negative hypotheticals. It shows what follows as true (
**T**), false (**F**), or undetermined (**?**), from the truth of any of them. Note well that the table is an outcome of the hypothetical relations, but does not constitute their definition.

Proposition | Given | P | Q | NonP | NonQ |

If P, then Q | |||||

P | T | T | F | F | |

Q | ? | T | ? | F | |

NonP | F | ? | T | ? | |

NonQ | F | F | T | T | |

If P, not-then Q | |||||

P | T | ? | F | ? | |

Q | ? | T | ? | F | |

NonP | F | ? | T | ? | |

NonQ | ? | F | ? | T |

- Hypotheticals are not only used in everyday reasoning, but also to develop logical theory; they express the formal connections between theses. The hypothetical relations validated by formal logic are not defined by mere denial of the occurrence of this or that conjunction in a specific instance, but by claiming the logical impossibility of it with any content.

We use them to indicate the oppositional relations between any propositional forms, or the inferences which can be drawn from one, or a conjunction of two or more, propositional forms. Premises are antecedents, valid conclusions are consequents; an argument is valid if the premises imply the conclusion, invalid if they do not. Likewise, when we speak of assumptions and predictions, we refer to such logical relations.

The psychology of assumption consists in mentally imagining as true a proposition not yet so established, or even which is already known false. In the latter case, we phrase our hypothetical as ‘If this had been true, that would have been true’. Because of logic’s ability to deal with form irrespective of content, even untrue contents may be considered and analyzed.

As will be seen, hypothetical relations are established through a process of ‘production’. Most, if not all, of the logical relations we intuit in everyday reasoning processes are in fact expressions of formal connections.

### 3. Strict or Material Implication.

Note well that the definitions of both the positive and negative hypothetical forms involve two essential factors. First, they refer to a conjunction of two theses, symbolized by ‘P’ and ‘nonQ’ (meaning, the negation of Q). Secondly, hypotheticals are essentially modal propositions; they refer to the logical impossibility or possibility of such a conjunction.

Many logicians have defined the ‘if P, then Q’ form as identical with the negative conjunction ‘not-{P and nonQ}’. They have called this ‘material’ implication to distinguish it from the above ‘strict’ implication. The suggestion being that implication is a relation which ranges from singular contextuality or actuality (material), to all contexts or necessity (strict).

It is true that we often for practical purposes, intend an implicative statement as merely applicable to the present context. However, since the ‘present context’ is notoriously difficult to identify precisely, this is a practice which cannot be subjected to formal treatment. Two propositions cannot be compared or combined, if it is unclear what parts of the ever-changing context they depend on. The unstated conditions may be different enough that their fluxes are not in harmony.

My position is therefore that the idea of ‘material implication’ is mistaken. There is no such thing as nonmodal implication, in the sense they intended. All implication is inherently modal, ‘strict’. The realization of implication is not a more restrictive implication, but simply a factual conjunction or nonconjunction.

One mere denial of the bracketed conjunction is not implication: such definition only seems to work because it conceals a repetitive denial, coming into force whenever we bring the definition to mind.

The reason why the error arose, is because negative conjunction, even on a factual level, is intrinsically indefinite. When we say ‘not-{P and nonQ}’, we think: ‘well, if P, then not nonQ, and if nonQ, then not P’. However, these seemingly implicit hypotheticals are not themselves assertoric: they are preconditioned by a tacit ‘*if not-{P and nonQ}, then*: if P, then not nonQ, and if nonQ, then not P’. There is a hidden nesting involved. The consequent hypothetical proposition is in fact quite modal; it only appears nonmodal, because the antecedent nonconjunction is taken for granted.

I very much doubt that the form ‘not-{P and nonQ} ever occurs in practice, except insofar as it is logically implied by a factual conjunction like ‘nonP and nonQ’ or ‘P and Q’ or ‘nonP and Q’, or by the modal form ‘if P, then Q’ (in the sense of ‘{P and nonQ} is impossible’). For example, even though the conjunction ‘{chickens have teeth} and {squares are round}’ is indeed false, we do not interpret this to mean that these two happenstances are at all linked; the proposition as a whole can only be constructed *as a result* of our foreknowledge (in this case) that both clauses are separately false, and would not be otherwise arrived at.

This misconception has caused the logicians in question to ignore the contradictory ‘if P, not-then Q’ form altogether, since that would be equivalent to the positive conjunction ‘P and nonQ’, according to that theory. Yet, we commonly reason in such terms, saying ‘it does not follow that’ or ‘it does not imply that’, without intending to affirm the theses categorically thereby [as in negation of conjunction].

The antecedent does not merely happen to precede the subsequent, as that theory suggests. In the ‘if P, then Q’ case, the consequent follows it as a logical necessity; it means effectively, ‘if P, necessarily Q’. In the ‘if P, not-then Q’ case, the inconsequent is denied such necessary subsequence, without affirming or denying that it may possibly happen to be conjoined; it means effectively ‘if P, possibly not Q’.

If we compare the truth-tables of ‘P strictly implies Q’ and ‘P materially implies Q’, we may be misled by the identity of the positive side (see the ‘if P, then Q’ half of table 24.1). But when we look at the negative side (i.e. the denial of ‘if P, then Q’), the difference between the two cases is glaring (for strict implication, see the second half of table 24.1; and for material implication, see row ‘K2’ of table 23.1).

That is to say, though strict and material implication seem to have the same truth-table, their negations have very different truth-tables, so their logical behaviors will be different. Moreover, the former is permanent (i.e. true for all time if true), whereas the latter (except when it is true by implication from the former) is temporary (i.e. true for a limited time if true).

Of course, we can invent any forms we please; but logical theory should reflect practice, and not be allowed to degenerate into an arbitrary game. What the proponents of material implication were looking for, the seed of truth they were trying to express, was, I suggest, the analogues of implication found in other types of modality — the natural, temporal or extensional. I will discuss these in detail later, and the truth of this statement will become more apparent then.

For all these reasons, I have not followed suit. I ignore so-called material implication (though not factual negative conjunction, of course), and limit hypotheticals to strict implication.

### 4. Full List of Forms.

Now, the forms ‘If P is true, then (or not-then) Q is true’ are paradigms. If we substitute in place of P and/or Q, their respective contradictories, that is, the antitheses nonP (P is false) and/or nonQ (Q is false), we obtain the following full list of eight possible relations. The symmetries involved ensure the completeness of our list of hypotheticals. Each hypothetical is defined by a modal conjunction, as shown, on the basis of our original definitions.

Form | Equivalent Modal Conjunction | Symb. |

If P, then Q | {P and nonQ} is impossible | H2n |

If P, not-then Q | {P and nonQ} is possible | K2p |

If P, then nonQ | {P and Q} is impossible | H1n |

If P, not-then nonQ | {P and Q} is possible | K1p |

If nonP, then Q | {nonP and nonQ} is impossible | H4n |

If nonP, not-then Q | {nonP and nonQ} is possible | K4p |

If nonP, then nonQ | {nonP and Q} is impossible | H3n |

If nonP, not-then nonQ | {nonP and Q} is possible | K3p |

- As earlier decided, hypotheticals with the ‘if, then’ operator, which posit a consequence, are classified as ‘positive’; these are fully defined by reference to the logical impossibility of a conjunction. Hypotheticals with the ‘if, not-then’ operator, which negate a consequence, are classified as ‘negative’; these are fully defined by the logical possibility of a conjunction. The unmentioned conjunctions in each case are of undetermined status; this means problematic, and should not be taken to mean logically contingent.

The oppositions between hypotheticals and factual conjunctives follow accordingly. Given the truth of a positive hypothetical, it follows that the conjunction which it by definition denies as possible is false,; and vice versa: so these are contraries. Given the falsehood of a negative hypothetical, *the negation of* the conjunction which it by definition admits as possible is true; so these are subcontraries. With regard to all other factual conjunctions, hypotheticals are neutral.

- There is another respect in which polar expressions might be applied to hypotheticals. We will reserve the labels
*‘*and**affirmative**’*‘*for this new division; here, unlike with categoricals, the terms must not be confused.**negatory**’

Thus, ‘If P, not-then not Q’, involving a double negation, is essentially as positive as ‘If P, then Q’ towards the subsequent Q; these forms, and their equivalents with nonP as antecedent, will therefore be classified as affirmative. Whereas ‘If P, then not Q’ or ‘If P, not-then Q’, which involve only one negation, effectively negate the subsequent thesis Q; so that they, and likewise the corresponding forms with nonP as antecedent will be said to be negatory hypotheticals. Such polarity considerations, also, as will be seen, clarify the basis of validity of certain hypothetical syllogisms.

- Although the hypotheticals included in our initial list of forms are all tenable and useful, half of them are somewhat artificial as they stand.

Forms involving a thesis ‘P’ as antecedent can be regarded as *‘ perfect’* in comparison to those involving an antithesis ‘not P’ as antecedent, labeled

*‘*, whether the forms are positive or negative, and whether the consequent or inconsequent is ‘Q’ or ‘not Q’.

**imperfect**’These characterizations are relative, and not of great importance, but they are useful. The significance of this division of hypotheticals will become more apparent in due course, when we deal with hypothetical inference. But the following are some explanations.

Those with the antecedent P are most ‘true to form’ and express a normal ‘movement of thought’, and may therefore be called perfect, whether P be in itself a thesis with an affirmative or negative content. But those with the antecedent notP, *qua* antithesis (and not because it may present a negative content), are not as such representative of a natural way of thinking. If notP is taken up as a thesis in itself (be it intrinsically affirmative or negative in form or content), rather than by virtue of its being the antithesis of P, the form is quite normally hypothetical, proceeding from a *posited* antecedent, which may happen to be of negative polarity, to some consequences or inconsequences. But if the focus or stress is on the anti-P aspect of our ‘nonP’, the form is relatively artificial, and so ‘imperfect’.

- The use of
*‘*, putting an antithesis in place of a thesis, or vice versa, is a theoretical device of the science of formal logic, rather than a process in the practical art of logic. The science of logic is built as a conceptual algebra, with ‘variables’ open to any content, related by selected ‘constants’. In categoricals, the variables are terms, the constants, the copula, the polarity, the quantity, and so on. In hypotheticals, the variables are propositions, the constants, the relational factors peculiar to them.**substitution**’

But the use of substitution, in the sense of putting specific ‘values’ in the place of logic’s variables, is a practical, rather than theoretical, process, and should be counted as a form, or at least stage, of inference. Here, the thinker is applying logical principles to a given situation, appealing to generally established processes to justify a particular act of thought. Such movement from knowledge of logical science to practical application, is in itself a reasoning process.

## 25. HYPOTHETICALS: OPPOSITIONS AND EDUCTIONS.

### 1. Connection and Basis.

We defined positive and negative hypothetical propositions in terms of the logical impossibility or possibility, respectively of a certain conjunction. This phenomenon refers to the logical *connection* between the theses concerned. Taken by itself, such a relation does not require that the theses be more than problematic; we need not know whether each of them is contingent, necessary or impossible.

However, in everyday discourse, we commonly regard the logical modality of the theses as tacitly, mutually understood. That is, we take for granted that the respondent has the same idea as the speaker with regard to the contingency, necessity or impossibility of each of the theses. This phenomenon refers to the logical *base(s)* of the theses, or the *basis* of a hypothetical proposition.

** Normally**, in most cases we ordinarily encounter, this underlying modality is logical contingency, for both the theses.

**, in rare cases of a usually philosophical nature, the modality of one or both of the theses is found to be logical necessity or impossibility. For this reason, we may refer to two broad classes of hypotheticals, the normal or the abnormal.**

*Abnormally*As we shall see, hypotheticals behave according to different logics. ‘Baseless’ hypotheticals, those with a problematic basis, representing only various connections, without specifying the logical modality of the theses — display what may be called the general or absolute or unconditional behavior patterns. Normal hypotheticals, which have contingent bases, and abnormal hypotheticals, which have one or both theses incontingent, each display slightly different patterns, their own particular or relative or conditional patterns.

Thus, we could develop considerably different logics for each variety of hypothetical. In this volume, we will try to highlight the main features of hypothetical logic, sometimes for unspecified basis, sometimes for specified bases, normal (fully contingent) or abnormal (partly or fully incontingent), as appropriate.

Note that we could similarly regard conjunctions as having a variety of bases. The logics would parallel those of hypotheticals of specified bases.

### 2. Oppositions.

- The absolute oppositions, between the forms of hypothetical proposition whose bases are unspecified, proceed from the definitions of connections as modal conjunctions. They are identical to the oppositions between the conjunctives
**H1n**,**H2n**,**H3n**,**H4n**,**K1p**,**K2p**,**K3p**,**K4p**, which we discussed in a previous chapter.

Here, our purpose is to identify the oppositions between hypotheticals, especially in cases where the logical modality of the theses is more specifically known. We will first deal with merely connective and/or normal hypotheticals, for which the theses may be assumed both contingent, and thereafter consider some of the differences in oppositional properties for abnormal hypotheticals.

- Normal hypotheticals are opposed as follows. Note well the unstated condition that the theses are logically contingent. Let us consider, to begin with, the four forms with a common antecedent P.

Since ‘If P, then Q’ and ‘If P, not-then Q’ inform that the conjunction ‘P and nonQ’ is, in the former case, impossible, and, in the latter case, possible, they are contradictory. Likewise for the other diagonal.

The contrariety of ‘If P, then Q’ and ‘If P, then nonQ’ is obtained by supposing them both true; in that case, if P was true, Q and nonQ would be both true; therefore, these hypotheticals are incompatible; on the other hand, supposing them both false yields no impossible result.

The subcontrariety of ‘If P, not-then Q’ and ‘If P, not-then nonQ’ follows, since if they were both false, their contradictories would be both true, though incompatible; on the other hand, supposing them both true yields no impossible result.

Finally, if ‘If P, then Q’ is true, then ‘If P, then nonQ’ is false, by contrariety; then ‘If P, not-then nonQ’ is true, by contradiction; whereas nothing can be shown concerning the latter if ‘If P, then Q’ is false; so their subalternative relation (downward) holds. The other subalternation can be likewise shown.

A similar square of opposition can be demonstrated for the forms with a common antecedent nonP, namely, ‘If nonP, then (or not-then) Q (or nonQ)’. We can show that hypotheticals with a common consequent Q, but different antecedents, P or nonP, fall into such a square of opposition, by contraposing the forms (see next section on eduction). Likewise, if the common consequent is nonQ, of course.

However, concerning propositions whose antecedents and consequents are both different, namely, ‘If P, then (or not-then) Q’ and ‘If nonP, then (or not-then) nonQ’, the same cannot be said. For their definitions as impossibility (or possibility) of the conjunctions ‘P and nonQ’ and ‘nonP and Q’, respectively, leave them quite compatible, and unconnected. Likewise, for opposite pairs of the forms ‘If P, then (or not-then) nonQ’ and ‘If nonP, then (or not-then) Q’

The oppositions of the eight forms of hypothetical could be illustrated by means of a cube. However, the following tables summarize all these results for us, just as well. (The numbering of forms and symbols for oppositions used in these tables is arbitrary.)

Table 25.1 Table of Oppositions between Hypotheticals.

Key to symbols: | Unconnected | [ | |

Implicant | z | Contradictory | M |

Subalternating | È | Contrary | Ê |

Subalternated | Ã | Subcontrary | Ç |

Form | No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

If P, then Q | 1 | z | Ê | Ê | [ | [ | È | È | M |

If P, then nonQ | 2 | Ê | z | [ | Ê | È | [ | M | È |

If nonP, then Q | 3 | Ê | [ | z | Ê | È | M | [ | È |

If nonP, then nonQ | 4 | [ | Ê | Ê | z | M | È | È | [ |

If nonP, not-then nonQ | 5 | [ | Ã | Ã | M | z | Ç | Ç | [ |

If nonP, not-then Q | 6 | Ã | [ | M | Ã | Ç | z | [ | Ç |

If P, not-then nonQ | 7 | Ã | M | [ | Ã | Ç | [ | z | Ç |

If P, not-then Q | 8 | M | Ã | Ã | [ | [ | Ç | Ç | z |

These relationships may be clarified by means of a truth-table, in which given the truth of a form under heading **T**, or the falsehood of one under heading **F**, the status of the others along the same row is revealed.

(key: T = true, F = false, . = undetermined.)

Form | T | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | F |

If P, then Q | 1 | T | F | F | . | . | T | T | F | 8 |

If P, then nonQ | 2 | F | T | . | F | T | . | F | T | 7 |

If nonP, then Q | 3 | F | . | T | F | T | F | . | T | 6 |

If nonP, then nonQ | 4 | . | F | F | T | F | T | T | . | 5 |

If nonP, not-then nonQ | 5 | . | . | . | F | T | . | . | . | 4 |

If nonP, not-then Q | 6 | . | . | F | . | . | T | . | . | 3 |

If P, not-then nonQ | 7 | . | F | . | . | . | . | T | . | 2 |

If P, not-then Q | 8 | F | . | . | . | . | . | . | T | 1 |

### 3. Hierarchy.

The square of opposition shown in the previous section, you will notice, is the familiar one encountered for the categorical propositions **A**, **E**, **I**, **O**. The analogy is not accidental. The contrariety between ‘If P, then Q’ and ‘If P, then nonQ’ is obviously similar in meaning to that between ‘All S are P’ and ‘All S are nonP’, and the diagonal contradictions can also obviously be likened.

This analogy suggests that normal positive and negative hypotheticals constitute a hierarchy, the former being ‘** uppercase**’ forms similar to general propositions and the latter ‘

**’ forms similar to particulars. Indeed, this is implicit in the definitions of hypotheticals.**

*lowercase*Thus, ‘If P, not-then notQ’ (note the double negation) is the lowercase form corresponding to the uppercase ‘If P, then Q’; likewise, ‘If P, not-then Q’ is the subaltern form of ‘If P, then notQ’, ‘If notP, not-then notQ’ is the subaltern form of ‘If notP, then Q’, and ‘If notP, not-then Q’ is the subaltern form of ‘If notP, then notQ’. Each positive hypothetical includes the negative hypothetical with like antecedent and unlike subsequent (i.e. consequent or inconsequent).

This uppercase/lowercase classification will be found useful in understanding of much hypothetical inference. By expressing the form ‘If P, then Q’ as a generality ‘All P occurrences are Q occurrences’, and the form ‘If P, not-then notQ’ as a particular ‘Some P occurrences are Q occurrences’, we will be able to understand why, for instance, the major premise in first figure hypothetical syllogism must be uppercase, and cannot be lowercase.

Now, what of the oppositions between the eight hypotheticals and the four factual conjunctions referred to in their definitions? First, we note that any pair of the four conjunctions are opposed to each other in the way of contraries; that is, they cannot be both true, but may be both false.

Secondly, we know that each uppercase hypothetical form is contrary to the conjunction which it denies as possible by definition; it is oppositionally neutral to the remaining three conjunctions, since, taken as a pair with any one of them, they may be both true or both false without problem. Thirdly, each lowercase form is subaltern to (implied by) the conjunction which it affirms as possible by definition; and unconnected oppositionally to the other conjunctions.

From this we may conclude that while, for example, ‘P and Q’ implies ‘If P, not-then notQ’, in the same way as a singular categorical implies a particular, the analogy stops there. For ‘P and Q’ is not in turn implied by ‘If P, then Q’, as analogy would require. That is, the conjunctions are not exactly ‘middle case’ forms, between the upper and lower cases.

This discussion of course serves to clarify the inter-relationships of the categories of logical modality. Uppercase is logical incontingency, lowercase is logical possibility or unnecessity; and conjunction is plain fact, lying in between. It concerns, of course, contingency-based hypotheticals, rather than hypotheticals with one or both theses incontingent. It applies to normal logic, rather than abnormal or general-case forms.

### 4. Eductions.

Here again, we will first consider normal hypotheticals, and then mention merely-connective hypotheticals and abnormals.

We need only, to begin with, deal with the primary hypothetical forms, ‘If P, then Q’ and ‘If P, not-then Q’, as our source propositions, to elucidate the processes. What is found valid for these, is mutatis mutandis applicable to forms involving ‘nonP’ and/or ‘nonQ’ as the source theses. The educed hypotheticals may have different polarity (‘not-then’ instead of ‘then’, the reverse never occurs), may involve the antithesis of one or both of the original theses as a new thesis, and may switch the positions of the theses. The valid processes are:

- Obversion. From P-Q to P-nonQ.

If P, then Q | implies | If P, not-then nonQ. |

- Conversion. From P-Q to Q-P.

Not applicable.

- Obverted Conversion. From P-Q to Q-nonP.

If P, then Q | implies | If Q, not-then nonP. |

- Conversion by Negation. From P-Q to nonQ-P.

If P, then Q | implies | If nonQ, not-then P. |

- Contraposition. From P-Q to nonQ-nonP.

If P, then Q | implies | If nonQ, then nonP. |

If P, not-then Q | implies | If nonQ, not-then nonP. |

- Inversion. From P-Q to nonP-nonQ.

Not applicable.

- Obverted Inversion. From P-Q to nonP-Q.

If P, then Q | implies | If nonP, not-then Q. |

The primary process here is (e) contraposition. These eductions are validated by reference to the forms’ definitions. Since ‘If P, then Q’ means that the conjunction ‘P and nonQ’ is impossible, and ‘If nonQ, then nonP’ that ‘nonQ and not-nonP’ is impossible, and these two conjunctions are equivalent, it follows that the two hypotheticals involved are also equivalent.

The same can be said with regard to the negative forms: they are defined by the same possibility of conjunction, and therefore equal. Contraposition is therefore a reversible process, and applicable as described to all hypotheticals without loss of power.

This process applies to unspecific hypotheticals and abnormals, as well as to contingency-based normals, because it only requires for its validity the connection implied by the defining modal conjunction.

The other processes, however, are only applicable to normal positive hypotheticals, if at all, and always yield a weaker, negative result. These processes are only applicable to normal hypotheticals, because they presume that the theses are to be understood as both logically contingent.

They are proved by reductio ad absurdum, combining the source proposition with the contradictory of the target proposition, to yield an inconsistency, in some cases after some contraposition(s).

Thus, given ‘If P, then Q’ to be true, (a) if ‘If P, then nonQ’ was true, it would follow that P implied both Q and nonQ, an absurdity, therefore the stated obverse must be valid; (c) if ‘If Q, then nonP’ was true, we could contrapose it and obtain the same absurdity, therefore the stated obverted converse must be valid; (d) if ‘If nonQ, then P’ was true, it would follow, after contraposing ‘If P, then Q’, that nonQ implied both P and nonP, an absurdity, therefore the stated converse by negation must be valid; and (g) if ‘If nonP, then Q’ was true, we could contrapose both it and the source proposition, and obtain the same absurdity, therefore the stated obverted inverse must be valid.

All processes with theses P, Q in the source propositions, excluded from the above list, cannot be likewise validated, and so are invalid.

By substituting the antitheses of P and/or Q in the above validated processes, we get the following full list of possible eductions, which is useful for reference purposes.

- Obversions.

If P, then Q | implies | If P, not-then nonQ. |

If P, then nonQ | implies | If P, not-then Q. |

If nonP, then Q | implies | If nonP, not-then nonQ. |

If nonP, then nonQ | implies | If nonP, not-then Q. |

- Conversion. Not applicable.
- Obverted Conversions.

If P, then Q | implies | If Q, not-then nonP. |

If P, then nonQ | implies | If nonQ, not-then nonP. |

If nonP, then Q | implies | If Q, not-then P. |

If nonP, then nonQ | implies | If nonQ, not-then P. |

- Conversion by Negations.

If P, then Q | implies | If nonQ, not-then P. |

If P, then nonQ | implies | If Q, not-then P. |

If nonP, then Q | implies | If nonQ, not-then nonP. |

If nonP, then nonQ | implies | If Q, not-then nonP. |

- Contrapositions.

If P, then Q | implies | If nonQ, then nonP. |

If P, then nonQ | implies | If Q, then nonP. |

If nonP, then Q | implies | If nonQ, then P. |

If nonP, then nonQ | implies | If Q, then P. |

If P, not-then Q | implies | If nonQ, not-then nonP. |

If P, not-then nonQ | implies | If Q, not-then nonP. |

If nonP, not-then Q | implies | If nonQ, not-then P. |

If nonP, not-then nonQ | implies | If Q, not-then P. |

- Inversion. Not applicable.
- Obverted Inversions.

If P, then Q | implies | If nonP, not-then Q. |

If P, then nonQ | implies | If nonP, not-then nonQ. |

If nonP, then Q | implies | If P, not-then Q. |

If nonP, then nonQ | implies | If P, not-then nonQ. |

A final comment. We may observe in the above that obversion of uppercase hypotheticals merely yields the corresponding lowercase form, so such eduction yields no more than the oppositional inference of a subaltern.

We could have regarded the obverse of ‘If P, then Q’ to be ‘If P, then not-nonQ’, rather than merely ‘If P, not-then nonQ’. This would obviously be correct, and analogous to the obversion of ‘All S are P’ to ‘No S are nonP’. Effectively, we would be introducing a relational operator ‘then-not’ (and its negation, ‘not-then-not’), to complement ‘then’ (and ‘not-then’). But I think such multiplication of ‘nots’ is without value.

## 26. DISJUNCTION.

One way to introduce the topic of ‘disjunction’, is to view it in contradistinction to ‘subjunction’. According to this approach, we may divide hypotheticals into two groups, with reference to the emphasis they put on their theses and antitheses.

### 1. Subjunction.

Hypotheticals which relate two theses as such, or two antitheses as such, may be called ‘subjunctive’. The reason these two sets are grouped into one class becomes clearer when their definitions are considered.

The primary form of subjunction is ‘If P, then Q’, which tells us that ‘{P and nonQ} is logically impossible’ (**H2n**). This is known as implication. Its negation is ‘if P, not-then Q’, meaning ‘{P and nonQ} is possible’ (**K2p**).

The other form of subjunction, ‘If nonP, then nonQ’, tells us that ‘{nonP and Q} is logically impossible’ (**H3n**), and so is equivalent to the statement ‘If Q, then P’, which has a similar meaning to ‘If P, then Q’, but in the anti-parallel direction. This could therefore be called reverse implication. The corresponding negative form is ‘if nonP, not-then nonQ’, meaning ‘{nonP and Q} is possible’ (**K3p**).

We may view implication and its reverse as forms of subjunction, and their contradictories as forms of nonsubjunction. Or we may conventionally broaden the sense of the word subjunction, and speak of positive and negative subjunction, respectively.

Now, taken individually, these various logical relations are indefinite. Hypotheticals are elementary forms, capable of various combinations, called compounds, which define relationships more definitely. The forms are intentionally left open, to allow expression of the maximum number of combinations using a minimum number of building blocks. These effects have already been encountered in the context of opposition theory, and will only be briefly reviewed here for the sake of thoroughness.

Implication and its reverse are oppositionally neutral to each other (likewise, therefore, their contradictories). They are therefore capable of four combinations: they may be both true, or one true and the other false, or both false. The hypotheticals conjoined in such combinations are called complementary, in that they together serve to define the relationship between the theses in both directions.

In such case as ‘If P, then Q’ and ‘If nonP, then nonQ’ are both true, the resulting relation is one of mutual or reciprocal implication of P and Q (or nonP and nonQ). This may be called implicance, and viewed as asserting the logical equivalence of these two theses (or of their antitheses).

In such case as ‘If P, then Q’ and ‘If nonP, not-then nonQ’ are both true, P is said to subalternate Q; in such case as ‘If P, not-then Q’ and ‘If nonP, then nonQ’ are both true, P is said to be subalternated by Q. Thus, subalternation, in contrast to implicance, is one-way subjunction, and not reversible.

In such case as ‘If P, not-then Q’ and ‘If nonP, not-then nonQ’ are both true, we are left with a relation which might be called ‘unsubjunction’. This is not a fully defining combination, unlike the preceding three compounds, in that it allows the possibility of disjunction.

### 2. Manners of Disjunction.

In contrast, we call ‘disjunctive’ those hypotheticals which relate a thesis with an antithesis, or an antithesis with a thesis. We usually express such relationship by means of the word ‘or’. Rephrasing a hypothetical in disjunctive form allows us to conceal the negative polarity of the antitheses involved, so that the statement is made purely in terms of theses. The two theses are known as the ‘alternatives’ (or disjuncts).

Two essential *manners* of disjunction may be distinguished. As usual in logic, we must adopt some clear-cut differences in terminology to facilitate treatment; but, although the underlying distinctions of meaning are indeed intended in practice, they are not always verbalized so exclusively.

(i) ‘P and/or Q’ (or ‘P or also Q’) signifying simply ‘If nonP, then Q’ (or ‘if nonQ, then P’), in other words, ‘{nonP and nonQ} is logically impossible’ (**H4n**). This is known as inclusive disjunction, and expresses the *exhaustiveness* of P and Q: one of them must be true. This is the more commonly intended sense of ‘P or Q’; it stresses the theses (P, Q), rather than the ‘or’ operator.

The negation of this form ‘not-{P and/or Q}’ (which could be written ‘P not-{and/or} Q’) means ‘If nonP, not-then Q’ (or ‘if nonQ, not-then P’); in other words ‘{nonP and nonQ} is not logically impossible’ (**K4p**). This of course signifies inexhaustiveness.

(ii) ‘P or else Q’ (or ‘P otherwise Q’) signifying simply ‘If P, then nonQ’ (or ‘if Q, then nonP’); in other words, ‘{P and Q} is logically impossible’ (**H1n**), suggesting a difference. This is known as exclusive disjunction, and expresses the *incompatibility* of P and Q: one of them must be false. This is a rarer sense of ‘P or Q’; it stresses the separation of the theses (P, Q), the ‘or’ operator.

The negation of this form ‘not-{P or else Q}’ (which could be written ‘P not-{or-else} Q’) means ‘If P, not-then nonQ’ (or ‘if Q, not-then nonP’); in other words, ‘{P and Q} is not logically impossible’ (**K1p**). This of course signifies compatibility.

We may view exhaustiveness and incompatibility as forms of disjunction, and their contradictories as forms of nondisjunction. Or we may conventionally broaden the sense of the word disjunction, and speak of positive and negative disjunction, respectively.

Note, sometimes when we say ‘P and/or Q’, we intend to admit of only two alternatives, ‘P and Q’ or ‘nonP and Q’, in advance excluding or not meaning to include ‘P and nonQ’, as well as ‘nonP and nonQ’. Sometimes, this is what we intend when we say ‘P or else Q’, for that matter; meaning, ‘at least Q, whether or not P’. Likewise, ‘P or also Q’ may be intended to mean: ‘P and Q’ or ‘P and nonQ’; that is, ‘at least P, possibly without Q but also possibly with it’. Sometimes, ‘P or Q’ is understood to mean ‘P and nonQ’ or ‘P and Q’.

Such implications are often obvious to us by virtue of the subject involved; the subject-content is well known to everyone to exclude certain alternatives, so that these exclusions are virtually formal. The logic of such forms can easily be derived from the logic of the forms here considered, so they will be ignored.

The recasting of a hypothetical form into disjunctive form, or vice versa, may be called ‘transformation’. This may be viewed as a form of inference, or of elucidation, insofar as the mind may favor such process to more fully understand the relationship under consideration.

Note that disjunctives, like hypotheticals, may each be dissected into their implicit connection and basis. The general case comprises only the ‘connective’ (a modal conjunction) for its definition, whereas normal and abnormal disjunctions specify the logical modalities of the theses in various ways. Many processes are only valid for contingency-based disjunctions.

Needless to say, the theses of disjunctions may be any kind or complex of proposition(s): categoricals, conjunctives, hypotheticals, or also disjunctive clauses. The logic involved becomes progressively more intricate and complicated, accordingly. Some such logical ‘compositions’ will be analyzed in the next two chapters.

Each of the forms of disjunction is, we note, nondirectional, unlike the forms of subjunction. By reference to their definitions, it is easy to see that: ‘If P, then nonQ’ is equivalent to ‘If Q, then nonP’; ‘If P, not-then nonQ’ is equivalent to ‘If Q, not-then nonP’; ‘If nonP, then Q’ is equivalent to ‘If nonQ, then P’; and ‘If nonP, not-then Q’ is equivalent to ‘If nonQ, not-then P’. These equations have already been encountered under the heading of contraposition.

The forms of elementary disjunction are complementary; any pair of them, other than contradictories of course, may be used in conjunction to define a compound relationship, as follows. Note that each of these relations is reversible.

Contradiction combines ‘If P, then nonQ’ and ‘If nonP, then Q’. We could assign to the disjunctive form ‘Either P or Q’ this specific meaning, comprising both incompatibility and exhaustiveness of P and Q. The proposition ‘Either nonP or nonQ’ is equivalent, note well.

Contrariety combines ‘If P, then nonQ’ and ‘If nonP, not-then Q’. Thus, contrariety means incompatibility without exhaustiveness.

Subcontrariety combines ‘If nonP, then Q’ and ‘If P, not-then nonQ’. Thus, subcontrariety means exhaustiveness without incompatibility.

‘Undisjunction’ might be used to label the combination of ‘If P, not-then nonQ’ and ‘If nonP, not-then Q’, which means inexhaustive and compatible. This is not a fully defining combination, unlike the preceding three compounds, in that it allows the possibility of subjunction.

The oppositions of all forms of subjunction and disjunction, elementary or compound, to each other, and the eductions feasible from each of them, are all easily inferred from the findings for the corresponding hypotheticals. I will not list them all, to avoid repetition, but a couple are worth highlighting.

Thus, note that ‘P and/or Q’ and ‘nonP or else nonQ’ are equivalent, and likewise, ‘P or else Q’ and ‘nonP and/or nonQ’ are equivalent. Also, ‘either P or Q’ and ‘either nonP or nonQ’ are identical.

### 3. Broadening the Perspective.

- Interface of Subjunction and Disjunction.

Since the conjunctive roots of subjunctions and disjunctions, namely **H2n**, **H3n**, and **H4n**, **H1n**, are neutral to each other, they are in principle combinable together. However, normally, subjunctions and disjunctions are contrary to each other and not combinable; this applies to formal logic, where the theses and antitheses are all granted the status of logical contingency, as in the theory of opposition. This further justifies their division into two classes.

In contrast, nonsubjunctions and nondisjunctions, namely **K2p**, **K3p**, and **K4p**, **K1p**, are generally combinable, since they are compatible both in absolute terms (neutral) and in formal situations (subcontrary).

In opposition theory (ch. 6), we identified seven fully defining logical relations. The six main ones — implicance, subalternating, being-subalternated, contradiction, contrariety, and subcontrariety — have been reviewed in the previous sections of the present chapter. The remaining one was, you will recall called ‘unconnectedness’ or ‘neutrality’, in formal logic discussions. This may be defined as a combination of ‘unsubjunction’ and ‘undisjunction’. Although each, taken alone, is still an indefinite compound, taken together they form a fully defining and reversible relationship.

In formal logic contexts, these 7 fully defining compounds are all mutually exclusive and constitute an exhaustive list of possibilities; if any one holds, the other six are out, and if any six are rejected, the remaining one must stand. The negation of any one of them means one or more of its constituent hypotheticals is false, without specification as to which one(s); so we must be careful not to make errors here.

In particular, note that the expression ‘neither P nor Q’ is normally equivalent to ‘both nonP and nonQ’, and should not be thought to be the logical negation of ‘either — or —’ in the above suggested sense, though it is sometimes so intended.

Beyond these definitions, we will not further discuss compound forms, so as not to complicate matters further. The inferences possible from them are all implicit in those concerning the constituent elementary forms, and can easily be derived.

- Vague Disjunctions.

The important thing is not to confuse the elementary forms with their compounds, and to be aware of the reducibility of compound forms to their elementary positive and negative constituent hypotheticals. Especially, disjunctive propositions are in practice often notoriously ambiguous.

Sometimes, when we say, ‘P and/or Q’ we only intend ‘if nonP, then Q’, sometimes an additional ‘if P, not-then nonQ’ is *sous-entendu*. The elementary case merely forbids ‘nonP and nonQ’, without specifically allowing or forbidding ‘P and Q’, whereas the compound case specifically allows for the latter. Similarly, mutatis mutandis, with regard to ‘P or else Q’.

The difficulty is due to the previously mentioned inductive rule for weak relations in logical modality: here, there is little distinction between the ‘open’ and the ‘possible’. Ultimately, a conjunction which is neither specifically allowed nor specifically forbidden, is effectively allowed. The difference is merely one of degree. If the open turns out to be impossible, it is just eliminated from the list of alternatives as a matter of course, without affecting the overall truth of the disjunctive proposition.

In practice, we often use a vague form of disjunction, ‘P or Q’, which might mean anything from an elementary inclusive or exclusive disjunction, to a compound like subcontrariety, contrariety, or even contradiction. It is thus relatively uninformative; nevertheless, it shows why we can class all these relations under the common heading of disjunctions.

The forms ‘P and/or Q’ and ‘P or else Q’ and ‘either P or Q’ all suggest that ‘P or Q’, though for different reasons. The form ‘P or Q’ in its broadest sense recognizes at least ‘P and nonQ’ or ‘Q and nonP’ as conceivable outcomes, without telling us at the outset whether ‘P and Q’ or ‘nonP and nonQ’ are allowed or forbidden, though it is understood that at least one of them (if not both) is forbidden.

The implicit questions are left open, unless the relation is further specified by ‘and’ or ‘else’ or ‘either’, in which case the additional allowance is made more firm (given a greater degree of eventuality) by what is specifically forbidden. If both the open questions are answered negatively, then ‘or’ means ‘either-or’.

The vague form ‘P or Q’ may thus be defined by the disjunction of all the clearer forms of disjunction. The following table shows the common ground between these forms. Note that the ‘allowances’ here should be interpreted minimally, as problemacies, though they are often in practice meant to be logical possibilities in the stricter sense.

Conjunction | P+Q | P+nonQ | nonP+Q | nonP+nonQ |

P and/or Q | allow | allow | allow | forbid |

P or-else Q | forbid | allow | allow | allow |

Either P or Q | forbid | allow | allow | forbid |

The negation of ‘P or Q’ may be stated as ‘not-{P or Q}’ (or ‘P not-or Q’). What we mean by that of course depends on what we intend by ‘P or Q’.

- Involving Antitheses.

We presented subjunction and disjunction as subdivisions of hypotheticals. But unlike subjunction, disjunction involves a distinct set of operators, ‘or’ and its derivatives. So disjunction deserves to be viewed as a logical relation in its own right. We can see from its name that we intend this relation as conceptually opposed to conjunction.

What this means is that, in addition to ‘P or Q’, we should consider ‘P or nonQ’, ‘nonP or Q’, ‘nonP or nonQ’. Similarly for the less vague operators ‘and/or’, ‘or-else’, and their compounds, including ‘either-or’: we can insert one or both antitheses, in place of the original theses, to obtain other forms, as we did for hypotheticals. And of course, all these have contradictories.

It is very easy to determine the conjunctive definition for each form, and then compare it to all the others. Since each operator gives rise to four impossible conjunctions and four possible ones, and these eight conjunctions are ubiquitous, there is bound to be a corresponding number of equations.

I will not go into this domain in any detail, so as not to expand this treatise unnecessarily. The reader is invited to explore it for him or her self.

## 27. INTRICATE LOGIC.

### 1. Organic Knowledge.

People think that logic is a linear enterprise, antithetical to the curvatures of poetic knowledge. But, viewed holistically, knowledge is not essentially a mechanical activity and product, but more akin to a living organism.

Just as any living organism functions on many levels, from the physical-chemical, through the biochemical and cellular, to the gross level of our sensory perceptions, and beyond that, as an intricate part of the natural environment as a whole, through the intellectual and spiritual dimension, the whole being sustained by the Creator — so knowledge ought to be viewed.

Logic is the way we establish the chemical bonds between the different data elements of our knowledge. These bonds vary in kind and effect, and can occur cooperatively in any number and complication of combinations. The result is a network, from the microscopic level of precise logical relations, to the less-magnified level of clusters of information, to the organic whole, to the cultural context.

This knowledge network is not stationary, but like an organism, pulses and glows with life, growing, ordering, clarifying, strengthening. This life has a mechanical level, a vegetative level, and a conscious and volitional level, which is animal and human, and therefore spiritual.

So viewed, logic and the poetic side of us are not in conflict, but in easy, friendly, fruitful togetherness. A balanced, healthy mind, requires some degree of rigor in observation and thought, and also some degree of freedom to move, some room to maneuver. Because knowledge is always in flux, and there is always some inconsistency involved.

One has to be able to flow with the tides of information, the momentary waves, and even the momentary storms, and remain patient and awake at one’s center. Logic maps for us the wide terrain of the mind, improving our research skills. Thus, ultimately, logic is an aspect of wisdom, knowing to navigate smoothly in the changing sea of information.

Logic teaches us to clarify information, by *engineering* tools for this purpose. Especially multiple-theses, mixed-form, modal logic provides us with ways to express ideas precisely, and thus construct and check them more rigorously. Let us look at some of the possible intricacies of logical relations between items of knowledge.

### 2. Conjunctives.

Let us now broaden our understanding of conjunctive logic, in different directions. Note that we refer to any specialized field of logic as ‘a logic’.

We have talked of conjunction with reference to two theses, because the logic of conjunctions of more than two theses is derivable from it.*Multiple-Theses Logic.*

Thus, we may inspect the theses of a proposition of the form ‘P and Q’, and find that, say, Q is itself composed of two theses ‘Q1 and Q2’; from this we conclude that ‘P and Q1 and Q2’ is also true. Likewise, though with diminishing statistical probability, any of the theses P, Q1, Q2, may in turn be found subdividable. Thus, conjunctives may have any number of theses.

It follows that a conjunctive clause within a conjunction, is equivalent to a larger conjunctive proposition, so that we need not think in terms of clauses. This process may be viewed, in analogy to mathematics, as ‘addition of conjuncts’; or we may refer to it as ‘logical composition’, the formation of composites out of elements or other components. For example:

‘P and {Q and R}’ is identical to ‘P and Q and R’.

A corollary of this is that we can isolate part of a conjunction as a clause, at will. All that should be obvious, since ‘P and Q..’. simply informs us that the theses are individually, as well as together, true. Since the order of the theses is irrelevant in the case of two-theses conjunction, meaning ‘P and Q’ equals ‘Q and P’, it can likewise be shown that order does not affect the logical relation of any number of conjuncts.

This begins for us the topic of multiple-theses logic.

It is well, as we shall see, to think of multiple conjunctions as forming a continuum. The number of conjuncts (ands) is one less than the number of theses. Here, a single thesis is the limiting case of the continuum, a conjunction without conjunct, as it were. Clearly, the more theses are conjoined, the more overall information we have. Thus:*Matrix Logic.*

P (one thesis)

P and Q (two theses)

P and Q and R (three theses)

P and Q and R and… (and so on).

The logic of nonconjunction should follow, though it is more complex. Thus, the negative conjunction ‘not-{P and Q}’, where the theses are entirely problematic, signifies that any of the positive conjunctions ‘P and nonQ’, ‘nonP and Q’, or ‘nonP and nonQ’ might be true, since they are formally the only conceivable alternatives to the negated one.

We can therefore think of negative conjunctions with reference to positive ones entirely. The existence of a negative is expressed only through positives; negation is a lesser, derivative expression of existence. It is useful therefore, to view negative conjunctions as equivalent to *‘matrixes’* of positive alternatives, as follows:

not-{P+Q} | not-{P+nonQ} | not-{nonP+Q} | not-{nonP+nonQ} |

P+nonQ | P+Q | P+Q | P+Q |

nonP+Q | nonP+Q | P+nonQ | P+nonQ |

nonP+nonQ | nonP+nonQ | nonP+nonQ | nonP+Q |

We may call each of the alternatives in a matrix, a *‘root’* conjunction of the nonconjunction. Such matrixes are very useful in clarifying the logic of negative conjunction, since we need only find the common ground of the positive alternatives (the roots) in each matrix, to know the properties of the corresponding negative. Thus, for instance, we can here too prove that the order of the theses is irrelevant, since all the alternatives of the matrix can be reversed.

It can thus be shown that clauses may be inserted or removed arbitrarily, with negative conjunctions as well. We can accordingly develop a logic of negations of multiple conjunctions, again thinking of all such conjunctions as forming a continuum. The difference here being that the more alternatives there are the wider, vaguer, and weaker, is their overall negation. That is, the more theses are involved in a negative conjunction, the less information we have; negation of one thesis being the most definite, limiting case. Thus, we have an upside-down continuum:

not-{P and Q and R and…} (and so on).

not-{P and Q and R} (three theses)

not-{P and Q} (two theses)

not-{P}…or P (one thesis)

The one-thesis case may be P, as well as nonP, if we understand these negations of conjunctions as effectively disjunctions, meaning ‘P or Q or..’., for then P is one of the ways the disjunction can be resolved (since Q or R may be negated instead).

In this way, this here continuum of negative conjunctions, can be attached to the previously described positive conjunctions continuum, resulting in a larger continuum, stretching from the negative forms with the most theses, through the central one-thesis case (the ‘P’ common to both positive and negative conjunction), up to the positive forms with the most theses.

The negative, say left, side is a virtual kind of knowledge, getting ever vaguer, a storehouse of possibilities. The positive, say right, side is a growing categorical knowledge, ever more precise. As we move from left to right, our knowledge becomes more specific; we have more information, a higher, wider, deeper view.

It is interesting to note, in passing, that in Hebrew the word for ‘and’ is *‘oo’* (spelt, vav; also pronounced as ‘ve’), and the word for ‘or’ is *‘o’* (spelt, alef-vav). This similarity confirms that the notions conjunction and disjunction are intuitively conceived as continuous, different degrees of the same thing.

Each of the multiple-theses negative conjunctions may be dissected into a matrix of positive conjunctions, the alternatives to the one negated. Negation of one thesis, P, leaves us with only one alternative, nonP. Effectively, every theses should be viewed as including the denial of its contradictory; P, say, may be taken as implying ‘P and not-{nonP}’; nonP likewise becomes ‘nonP and not-{P}’.

Negation of two theses, as we saw above, leaves us with three alternatives. Since three theses and their negations are combinable in eight ways, negation of a conjunction of three theses, leaves us with seven positive alternatives out of the eight:

P + Q + R | nonP + Q + R |

P + Q + nonR | nonP + Q + nonR |

P + nonQ + R | nonP + nonQ + R |

P + nonQ + nonR | nonP + nonQ + nonR |

Beyond that, the general formula is clearly, for n theses, there are: two to the nth power combinations, and therefore that number minus one positive alternatives to the negation of any root. For instance, for 5 theses, there are 2X2X2X2X2 = 32 possible combinations.

Accordingly, the positive conjunction of two (or more) negative conjunctive clauses, may be also expressed by reference to the leftover positive combinations. We can thus develop a general logic of conjunction; that is, any complex of positive and negative conjunctions can be interpreted in positive terms. I will not go into such detail here, however.

The above concerns factual conjunction; modal conjunction has yet to be considered. To say that a conjunction is logically necessary means that it holds, no matter what the surrounding conditions. In contrast, a logically contingent conjunction depends for its eventual realization on certain conditions. If those conditions are unspecified, we have a nonhypothetical modal proposition; if sufficient conditions are specified, we have a precise hypothetical relation. All this applies to positive and negative conjunctions.*Modal Logic.*

It follows that modal conjunctions can always be understood in terms of factual ones, whether the latter are framed by conditions, specified or unspecified, or unconditional. In conjoining modal conjunctions, we must however be careful, and consider whether the conditions under which each clause is realized are compatible with the conditions for the other clause(s) to become factual.

Consider, for instance, the following complex: {P and Q} is possible and {Q and R} is possible. Does it follow that: {P and Q and R} is possible? The answer is clearly, No! It is conceivable that, though these possibilities are compatible as modal propositions, they are incompatible in their factual embodiments. That is, it may be that: {P and Q and R} is impossible, and the given possibilities can only be realized separately, through {P and Q and nonR} or {nonP and Q and R}, respectively.

In this way, by focusing on the underlying factual conjunctions, we can develop a detailed logic of modal conjunction. In formal logic, using variables for terms or propositions, whatever is conceivable is logically possible. But in practice, when dealing with specific terms and specific relations, we must be careful to distinguish between problemacy and logical contingency.

In the above example, for instance, if the given two ‘possibilities’ are mere problemacies, then any combination is conceivable; and we can say (also problematically) that {P and Q and R} might well be true. But if the premises are logical possibilities, we cannot conclude that the {P and Q and R} conjunction is also logically possible.

- Thus, a complete logic of conjunction, whether positive or negative, factual or modal, evolves entirely from the logic of positive conjunctions.

Since hypothetical and disjunctive propositions are in turn defined with reference to conjunctions, the logic of all mixtures of logical relations is likewise reducible to the logic of positive conjunctions.

Any statement, whatever its mix of logical relations — of whatever modalities and polarities — can thus be analyzed through matrixes, and compared to any other statement similarly analyzed.

### 3. Hypotheticals.

We present an argument by listing its premises and conclusions as follows. There are of course arguments with one premise (eductions), and arguments with more than two premises (as in sorites), and some with more than one conclusion, but the typical unit of deduction is two premises, one conclusion.*The form of argument.*

P,

and Q,

therefore R.

A valid categorical, Aristotelean syllogism, for instance, may be regarded as establishing a hypothetical link between premises and conclusion, *by way of* the common terms in these propositions, in specific figures and with precise polarity, quantity and modality specifications.

Thus, although we cannot say generally of any group of propositions that P and Q imply R, we do know that under specific conditions (where for instance P means ‘X is Y’, Q means ‘Y is Z’, and R means ‘X is Z’), such a bond can be established, for all cases of that form. Thus, categorical syllogism may be viewed as one condition under which the form ‘if P and Q, then R’ may be viewed as universally true.

Now, this can be interpreted as a hypothetical proposition with a conjunctive antecedent: (a) ‘If {P and Q}, then R’. Alternatively, we tend to interpret it as a hypothetical proposition with a hypothetical clause as its consequent: (b) ‘if P, then {if Q, then R}’, meaning that, under the condition P, Q implies R. The former states that ‘{P and Q and nonR} is impossible’, whereas the latter states that ‘{P and possibly[Q and nonR]} is impossible’.

At first sight, the two statements may seem significantly different, yet if we analyze them with reference to the underlying positive conjunctions, it is seen that they make identical allowances. The form ‘{P and Q and nonR} is impossible’ obviously allows for seven alternative positive conjunctions. The form ‘{P and possibly[Q and nonR]} is impossible’ allows for:

(i) ‘P and impossibly{Q and nonR}’, implying the factual ‘P and not-{Q and nonR}’, which might be realized as ‘P and {Q and R}’, ‘P and {nonQ and R}’, or ‘P and {nonQ and nonR}’;

(ii) ‘nonP and possibly{Q and nonR}’, which grants the realizability of ‘nonP and {Q and nonR}’;

(iii) ‘nonP and impossibly{Q and nonR}’ implying the factual ‘nonP and not-{Q and nonR}’, which might be realized as ‘nonP and {Q and R}’, ‘nonP and {nonQ and R}’, or ‘nonP and {nonQ and nonR}’.

Clearly, here again all seven alternatives to ‘P and {Q and nonR}’ are eventually permitted. Thus, the two expressions compared are equal: they have the same root conjunctions. This is an important finding for hypothetical logic.

The allowances in all cases are of course problemacies. In purely formal contexts, these problemacies do ordinarily signify that there are unspecified contents fitting the various alternatives. But in contexts of specified content, these problemacies should not be taken as formally logical possibilities, since some of the alternatives may well be excluded by additional statements.

The definition of hypotheticals accurately reflects our formation of such thoughts. Assuming the antecedent clause allows us to hold it mentally in place, so that we can be free to deal with other matters, namely the relative status of the subsequent clause. This process may be called ‘nesting’, or ‘framing’. It is similar to the technique of control in the experimental sciences, where, while keeping all other things equal, we observe the effects on our subject, of a precise change in the single remaining factor.*Nesting.*

In the case of two theses, appropriately related, we frame the one by means of the other, in a simple hypothetical proposition, ‘If P, then Q’. In the case of three theses, we can say ‘If P, then {if Q, then R}’, meaning that P is a context or framework for Q implying R. Likewise, for four theses, ‘If P and Q and R, then S’ can be reformed as ‘If P, then {if Q, then [if R, then S]}’.

We can in this manner nest any number of hypotheticals within each other. In practice, much of the framework is often left tacit, note. Such multiple-theses hypotheticals serve to express partial or conditional antecedence. They may be viewed as forming a continuum, ranging from a single, unconditional thesis, to one framed by more and more difficult demands.

The value of such successive framing by hypotheticals can be seen in analysis of the process of *reductio ad absurdum* used in validation of syllogisms. To prove that ‘If {P and Q}, then R’; we infer ‘If P, then {if Q, then R}’ by framing; then we contrapose the inner hypothetical to obtain ‘If P, then {if nonR, then nonQ}’; then we remove the frame to obtain ‘If {P and nonR}, then nonQ’; thus showing that denial of the conclusion leads to denial of a premise.

Just as the antecedent of a hypothetical may be composite, so may the consequent be, as in ‘if P, then {Q and R}’; this is equivalent to the conjunction of ‘if P, then Q’ and ‘if P, then R’. Just as the consequent may be hypothetical, so may the antecedent be, as in ‘if {if P, then Q}, then R’; this is not equivalent to ‘if {P and Q}, then R’, note well.*Mixed-Form Logic.*

We can also use the disjunctive format in complicated propositions, which present alternative antecedents and/or consequents. For example, ‘If {P or Q}, then R’ (which is ordinarily taken to imply ‘if P, then R’ and ‘if Q, then R’); or again, ‘if P, then {Q or R}’, which, though not incompatible with ‘if P, then Q’ or ‘if P, then R’, does not imply them. Those methods are used to find alternate conclusions from weaker premises (as seen in transitive syllogism), or weaker conclusions from alternate premises (as we shall see with ‘double syllogism’).

More broadly still, any kind of conjunction, hypothetical, or disjunction, positive or negative, may be involved with any other(s), in countless, intricate relations. Of course, it is wise not to get too carried away, it must be possible for the mind to unravel the meaning with relative ease. Going into the mechanics of all these relations in detail is beyond the scope of this book, but it can be expected to be an interesting field.

### 4. Disjunctives.

Disjunctions may involve more than two alternatives, as in ‘P or Q or R or..’.. We tend to use the general operator ‘or’, rather than the more specific ‘and/or’, ‘or else’, and ‘either-or’, because with three or more alternatives, disjunction has more nuances in meaning. Indeed, we need not specify any disjunctive operator at all, but could just list the theses under consideration (P, Q, R, etc.) and verbally specify their collective relations (as explained below).*Multiple Disjunctions.*

Usually, of course, the inclusive form ‘P and/or Q and/or R and/or…’ may be supposed to mean ‘at least one of P, Q, R, etc. must be true’ (leaving open whether each of the others can or must be true or false). Similarly, the exclusive form ‘P or else Q or else R or else…’ may be supposed to mean ‘all but one of P, Q, R, etc. must be false’ (implying only one can be true, but leaving open whether it can be false or must be true; note too that any pair of theses are incompatible). If both these disjunctions are affirmed, the two or more theses involved may be said to be both exhaustive and incompatible.

More generally conceived, a multiple disjunction depends for its definition on *how many theses, out of the total number listed, must be true, and/or how many must be false.* These components specify the degrees of exhaustiveness and/or incompatibility of the alternatives. In some cases they are independent variables, in others, they affect each other, according to the total number of theses available.

Strictly, we should specify the definitions of our disjunction parenthetically; though in practice they are often left unsaid, when we do not know them precisely, or when we consider them as obvious in the context. Note well that the definitions do not tell us exactly which of the theses are true and which false; they only tell us that some stated number are this or that.

With two theses, as already seen, ‘one must be true’ signifies that both cannot be false, ‘one must be false’ signifies that both cannot be true, and those two specifications may occur without each other or together.

With three theses, the specifications ‘one must be true’, ‘two must be true’, ‘one must be false’, ‘two must be false’, can be combined every which way, except for ‘two must be true and two must be false’ together, normally (though in abnormal logic, this is not excluded).

With four (or more) theses, likewise, we can specify that one to three (or more) of the theses must be true and/or that one to three (or more) of the theses must be false, though the total number of theses so specified should not normally exceed the total number of theses available.

Whatever the number of theses, it is clear that the more of them are specified as having to be true or false, the firmer the implied bond between them. For instance, ‘two must be true’ is a more forceful relation than ‘one must be true’. The more definite the bond, the more restrictive the relation, but also the more informative.

In the maximalist case, where we are given that all the theses must be true, or all must be false, or exactly which are true and which false is specified, we are left with no degree of freedom, and no ignorance. In the minimalist case, where any number may be true or false, in any combination, there is no link between the theses, and all the issues remain unresolved. The various degrees of disjunction lie in between these extremes.

Inversely, the greater the total number of theses listed in a disjunction, the looser the bond implied by the ‘or’ operators in it. For instance, ‘one thesis must be true’ represents a weaker relation with reference to a total of three or four theses than to a total of two theses. The more alternatives are available, the more of them we have to eventually eliminate to arrive at categorical knowledge, therefore the less we know so far.

Thus, the operator ‘or’ has many gradations of meaning, depending on various factors. However, we can think of all disjunctives as aligned in a continuum, ranging from one to any number of theses *in toto*, and from one to any number among them specified as having to be true or false. In some cases the inclusive and exclusive specifications diverge, in some cases they converge. Ultimately, all disjunctives are part of the same continuum as conjunctives.

It is best, when faced with such multiplicity of alternatives, to think in terms of the underlying possible outcomes of positive conjunction. For example, ‘one of {P or Q or R} must be true, and two must be false’ may be interpreted as ‘{P and nonQ and nonR}, {nonP and Q and nonR}, and {nonP and nonQ and R}, are possible (that is, at least problematic) conjunctions of the given theses’.*Matrixes.*

This format is least ambiguous, because we may on formal grounds understand the disjunction of the factual conjunctions listed to be formally of the ‘one must be true and all the others must be false’ degree, without having to say so, no matter what the original number of theses. We earlier referred to this as matrix logic.

Note that any of the underlying positive conjunctions involving a negative thesis, may themselves conceal an internal disjunction. For negation is often a shorthand expression of a number of positive alternatives; thus, nonP might mean ‘P1 or P2’, if it so happens that P, P1, and P3 are exhaustive. This is applicable even to elements, and all the more so to compounds and all composites.

Thus, we may find disjunctions within disjunctions within disjunctions; these may be referred to as different levels of disjunction. This phenomenon is interesting, because it illustrates the complexities of stratification which occur among propositions. There is an enormous wealth of possible relations among propositions.

Disjunctions may also may be expressed in hypothetical form, and vice versa. For instance, ‘P or Q or R’ (as defined in the above example) can be reformulated as: ‘If nonP and nonQ, then R, and if nonP and nonR, then Q, and if nonQ and nonR, then P’ (the ‘one thesis is true’ component), and ‘If P, then nonQ and nonR, and if Q, then nonP and nonR, and if R, then nonP and nonQ’ (the ‘two theses are false’ component). But such formulas can get pretty intricate and confusing. This is what justifies disjunction as a valuable form in itself.

But it follows anyway that the laws of intricate logic for hypotheticals may be used to obtain analogous laws for disjunctives; and vice versa. Thus, for instance, the case of a disjunctive proposition with a disjunctive clause as one of its theses, corresponds to the case of premise nesting we encountered in an earlier section.

We found that a modal conjunction within a larger modal conjunction, is equivalent to a factual clause. That is, since ‘nonP and {nonQ and nonR} is impossible’, and ‘nonP and possibly{nonQ and nonR} is impossible’, yield the same matrix of seven alternative conjunctions, they have the same logical properties. It follows that the corresponding disjunctives ‘P or Q or R’ and ‘P or {Q or R}’, intended in the ‘one thesis must be true’ sense, are equivalent.

A disjunction may be taken as a gross unit, as well as with reference to the alternatives it lists. We may focus on the whole or the parts, and determine the one or the others as our clause(s).

Such intricacies will not be covered in any great detail in this work, though interesting. All this is part of a yet broader field of research. The nesting case concerns a possible conjunction within impossible conjunction. But other combinations of ‘modality within modality’ can also be worked out.

- Another direction of development for disjunctive logic, is the introduction of
*modalities of disjunction**.*The concepts of connection and basis are applicable to disjunction. Purely connective disjunction has entirely problematic bases; if the base of each thesis is specified, whether as logical contingency (normally) or as incontingency (abnormally), special logics may apply.

The ‘connection’ of the disjunction is the impossibility of the conjunction(s) which are excluded from the underlying matrix. Here, the law of contradiction is that at least one of all the possible conjunctions in a matrix, for the given number of theses, must be true; the law of the excluded middle is that all but one of them must be false. Thus, connection is inherently incontingent.

One could argue that, since we can place a disjunction as the consequent of a hypothetical statement, we can think of conditional levels of disjunction as well. In that event, the connection may be logically contingent, valid in some specific (though not always specified) context(s). It follows that we can also think of a factual level of disjunction (loosely speaking), signifying that it is operative in the presently held context.

A more modal logic of disjunction may accordingly be developed, and here again basis may come into play. Possible disjunction implies that the disjunction is consequent to certain conditions, and therefore can be made factual by revealing the implicit antecedent. Problematic or logically possible disjunctives, underlie hypothetical propositions with a disjunction as antecedent or consequent. Disjunctions may of course also appear within larger disjunctions.

However, factual (contextual) versus incontingent (unconditional) disjunction, may be compared to material versus strict implication. So these concepts may be used to some extent, if we remain conscious of their main pitfall — namely, the difficulty of pin-pointing precisely just which parts of the overall context frame our propositions, making up our effective so-called ‘context’. In practice, we wordlessly ‘know’ the intended context, but in formal work this vague knowledge is not a useable capital.

In conclusion, the concept of modality provides us with a means of clarifying thoughts to a much greater degree than purely factual logic, giving us a new/improved tool of analysis of data. I leave it to you, to explore this field more thoroughly; this may be compared to presenting you with an object for inspection under your own microscope, using the techniques developed in this treatise.

## 28. LOGICAL COMPOSITIONS.

### 1. Symbolic Logic.

This chapter very briefly describes various processes having to do with the logical composition of conjunctions and disjunctions. The equations developed here, are selected to enable us to deal efficiently, in later chapters, with factorial formulas, especially. They are only a tip of an enormous iceberg, comprising hypothetical relations as well, consideration of all modalities of connections and bases, and full analysis of the negative side.

Although I personally avoid symbolic logic as much as possible, so as not to obscure for myself and others the meaning of what I am doing — in the case of the theorems below, I find that symbolization of logical relations does indeed bring out the processes more clearly. I will use a nomenclature and symbolic representation, which I personally find more comfortable, but which differs slightly from that adopted by modern logic (Copi, 319). It is, of course, to modern logicians that we owe these valuable clarifying formulas.

Let **p**, **q**, **r**, **s** be any theses, which will be conjoined or disjoined; their antitheses **not**p, **not**q, etc., may be symbolized by a so-called ‘curl’ (a curved minus sign like this: **~**), as in ~p, ~q, etc. We may write: ‘p **and** q’ symbolically as ‘p **+** q’ (with plus sign) or as ‘pq’ (with no separation) or as ‘p**.**q’ (the dot suggesting a product). Also, we may write ‘p **or** q’, taken in the weakest ‘and/or’ sense, that ‘one of the theses must be true’, as ‘p **v** q’ (v for versus, supposedly) or as ‘p**,**q’ (note the comma); note that some computer programming languages use a vertical bar (like this: ½), instead. Brackets ‘{}’ are used to signify a clause within a larger sentence.

Note that the results for other forms of disjunction, like ‘or else’ or ‘either-or’, are often different; these will not be discussed here.

There are two sets of logical composition processes for us to consider:

*‘*, which is merging of conjunctions, or of disjunctions, with others of the same sign; the reverse process of separation, where it is feasible, might be called ‘subtraction’ (symbolized by a minus sign:**Addition**’**–**); and*‘*, which is merging a mix of conjunctions and disjunctions with each other; the reverse direction, where it is feasible, might be called ‘division’.**Multiplication**’

Two or more propositions which are added together are said to form a single, composite proposition. A proposition which is not itself a composite of others is called elementary. Specifically, a proposition consisting of a conjunction of others is called a compound; this is one form of composition.

Note well that this logic is limited to the one sense of disjunction, and to fully problematic bases. Other manners of disjunction, such as ‘two (or more) theses must be true’ and ‘one (or more) theses must be false’, and more specific bases, each have their own logic. Also, we are here going to deal with disjunction with minimal reference to modality, although a more modal approach would be more precise and interesting. Consideration of conditional disjunction would cause the study to spill over into the interplay of hypotheticals with the processes here considered.

So the present research is limited, because its purpose is utilitarian. A fuller theory of logical composition requires additional work. However, the material dealt with here has indirect applications. Many theorems can be derived from those here described. We can, for instance, change the polarity of theses or logical relations in various ways. We can also expect that some of the laws of ‘at least one thesis must be true’ logic, carry over into ‘more than one thesis must be true’ logic.

The theorems below are presented as usually reversible eductions, meaning that given the form on the left, the form on the right follows at will, and usually vice versa. However, many of them can also be classed as deductive processes, and will reappear in later chapters in that guise. Another way to view them is, as rules of transformation, like changes in what we regard as clauses.

The analogies between mathematics and logic should not be overrated; they only go so far. Logic may be regarded as the manipulation of concepts of any kind, whereas mathematics concerns specifically numerical concepts. Although there is some intervention of mathematics in logic, for the resolution of quantitative issues, and we may be said to think logically when engaged in mathematics, these sciences are very different fields of interest. They have rationalism in common, but their scopes are different and neither is really a subsidiary of the other.

### 2. Addition.

- Addition of conjuncts:

p + {q + r} = p + q + r

{p + q} + {r + s} = p + q + r + s

…and so on for any number of conjuncts. Proof is that p, q, r, s are all independently true, anyway, on both sides of the equations. The elimination of repetitives is a special case of addition: since p + p = p, it follows that p + {p + q} = p + q.

- Addition of alternatives:

p v {q v r} = p v q v r

{p v q} v {r v s} = p v q v r v s

…and so on for any number of disjuncts. Proof is in such cases best sought by matrixual analysis; that is, *testing* each and every eventual combination of theses and antitheses, to see whether or not it obeys the demands of the given composite, then comparing the results on the two sides of the equation. Thus, with three theses:

p v {q v r} | p v q v r |

p + q + r | p + q + r |

p + q + notr | p + q + notr |

p + notq + r | p + notq + r |

p + notq + notr | p + notq + notr |

notp + q + r | notp + q + r |

notp + q + notr | notp + q + notr |

notp + notq + r | notp + notq + r |

(notp + notq + notr) | (notp + notq + notr) |

The conjunctions shown in brackets are those which, having been tried out on the given disjunctions, failed the test. Both sides evidently mean that, so far, any conjunction of p, q, r and their antitheses is a conceivable outcome, to the exception of ‘notp and notq and notr’. Equations involving more theses are similarly dealt with.

As with conjuncts, the elimination of repetitives is a special case of addition. Since p v p = p (meaning, p will be affirmed in either case), it follows that p v {p v q} = p v q. Note that in the case of ‘p or else p’, notp would follow; the form ‘either p or p’ is of course inconceivable.

- With regard to subtraction, the equations above are to be reread from right to left.

Addition followed by subtraction is useful to remove a common factor from two brackets:

{p + q} + {p + r} = p + p + q + r = p + {q + r}

{p v q} v {p v r} = p v p v q v r = p v {q v r}

…or to reshuffle brackets:

{p + q} + {r + s} = p + q + r + s = {p + r} + {r + s}

{p v q} v {r v s} = p v q v r v s = {p v r} v {r v s}

- But the idea of ‘subtraction’ more precisely fits the equation ‘p + {~q} = p – q’, of course. This suggests implications like the following, which shall be seen again in the context of ‘logical apodosis’:

{p v q} – q implies p

{p v q v r} – r implies p v q

…and so on for any number of theses. Note that these implications are valid only in one direction. For instance, in the first case, p alone cannot tell us whether q or notq is true, and therefore cannot yield the conclusion that ‘{p v q} and notq’ are both true.

Nor may one push the analogy to mathematics so far as to move q to the other side of the implication and claim that ‘p v q’ and ‘p + q’ are equal. It only follows that ~p + ~q together imply ~{p v q}, and ~p + {p v q} together imply ‘q’.

### 3. Multiplication.

The significance of multiplication in practice, is to clarify the logically possible combinations of theses, into compounds or other composites, which are implied by various interplays of conjunction and disjunction. This is mixed-form logic. Any impossible combinations are put aside.

- Conjunctive multiplication:

p + {q v r} = {p + q} v {p + r} = pq v pr

{p v q} + {r v s} = pr v ps v qr v qs

Proof is best sought by matrixual analysis. Thus, with three theses, we find that only three of the conjunctions in each matrix are allowed so far, and those three are the same on both sides. Each side excludes the five bracketed conjunctions. So the two statements are equivalent.

p + {q v r} | {p + q} v {p + r} |

p + q + r | p + q + r |

p + q + notr | p + q + notr |

p + notq + r | p + notq + r |

(p + notq + notr) | (p + notq + notr) |

(notp + q + r) | (notp + q + r) |

(notp + q + notr) | (notp + q + notr) |

(notp + notq + r) | (notp + notq + r) |

(notp + notq + notr) | (notp + notq + notr) |

Equations involving more theses are similarly dealt with. Note well that if the result of such a multiplication contains an inconsistent clause, it is simply canceled out; for instance, if q = ~p, then the ‘p + q + r’ and ‘p + q + notr’ combinations are automatically eliminated, leaving only ‘p + notq + r’ as possible.

Also note that ‘p + {p v q}’ implies p (with the status of q left open), since p is already affirmed independently; this incidentally limits the ‘p v q’ clause to the two roots ‘p + q’ and ‘p + ~q’. As for ‘{p v q} + {p v r}’, it does not imply p, since notp may concur with q and r, without disobeying the premise.

- Disjunctive multiplication:

p v {q + r} = {p v q} + {p v r} = p,q + p,r

{p + q} v {r + s} = p,r + p,s + q,r + q,s

Proof is again best sought by matrix logic. The two sides of the equation yield five identical allowances, and three identical exclusions (in brackets).

p v {q + r} | {p v q} + {p v r} |

p + q + r | p + q + r |

p + q + notr | p + q + notr |

p + notq + r | p + notq + r |

p + notq + notr | p + notq + notr |

notp + q + r | notp + q + r |

(notp + q + notr) | (notp + q + notr) |

(notp + notq + r) | (notp + notq + r) |

(notp + notq + notr) | (notp + notq + notr) |

Note that the special composite ‘p v {p + q}’ implies p (with the status of q left open), since p v p = p. In contrast, the special composite ‘{p v q} + {p v r}’ does not imply p, since notp may concur with q and r,

More complex cases are proved similarly, by testing the various roots, by exposing implied possibilities of conjunctions between all the theses and antitheses, and seeing if they correspond on both sides of the equations. In practice, multiplication of more than two clauses is best dealt with by successive multiplication of pairs of clauses.

Observe, incidentally, that the matrix of ‘p v {q + r}’ includes the three roots of ‘p + {q v r}’, and an additional two alternatives.

- With regard to division, the equations above are to be reread from right to left. The idea of division lies in our seeming to take the common factor p out of the brackets, as in mathematics.

### 4. Expansions.

- The various equations developed thus far can be used to analyze more complex mixtures of conjunction and disjunction. Processes like the following may be called ‘expansions’:

p = p + {q v ~q} = {p.q} v {p.~q}

This equation teaches us that any proposition p may be logically composed, in the way of conjunctive multiplication, with any other meaningful proposition q and its negation ~q, since the proposition ‘q or notq’ is always true by the law of the excluded middle. We can repeat the process as often as we wish, as in:

p = p + {q v ~q} + {r v ~r} = {p.q.r} v {p.q.~r} v {p.~q.r} v {p.~q.~r}

- The purpose of logical composition, is
*to reduce any given formula to a disjunction of conjunctions*. It appears that our faculty of understanding requires such reduction, to fully grasp the significance of any complex formula. This means that the results we obtained earlier for multiplication are not final, because they do not satisfy the mind’s requirement.

Expressions like ‘pq v pr’ or ‘p,q + p,r’ are not satisfactory, because they do not specify the truth or falsehood of every proposition involved. They must be expanded further, as follows:

(i) Conjunctive multiplication.

p + {q v r} = {p + q} v {p + r} = pq v pr

but, pq = {p.q.r} v {p.q.~r}

pr = {p.q.r} v {p.~q.r}

therefore, p + {q v r} = {p.q.r} v {p.q.~r} v {p.~q.r}

(ii) Disjunctive multiplication.

p v {q + r} = {p v q} + {p v r} = pp v pr v qp v qr = p v pq v pr v qr

but, p = {p.q.r} v {p.q.~r} v {p.~q.r} v {p.~q.~r}

pq = {p.q.r} v {p.q.~r}

pr = {p.q.r} v {p.~q.r}

qr = {p.q.r} v {~p.q.r}

therefore, p v {q + r} = {p.q.r} v {p.q.~r} v {p.~q.r} v {p.~q.~r} v {~p.q.r}

Notice the elimination of repetitive conjuncts or disjuncts, and the reordering of clauses, in accordance with the principles of addition earlier established. (That is, since xx = x and x,x = x and xy = yx and x,y = y,x.)

The above expansions are the ultimate solutions of the problems of multiplication: the most informative interpretations. The object of such process is to express the original formula in less ambiguous form. The preceding results do not clearly define the status of each of the propositions p, q, r. The components have to always be fully expanded to become comprehensible.

We see that the final equations for multiplications, are simply restatements of *the matrixes* of {p + [q v r]}, and {p v [q + r]}, respectively. They provide us with a set of roots, like ‘p and q and r’ or ‘p and q and notr’ or ‘p and notq and r’.

The various conjunctions in disjunction represent all the possible outcomes of the original formula. They tell us the various ways it can be read, making a list of its alternative meanings. These are the eventual inferences which can be drawn from it. Any combination which is not mentioned in the conclusion, is not inferable from the premise.

Thus, just as ordinary disjunction is best understood with reference to a matrix, so in more complex situations we must reassemble the components of our proposition into more mentally accessible results. Two formulas with the same matrix, are logically equal.

- These findings allow us to deal with still more intricate combinations of addition and multiplication. Consider, for instance, the puzzle: What does ‘p and q or r’ mean? Using the symbolic techniques introduced thus far, we can ‘expand’ that proposition as follows. q is in an ambiguous position, between an ‘and’ and an ‘or’, so:

p + q v r may mean p + {q v r}, or may mean {p + q} v r

that is, conjunctive or disjunctive multiplication.

so, {p + q v r} = {p + [q v r]} v {[p + q] v r}

but, {p + [q v r]} = {p.q.r} v {p.q.~r} v {p.~q.r}

and, {[p + q] v r} = {p.q.r} v {p.q.~r} v {p.~q.r} v {~p.q.r} v {~p.~q.r}

then, by addition of alternatives and elimination of repetitives, it follows that:

{p + q v r} = {p.q.r} v {p.q.~r} v {p.~q.r} v {~p.q.r} v {~p.~q.r}

This teaches us incidentally that, since the roots of {p + q v r} are all among the roots of {[p + q] v r}, and vice versa, these two composites are no more nor less informative than each other. That is, the following equation is valid:

p + q v r = {p + q} v r

On the other hand, the composite {p + [q v r]} is more specific and restrictive than either of the composites {p + q v r} or {[p + q] v r}, because it makes allowance for less possibilities. It writes off the alternative outcomes ‘notp and q and r’ and ‘notp and notq and r’, at the outset.

- Still more complex puzzles can be resolved. These are interesting training exercises, like ladders. The easiest course is to apply already known and simpler processes,
*successively*.

For instance, to expand the formula: {p v q} + {r v s}, let {p v q} = x, say. Then, by substitution and conjunctive multiplication, x + {r v s} = xr v xs. This means {[p v q] + r} v {[p v q] + s}. These clauses can now be expanded, and the resulting alternatives added together. Similarly, we can clarify the formula: {p + q} v {r + s} in stages. Try doing it.

### 5. Utility.

In conclusion, we see that symbolic logic can be a valuable tool for untangling perplexing statements. Modern logicians have also developed similar techniques for compositions involving hypothetical relations, as already mentioned.

However, it should also be apparent that the more intricate the formula, the less likely are we to come across it in practice. This is why modern, symbolic logic tends to degenerate into irrelevancy, and give logic as a whole a bad name.

The value of the main equations, is to show us that our sentences should be clearly formulated, so that the phrases we intend as our clauses are apparent to all. Otherwise, we might be misunderstood. This is especially important when drawing up legal documents, or making scientific statements. Perhaps the best practical applications are in computer and robot programming.

However, beyond a certain point, there is no utility in studying complex formulas, because they are sure to be misinterpreted by the uninitiated, anyway. Likewise, when interpreting texts written by other people, we cannot always be sure that *they* formulated them with expert knowledge and total awareness of their logical significance.[3]

Even if overly intricate logic is of limited practical utility, it is an important enough doctrine. It is a part of the grand enterprise of pursuit of consistency in Knowledge as a whole. It describes for us, how to make peace within or between large bodies of information.

## 29. HYPOTHETICAL SYLLOGISM AND PRODUCTION.

There are several kinds of deductive argument involving hypothetical propositions or their derivatives. They are distinguished according to whether they involve only hypotheticals, or hypotheticals mixed with categorical forms. The main kinds are syllogism, production, apodosis and dilemma. Note that the valid moods are not here listed in symbolic terms, as we did with categoricals, to avoid obscuring their impact.

### 1. Syllogism.

Hypothetical syllogism is argument whose premises and conclusion are all hypotheticals. It is mediate inference, with minor (symbol P), middle (M), and major (Q) theses, deployed in figures, as was the case in categorical syllogism.

Its *most primary valid mood*, from which *all* others may be derived by direct or indirect reduction, is as follows. It tells us, as for the analogue in categorical syllogism, that, as H.W.B. Joseph would say, ‘whatever falls under the condition of a rule, follows the rule’.

This primary mood is valid irrespective of whether the hypotheticals involved are of unspecified base, normal (contingency-based), or abnormal. That is generally true for its primary derivatives, too; but subaltern derivatives are only applicable in cases where both theses are known to be logically contingent (and not just problematic), because the subalterns require eductive processes which depend on this condition for their validity.

If M, then Q

if P, then M

so if P, then Q

This is a first figure syllogism. Its validity obviously follows from the meaning of the operator ‘if-then’ involved. Although the connection in hypotheticality is expressed by modal conjunctive statements, ‘if-then’ underscores an additional, not-tautologous, sense, occurring on a finer level. This teaches us a purely conjunctive argument, from which many laws for the logic of conjunction may be inferred, that:

The premises: {M and nonQ} is impossible,

and {P and nonM} is impossible, together

yield the conclusion: {P and nonQ} is impossible.

This could be written symbolically as **1/H2nH2nH2n**, note.

- Figure One.

(i) From the primary valid mood, we can draw up the following full list of valid, **uppercase, perfect** moods, in first figure, by substituting antitheses for theses in every possible combination.

If M, then Q | If nonM, then Q |

if P, then M | if P, then nonM |

so, if P, then Q | so, if P, then Q |

If M, then nonQ | If nonM, then nonQ |

if P, then M | if P, then nonM |

so, if P, then nonQ | so, if P, then nonQ |

If M, then Q | If nonM, then Q |

if nonP, then M | if nonP, then nonM |

so, if nonP, then Q | so, if nonP, then Q |

If M, then nonQ | If nonM, then nonQ |

if nonP, then M | if nonP, then nonM |

so, if nonP, then nonQ | so, if nonP, then nonQ |

(ii) Next, from one of the valid, uppercase, perfect moods, we derive the primary, valid, **lowercase, perfect** mood, by reductio ad absurdum, as follows. Note that the major premise is uppercase, and the minor premise and conclusion are lowercase.

If M, then Q | contrapose major: | If nonQ, then nonM |

if P, not-then nonM | deny conclusion: | if P, then nonQ |

so, if P, not-then nonQ | get anti-minor | if P, then nonM |

From this primary mood, we can draw up the following full list of valid, lowercase, perfect moods, in the first figure, by substituting antitheses for theses in every possible combination.

If M, then Q | If nonM, then Q |

if P, not-then nonM | if P, not-then M |

so, if P, not-then nonQ | so, if P, not-then nonQ |

If M, then nonQ | If nonM, then nonQ |

if P, not-then nonM | if P, not-then M |

so, if P, not-then Q | so, if P, not-then Q |

If M, then Q | If nonM, then Q |

if nonP, not-then nonM | if nonP, not-then M |

so, if nonP, not-then nonQ | so, if nonP, not-then nonQ |

If M, then nonQ | If nonM, then nonQ |

if nonP, not-then nonM | if nonP, not-then M |

so, if nonP, not-then Q | so, if nonP, not-then Q |

(iii) Next, from one of the valid, uppercase, perfect moods, we derive the primary, valid, **imperfect** mood, by reductio ad absurdum, as follows. Note the change in polarity of the minor thesis in the conclusion, which defines the moods as imperfect, and the distinct mixed polarity of the middle thesis in the two premises. Note also that the minor premise is uppercase, and the major premise and conclusion are lowercase.

If M, not-then Q | deny conclusion: | If nonP, then Q |

if P, then nonM | contrapose minor: | if M, then nonP |

so, if nonP, not-then Q | get anti-major: | if M, then Q |

From this primary mood, we can draw up the following full list of valid, imperfect moods, in the first figure, by substituting antitheses for theses in every possible combination.

If M, not-then Q | If nonM, not-then Q |

if P, then nonM | if P, then M |

so, if nonP, not-then Q | so, if nonP, not-then Q |

If M, not-then nonQ | If nonM, not-then nonQ |

if P, then nonM | if P, then M |

so, if nonP, not-then nonQ | so, if nonP, not-then nonQ |

If M, not-then Q | If nonM, not-then Q |

if nonP, then nonM | if nonP, then M |

so, if P, not-then Q | so, if P, not-then Q |

If M, not-then nonQ | If nonM, not-then nonQ |

if nonP, then nonM | if nonP, then M |

so, if P, not-then nonQ | so, if P, not-then nonQ |

(iv) *Subaltern moods**.* These are valid only with normal hypotheticals, unlike the preceding, because they are derived from the latter by subalternating a lowercase premise or being subalternated by an uppercase conclusion. Their premises are always both uppercase, and their conclusion lowercase.

The following sample can be derived from moods of type (i) by obverting the conclusion, or equally well from moods of type (ii) by replacing the minor premise with its obvertend. On this basis, 8 subaltern moods can be derived in the usual manner. These are perfect in nature.

If M, then Q

if P, then M

so, if P, not-then nonQ.

The following sample can be derived from moods of type (i) by obvert-inverting the conclusion, or equally well from moods of type (iii) by replacing the major premise with its obvertend. On this basis, 8 subaltern moods can be derived in the usual manner. These are imperfect, since the minor thesis changes polarity in the conclusion.

If M, then Q

if P, then M

so, if nonP, not-then Q.

In summary, we thus have a total of 3X8 = 24 primary valid moods in the first figure, plus 2X8 = 16 subaltern valid moods. Or a total of 40 valid moods, out of 8X8X8 = 512 possibilities.

- Figure Two.

(i) From one of the valid, lowercase, perfect moods, of the first figure, we derive the primary, valid, **uppercase, perfect** mood, of the second figure, by reductio ad absurdum, as follows. Alternatively, we could have used direct reduction, by contraposing the major premise, through a valid, uppercase, perfect mood, of the first figure.

If Q, then M | with same major: | If Q, then M |

if P, then nonM | deny conclusion: | if P, not-then nonQ |

so, if P, then nonQ | get anti-minor: | so, if P, not-then nonM |

From this primary, valid mood, we can draw up the following full list of valid, uppercase, perfect moods, in the second figure, by substituting antitheses for theses in every possible combination.

If Q, then M | If Q, then nonM |

if P, then nonM | if P, then M |

so, if P, then nonQ | so, if P, then nonQ |

If nonQ, then M | If nonQ, then nonM |

if P, then nonM | if P, then M |

so, if P, then Q | so, if P, then Q |

If Q, then M | If Q, then nonM |

if nonP, then nonM | if nonP, then M |

so, if nonP, then nonQ | so, if nonP, then nonQ |

If nonQ, then M | If nonQ, then nonM |

if nonP, then nonM | if nonP, then M |

so, if nonP, then Q | so, if nonP, then Q |

(ii) Next, from one of the valid, uppercase, perfect moods, of the first figure, we derive the primary, valid, **lowercase, perfect** mood, of the second figure, by reductio ad absurdum, as follows. Alternatively, we could have used direct reduction, by contraposing the major premise, through a valid, lowercase, perfect mood, of the first figure. Note that the major premise is uppercase, and the minor premise and conclusion are lowercase.

If Q, then M | with same major: | If Q, then M |

if P, not-then M | deny conclusion: | if P, then Q |

so, if P, not-then Q | get anti-minor: | if P, then M |

From this primary mood, we can draw up the following full list of valid, lowercase, perfect moods, in the second figure, by substituting antitheses for theses in every possible combination.

If Q, then M | If Q, then nonM |

if P, not-then M | if P, not-then nonM |

so, if P, not-then Q | so, if P, not-then Q |

If nonQ, then M | If nonQ, then nonM |

if P, not-then M | if P, not-then nonM |

so, if P, not-then nonQ | so, if P, not-then nonQ |

If Q, then M | If Q, then nonM |

if nonP, not-then M | if nonP, not-then nonM |

so, if nonP, not-then Q | so, if nonP, not-then Q |

If nonQ, then M | If nonQ, then nonM |

if nonP, not-then M | if nonP, not-then nonM |

so, if nonP, not-then nonQ | so, if nonP, not-then nonQ |

(iii) ** Subaltern moods**. These are valid only with normal hypotheticals, unlike the preceding, because they are derived from the latter by subalternating a lowercase premise or being subalternated by an uppercase conclusion. Their premises are always both uppercase, and their conclusion lowercase.

The following sample can be derived from moods of type (i) by obverting the conclusion, or equally well from moods of type (ii) by replacing the minor premise with its obvertend. On this basis, 8 subaltern moods can be derived in the usual manner. These are perfect in nature.

If Q, then M

if P, then nonM

so, if P, not-then Q.

The following sample can be derived from moods of type (i) by obvert-inverting the conclusion. On this basis, 8 subaltern moods can be derived in the usual manner. These are imperfect, since the minor thesis changes polarity in the conclusion.

If Q, then M

if P, then nonM

so, if nonP, not-then nonQ.

The following sample can be derived from moods of type (ii) by replacing the minor premise with its obvert-invertend. On this basis, 8 subaltern moods can be derived in the usual manner. These are imperfect, since the minor thesis changes polarity in the conclusion. Note the distinct uniform polarity of the middle thesis in the two premises.

If Q, then M

if P, then M

so, if nonP, not-then Q.

In summary, we thus have a total of 2X8 = 16 primary valid moods in the second figure, plus 3X8 = 24 subaltern valid moods. Or a total of 40 valid moods, out of 8X8X8 = 512 possibilities.

- Figure Three.

(i) From one of the valid, uppercase, perfect moods, of the first figure, we derive the primary, valid, **perfect** mood, with **lowercase major** premise, of the third figure, by reductio ad absurdum, as follows. Alternatively, we could have used direct reduction, by contraposing the major premise, and transposing, through a valid, lowercase, perfect mood, of the first figure. The conclusion is of course lowercase.

If M, not-then nonQ | deny conclusion: | If P, then nonQ |

if M, then P | with same minor: | if M, then P |

so, if P, not-then nonQ | get anti-major: | if M, then nonQ |

From this primary, valid mood, we can draw up the following full list of valid, perfect moods, with lowercase major premise, in the third figure, by substituting antitheses for theses in every possible combination.

If M, not-then nonQ | If nonM, not-then nonQ |

if M, then P | if nonM, then P |

so, if P, not-then nonQ | so, if P, not-then nonQ |

If M, not-then Q | If nonM, not-then Q |

if M, then P | if nonM, then P |

so, if P, not-then Q | so, if P, not-then Q |

If M, not-then nonQ | If nonM, not-then nonQ |

if M, then nonP | if nonM, then nonP |

so, if nonP, not-then nonQ | so, if nonP, not-then nonQ |

If M, not-then Q | If nonM, not-then Q |

if M, then nonP | if nonM, then nonP |

so, if nonP, not-then Q | so, if nonP, not-then Q |

(ii) Next, from one of the valid, lowercase, perfect moods, of the first figure, we derive the primary, valid, **perfect** mood, with **lowercase** **minor** premise, of the third figure, by reductio ad absurdum, as follows. Alternatively, we could have used direct reduction, by contraposing the minor premise, through a valid, lowercase, perfect mood, of the first figure. The conclusion is of course lowercase.

If M, then Q | deny conclusion: | If P, then nonQ |

if M, not-then nonP | with same minor: | if M, not-then nonP |

so, if P, not-then nonQ | get anti-major: | if M, not-then Q |

From this primary, valid mood, we can draw up the following full list of valid, perfect moods, with lowercase minor premise, in the third figure, by substituting antitheses for theses in every possible combination.

If M, then Q | If nonM, then Q |

if M, not-then nonP | if nonM, not-then nonP |

so, if P, not-then nonQ | so, if P, not-then nonQ |

If M, then nonQ | If nonM, then nonQ |

if M, not-then nonP | if nonM, not-then nonP |

so, if P, not-then Q | so, if P, not-then Q |

If M, then Q | If nonM, then Q |

if M, not-then P | if nonM, not-then P |

so, if nonP, not-then nonQ | so, if nonP, not-then nonQ |

If M, then nonQ | If nonM, then nonQ |

if M, not-then P | if nonM, not-then P |

so, if nonP, not-then Q | so, if nonP, not-then Q |

(iii) Next, from one of the valid, lowercase, perfect moods, of the first figure, we derive the primary, valid, **imperfect** mood, of the third figure, by direct reduction, as follows. Note the change in polarity of the minor thesis in the conclusion, which defines the mood as imperfect, and the distinct mixed polarity of the middle thesis in the two premises. Note also that both premises and the conclusion are uppercase.

If M, then Q | with same major: | If M, then Q |

if nonM, then P | contrapose minor: | if nonP, then M |

so, if nonP, then Q | get conclusion: | so, if nonP, then Q |

From this primary mood, we can draw up the following full list of valid, imperfect moods, in the third figure, by substituting antitheses for theses in every possible combination.

If M, then Q | If nonM, then Q |

if nonM, then P | if M, then P |

so, if nonP, then Q | so, if nonP, then Q |

If M, then nonQ | If nonM, then nonQ |

if nonM, then P | if M, then P |

so, if nonP, then nonQ | so, if nonP, then nonQ |

If M, then Q | If nonM, then Q |

if nonM, then nonP | if M, then nonP |

so, if P, then Q | so, if P, then Q |

If M, then nonQ | If nonM, then nonQ |

if nonM, then nonP | if M, then nonP |

so, if P, then nonQ | so, if P, then nonQ |

(iv) *Subaltern moods**.* These are valid only with normal hypotheticals, unlike the preceding, because they are derived from the latter by subalternating a lowercase premise or being subalternated by an uppercase conclusion. Their premises are always both uppercase, and their conclusion lowercase.

The following sample can be derived from moods of type (i) by replacing the major premise with its obvertend, or equally well from moods of type (ii) by replacing the minor premise with its obvertend. On this basis, 8 subaltern moods can be derived in the usual manner. These are perfect in nature.

If M, then Q

if M, then P

so, if P, not-then nonQ.

The following sample can be derived from moods of type (i) by replacing the major premise with its obvert-invertend, or equally well from moods of type (iii) by obvert-inverting the conclusion. On this basis, 8 subaltern moods can be derived in the usual manner. These are perfect in nature, but note the distinct mixed polarity of the middle thesis in the two premises.

If M, then Q

if nonM, then P

so, if P, not-then Q.

The following sample can be derived from moods of type (ii) by replacing the minor premise with its obvert-invertend, or equally well from moods of type (iii) by obverting the conclusion. On this basis, 8 subaltern moods can be derived in the usual manner. These are imperfect, since the minor thesis changes polarity in the conclusion. Note the distinct mixed polarity of the middle thesis in the two premises.

If M, then Q

if nonM, then P

so, if nonP, not-then nonQ.

In summary, we thus have a total of 3X8 = 24 primary valid moods in the third figure, plus 3X8 = 24 subaltern valid moods. Or a total of 48 valid moods, out of 8X8X8 = 512 possibilities.

- With regard to the
*fourth figure*, it can be ignored in hypothetical syllogism. Since the first figure here (unlike with categorical syllogism) includes imperfect moods, the fourth figure here would introduce no new valid moods for us. Its valid moods can of course all be reduced directly to the first figure, by transposing or contraposing the premises, but they do not represent a movement of thought of practical value.

We therefore have, in the three significant figures taken together, a total of 24+16+24 = 64 primary valid moods, plus 16+24+24 = 64 subaltern valid moods. Or a total of 128 valid moods, out of 3X512 = 1536 possibilities; meaning a validity rate of 8.33%.

### 2. Other Derivatives.

The chaining of syllogisms into a series forming a *sorites* is possible with hypothetical syllogism, similarly to categorical syllogism. This is used in practice, of course, and applies irrespective of basis. The typical sorites looks as follows:

If A, then B

if B, then C

…

if G, then H

therefore, if A, then H.

Note that we are in the figure one, and we state the most minor premise first, and successively work up to the most major premise, and lastly the conclusion. A sorites should be reducible to valid syllogisms to be valid.

Of course, sorites is only the most regular form of continuous argument, the easiest to think without aid of paper and pencil. More broadly, any succession of premises, in any combination of figures, yielding a valid final conclusion, may be viewed as continuous, even though we have to think out the intermediate conclusions, zigzagging from figure to figure, to reach the result.

We can readily reformulate all the above syllogisms using derivative forms, such as simple disjunctions. For examples, the following arguments, taken at random, are easily validated by transforming the disjunctives into standard hypotheticals:

M and/or Q | Q or else M |

P or else M | P not and/or nonM |

P or else nonQ | P not and/or Q. |

Here again, I would not regard these as distinct valid moods. Even if they are used in practice, we are mentally required to restate them in ‘If/then’ form to understand them. It will however be seen, in the context of dilemma, that there are certain arguments, which mix ‘If/then’ forms with disjunctives, which are comprehensible on their own merit, and used in everyday discourse.

Such arguments may also be regarded as ‘logical compositions’. With multiple alternatives, the possible number of arguments increases and so does the mental confusion. When translating the given disjunctions into ‘If-then’ statements causes us as much confusion, the best course is to express each proposition in terms of the conjunctions is allows and forbids; then we can best see what conclusion, if any, may be drawn.

We can also, it is noted, appeal to the above valid moods of the syllogism to clarify reasoning involving compound forms. That is, when one or both premises signifies implicance or subalternation or contradiction or contrariety or subcontrariety, we may be able to fuse the results of two or more simple syllogisms, and get a compound conclusion.

Lastly, arguments may be fashioned in conditional frameworks, so that we have nested hypotheticals for premise(s) and conclusion. This may be viewed as a wider logic, concerning composite antecedents or consequents, conjunctive or even disjunctive ones. Researching the mechanics of partial or alternative theses is an area that deserves eventual attention, but presumably it can be reduced to the findings of unconditional logic.

Subaltern moods are implicitly conditional; they have as hidden premises, the categorical propositions that the theses are logically contingent, rather than merely problematic or partly or wholly incontingent. The tacitly understood premises are: ‘P (and nonP) is contingent, and Q (and nonQ) is contingent’. I have made no effort to develop subaltern moods with abnormal bases, because once a thesis is known to be incontingent it is rarely thereafter used in hypothetical propositions.

### 3. Production.

How are hypothetical propositions produced? By their very nature they do not presuppose the reality of their theses, so how do we know that the antecedent does (or does not) engage the consequence? This question will be answered in this section.

Hypothetical propositions signify a logical connection between the theses, so that any argument which is logically valid may be recast in hypothetical form.

The theses involved may of course have any form, including themselves hypothetical. The term ‘connection’ here is to be understood in its widest sense, including any logical relationship, positive or negative, normal or abnormal. Thus, all oppositions, eductions, deductions, are included here; overall, a valid inference of any kind produces a positive hypothetical, an invalid inference produces a negative hypothetical.

Also, the expression ‘logically valid’ should be taken as comprehensive of the known and the unknown; there is no presumption here that the science of logic as we know it to date is complete. It is important to stress this; while all established logical truths are capable of producing hypotheticals, it does not follow that hypotheticals cannot be produced by means not yet clarified by this science. No claim to omniscience is required.

An example of production would be recasting a categorical syllogism in hypothetical form: e.g. ‘If all S are M and all M are P, then all S are P’. This is a conclusion, whose premises are the process of validation of that mood of the syllogism via the laws of logic.

If we instead produced the briefer conclusion ‘If all S are M, all S are P’, the process to be valid must have included, after the above, a nesting (to ‘If all M are P, then if all S are M, all S are P’) and an apodosis (with minor premise ‘All M are P’). Thus enthymeme need not be viewed as merely syllogism with a suppressed (tacit) premise, but as the end product of a series of definite arguments.

However, production is not limited to relationships in terms of variables, but is especially useful for application to specific values. Using a formal relationship as major premise, we may, through the act of substitution as minor premise, produce a hypothetical with particular contents as conclusion. Continuing the above example, we might for instance produce, ‘If all men were wise, they would not make war’.

In short, any logical series which is incomplete, may be made to at least yield a hypothetical conclusion, and thus constitute a productive process.

The missing information may simply be the exact quantity involved. Thus, if in the above example we do not know whether all or only some S are M, we can still conclude from ‘All M are P’ that ‘If any S is M, it is P’. This produces a hypothetical proposition which seems general, but in fact only suggests that some S may be M. Incidentally, the expression ‘whether’ may itself be viewed as a derivative form of hypothetical, concealing a dilemma.

Similarly, a negative hypothetical would express a nonsequitur. For example, ‘If no S are M and all M are P, it does not follow that no S are P’. Likewise, with particular contents or indefinite quantities, as above.

Clearly, the possibilities are virtually infinite. Any formal or informal sequence permitted or forbidden by the laws of logic constitutes a productive process. Ordinarily, a hypothetical would not be formed, unless information was missing or already known wrong, and only problematic elements would be included in it as theses; but there is nothing illicit in forming one even with definite theses of known truth.

## 30. LOGICAL APODOSIS AND DILEMMA.

### 1. Apodosis.

Apodosis is argument involving a hypothetical proposition as major premise, and the affirmation or denial, of one thesis as minor premise, and of the other as conclusion. Needless to say, the two premises must be true, for the conclusion to follow, as in all argument. There are essentially two valid moods, as follows:

If P, then Q | If P, then Q |

and P | but not Q |

hence, Q | hence, not P |

We see that the major premise has to be a positive hypothetical, the minor premise must either affirm the antecedent or deny the consequent, and the conclusion can only, accordingly, affirm the consequent or deny the antecedent. Such argument merely activates, as it were, the dormant power of the hypothetical, when the minor premise is independently found true.

In the first case, the conclusion follows directly; the second case could be reduced to the first, by contraposition of the major premise. The validity of these moods can be demonstrated by reference to the definition of the major premise as ‘the conjunction of P and nonQ is impossible’; it follows that, if P was true without Q being so, or Q was false without P being so, this impossibility fail to be upheld.

We can also present as valid moods, and in like manner validate, moods involving the remaining positive hypothetical forms. These are to some extent interesting in themselves, showing us still different ways otherwise inaccessible information might be indirectly arrived at, or that assumptions might be eliminated. But they are also valuable for the validation of certain arguments involving disjunctive or compound, derivative forms.

If P, then not Q | If P, then not Q |

and P | but Q |

hence, not Q | hence, not P |

If not P, then Q | If not P, then Q |

and not P | but not Q |

hence, Q | hence, P |

If not P, then not Q | If not P, then not Q |

and not P | but Q |

hence, not Q | hence, P |

Other moods of apodosis, including those with a lowercase major premise, such as ‘If P, not-then Q’, have no demonstrable basis or lead to inconsistency, and so are invalid. The following two are specially noteworthy, because they represent oft-made errors of judgement.

If P, then Q | If P, then Q |

and Q | but not P |

hence, P. | hence, not Q. |

These are, to repeat, formally invalid. However, note that they are used for purposes of ‘adduction’, the provision of evidence or counterevidence to support or discredit theories. The positive mood tells us that the more a theory P makes successful predictions, such as Q, the more credible it becomes (though it is not proved); the negative mood tells us that when the presuppositions, such as P, of a theory Q collapse, it is undermined (though it is not disproved). This topic is dealt with in detail in a later chapter.

We can interpret the following expectative propositions as abridged, ‘failed’ apodoses of this kind. ‘Even if P were true, Q would be true’ implies ‘if P, then Q’ and ‘P is false, and Q is true’. ‘Though P is true, Q is true’ implies ‘if not P, then Q’ and ‘P is true and Q is true’.

The following invalid moods also typify fallacious attempts at apodosis; note the negative hypothetical major premises, lowercase forms implying a link yet too weak to yield any definite conclusion. People often fail to first establish a bond between the theses, because they hurriedly assume that no contrary hypotheticals can be put forward.

If P, not-then Q | If P, not-then Q |

and P | but not Q |

hence, not Q. | hence, P. |

In summary, there are 8 valid moods of apodosis. That being out of 8X4X4 = 128 possible moods, the validity rate is 6.25%. Compounds of these premises would yield compound conclusions.

The above described valid moods of logical apodosis involve a factual minor premise: P or Q is true or false. The concept can be broadened to involve minor premises which are modal (this refers of course to logical modality). The following four valid moods are derivable from the basic two by exposition, as usual:

If P, then Q | If P, then Q |

and necessarily P | but impossibly Q |

so, necessarily Q | so, impossibly P |

If P, then Q | If P, then Q |

and possibly P | but unnecessarily Q |

so, possibly Q | so, unnecessarily P |

Think of the major premises as modal conjunctions. For example, in the first of these moods, ‘{P and nonQ} is impossible’ plus ‘P is necessary’ imply that ‘Q is necessary’, for if nonQ ever occurred in any context, we would be faced with a contradiction.

The remaining six factual moods can likewise be used to construct another twelve valid modal moods. The significance of these arguments is of course the transmissibility of logical modality across the hypothetical relation. Hypotheticals per se have problematic theses; under the appropriate conditions, this merely prima facie thinkability is up or down graded to a more definite logical status.

Modal apodosis suggests that ‘if P, then Q’ implies ‘if P**n**, then Q**n**’ and ‘if P**p**, then Q**p**’, where the suffixes **n** and **p** refer to logical necessity and possibility respectively. However, note well that the reverse does not follow; the version with nonmodal theses cannot be inferred from either of the modal-theses versions, since the relation might conceivably only apply in the collective case or in the indefinite case without implying a singular and specific equivalent.

Moreover, I see no point in treating hypotheticals with modal theses independently, and I do not think that we ever do so in practice. Since an ordinary hypothetical, with nonmodal theses, contains within it all the requisite information for the solution of problems of a modal nature, we have no need of these implicit forms, and introducing them would be very artificial.

As already mentioned, the valid moods of apodosis can be reformulated to describe arguments involving derivative forms. In particular, note the following examples:

P and/or Q | P and/or Q |

but not P | but not Q |

hence, Q | hence, P |

P or else Q | P or else Q |

and P | and Q |

hence, not Q | hence, not P |

I would not count the rewriting of valid moods in derivative forms as yielding additional valid moods. But they become more significant with multiple disjunctions, which yield more complex conclusions, so long as more than one alternative have not been eliminated.

P and/or Q and/or R and/or S | P or else Q or else R or else S |

but not P | and P |

so, Q and/or R and/or S. | so, not Q and not R and not S. |

In the mood with inclusive disjunction (left), we are given that at least one of the theses listed must be true (i.e. they cannot all be false); so if one is found false, we can conclude that at least one of the remaining ones must be true. In the mood with exclusive disjunction (right), we are given that all but one of the theses listed must be false (i.e. only one can be true); so if one is found true, we can conclude that all the remaining ones must be false. Note that if both major premises are true, i.e. if the theses are both ‘exhaustive’ and ‘mutually exclusive’, then a conclusion is possible from the truth or falsehood of any of the theses, as shown in these two moods.

Most of what we have said about apodosis concerns all hypotheticals, whether of unknown logical basis, normal or abnormal. However, apodosis with a necessary or impossible minor premise and conclusion (as shown earlier) obviously concerns abnormal hypotheticals in particular, because the basis is implied to be not contingent at all. In contrast, apodosis with a possibility or unnecessity as its minor premise, teaches us the logic specific to normal hypotheticals, which are contingency-based.

Thus, we have here a foundation for the specialized study of normal or abnormal hypotheticals, an entry point into the topic; I will not however here pursue the matter further. The same can be said for disjunctives.

### 2. Dilemma.

Colloquially, we call a ‘dilemma’, any impossible choice. ‘If I do this, I’ve had it; if I do that, I’ve had it — so I’ve had it anyway (and it is no use my doing this or that)’. This is indeed a case of dilemma, but in logic the expression is understood more broadly, to cover more positive situations. Thus, often, in action contexts, when we are faced with a choice of means to get to a goal, we might resolve the dilemma by using all available means, even at the cost of redundancies, so as to ensure that the goal is attained one way or the other.

Although dilemmatic argument may be derived from apodosis and syllogism, it has a certain autonomy of cogency and is commonly used in practice, so it deserves some analysis. Note well first that the disjunction used in dilemma is the ‘and/or’ type (not the ‘or else’ type), even if in practice this is not always made clear.

The hypotheticals which constitute the major premise of a dilemma are called its ‘horns’; they give an impression of presenting us with a predicament. The minor premise is a disjunction; it is said to ‘take the dilemma by its horns’. The conclusion is said to ‘resolve’ the dilemma.

*Simple*dilemma consists of a conjunction of subjunctives as major premise, a disjunctive as minor premise, and a (relative) categorical as conclusion. It normally involves three theses. Tradition has identified two valid moods.

(i) The simple *constructive* dilemma.

If M, then P — and — if N, then P

but M and/or N

hence, P

This is proved by reduction ad absurdum through two negative apodoses, as follows:

If M, then P — and — if N, then P (original major premise)

and not P (denial of conclusion)

so, not M and not N (contrary of minor).

Alternatively, we could regard the simple constructive dilemma as summarizing a number of positive apodoses, with reference to the matrix of alternative conjunctions underlying the minor premise:

If M, then P — and — if N, then P | (common major) | |||||

but ‘M (and not N)’ | or ‘N (and not M)’ | or ‘M and N’ | (alternative minors) | |||

whence, P | whence, P | whence, P and P | (common conclusion). | |||

This shows the essential continuity between the concepts of apodosis and dilemma, note.

(ii) The simple *destructive* dilemma.

If P, then M — and — if P, then N

but not M and/or not N

hence, not P

This is proved by reduction ad absurdum through two apodoses, as follows:

If P, then M — and — if P, then N (original major premise)

and P (denial of conclusion)

so, M and N (contrary of minor).

In contrast, the following two arguments would be fallacious:

If M, then P — and — if N, then P | If P, then M — and — if P, then N |

but not M and/or not N | but M and/or N |

hence, not P | hence, P |

*Complex*dilemma consists of a conjunction of subjunctives as major premise, and disjunctives as minor premise and conclusion. Tradition has identified two valid moods. It normally involves four theses, though two are occasionally merely mutual antitheses.

(i) The complex *constructive* dilemma.

If M, then P — and — if N, then Q

but M and/or N

hence, P and/or Q

This can be proved by reductio ad absurdum, as in simple dilemma. Alternatively, we may analyze it through a sorites, as follows:

If not P, then not M (contrapose left horn)

if not M, then N (from minor)

if N, then Q (right horn)

therefore, if not P, then Q (transform to conclusion).

(ii) The complex *destructive* dilemma.

If P, then M — and — if Q, then N

but not M and/or not N

hence, not P and/or not Q

This can be proved by reductio ad absurdum, as in simple dilemma. Alternatively, we may analyze it through a sorites, as follows:

If not not P, then P (axiomatic)

if P, then M (left horn)

if M, then not not M (axiomatic)

if not not M, then not N, (from minor)

if not N, then not Q (contrapose right horn)

therefore, if not not P, then not Q (transform to conclusion).

In contrast, the following two arguments would be fallacious:

If M, then P — and — if N, then Q | If P, then M — and — if Q, then N |

but not M and/or not N | but M and/or N |

hence, not P and/or not Q | hence, P and/or Q |

- Concerning both the simple and complex valid moods, note that, formally speaking, we could use as minor premises the equivalent forms ‘not M or else not N’ and ‘M or else N’, respectively, in the valid constructive and destructive moods. But this would not reflect the true format of dilemma. The goal here is only to describe actual thought processes, not to accumulate useless formulas. However, in view of the similarity in appearance between these valid substitutes, and the minor premises of the invalid moods, it is well to be aware of the possibility of confusion.

A *special case* of complex constructive dilemma is worthy of note, because people sometimes argue in that way. Its form is:

If M, then {P and nonQ} — and — if N, then {nonP and Q}

but M and/or N

hence, either P or Q.

We may understand this argument as follows: contrapose the left horn to ‘if not-{P and nonQ}, then nonM’; the minor premise means ‘if nonM, then N’; these propositions, together with the right horn, form a sorites whose conclusion is ‘if not-{P and nonQ}, then {nonP and Q}’. But we know on formal grounds, for any two propositions, that ‘if {P and nonQ}, then not-{nonP and Q}’. Therefore, ‘either {P and nonQ} or {nonP and Q}’ is true, which can in turn be rephrased as ‘either P or Q’.

Thus, what this argument achieves is the elimination of the remaining two formal alternatives, {P and Q} and {nonP and nonQ}; the combinations {P and nonQ} and {nonP and Q} become not merely incompatible, but also exhaustive. There is no destructive version of this argument, because its result would only be ‘if {P and nonQ}, then not-{nonP and Q}’, which is formally given anyway.

There is also no equivalent argument in simple dilemma. But note that if we substitute nonM for N in the one above, we obtain something akin to it: if M, then {P and nonQ}, and if nonM, then {nonP and Q}; but either M or nonM; hence, either P or Q. This is not really simple dilemma because the antecedents are not identical; but there is a resemblance, in that only three theses are involved. Also, the minor premise here is redundant, since formally true, so the conclusion may be viewed as an eduction from the compound major premise.

Also note, simple and complex dilemmas may consist of more than two horns. The following are examples of *multi-horned* simple dilemma:

Constructive:

If B and/or C and/or D… is/are true, then A is true

but B and/or C and/or D…etc. is/are true

therefore A is true.

Destructive:

If A is true, then B and C and D …etc. are true

but B and/or C and/or D…etc. is/are false

therefore A is false’.

Similarly with other sorts of arrays. This shows that we can view the horns of dilemmas as forming a single hypothetical proposition whose antecedent and/or consequent is/are conjunctive or disjunctive. It follows that simple and complex dilemma should not be viewed as essentially distinct forms of argument; rather, simple dilemma is a limiting case of complex dilemma, the process involved being essentially one of purging our knowledge of extraneous alternatives.

The commonly employed form *‘Whether P or Q, R’* is normally understood as an abridged simple constructive dilemma, meaning ‘If P, then R, and if Q, then R, but P and/or Q, hence R anyway’. However, we should be careful with it, because in some cases we intend it to dissociate R from P and Q, meaning ‘If P not-then R, and if Q not-then R, but R’. Note well the difference. In the former case, the independence is an outcome of multiple dependence; in the latter case, the independence signals lack of connection.

Dilemma, especially its ultimate, simple version, is a very significant form of reasoning, in that *it is capable of yielding factual results from purely problematic theses* (implicit in hypotheticals or disjunctives). Like the philosopher’s stone of the alchemist, it turns lead into gold. Without this device, knowledge would ever be conjectural, a mass of logically related but unresolvable problems.

Note however that the conclusion of a simple dilemma is still, logically, only factual in status. A thesis only acquires the status of logical necessity or impossibility, when it is implied or denied by all eventualities; this means, in dilemma, when the exhaustiveness of the alternatives in the premises is itself logically incontingent (rather than a function of the present context of knowledge). The significance of this will become more transparent as we proceed further, and deal with paradoxical logic.

### 3. Rebuttal.

The so-called ‘*equally cogent* rebuttal’ is a special case of dilemma, worthy of analysis in this context. It happens in debate that a seemingly cogent dilemma may be rebutted by a seemingly equally cogent dilemma.

- With regard to complex dilemma, though the arguments are indeed equally cogent, the impression of ‘rebuttal’ is illusory, due to a misconception of the opposition between the conclusions.

If M, then P — and — if N, then Q

but M and/or N

hence, P and/or Q

If P, then M — and — if Q, then N

but not M and/or not N

hence, not P and/or not Q

Clearly, the major premises are compatible; taken together, they signify two reciprocal subjunctions. The minor premises are also compatible, since they mean, respectively, ‘if nonM, then N’ and ‘if not nonM, then nonN (i.e. if M, then nonN)’; taken together, they signify contradictive disjunction between M and N.

Likewise, for the conclusions: they are not inconsistent with each other, but taken together mean that P and Q are contradictory. So in fact the two dilemma do not exclude each other, it is formally quite possible for them to be both true. If indeed all the propositions involved are true, they merely together constitute a compound dilemma which is quite valid.

We seem to be faced with equally cogent arguments yielding conflicting conclusions, but this is an erroneous impression, because in fact the conclusions are consistent. They may seem to conflict, because they refer to contradictory theses, P and Q, nonP and nonQ; but the disjunctive way in which these theses are connected, makes the conclusions complementary, rather than inconsistent.

Restating the entire arguments in standard hypothetical syllogisms can be helpful. The conclusions should be viewed as ‘If nonP, then Q’ and ‘If P, then nonQ’, respectively, to avoid confusion. If the result persists in seeming unintelligible, the wording may be misleading or there may be a factually erroneous premise.

The frustration underlying such arguments, why they are experienced as somehow in conflict — is due to the fact that each party assumed the contradictory of the other’s assumption to be tacitly included in his or her own premises. Thus, it is the compound each implicitly assumed, rather than the explicit elements, which each finds rightly denied by the other.

In some cases, the presumptions are inductively legitimate for the context each has at hand, following the principle that what is not found connected may be assumed unconnected, so that the face-off with the rebuttal view indeed intimates a possible error somewhere in one’s own views. Someone with an open mind does not feel threatened by such an eventuality, but may give some attention to the problem without resentment, if the issue is sufficiently interesting.

- With regard to simple dilemma, the rebuttal is, on formal grounds,
*never*‘equally cogent’, so it should not surprise us that the conclusions are contradictory.

If M, then P — and — if N, then P

but M and/or N

hence, P

If P, then M — and — if P, then N

but not M and/or not N

hence, not P.

Although the two major premises are formally compatible with each other, and the two minor premises are formally compatible with each other, the conclusions are indubitably incompatible with each other. What this tells us is that the premises, though severally consistent, are taken together inconsistent. They are not, therefore, equally cogent dilemmas; one or both must contain a factual error.

In other words, a simple dilemma is not logically valid, if the horns of the major premise are reversible hypotheticals and the minor premise is a contradictive disjunctive. The compound propositions ‘Only if M, then P — and — only if N, then P’ and ‘Either M or N’ cannot coexist. This may be shown as follows:

The first minor ‘M and/or N’ taken alone allows for the conjunction ‘M and N’, while excluding ‘nonM and nonN’. The second minor ‘not M and/or not N’ taken alone allows for the conjunction ‘nonM and nonN’, while excluding ‘M and N’. When these disjunctions are conjoined together, they mean ‘either M or N’ which still allows for ‘M and nonN’ or ‘nonM and N’, but now formally excludes both ‘M and N’ and ‘nonM and nonN’.

Yet, in the case of M and nonN being both true, the left horn of the first major and the right horn of the second, would yield conflicting conclusions: P and nonP; and, in the case of nonM and N being both true, the left horn of the second major and the right horn of the first, would yield conflicting conclusions: P and nonP.

Thus, rebuttal of simple dilemma is formally unfeasible with contingency-based hypotheticals. With an incontingent theses P or nonP, this paradox is acceptable, because if P is necessary or impossible, the arrival at its negation does not cause a serious conflict, since then the necessary theses is implied by its impossible antithesis. Equally cogent simple dilemmas are therefore feasible in abnormal logic specifically, even though they cannot arise in normal logic. It follows that in the logic of unspecified-basis hypotheticals, these are conditionally possible.

The foregoing means that the valid moods of simple dilemma given initially were not as fully defined and unconditional as they should have been, in other respects, besides.

For a simple dilemma to be valid, one or both of the horns in the major premise must be implicitly a subalternation, rather than an implicance (whereas we left them open as implications); and/or the minor premise must be implicitly a subcontrariety (if constructive) or contrariety (if destructive) between the theses in question, rather than a contradiction (whereas we left it open as a not fully defined disjunction).

## 31. PARADOXES.

A very important field of logic is that dealing with paradox, for it provides us with a powerful tool for establishing some of the most fundamental certainties of this science. It allows us to claim for epistemology and ontology the status of true sciences, instead of mere speculative digressions. This elegant doctrine may be viewed as part of the study of axioms.

### 1. Internal Inconsistency.

Consider the hypothetical form ‘If P, then Q’, which is an essential part of the language of logic. It was defined as ‘P and nonQ is an impossible conjunction’.

It is axiomatic that the conjunction of any proposition P and its negation nonP is impossible; thus, a proposition P and its negation nonP cannot be both true. An obvious corollary of this, obtained by regarding nonP as the proposition under consideration instead of P, is that the conjunction of any proposition nonP and its negation not-nonP is impossible; thus, a proposition P and its negation nonP cannot be both false.

So, the Law of Identity could be formulated as, “For any proposition, ‘If P, then P’ is true, and ‘If nonP, then nonP’ is true”. The Laws of Contradiction and of the Excluded Middle could be stated: “For any proposition, ‘If P, then not-nonP’ is true (P and nonP are incompatible), and ‘If not-nonP, then P’ is true (nonP and P are exhaustive)”.

Now, consider the paradoxical propositions ‘If P, then nonP’ or ‘If nonP, then P’. Such propositions appear at first sight to be obviously impossible, necessarily false, antinomies.

But let us inspect their meanings more closely. The former states ‘P and (not not)P is impossible’, which simply means ‘P is impossible’. The latter states ‘nonP and not P is impossible’, which simply means ‘nonP is impossible’. Put in this defining format, these statements no longer seem antinomial! They merely inform us that the proposition P, or nonP, as the case may be, contains an intrinsic flaw, an internal contradiction, a property of self-denial.

From this we see that there may be propositions which are logically self-destructive, and which logically support their own negations. Let us then put forward the following definitions. A proposition is ** self-contradictory** if it denies itself, i.e. implies its own negation. A proposition is therefore

**if its negation is self-contradictory, i.e. if it is implied by its own negation.**

*self-evident*Thus, the proposition ‘If P, then nonP’ informs us that P is self-contradictory (and so logically impossible), and that nonP is self-evident (and so logically necessary). Likewise, the proposition ‘If nonP, then P’ informs us that nonP is self-contradictory, and that P is self-evident.

The existence of paradoxes is not in any way indicative of a formal flaw. The *paradox*, the hypothetical proposition itself, is not antinomial. It may be true or false, like any other proposition. Granting its truth, it is its antecedent thesis which is antinomial, and false, as it denies itself; the consequent thesis is then true.

If the paradoxical proposition ‘If P, then nonP’ is true, then its contradictory ‘If P, not-then nonP’, meaning ‘P is not impossible’, is false; and if the latter is true, the former is false. Likewise, ‘If nonP, then P’ may be contradicted by ‘If nonP, not-then P’, meaning ‘nonP is not impossible’.

The two paradoxes ‘If P, then nonP’ and ‘If nonP, then P’ are contrary to each other, since they imply the necessity of incompatibles, respectively nonP and P. Thus, although such propositions taken singly are not antinomial, double paradox, a situation where both of these paradoxical propositions are true at once, is unacceptable to logic.

In contrast to positive hypotheticals, negative hypotheticals do not have the capability of expressing paradoxes. The propositions ‘If P, not-then P’ and ‘If nonP, not-then nonP’ are *not *meaningful or logically conceivable or ever true. Note this well, such propositions are formally false. Since a form like ‘If P, not-then Q’ is defined with reference to a positive conjunction as ‘{P and nonQ} is possible’, we cannot without antinomy substitute P for Q here (to say ‘{P and nonP} is possible’), or nonP for P and Q (to say ‘{nonP and not-nonP} is possible’).

It follows that the proposition ‘if P, then nonP’ does not imply the lowercase form ‘if P, not-then P’, and the proposition ‘if nonP, then P’ does not imply the lowercase form ‘if nonP, not-then nonP’. That is, in the context of paradox, hypothetical propositions behave abnormally, and not like contingency-based forms.

This should not surprise us, since the self-contradictory is logically impossible and the self-evident is logically necessary. Since paradoxical propositions involve incontingent theses and antitheses, they are subject to the laws specific to such basis.

The implications and consistency of all this will be looked into presently.

### 2. The Stolen Concept Fallacy.

Paradoxical propositions actually occur in practice; moreover, they provide us with some highly significant results. Here are some examples:

- denial, or even doubt, of the laws of logic conceals an appeal to those very axioms, implying that the denial rather than the assertion is to be believed;
- denial of man’s ability to know any reality objectively, itself constitutes a claim to knowledge of a fact of reality;
- denial of validity to man’s perception, or his conceptual power, or reasoning, all such skeptical claims presuppose the utilization of and trust in the very faculties put in doubt;
- denial on principle of all generalization, necessity, or absolutes, is itself a claim to a general, necessary, and absolute, truth.
- denial of the existence of ‘universals’, does not itself bypass the problem of universals, since it appeals to some itself, namely, ‘universals’, ‘do not’, and ‘exist’.

More details on these and other paradoxes, may be found scattered throughout the text. Thus, the uncovering of paradox is an oft-used and important logical technique. The writer Ayn Rand laid great emphasis on this method of rejecting skeptical philosophies, by showing that *they implicitly appeal to concepts which they try to explicitly deny*; she called this ‘the fallacy of the Stolen Concept’.

A way to understand the workings of paradox, is to view it in the context of dilemma. A self-evident proposition P could be stated as ‘Whether P is affirmed or denied, it is true’; an absolute truth is something which turns out to be true whatever our initial assumptions.

This can be written as a constructive argument whose left horn is the axiomatic proposition of P’s identity with itself, and whose right horn is the paradox of nonP’s self-contradiction; the minor premise is the axiom of thorough contradiction between the antecedents P and nonP; and the conclusion, the consequent P’s absolute truth.

If P, then P — and — if nonP, then P

but either P or nonP

hence, P.

A destructive version can equally well be formulated, using the contraposite form of identity, ‘If nonP, then nonP’, as left horn, with the same result.

If nonP, then nonP — and — if nonP, then P

but either not-nonP or nonP

hence, not-nonP, that is, P.

The conclusion ‘P’ here, signifies that P is logically necessary, not merely that P is true, note well; this follows from the formal necessity of the minor premise, the disjunction of P and nonP, assuming the right horn to be well established.

Another way to understand paradox is to view it in terms of knowledge contexts. Reading the paradox ‘if nonP, then P’ as ‘all contexts with nonP are contexts with P’, and the identity ‘if P, then P’ as ‘all contexts with P are contexts with P’, we can infer that ‘all contexts are with P’, meaning that P is logically necessary.

We can in similar ways deal with the paradox ‘if P, then nonP’, to obtain the conclusion ‘nonP’, or better still: P is impossible. The process of resolving a paradox, by drawing out its implicit categorical conclusions, may be called *dialectic*.

Note in passing that the abridged expression of simple dilemma, in a single proposition, now becomes more comprehensible. The compound proposition ‘If P, then {Q and nonQ}’ simply means ‘nonP’; ‘If nonP, then {Q and nonQ}’ means ‘P’; ‘If (or whether) P or nonP, then Q’ means ‘Q’; and ‘If (or whether) P or nonP, then nonQ’ means ‘nonQ’. Such propositions could also be categorized as paradoxical, even though the contradiction generated concerns another thesis.

However, remember, the above two forms should not be confused with the lesser, negative hypothetical, relations ‘Whether P or nonP, (not-then not) Q’ or ‘Whether P or nonP, (not-then not) nonQ’, respectively, which are not paradoxical, unless there are conditions under which they rise to the level of positive hypotheticals.

### 3. Systematization.

Normally, we presume our information already free of self-evident or self-contradictory theses, whereas in abnormal situations, as with paradox, necessary or impossible theses are formally acceptable eventualities.

A hypothetical of the primary form ‘If P, then Q’ was defined as ‘P and nonQ are impossible together’. But there are several ways in which this situation might arise. Either (i) both the theses, P and nonQ, are individually contingent, and only their conjunction is impossible — this is the normal situation. Or (ii) the conjunction is impossible because one or the other of the theses is individually impossible, while the remaining one is individually possible, i.e. contingent or necessary; or because both are individually impossible — these situations engender paradox.

Likewise, a hypothetical of the contradictory primary form ‘If P, not-then Q’ was defined as ‘P and nonQ are possible together’. But there are several ways this situation might arise. Either (i) both the theses, P and nonQ, and also their conjunction, are all contingent — this is the normal situation. Or (ii) one or the other of them is individually not only possible but necessary, while the remaining one is individually contingent, so that their conjunction remains contingent; or both are individually necessary, so that their conjunction is also not only possible but necessary — these situations engender paradox.

These alternatives are clarified by the following tables, for these primary forms, and also for their derivatives involving one or both antitheses. The term ‘possible’ of course means ‘contingent or necessary’, it is the common ground between the two. We will here use the symbols ‘**N**’ for necessary, ‘**C**’ for contingent (meaning possible but unnecessary), and ‘**M**’ for impossible. The combinations are numbered for ease of reference. The symmetries in these tables ensure their completeness.

No. | Theses | Conjunctions | ||||||

P | nonP | Q | nonQ | PandQ | PandnonQ | nonPandQ | nonPandnonQ | |

Normal (P,Q both contingent) | ||||||||

1. | C | C | C | C | C | C | C | C |

2. | C | C | C | C | M | C | C | C |

3. | C | C | C | C | C | M | C | C |

4. | C | C | C | C | C | C | M | C |

5. | C | C | C | C | C | C | C | M |

6. | C | C | C | C | C | M | M | C |

7. | C | C | C | C | M | C | C | M |

Abnormal (one or both of P, Q not contingent) | ||||||||

8. | M | N | C | C | M | M | C | C |

9. | N | M | C | C | C | C | M | M |

10. | C | C | M | N | M | C | M | C |

11. | C | C | N | M | C | M | C | M |

12. | N | M | N | M | N | M | M | M |

13. | N | M | M | N | M | N | M | M |

14. | M | N | N | M | M | M | N | M |

15. | M | N | M | N | M | M | M | N |

The following table follows from the preceding. ‘**Yes**’ indicates that an implication and its contraposite are implicit in the form concerned, while ‘**no**’ indicates that they are excluded from it. ‘à‘ here means implies, and ‘ß‘ means is implied by.

No. | Name | Implications (à) and Contraposites (ß) | ||||||

PànonQ | PàQ | nonPànonQ | nonPàQ | |||||

nonPßQ | nonPßnonQ | PßQ | PßnonQ | |||||

Normal (P,Q both contingent) | ||||||||

1. | Neutral | no | no | no | no | |||

2. | Contrary | yes | no | no | no | |||

3. | Subalternating | no | yes | no | no | |||

4. | Subalternated | no | no | yes | no | |||

5. | Subcontrary | no | no | no | yes | |||

6. | Implicant | no | yes | yes | no | |||

7. | Contradictory | yes | no | no | yes | |||

Abnormal (one or both of P, Q not contingent) | ||||||||

8. | P impossible,Q contingent | yes | yes | no | no | |||

9. | P necessary,Q contingent | no | no | yes | yes | |||

10. | P contingent,Q impossible | yes | no | yes | no | |||

11. | P contingent,Q necessary | no | yes | no | yes | |||

12. | P, Q bothnecessary | no | yes | yes | yes | |||

13. | P necessary,Q impossible | yes | no | yes | yes | |||

14. | P impossible,Q necessary | yes | yes | no | yes | |||

15. | P, Q bothimpossible | yes | yes | yes | no | |||

Normal hypothetical logic thus assumes the theses of hypotheticals always both contingent, and so limits itself to cases Nos. **1 **to** 7** in the above tables. However, the abnormal cases Nos. **8 **to** 15**, in which one or both theses are not contingent (that is, are self-evident or self-contradictory), should also be considered, to develop a complete logic of hypotheticals.

The definition of the primary positive form ‘If P, then Q’, while remaining unchanged as ‘P plus nonQ is not possible’, is now seen to more precisely comprise the following situations: Nos. **3**, **6**, **8**, **11**, **12**, **14**, or **15**, that is, all the cases where ‘P and nonQ’ is impossible (‘M’), or ‘P implies Q’ is marked ‘yes’.

The definition of the primary negative form ‘If P, not-then Q’, while remaining unchanged as ‘P plus nonQ is not impossible’, is now seen to more precisely comprise the following situations: Nos. **1**, **2**, **4**, **5**, **7**, **9**, **10**, or **13**, that is, all the cases where ‘P and nonQ’ is contingent (C), or ‘P implies Q’ is marked ‘no’.

The other six hypothetical forms, involving the antitheses of P and/or Q, can likewise be given improved definitions, by reference to the above tables.

Notice the symmetries in these tables. In case No. **1**, all conjunctions are ‘C’ and all implications are ‘no’. In cases Nos. **2**–**5**, one conjunction is ‘M’, and one implication is ‘yes’. In cases **6**–**11**, two conjunctions are ‘M’, and two implications are ‘yes’. In cases Nos. **12**–**15**, three conjunctions are ‘M’, and three implications are ‘yes’. Note the corresponding statuses of individual theses in each case.

The process of contraposition is universally applicable to all hypotheticals, positive or negative, normal or abnormal, for it proceeds directly from the definitions. For this reason, in the above tables, each implication is firmly coupled with a contraposite. Likewise, the negation of any implication engenders the negation of its contraposite, so that the above tables also indirectly concern negative hypotheticals, note well.

We must be careful, in developing our theory of hypothetical propositions, to clearly formulate the breadth and limits of application of any process under consideration, and specify the exceptions if any to its rules. The validity or invalidity of logical processes often depends on whether we are focusing on normal or abnormal forms, though in some cases these two classes of proposition behave in the same way. If these distinctions are not kept in mind, we can easily become guilty of formal inconsistencies.

### 4. Properties.

Paradoxical propositions obey the laws of logic which happen to be applicable to all hypotheticals, that is, to hypotheticals of unspecified basis. But paradoxicals, being incontingency-based hypotheticals, have properties which normal hypotheticals lack, or lack properties which normal hypotheticals have. In such situations, where differences in logical properties occur, general hypothetical logic follows the weaker case.

The similarities and differences in formal behavior have already been dealt with in appropriate detail in the relevant chapters, but some are reviewed here in order to underscore the role played by paradox.

- Opposition.

In the doctrine of opposition, we claimed that ‘If P, then Q’ and ‘If P, then nonQ’ must be contrary, because if P was true, Q and nonQ would both be true, an absurdity. However, had we placed these propositions in a destructive dilemma, as below, we would have obtained a legitimate argument:

If P, then Q — and — if P, then nonQ

but either nonQ or Q

hence nonP

Likewise, ‘If P, then Q’ and ‘If nonP, then Q’ could be fitted in a valid simple constructive dilemma, yielding Q, instead of arguing as we did that they must be contrary because their contrapositions result in the absurdity of nonQ implying nonP and P.

It follows that these contrarieties are only valid conditionally, for contingency-based hypotheticals. There are exceptional circumstances in which they do not hold, namely relative to abnormal hypotheticals (including paradoxicals).

This is also independently clear from the observation of ‘yes’ marks standing parallel, in cases Nos. **8**, **14**, **15** (allowing for both ‘P implies nonQ’ and ‘P implies Q’, where P is impossible), and in cases Nos. **11**, **12**, **14** (allowing for both ‘P implies Q’ and ‘nonP implies Q’, where Q is necessary).

Similar restrictions follow automatically for the subcontrariety between ‘If P, not-then nonQ’ and ‘If P, not-then Q’, and likewise for the subalternation by the uppercase ‘If P, then Q’ of the lowercase ‘If P, not-then nonQ’ (which corresponds to obversion). These oppositions only hold true for normal hypotheticals; when dealing with abnormal hypotheticals (and therefore in general logic), we must for the sake of consistency regard the said propositions as neutral to each other.

- Eduction.

Similarly with the derivative eductions. The primary process of contraposition is unconditional, applicable to all hypotheticals, but the other processes can be criticized in the same way as above, by forming valid simple dilemmas, using the source proposition and the denial of the proposed target, or the contraposite(s) of one or the other or both, as horns.

Alternatively, these propositions can be combined in a syllogism, yielding a paradoxical conclusion. Thus:

In the case of obversion or obverted conversion (in the former, negate contraposite of target):

If Q, then nonP (negation of target)

if P, then Q (source)

so, if P, then nonP (paradox = nonP)

In the case of conversion by negation or obverted inversion (in the latter, negate contraposite of target):

If P, then Q (source)

if nonQ, then P (negation of target)

so, if nonQ, then Q (paradox = Q)

Thus, eductive processes other than contraposition are only good for contingency-based hypotheticals, and may not be imitated in the abnormal logic of paradoxes. This is made clear in the above tables, as follows.

Consider the paradigmatic form ‘If P, then Q’. If we limit our attention to cases Nos. 1-7, then it occurs in only two situations, subalternating (**3**) or implicance (**6**). In these two situations, ‘P implies nonQ’ is uniformly ‘no’, so the obverse, ‘If P, not-then nonQ’ is true; and the contraposite ‘Q implies nonP’ is also ‘no’, so the obverted converse, ‘If Q, not-then nonP’ is true; ‘nonP implies Q’ is uniformly ‘no’, so the obverted inverse ‘If nonP, not-then Q’ is true; and the contraposite ‘nonQ implies P’ is also ‘no’, so the converse by negation ‘If nonQ, not-then P’ is true. With regard to inversion and conversion, they are not applicable, because ‘nonP implies nonQ’ and ‘Q implies P’ are ‘no’ in one case, but ‘yes’ in the other.

However, if now we expand our attention to include cases Nos. **8**–**15**, we see that ‘If P, then Q’ occurs additionally if P is self-contradictory and Q is contingent (**8**) or P is contingent and Q is self-evident (**11**) or P,Q are each self-evident (**12**) or P is self-contradictory and Q is self-evident (**14**) or P,Q are each self-contradictory (**15**). The above mentioned uniformities, which made the stated eductions feasible, now no longer hold. There is a mix of ‘no’ and ‘yes’ in the available alternatives which inhibits such eductions.

- Deduction.

With regard to syllogism, the *nonsubaltern moods*, validated by reductio ad absurdum, remain universally valid, since such indirect reduction is essentially contraposition, and no other eductive process was assumed. But the *subaltern moods* in all three figures, are only valid for normal hypotheticals. Since these moods presuppose subalternations for their validation, i.e. depend on direct reductions through obversion or obverted inversion, they are not valid for abnormal hypotheticals.

With regard to apodosis, the moods with a modal minor premise provide us with the entry-point into abnormal logic. As for dilemma, it is the instrument *par excellence* for unearthing paradoxes in the course of everyday reasoning. If we put *any* simple dilemma, constructive (as below) or destructive (mutatis mutandis), in syllogistic form, we obtain a paradoxical conclusion:

If P, then R — and — if Q, then R

but P and/or Q

hence, R

This implies the sorites:

If nonR, then nonP (contrapose left horn)

if nonP, then Q (minor)

if Q, then R (right horn)

hence, if nonR, then R (paradoxical conclusion = R)

Thus, paradoxical propositions are an integral part of general hypothetical logic, not some weird appendix. They highlight the essential continuity between syllogism and simple dilemma, the latter being reducible to the former.

It follows incidentally that, since (as earlier seen) apodosis may be viewed as a special, limiting case of simple dilemma, and simple dilemma as a special, limiting case of complex dilemma — all the inferential processes relating to hypotheticals are closely related.

The paradox generated by simple dilemma of course depends for its truth on the truth of the premises. We should not hurriedly infer, from the paradox inherent in every simple dilemma, that all truths are ultimately self-evident, and all falsehoods ultimately self-contradictory. Knowledge is not a purely rational enterprise, but depends largely on empirical findings.

As already pointed out, simple dilemma yields a categorical necessity or impossibility as its conclusion, only if all its premises are themselves indubitably incontingent. Should there be tacit conditions for, or any doubt regarding the unconditionality of, the hypotheticals (the horns) and/or the disjunction (the minor premise), then the conclusion would be proportionately weakened with regard to its logical modality.

Thus, with reference to the foregoing example, *granting the horns* of the major premise: in the specific case where our minor premise is a formally given disjunction — if, say, P and Q are contradictory to each other (P = nonQ, Q = nonP) — then the R conclusion is indeed necessary. But usually, the listed alternatives P and Q are only contextually exhaustive, so that the R conclusion is only factually true.

So, although every logical necessity is self-evident, and every logical impossibility is self-contradictory, formally speaking, according to our definitions, we might be wise to say that these predications are not in practice reciprocal, and make a distinction between apodictic and factual paradox. The former is independently obvious; the latter derives from more empirical data, and therefore, though contextually trustworthy, has a bit less weight and finality.

Note lastly, the inconsistency of two ‘equally cogent’ simple dilemmas can now be better understood, as due to their implying contrary paradoxes.

## 32. DOUBLE PARADOXES.

### 1. Definition.

We have seen that logical propositions of the form ‘if P, then nonP’ (which equals to ‘nonP’) or ‘if nonP, then P’ (which equals to ‘P’), are perfectly legal. They signify that the antecedent is self-contradictory and logically impossible, and that the consequent is self-evident and logically necessary. As propositions in themselves, they are in no way antinomial; it is one of their constituents which is absurd.

Although either of those propositions, occurring alone, is formally quite acceptable and capable of truth, they can never be both true: they are irreconcilable contraries and their conjunction is formally impossible. For if they were ever both true, then both P and nonP would be implied true.

We must therefore distinguish between *single paradox*, which has (more precisely than previously suggested) the form ‘if P, then nonP; but if nonP, not-then P; whence nonP’, or the form ‘if nonP, then P; but if P, not-then nonP; whence P’ — and *double paradox*, which has the form ‘if P, then nonP, *and* if nonP, then P’.

Single paradox is, to repeat, within the bounds of logic, whereas double paradox is beyond those bounds. The former may well be true; the latter always signifies an error of reasoning. Yet, one might interject, double paradox occurs often enough in practice! However, that does not make it right, anymore than the occurrence of other kinds of error in practice make them true.

Double paradox is made possible, as we shall see, by a hidden *misuse of concepts*. It is sophistry par excellence, in that we get the superficial illusion of a meaningful statement yielding results contrary to reason. But upon further scrutiny, we can detect that some fallacy was involved, such as ambiguity or equivocation, which means that in fact the seeming contradiction never occurred.

Logic demands that *either or both* of the hypothetical propositions which constituted the double paradox, or paradox upon paradox, *be false*. Whereas single paradox is *resolved*, by concluding the consequent categorically, without denying the antecedent-consequent connection — double paradox is *dissolved*, by showing that one or both of the single paradoxes involved are untrue, nonexistent. Note well the difference in problem solution: resolution ‘explains’ the single paradox, whereas dissolution ‘explains away’ the double paradox.

The double paradox *serves to show* that we are making a mistake of some kind; the fact that we have come to a contradiction, is our index and proof enough that we have made a wrong assumption of sorts. Our ability to intuit logical connections correctly is not put in doubt, because the initial judgment was too rushed, without pondering the terms involved. Once the concepts involved are clarified, it is the rational faculty itself which pronounces the judgment against its previous impression of connection.

It must be understood that every double paradox (as indeed every single paradox), is *teaching us something*. Such events must not be regarded as threats to reason, which put logic as a whole in doubt; but simply as lessons. They are sources of information, they reveal to us certain logical rules of concept formation, which we would otherwise not have noticed. They show us the outer limits of linguistic propriety.

We shall consider two classical examples of double paradox to illustrate the ways they are dissolved. Each one requires special treatment. They are excellent exercises.

### 2. The Liar Paradox.

An ancient example of double paradox is the well-known ‘Liar Paradox’, discovered by Eubulides, a 4th cent. BCE Greek of the Megarian School. It goes: ‘does a man who says that he is now lying speak truly?’ The implications seem to be that if he is lying, he speaks truly, and if he is not lying, he speaks truly.

Here, the conceptual mistake underlying the difficulty is that the proposition is *defined by reference to itself*. The liar paradox is how we discover that such concepts are not allowed.

The word ‘now’ (which defines the proposition itself as its own subject) is being used with reference to something which is not yet in existence, whose seeming existence is only made possible by it. Thus, in fact, the word is empty of specific referents in the case at hand. The word ‘now’ is indeed usually meaningful, in that in other situations it has precise referents; but in this case it is used before we have anything to point to as a subject of discourse. It looks and sounds like a word, but it is no more than that.

A more modern and clearer version of this paradox is ‘this proposition is false’, because it brings out the indicative function of the word ‘now’ in the word ‘this’.

The word ‘this’ accompanies our pointings and presupposes that there is something to point to already there. It cannot create a referent for itself out of nothing. This is the useful lesson taught us by the liar paradox. We may well use the word ‘this’ to point to another word ‘this’; but not to itself. Thus, I can say to you ‘this “this”, which is in the proposition “this proposition is false”‘, without difficulty, because my ‘this’ has a referent, albeit an empty symbol; but the original ‘this’ is meaningless.

Furthermore, the implications of this version seem to be that ‘if the proposition is true, it is false, and if it is false, it is true’. However, upon closer inspection we see that the expression ‘the proposition’ or ‘it’ has a different meaning in antecedents and consequent.

If, for the sake of argument, we understand those implications as: if this proposition is false, then this proposition is true; and if this proposition is true, then this proposition is false — taking the ‘this’ in the sense of *self*-reference by every thesis — then we see that the theses do not in fact have one and the same subject, and are only presumed to be in contradiction.

They are not formally so, any more than, for any P1 and P2, ‘P1 is true’ and ‘P2 is false’ are in contradiction. The implications are not logically required, and thus the two paradoxes are dissolved. There is no self-contradiction, neither in ‘this proposition is false’ nor of course in ‘this proposition is true’; they are simply meaningless, because the indicatives they use are without reference.

Let us, alternatively, try to read these implications as: if ‘this proposition is false’ is true, then that proposition is false; and if that proposition is false, then that proposition is true’ — taking the first ‘this’ as self-reference and the ‘thats’ thereafter as all pointing us backwards to the original proposition and not to the later theses themselves. In other words, we mean: if ‘this proposition is false’ is true, then ‘this proposition is false’ is false, and if ‘this proposition is false’ is false, then ‘this proposition is false’ is true.

Here, the subjects of the theses are one and the same, but the implications no longer seem called for, as is made clear if we substitute the symbol P for ‘this proposition is false’. The flavor of paradox has disappeared: it only existed so long as ‘this proposition is false’ seemed to be implied by or to imply ‘this proposition is true’; as soon as the subject is unified, both the paradoxes break down.

We cannot avoid the issue by formulating the liar paradox as a generality. The proposition ‘I always lie’ can simply be countered by ‘you lie sometimes (as in the case ‘I always lie’), but sometimes you speak truly’; it only gives rise to double paradox in indicative form. Likewise, the proposition ‘all propositions are false’ can be countered by ‘some, some not’, without difficulty.

However, note well, both the said general propositions are indeed self-contradictory; they do produce single paradoxes. It follows that both are false: one cannot claim to ‘always lie’, nor that ‘there are no true propositions’. This is ordinary logical inference, and quite legitimate, since there are logical alternatives.

With regard to those alternatives. The proposition ‘I never lie’ is not in itself inconsistent, except for the person who said ‘I always lie’ intentionally. The proposition ‘all propositions are true’ is likewise not inconsistent in itself, but is inconsistent with the logical knowledge that some propositions are inconsistent, and therefore it is false; so in this case only the contingent ‘some propositions are true, some false’ can be upheld.

### 3. The Barber Paradox.

The Barber Paradox may be stated as: ‘If a barber shaves everyone in his town who does not shave himself, does he or does he not shave himself? If he does, he does not; if he does not, he does’.

This double paradox arises through confusion of the expressions ‘does not shave himself’ and ‘is shaved by someone other than himself’.

We can divide the people in any town into three broad groups: (a) people who do not shave themselves, but are shaved by others; (b) people who do not shave themselves, and are not shaved by others; (c) people who shave themselves, and are not shaved by others. The given premise is that our barber shaves all the people who fall in group (a). It is tacitly suggested, but not formally implied, that no one is in group (b), so that no one grows a beard or is not in need of shaving. But, in any case, the premise in fact tells us nothing about group (c).

Next, let us subdivide each of the preceding groups into two subgroups: (i) people who shave others, and (ii) people who do not shave others. It is clear that each of the six resulting combinations is logically acceptable, since who shaves me has no bearing on whom I can shave. Obviously, only group (i) concerns barbers, and our premise may be taken to mean that our barber is the only barber in town.

Now, we can deal with the question posed. Our barber cannot fall in group (a)(i), because he is not shaved by others. He might fall in group (b)(i), if he were allowed to grow a beard or he was hairless; but let us suppose not, for the sake of argument. This still allows him to fall in group (c)(i), meaning that he shaves himself (rather than being shaved by others), though he shaves others too.

Thus, there is no double paradox. The double paradox only arose because we wrongly assumed that ‘he shaves all those who *do not* shave themselves’ excludes ‘he shaves some (such as himself) who *do* shave themselves’. But *‘X shaves Y’ does not formally contradict ‘X shaves nonY’*; there is no basis for assuming that the copula ‘to shave’ is obvertible, so that ‘X shaves Y’ implies ‘X does not shave nonY’.

If the premise was restated as ‘he shaves all those *and* *only* those who do not shave themselves’ (so as to exclude ‘he shaves himself’), we would still have an out by saying ‘he does not shave at all’. If the premise was further expanded and restricted by insisting that ‘he somehow shaves or is shaved’, it would simply be self-contradictory (in the way of a single paradox).

Further embellishments could be made to the above, such as considering people who shave in other towns, or making distinctions between always, sometimes/sometimes-not, and never. But I think the point is made. The lesson learned from the barber ‘paradox’ is that without clear categorizations, equivocations can emerge (such as that between ‘shaves’ and ‘is shaved’), which give the illusion of double paradox.

# PART IV. DE RE CONDITIONING.

## 33. CONDITIONAL PROPOSITIONS.

### 1. *De-Re* Conditioning.

Logic has traditionally been focused on two types of proposition, the actual categorical of Aristotle, and the logical hypothetical or disjunctive of later logicians. Categorical propositions (including their factual, positive conjunctions) were seen as essentially ‘*de-re*’, telling us about things in themselves. Hypothetical and disjunctive propositions (essentially, modal or negative conjunctions) were seen as essentially ‘*de-dicto*’, telling us about connections between thoughts.

However, we will now develop a more accurate, broader theory of conditioning, which acknowledges not only logical conditioning, but also ‘*de-re* conditioning’, constructed with reference to other *types* of modality. (The reader is referred to all our previous definitions of the different types of modality.)

This does not mean to imply that logical conditioning is any less ‘real’ than *de-re* conditioning. But rather, only that the type of modality qualifying the connection and basis is different in each case. As we shall see, each type of conditioning has to do with a distinct type of causality.

The following should serve to illustrate the distinction between types of conditioning:

*Logical*: ‘if this, then that’, meaning: in such context as this is true, that is also true.

*Natural*: ‘when this, that’, meaning: in such natural circumstance as this is actual, that is also actual.

*Temporal*: ‘when this, that’, meaning: at such times as this is actual, that is also actual.

*Extensional*: ‘where this, that’, meaning: in such cases as this occurs, that also occurs. (By ‘cases’ we here refer to instances of a universal.)

Whereas every *de-re* conditional implies some kind of *de-dicto* conditional, the reverse does not always hold. This is because *de-re* propositions are formally more demanding than logical statements; we need more information to be able to formulate them.

For example, I can formulate an argument like ‘if nothing is knowable, then…’ without thereby suggesting that I acknowledge the antecedent as even logically possible, whereas with other types of conditioning such speculative freedom is lacking. However, note, the rephrasing of *de-re* into *de-dicto* will not be studied in detail here.

Natural, temporal and extensional conditionals and disjunctives, are essentially as *de-re* as two-term, single categoricals, even though they may tell us about connective relationships between three or more terms, or two or more categoricals.

Conditionals have many forms, but we will give most of our attention to those with three terms: the subject and the antecedent and consequent predicate, which best highlight the nature and properties of this family of propositions.

I do not intend to analyze natural, temporal, and extensional conditioning, in as much detail as categorical propositions were and will be treated. I will especially not attempt to develop theories of factorial analysis, and induction by factor selection and formula revision, relating to conditionals. The work done in later chapters on categoricals should be viewed as prototypical, a model for future investigations of the same kind in the field of conditionals.

Each type of modality has its own specific disjunctive propositions, distinguished by their bases and connectives. Relatively little attention will be devoted in this volume to disjunction, although it is in itself valuable, because its logic is derivable from that of conditionals. But some introductory comments will be made in their proper place.

Incidentally, one of the utilities of studying disjunction, is that it clarifies the logic of *degrees*. The various degrees or measures of any thing X may be viewed as standing in a disjunction ‘X1 or X2 or X3 or…’, of whatever modal type is appropriate. Each degree is a logical, natural, temporal or extensional alternative, and they usually range from some maximum to some minimum.

Disjunctive logic teaches us, for instance, not to confuse the affirmation of X as such (which is indefinite as to degree) with the affirmation of its extreme or most typical manifestation (a specific degree or range, say X1). Likewise, denial of X should mean negation of all its degrees (X1, X2, X3,…), and not mere negation of the more extreme or typical degree or range (as we often intend in practice). The fallaciousness of many an argument is explained with reference to such confusions.

### 2. Types of Causality.

Our expansion of the theory of conditioning is the gateway through which Logic enters into the field of ‘material’ causality.

Hypotheticals are concerned with logical causes; they show us the ‘reasons why’ of items of knowledge, with reference to the contextuality of information. Non-logical conditionals are concerned with more ‘substantive’ causation, occurring in the objective realms of matter or mind, irrespective of the stage of development of our knowledge.

Whereas hypotheticals tell us that ‘In all or this or some knowledge contexts, two theses P and Q both logically arise’, other conditionals tell us that ‘In all or this or some circumstances or times or cases, two events SP and SQ both really happen’.

The various types of conditioning are differentiated by the type of modality intended, in the connection (which qualifies the whole relation of antecedent and consequent), and in the basis (the underlying possibilities), which they respectively imply.

In typical hypotheticals, of the form ‘if P, then Q’, the connection is a logical incontingency and the basis is a problemacy or logical possibility of truth.

In contrast, typically, for natural conditionals like ‘when P is, Q must be’, the connection is a natural necessity and the basis is a potentiality of actualization in some circumstances.

For temporal conditionals like ‘when X is, Y always is’, the connection is a temporal constancy and the basis is the sometime occurrence of the events concerned.

For extensional conditionals ‘where X applies, Y applies’, the connection is a generality and the basis is applicability to part of the subject’s instances.

Through such formal analysis of conditioning, using the tools of modal logic, we can begin to understand and seriously examine the concept of causality.

Causality is of various types, in parallel to the types of modality. We can talk of logical causality, natural causality, temporal causality, and extensional causality. These are distinct, yet not unrelated, types of determinism. Making this distinction allows for more accurate and efficient reasoning processes.

Each type of causality orders reality in a special way. Logic determines reality in accordance with the order of development of knowledge; nature and time order individual external events as such; extension refers to the classification of universals. These represent distinct methods of explanation.

When we say that X causes Y, or Y is caused by X, we must first establish the type of causality intended Expressions like ‘because of’ or ‘as a result of’ or ‘depends on’, and such, are in everyday discourse used indiscriminately, without awareness of the modality type involved. Yet, epistemologically and ontologically the difference is important.

The various types of causality display both some similarities and some differences in structure and in logical behavior patterns. The common properties of all types of causality may be seen as the general laws of causality. The distinctive uniformities within each type give rise to a special logic for that specific modality. Thus, both the similarities and differences are significant.

The field of aetiology, the study of causality, is not intended to be within the scope of this dissertation. I have personally already done the needed logical work, so I know how vast and interesting it is. But these results belong in a separate volume. My purpose here is to give one more justification for my theory of modality, to highlight how useful to logic and all areas of knowledge this tool is. My policy here will therefore be to focus on information most relevant to this purpose, the bare essentials.

The differentiation of modality into types and categories allows for hitherto unmatched clarity and precision in the development of conceptual knowledge. Not that modality is something new to human thinking, but its systematic study greatly improves our understanding of it and our reasoning processes.

For the concept of modality, as indeed that of causality, transcends any specific content of knowledge, and is equally valuable in physical sciences, psychology, politics, religious discourse, or personal deliberations. It is not attached to any particular theory of the universe, or of any domain within it. It is grounded in common overall experience and logical consistency.

### 3. Laws of Causality.

I should perhaps, however, say a word or two about the so-called *‘*Laws of Causality’ which some philosophers have advanced.

- Some claim that ‘cause and effect must be substantially the same’. Thus, they deny that G-d could have created the world, ‘because’ a purely spiritual entity (G-d) cannot generate a material and mental one (the world we commonly experience). But there is no formal justification for such an argument, for spirit and matter still have in common one thing, namely
*existence*, so that the conclusion is not inferable from the premise. In any case, we commonly regard material and mental phenomena, though substantially different, as having mutual causal relations, whether in acts of will or in more reactive psychological situations, so that even within the empirical world such a ‘law of causality’ is untenable.

- Some claim that ‘something static cannot cause a motion’, in order to prove that G-d, who is unchanging, cannot have created a world of change, or to deny that human volition is initiation of motion by an unmoved soul. But this is contrary to common-sense intuition, so that aetiology may not
*ab-initio*reject this from formal possibility. One may seek to prove it eventually, but not posit it as a logical principle from the start.

- Some claim that ‘everything must have a cause,
*ad infinitum*’. They say that there are no prime movers, that everything is mechanistically determined, and from thence argue that G-d, and likewise human action, must also have a cause. Here again, there is no formal basis for such a claim. As we proceed, it will become clear that causality is quite definable without reference to such ‘laws’. We might posit such infinite regression as a generalization, an*inductive*principle, but there is no conceptual necessity in it.

## 34. NATURAL CONDITIONALS: FEATURES.

### 1. Basis and Connection.

There are six singular forms of natural conditionals with three terms, as follows. These forms are so structured that we can analyze the behavior of individual subjects, the relationships between their predicates, independently of other individuals. Note the three categories of modality and two polarities they feature.

(We could if need be use the same symbolic conventions as we did for categoricals, only perhaps prefix them with, say, a paragraph (**§**), to remind us of the differences.)

**Rn**: When this S is P, it must be Q**Gn**: When this S is P, it cannot be Q**R**: This S is P and Q**G**: This S is P and not Q**Rp**: When this S is P, it can be Q**Gp**: When this S is P, it can not-be Q

Let us examine the structure of these forms in more detail:

**The expression**used here signifies a conditionality of the type*‘when’**‘in such circumstances as’*; and it is intended to imply that the condition ‘this S is P’ is potential. Note well that the reference here is to*natural*circumstances; we are dealing with a real, objective type of causality.

‘When’ suggests that the underlying ‘this S can be P’ is an established fact, and not merely something logically conceivable. Thus, it is not equivalent to the ‘if’ of hypothetical propositions, which only signifies that the condition might turn out to be true, not being so far inconsistent with the context of knowledge.

Needless to say, by now, we are not always careful, in everyday discourse, to use ‘when’ (instead of ‘if’ or similar expressions) wherever natural conditioning is intended, or ‘must’ (instead of ‘is’) wherever necessity is intended. There is no harm in confusing words in practice, provided we know what we mean.

The S being P condition is called the antecedent; it is only operative when actual, and needs be at least potential to fit in this formal position. The S being or not-being Q conjunction is called the consequent; here too, the relevant modality is actuality, and potentiality is formally implied. These two actualities may be called ‘events’.

The implied potential of the events and their conjunction is called the ‘basis’, and the natural modality qualifying the conjunction as a whole specifies the ‘connection’ involved.

Every natural conditional proposition may be said to be ‘based on’ the natural possibility, the potentiality, of the antecedent’s eventual actualization. Each of the six forms introduced above logically implies the categorical proposition ‘this S can be P’.*Basis.*

Likewise, since when the condition is actualized, the consequence will also be actualized, whether unconditionally or under certain unspecified additional conditions, it follows that the consequent is also logically implied to be potential. That is, ‘this S can be (or can not-be) Q’ may be educed from these same forms (with the appropriate polarity).

More precisely, the full basis of these forms is the conjunctive categorical ‘this S can be both P and Q (or can be both P and nonQ, in negative cases)’, which incidentally implies the two above-mentioned separate potentialities. The conditional proposition implicitly guarantees that the said base potentiality exists. This joint potentiality underlying every natural conditional is the foundation on which the subjunction is built.

Potentiality signifies that certain unspecified surrounding circumstances, may underlie the specified event. This refers to the various postures of the real world, the situation of the rest of the material, mental, and even spiritual world. Since potentiality is compatible with both necessity and contingency, items in the wider environment may or not be responsible for triggering the reaction or inhibiting it.

Although the original function of the form is to capture actualization of naturally contingent phenomena, it is so engineered that one or both of the events could in fact be naturally necessary. The formal basis of any natural conditional is the potentiality of the events, not their natural contingency.

The precise function of a natural conditional is thus only to point out to us the *intersections, inclusions or exclusions, between the circumstances surrounding the two events.* This may be compared to the doctrine of ‘distribution of terms’, in categorical propositions.

That is, though each form is based on the potentiality of antecedent and consequent and their conjunction, this does not logically necessitate that both the events be conditional, but admits as logically possible that one or both of the events exist(s) under all natural circumstances.

Thus, though for instance the necessary form ‘When this S is P, it must be Q’ implies ‘This S can be P and Q’, it is still logically compatible with any of the conjunctions ‘This S can be nonP and nonQ’ (double contingency), or ‘This S must be P and must be Q’ (double necessity), or ‘This S can not-be P, and must be Q’ (contingency with necessity).

However, that necessary form is logically incompatible with the conjunction ‘This S must be P, and can not-be Q’ (necessity with contingency), because of the connection, as we shall see. Similarly, with a negative consequent (substitute nonQ for Q throughout).

In contrast, the corresponding actual and potential forms, allow for all those eventual modal conjunctions, though only the said basic joint potentiality is formally implied.

If one or both of the events is necessary, the conditioning is admittedly effectively redundant, since a necessary event exists independently, it is ‘already there’; but the relationship is still formally true.

Although the modal qualification (the ‘must’, ‘cannot’, ‘can’, or ‘can not’ modifier) is placed on the side of the consequent — it is not part of the consequent, but properly concerns*Connection.**the relation*between it and the preliminary condition, that is, the subjunction as a whole. This should be grasped clearly: the antecedent and consequent of natural conditional propositions can only be actualities or actualizations.

That is, ‘When this S is P, it must be Q’ does not say that the phenomenon ‘this S is P’ will be followed by the phenomenon of natural necessity ‘this S must be Q’, for it admits that ‘this S can not-be Q’ might be true. Rather ‘this S is P’ will, *whatever the surrounding circumstances*, be followed by the phenomenon of actuality ‘this S is Q’. Likewise for a negative consequence.

Similarly, ‘When this S is P, it can be Q’ does not say that the phenomenon ‘this S is P’ will be followed by the phenomenon of potentiality ‘this S can be Q’, for that is already given as part of the basis. Rather ‘this S is P’ will, *in some unspecified surrounding circumstances*, be followed by the phenomenon of actuality ‘this S is Q’. Likewise for a negative consequence.

It is thus very appropriate to regard the antecedent actuality and the consequent actuality, as the two ‘events’ referred to by the proposition. The modality merely acts as a bridge between them.

Note well that, even in the case of necessary conditioning, the natural circumstances in which the antecedent is actualized are not specified. What is specified, is that the conditions which suffice to actualize the antecedent, whatever they be, will also be sufficient to actualize the consequent.

That directional link between the events is formally expressed by saying that ‘When this S is P, it must be Q’ implies ‘This S cannot be {P and nonQ}’; and ‘When this S is P, it cannot be Q’ implies ‘This S cannot be {P and Q}’. These implications, in the form of naturally impossible conjunctions, are the connections between the events.

Thus, to define a necessary conditional, we must specify two categorical conjunctions (with appropriate polarities): the basis ‘this S can be both P and Q (or nonQ)’, and the connection ‘this S cannot be both P and nonQ (or Q)’. We cannot, with such natural conditioning (unlike with logical conditioning), ignore one or the other of these specifications; both must be kept in mind.

In the case of potential conditioning, the link between the events is formally expressed by contradicting the above necessary connections, and saying that ‘When this S is P, it can not-be Q’ implies ‘This S can be {P and nonQ}’; and ‘When this S is P, it can be Q’ implies ‘This S can be {P and Q}’. We see that, here, the implied basis and connection are one and the same naturally possible conjunction.

In merely potential conditionals, the (unspecified) conditions for actualization of the antecedent will not be enough to bring about the consequent; some additional (also unspecified) conditions are required for that. Clearly, these propositions enable us to express cases of partial, instead of complete, causality of natural phenomena; their subjunctive form is not artificial.

It is understood that there are some sets of circumstances, like say R, which in conjunction with P will suffice to cause Q (or nonQ, as the case may be) in this S. That is, for instance, ‘When this S is P, it can be Q, and when it is not P, it can not-be Q’ minimally implies ‘When this S is P and R, it must be Q’, for at least one (known or unknown) ‘R’.

However, that specifically concerns fully *deterministic* systems, and does not take *free will* into account. Indeed, denying such implication altogether is the way we can begin to formally develop the topic of spontaneous events. For this reason, I will not go into these issues in greater detail in the present study.

In summary, we can define modal natural conditionals entirely through categorical conjunctions, but all the implied categoricals must be specified.*Definitions.*

Thus, ‘When this S is P, it must be Q’ means ‘This S can be both P and Q, but cannot be P without being Q’; similarly, ‘When this S is P, it cannot be Q’ means ‘This S can be P without being Q, but cannot be both P and Q’. In contrast, ‘When this S is P, it can be Q’ means no more than ‘This S can be both P and Q’; and ‘When this S is P, it can not-be Q’ only means ‘This S can be both P and nonQ’.

It follows from these understandings that each of the necessary forms subalternates the potential form of like polarity (identical with their basis). Natural conditionals thus constitute a modal continuum, as did categoricals.

The *actual* forms, ‘This S is P and Q’ and ‘This S is P and not Q’, refer to conjunctions of events existing *‘in the present natural circumstances’*. They obviously imply, as their bases, the propositions ‘This S can be both P and Q’ and ‘This S can be P and nonQ’, respectively, since what is true of ‘one specified circumstance’ is equally true of ‘some unspecified circumstance(s)’.

The position of these actual conjunctions in the modal hierarchy of conditionals, to some extent parallels the position of single actuals among modal categoricals, since they are the way the potential conjunctions, which are the basis of all modal conditionals, are actualized. However, the analogy is limited, because in the field of conditionals, natural necessity does not imply actuality, though both necessity and actuality do imply potentiality.

This is obvious from the greater complexity of the necessary forms. A connective like ‘This S cannot be both P and Q’ remains *problematic* with respect to which of the alternative positive conjunctions ‘P and nonQ’, ‘nonP and Q’, ‘nonP and nonQ’ will actually take the place of the excluded ‘P and Q’. We cannot even be sure that all these conjunctions are even potential; the only one formally given as potential is the one serving as basis, namely ‘P and nonQ’, the others may or not be so. Similarly, with appropriate polarity changes, for ‘This S cannot be both P and nonQ’.

Thus, a naturally impossible conjunction involves a certain amount of leeway, like a logically impossible conjunction. It does not by itself formally fully determine any actuality or even all potentialities. However, to repeat, a natural connective is not by itself ground enough to form a conditional proposition; an adequate basis is also required for that (whereas in the case of hypotheticals, logical basis is varied and optional).

Note, however, in exceptional cases, our use of the expression ‘when and if’ to suggest that we know the natural connection to apply (as suggested by the ‘when’), but we do not know the natural basis to be applicable (whence the ‘if’ proviso). But this expression may have other meanings (see section 3b further on).

- Note well that
**actual**‘conditionals’ are in fact conjunctions, and cannot meaningfully be written in conditional form, with a ‘when’. With regard to the seemingly nonmodal conditional form ‘When this S is P, it*is*Q’, which we commonly use to describe habitual, voluntary actions or events, the following may be said:

A proposition such as ‘When she is happy, she sings’, should not be regarded as an actual conditional, but rather as a form vaguely expressing a degree of natural *probability* below necessity. It means, in ordinary circumstances, so and so is very likely, but in extraordinary circumstances, it is less to be expected. Alternatively, the intention may be to express a temporal modality, as in ‘When this S is P, it is always (or usually or sometimes) Q’; in which case the form properly belongs under the heading of temporal conditionals.

Ultimately, volitional conditioning involves a type of modality different from natural conditioning. Note that the antecedent of a natural conditional proposition may be voluntary, since even something freely willed may have naturally necessary consequences. What distinguishes volitional conditioning is that, whether the antecedent is emerges naturally or voluntarily, the consequence is voluntarily chosen and brought about. For example, ‘If you do this, I will do that’ involves two voluntary actions.

Volitional conditionals are thus statements of conditional intention. The ‘will’ involved, concerns another type of causality, than the ‘must’ of naturals. Volition is a special domain within Nature (in the broadest sense), where otherwise common relations (those of natural modality) do not all apply. Volition denotes a greater than usual degree of agency.

However, this type of modality will not be dealt with in this treatise, but belongs in a work on aetiology.

### 2. Quantification.

Quantification of the six prototypes expands the list of such natural conditional propositions to 18.

**An**: When any S is P, it must be Q**En**: When any S is P, it cannot be Q**In**: When certain S are P, they must be Q**On**: When certain S are P, they cannot be Q**A**: All S are P and Q**E**: No S is P and not Q**I**: Some S are P and Q**O**: Some S are P and not Q**Ap**: When any S is P, it can be Q**Ep**: When any S is P, it can not-be Q**Ip**: When certain S are P, they can be Q**Op**: When certain S are P, they can not-be Q

We have already analyzed the expression ‘when’, signifying the natural conditionality, and the features of polarity of consequent (is or is not Q), and modality (in all, the given, or some circumstances). Here, we introduce plural quantity (any, certain), in place of the singular indicative (this).

The first thing to note is that the quantifiers are here intended as dispensive, and not collective or collectional. They refer to the instances of the subject severally, each one singly, so that the plural forms are merely a shorthand rendition of a number of singular propositions. The all or some units of the subject-concept do not have to simultaneously fulfill the condition for the consequence to follow, and the two predicates apply to the individual units, and not to a group of such units as a whole.

For this reason, the words ‘any’ and ‘certain’ are preferably used in this context, less misleading (however, the word ‘certain’ should understood as meaning ‘at least some’, and not ‘only some’).

Secondly, the basis of the general conditional propositions should be a categorical generality. For instance, ‘When any S is P, it must be Q’ implies that ‘all S can be P and Q’. In practice, we tend to confuse or mix the methodology of natural and extensional modality (see the discussion of the latter, in a later chapter), and often intend only a particular basis for a seemingly general natural conditional; however, here, the stated generality will be regarded as genuine. The corresponding particular conditionals only have particular bases, obviously.

Note that, although we have dealt with forms with a negative consequent, we did not so far mention forms with a *negative antecedent*, like ‘When this S is not P,….’ Obviously, we could construct another 18 forms (6 singulars and 12 plurals; or 6 actuals and 12 modals), with this added feature in mind.

I will not devote much attention to these extra forms, because their logic is easily derived. With reference to the eductions feasible from forms with positive antecedents, we can infer their oppositions to those with negative antecedents. And all the inferences feasible with the former can be duplicated with the latter, by simply substituting ‘nonP’ for ‘P’ throughout.

I do not here mean to underrate negative antecedents. Taking the antecedent as a whole, its polarity is of course logically irrelevant. Undeniably, forms with antithetical antecedents are important, because they complement each other.

For instance, a form like ‘When this S is P, it must be Q’ does not by itself communicate change, but combined with ‘When this S is not P, it cannot be Q’, we get a sense of the dynamics involved. That is, not merely is the static actuality of P accompanied by that of Q (or nonP by nonQ), but the actualization of P brings about that of Q (or nonP, nonQ).

We may view this as a formal implication, by certain combinations of conditionals involving actualities and inactualities, of similar conditionals concerning the triggering or prevention of actualizations (using the transitive copula, ‘gets to be’).

Besides natural conditionals with three terms, there are other varieties: those with four terms, such as ‘When this S1 is P, that S2 is Q’. Quite often used and important, is the case: ‘When any S1 is P, *the corresponding (or some unspecified)* S2 is Q’. Here, the mediation provided by an explicit common subject is lacking, though some hidden thread links the two events. For examples, ‘When a car runs out of fuel, its motor stops’ or ‘When evil is let loose, somebody somewhere suffers’.

Often, of course, we use still more complex versions, involving composite antecedent and/or consequent, such as ‘When {S is P1 and P2} and {S2 is P3}, {S3 is Q1 and Q2}’, say.

The logical mechanisms applicable to these more complex varieties should be similar to those for the standard three-term forms we are focusing on. So long as we clearly understand which individual subjects are denoted, so that we know precisely which one affects which, there should be no logical confusion.

### 3. Other Features.

The forms mentioned thus far deal with most natural conditioning situations. In this section, we will mention various notable departures from these norms.

- The order of
, or chronology, of the antecedent and consequent events must be kept in mind to avoid errors of judgment with regard to natural conditionals. Here, I assume that a consequence takes place as soon as and so long as its antecedent. More broadly:*sequence*

The antecedent may accompany the consequent immediately (and thus be simultaneous), or later in time, or earlier in time; and the time lapse between them may be mentioned explicitly, or tacitly understood (as we do here).

In the case of simultaneity, the events may happen at the same time, and yet not be contemporaneous, that is, not last for equal lengths of time. All the more, in cases of nonsimultaneity, the lasting power of the events may be very different; for instances, a flash of lightning may cause permanent damage, a long burning fuse may end in a sudden explosion. These issues should be taken into consideration in reasoning from natural conditionals.

In real causality, the cause is immediately or after some time followed by the effect; if we place the effect temporally before the cause, we are considering it as an ‘index’, ‘sign’ or ‘symptom’ of the cause’s presence or absence. Natural conditionals mirror reality if expressed in the right sequence, otherwise they are a logical artifice (wherein, instead of cause causing effect, the knowledge of the effect ‘causes’ the knowledge of the cause).

In any case, the temporal qualification of the events is usually relative; the time of one event is defined as before or after the other’s time, by so much. In some cases, we refer to ‘absolute’ time — that is, date and o’clock; the relative time follows by inference.

Note also the different ways time may be specified: we can say that an event does or does not happen at some (stated or undefined) point or segment of time; or permanently, in past and/or future areas of time.

The ‘when’ should be taken in its weakest sense, as suggested in the expression ‘when and if’ or ‘if ever’, and not as implying the inevitability of actualization of the condition. This sense of ‘it can happen, but is not bound to happen’ is to be preferred as our standard because it is broader, more generally applicable.*Modalities of Actualization.*

We could work out a specialized logic for inevitable antecedents. A complex proposition like ‘When S gets to be P, and it is bound to be P eventually, it must be Q’, would have as its first base that ‘although this S can be and can not-be P, sooner or later it is must change from nonP to P’, and imply the inevitability of Q too (unless already actual through other causes).

Note well that inevitability of actualization signifies an underlying natural contingency of actuality, it is only the transition from nonP to P which is naturally necessary. Obviously, this should not be confused with the more static natural necessity of actuality, which we are usually concerned with, which is the antithesis of contingency.

All this brings to mind the wider field of natural conditionals for transitive events, incidentally. No one has researched it.

However, these are relatively narrow topics, and will not be discussed further here.

- Within natural modality, we also need to recognize the phenomenon of
. The concept of a power is rather difficult to define. By a power, we mean a potential close to actualization; something readily available, without too many preparatory measures. But this definition is too vague for formal work.*acquisition or loss of ‘powers’*

Anyway, something may remain outside the powers of a subject for a part of its existence and then eventually appear (e.g. through the maturing of an organism); or a subject may initially have a power and then lose it irrecoverably (e.g. the use of a hand which is cut off). Thus, we can talk of actualization of the presence or absence of powers.

Obviously, an ‘acquired power’ was always potential, even before it became more accessible; so the concept of a ‘acquired power’ is subsidiary to the concept of a potentiality, and included in it as a special case. However, a ‘lost power’ is something previously potential and henceforth naturally impossible; so this concept introduces a serious complication into modal logic, namely the logical possibility for changes in bases and connections.

Thus, in some cases, the modality given within a natural conditional, may be intended to be an intrinsic part of the antecedent or consequent. Such modal specifications are effectively actualities, as far as the conditional proposition as a whole is concerned, and should not be confused with the modality of the relation between them.

Powers may be indicated by use of modal expressions like ‘is able to be’ (which is less demanding than ‘is’, but more specific than ‘can be’) or ‘is unable to be’ (which lies between ‘cannot be’ and ‘is not’); or more dynamically, ‘is henceforth able to be’ or ‘is no longer able to be’ (more explicitly implying a change in powers). Likewise for ‘not-be’.

Thus, ‘When this S is able to be P, it is Q’ would mean ‘when this S has the (actual) power to be P, it is Q’. Likewise, ‘When this S is P, it is able to be Q’ would mean ‘When this S is P, it has the (actual) power to be Q’. More precisely, the latter statement should be modal, like all conditionals; that is, we mean ‘it must be able to be Q’ or ‘it can be able to be Q’, where ‘must’ or ‘can’ define the connection, while ‘is able to be’ signifies an actuality of power. Similarly for the interpretation of negatives.

This topic requires further study, but will not be pursued further here.

- Note that in practice if one finds natural modality expressions, like ‘can’ or ‘must’ (or their negative equivalents), appearing in the antecedent or being intended as an intrinsic feature of the consequent — it does not follow that the conditioning is of the natural type.

On the contrary, this usually signifies that the conditional proposition is of the logical or extensional type. For examples, ‘If S must be P, then it can be Q’ is supposedly a hypothetical, and ‘In cases where S can be P, it must be Q’ is supposedly an extensional conditional, even though the antecedents and consequents are in natural modality.

As earlier pointed out, in practice the words we use are not always consistent with the intended modality of conditioning. One should therefore be careful to identify just what type of conditioning is intended, because their logics are considerably different.

### 4. Natural Disjunction.

Disjunction has traditionally been approached as an essentially logical relation. But our analysis of the types of modality shows clearly that disjunction also exists in nature. It can be understood with reference to natural conditioning.

- There are various
**modalities and polarities**of natural disjunction. Consider the simplest case of three terms, in the singular:

The necessary form ‘This S must be P or Q’, can be taken to mean that ‘When this S is not P, it must be Q, and when it is not Q, it must be P’, it follows that the implied connection is that ‘This S cannot be both nonP and nonQ’, and the implied basis is that ‘This S can be nonP and Q, and it can be P and nonQ’, which in turn imply that ‘This S can be and can not-be P, and can be and can not-be Q’. Note well the implied natural contingency of the individual events.

The corresponding potential form ‘This S can be P or Q’ accordingly means ‘When this S is not P, it can be Q, and when it is not Q, it can be P’ (same as the above basis).

As for the parallel negative forms: ‘This S can not-be P or Q’ has to mean ‘This S can be both nonP and nonQ’ (contradicting the above connection), and ‘This S cannot be P or Q’ may therefore be understood as ‘This S must be both nonP and nonQ’ (subalternating the preceding).

These various forms can of course be quantified.

**Other manners**of disjunction may also be used:

To describe a specifically ‘P and/or Q’ situation, we would have to add to the said ‘This S must be P or Q’ definitions, that ‘This S can be both P and Q’.

The natural disjunction ‘This S must be nonP or nonQ’ can be similarly interpreted, by substituting antitheses for theses throughout; briefly put, it means ‘When P, nonQ; when Q, nonP’. To describe a specifically ‘P or else Q’ situation, we would have to add to the said ‘nonP or nonQ’ definitions, that ‘This S can be both nonP and nonQ’.

An ‘either-or’ situation would be represented by a compound of the two disjunctions ‘P or Q’ and ‘nonP or nonQ’, meaning four natural conditional propositions.

- Also, analogous forms involving
**more than three terms**can be constructed, constituting multiple natural disjunctions. Their connections can be defined like multiple logical disjunctions, except with reference to numbers of actualities or inactualities, instead of truths or falsehoods.

However, here, note well, every one of the alternatives must be, taken individually, naturally contingent, as the two-alternative paradigm makes clear. Otherwise, the basis of disjunction is not properly, entirely natural, but closer to merely logical. Natural disjunction has a very different basis from logical disjunction; much more information is demanded of us, before we can formulate a natural one.

Note in any case that a logical ‘cannot’ implies, but is not implied, by a natural ‘cannot’; and therefore potentiality implies, but is not implied by, logical possibility.

After thus defining the various types of natural disjunction through naturally modal, categorical and conjunctive propositions, their logical interrelationships and processes can be worked out with little difficulty. The reader is invited to do this work.

In practice, it is not always clear whether we intend a disjunctive proposition looking like the above as natural or as logical. For instance, even though there is no such thing as actual natural disjunction, a proposition of the form ‘S is P or Q’ might be intended to mean ‘S must be P or Q’, rather than imply mere logical disjunction. But such ambiguities need not deter us from investigating the respective logical properties of these two types, and learning their differences. Some more comments will be made on this topic, in the chapter on condensed propositions.

## 35. NATURALS CONDITIONALS: OPPOSITIONS AND EDUCTIONS.

### 1. Translations.

We may call ‘translation’ the reformulation of a proposition in other form, such as the change from conditional to categorical, or vice versa.

Natural conditionals are reducible to their categorical definitions, their implicit bases and connections, of course. Thus, for instance, ‘When any S is P, it must be Q’ implies ‘All S can be P’ and ‘No S can be P and not Q’, and therefore ‘All S can be P and Q’ and ‘All S can be Q’. These implications could be viewed as distinct immediate inferences, which are collectively though not individually reversible.

Another way to translate such natural conditionals into categoricals would be by joining the antecedent predicate to the subject, to form a new, narrower, subject. Thus, for instance, ‘When any S is P, it must be Q’ would become ‘All SP must be Q’. However, the new class ‘SP’ would have to be actual, or such a necessary categorical must be regarded as not implying actuality and so tacitly still conditional.

Modern logicians tend to regard all categoricals as involving a conditional subject, and so would regard such translation of conditionals into categoricals as formally true. However, I beg to differ with current opinion on this point. My contention is that, logically, there has got to be categoricals which are genuinely so, before we can build up conditional forms; categoricals are logically prior to conditionals, since the latter correlate the former.

Cases where the subject is not actual are only artificially categorical; they are made to seem so, but in fact are still conditional. (This argument also holds for imaginary subjects, where there is a hidden hypothesis ‘Though the subject is nonexistent, if it existed, so and so would follow’.)

Thus, the hidden conditionality in some categoricals is an exception, rather than the rule. The position taken by certain logicians to the contrary is not logically tenable, in my view. This issue is further discussed in the chapter on modalities of subsumption.

### 2. Oppositions.

The form ‘When this S is P, it must be Q’ means ‘this S can be P, but it cannot be P without being Q’, which implies that ‘this S can be P and Q’. It follows that the logical contradictory of this form is ‘This S cannot be P, *or* it can be P without being Q’, and not merely ‘This S can be P without being Q’. That is, ‘When this S is P, it can not-be Q’ is not formally contradictory, but only contrary; it is contradictory only if we take for granted that ‘This S can be P’.

Similarly, ‘When this S is P, it cannot be Q’ is on an absolute level merely contraried by ‘When this S is P, it can be Q’, and becomes contradicted only in such case as ‘This S can be P’ is already given.

On the other hand, the form ‘When this S is P, it must be Q’ implies that ‘When this S is P, it can be Q’, since the latter means no more than ‘This S can be P and Q’, which is the tacit basis of the former. Likewise, ‘When this S is P, it cannot be Q’ implies ‘When this S is P, it can not-be Q’.

It follows that ‘When this S is P, it must be Q’ and ‘When this S is P, it cannot be Q’ are invariably contrary to each other, since they imply each other’s contraries.

As for ‘When this S is P, it can be Q’ and ‘When this S is P, it can not-be Q’, they may be both be true, since ‘This S can be P, with or without Q’ occurs in some cases; and they may both be false, since it is conceivable that ‘this S neither can be P and Q, nor can be P and not Q’, as occurs in the case of ‘this S cannot be P’ being true. Thus, these two bases are normally neutral to each other, though if ‘This S can be P’ is granted, they become subcontrary.

With regard to actuality, ‘When this S is P, it must be Q’ does not imply, nor exclude, that ‘this S is P (and thereby Q)’, although ‘This S is P and Q’ does imply that ‘when this S is P, it can be Q’. Thus, the necessary form is ontologically a relationship which exists potentially, even when not actually operative. It is, of course, conceivable that ‘This S is P and Q’ in the actual circumstance but not in all circumstances, or in some circumstance(s) but not the actual one. The same can be said about the forms negating the consequent.

As for the parallel forms which negate the antecedent, their basis is different, namely ‘This S can not-be P and be (or not-be) Q at once’.

Therefore, ‘When this S is not P, it must be Q’ is compatible with ‘When this S is P, it must be Q’ (these together would imply that ‘this S must be Q’), and likewise with ‘When this S is P, it cannot be Q’ (in which case, we have a *sine-qua-non* situation every which way). All the more, the potential versions are all compatible. We need not, for our present purposes, go beyond this degree of detail.

These oppositions concern singular forms, note well; the corresponding oppositions for plural forms follow automatically, in accordance with the general rules of ‘quantification of oppositions’, which we dealt with in the chapter on opposition of modal categoricals. Thus, for example, ‘When any S is P, it must be Q’ is ordinarily contrary to ‘When certain S are P, they can not-be Q’; but if it is established that ‘All S can be P’, they become contradictory.

### 3. Eductions.

Eduction from conditionals consists in changing the position and/or polarity of antecedent and consequent.

- With regard to
, suffices to say that ‘This S is P and Q’ and ‘This S is Q and P’ are, from our point of view, equivalent. For the rest:*actuals*

*Obversion* obviously applies to all the forms, without loss of modality. Thus ‘When this S is P, it can be (or must be) Q’ imply ‘When this S is P, it can not-be (or cannot be) nonQ’; likewise, ‘When this S is P, it can not-be (or cannot be) P’ imply ‘When this S is P, it can be (or must be) nonQ’.

‘When this S is P, it can be or must be Q’ *convert* to ‘When this S is Q, it can be P’, since ‘This S can be P and Q’ is implicit basis of the source.

‘When this S is P, it can not-be or cannot be Q’ *convert by negation* to ‘When this S is not Q, it can be P’, since the latter target means ‘This S can be P and not Q’, which is given in the original proposition; note well, they are not convertible to ‘When this S is Q, it can not-be P’, since the source contains no basis for ‘This S can be Q and not P’.

These results are of course in turn obvertible.

We note that these simple eductions, other than obversion, yield a potential conclusion, even from a necessary premise.

- However, a necessary conclusion may be drawn, if we are granted that the negation of the consequent is potential. This process may be called
, and viewed either as a deduction from two premises, or as an eduction from a compound premise. The following is the primary valid mood:*complex contraposition*

When this S is P, it must be Q

and This S can not-be Q

hence, When this S is not Q, it cannot be P

The proof of this argument is by reduction ad absurdum. The denial of the conclusion implies either that ‘This S must be Q’ (base denied) or that ‘This S can be nonQ and P’ (connection denied); but either way this results in the denial of the minor or major premises; therefore, the conclusion is valid.

From this mood we may derive the following, by obversion:

When this S is P, it cannot be Q

and This S can be Q

hence, When this S is Q, it cannot be P

Thus, full contraposition is feasible, but only on the proviso that the basis of the conclusion is in advance given as true; without this additional information, it is not permissible. The reason for this is that the original conditional is in principle compatible with the categorical necessity of its consequent.

Note that the above arguments incidentally yield the conclusion that ‘This S can not-be P’. This may be viewed as modal apodosis from the given premises.

- When
is introduced into all these equations, it is important to note that it is unaffected, unlike the modality. That is, a general natural conditional, is general for both the antecedent and consequent, implying that ‘all S can be Q’ as well as ‘all S can be P’.*quantity*

So ‘When any S is P, it can be or must be Q’ converts to ‘When any S is Q, it can be P’, and ‘When any S is P, it can not-be or cannot be Q’ converts by negation to ‘When any S is not Q, it can be P’. Similarly, a particular premise is convertible, though to a particular conclusion.

Likewise, given that ‘All S can not-be Q’, the necessary ‘When any S is P, it must be Q’ contraposes to ‘When any S is not Q, it cannot be P’. Also, note well, when only one of the premises is general, whichever one — that is, given ‘When any S is P, it must be Q’ and ‘Some S can not-be Q’, or given that ‘When certain S are P, they must be Q’ and ‘All S can not-be Q’ — we can still infer that ‘When certain S are not Q, they cannot be P’ (and so that ‘Some S can not-be P’). However, if both premises are particular, contraposition is not permitted. Similarly, throughout, for propositions with negative predicates.

Derivative processes behave accordingly. For instance, inversion, being contraposition followed by conversion or vice-versa, requires two premises at least one of which is general, and always results in a potential conclusion.

Lastly note, these changes all essentially concern the predicates of natural conditionals. We might additionally have considered changes affecting the subject, such as conversions within the antecedent or consequent clause. But the idea seems somewhat artificial in this context, unlike in hypotheticals.

## 36. NATURAL CONDITIONAL SYLLOGISM AND PRODUCTION.

### 1. Syllogism.

Syllogism in this context involves three natural conditional propositions, all having a common subject, and whose three predicates are positioned in figures analogous to those found in categorical syllogism. Although the rules of modality, polarity, and quantity are essentially similar, there are interesting differences of detail in the results obtained.

- The premier valid mood of syllogism involving natural conditionals is the following
singular necessary argument, where M is the middle term. From this mood all others are derivable.*first figure*

1/nnn

When this S is M, it must be Q

When this S is P, it must be M

so, When this S is P, it must be Q.

This is validated by exposition: consider any random circumstance in which this S is actually P; then, by apodosis from the minor premise, it is also M; and, by apodosis with that from the major premise, it is also Q.

By substituting nonQ for Q, we derive a similar negative-consequent version:

When this S is M, it cannot be Q

When this S is P, it must be M

so, When this S is P, it cannot be Q.

Next, a potential version may be constructed:

1/npp

When this S is M, it must be Q

When this S is P, it can be M

so, When this S is P, it can be Q.

This mood can be validated by reductio ad absurdum to the previous. If the conclusion were denied, then ‘this S cannot be P and Q’ would be true; but the original major premise implies as its basis that ‘this S can be Q’; it follows that:

When this S is Q, it cannot be P;

but When this S is M, it must be Q,

therefore, When this S is M, it cannot be P.

The connection implied by this result, being ‘this S cannot be M and P’, causes the original minor premise to be denied. Ergo, the original conclusion is undeniable.

The negative-consequent version of this mood is the following:

When this S is M, it cannot be Q

When this S is P, it can be M

so, When this S is P, it can not-be Q.

Needless to say, any modes subaltern to the above are also valid. Thus, **nnp** is implied valid, by **nnn** or **npp**.

Syllogism in this figure with a potential major premise are not valid. Consider, for example, the mood below:

1/pnp

When this S is M, it can be Q

When this S is P, it must be M

so, When this S is P, it can be Q.

Although this S is M in all the circumstances relating to this S being P (minor premise), it remains conceivable that there be circumstances in which this S is M without being P (as conversion attests); these latter circumstances may be precisely among the only ones in which this S is Q, as well as M (major premise); so there is no guarantee that this S can be P and Q together (as in the attempted conclusion), indeed it may well be that this S must cease to be P before it is allowed to be Q (in which case, when this S is P, it *becomes* Q).

*A-fortiori*, this invalidation also applies to the mode **1/ppp**. The argument is essentially that denying the attempted conclusion, by saying ‘This S cannot be P and Q’, does not result in the inconsistency of a denied major or minor premise. Analogous negative-consequent versions are equally spurious, of course.

We can also construct parallel actual moods. But, the following one might be regarded as more akin to apodosis than syllogism, though valid:

1/naa

When this S is M, it must (or cannot) be Q

This S is P and M

so, This S is P and Q (or nonQ).

As for the mood below, it concerns the mechanics of categorical conjunction, and hardly any longer qualifies as conditional argument in the narrow sense.

1/aaa

This S is M and Q (or nonQ), in actual circumstance,

This S is P and M, in the same circumstance,

so, This S is P and Q (or nonQ).

What we have here, of course, are interface situations, where different domains of logic meet.

Note that the mode **naa** is subaltern to **aaa** (even though necessity does not imply actuality here), because we can also infer that ‘This S is M and Q (or not Q)’ from the combination of major and minor premise. However, an actual conclusion from a necessary minor premise (as in **1/nna** or **1/ana**), and modes involving a mix of actual and potential premises (**ap** or **pa**), are invalid. This is easily demonstrated.

So much for the first figure. The parallels to categorical syllogism should be obvious; and indeed, categorical syllogism can be viewed as a special case of conditional syllogism, where the subject is ‘thing’ instead of a specific ‘S’.

Note in passing that sorites are possible with natural conditionals, as with categoricals.

- The valid singular moods of the
can easily be derived from those given so far, using the methods of reduction developed in other contexts. The primary ones are listed below, for the record, without little further discussion, for the sake of brevity.*other figures*

For the *second* figure:

2/nnn

When this S is Q, it must be M

When this S is P, it cannot be M

so, When this S is P, it cannot be Q.

When this S is Q, it cannot be M

When this S is P, it must be M

so, When this S is P, it cannot be Q.

2/npp

When this S is Q, it must be M

When this S is P, it can not-be M

so, When this S is P, it can not-be Q.

When this S is Q, it cannot be M

When this S is P, it can be M

so, When this S is P, it can not-be Q.

Note the change of polarity of the major event, in this figure. Mode **nnp** is subaltern to **nnn** or **npp**; but **pnp** is not valid. Also valid, in the fig. 2, is mode **2/naa**; though not **nna**, **ana**. Two actual premises (**aa**), with the polarities of the events as shown above, are naturally impossible, since the middle term would have mixed polarity; however, if the middle event has exceptionally the same polarity in the two premises, **aaa** becomes feasible, though the minor premise is useless to the inference. Also invalid, as before, are **ap**, **pa** or **pp**.

For the *third* figure:

3/npp

When this S is M, it must be Q

When this S is M, it can be P

so, When this S is P, it can be Q.

When this S is M, it cannot be Q

When this S is M, it can be P

so, When this S is P, it can not-be Q.

3/pnp

When this S is M, it can be Q

When this S is M, it must be P

so, When this S is P, it can be Q.

When this S is M, it can not-be Q

When this S is M, it must be P

so, When this S is P, it can not-be Q.

Subaltern to **npp** or **pnp**, is mode **3/nnp**; but mode **nnn** is invalid. Also valid, in the fig. 3, is mode **aaa**; and its subalterns **naa** and **ana**, though not **nna**. Also invalid, are **ap**, **pa** or **pp**, as always.

For the *fourth* figure (significant mood):

4/npp

When this S is Q, it cannot be M

When this S is M, it can be P

so, When this S is P, it can not-be Q.

Note the change of polarity of the major event, in this figure; also, the mixed polarity of the middle event. Mode **nnp** is subaltern to **npp**; but **nnn** or **pnp** are not valid. Also valid, in the fig. 4, is mode **4/naa**; though not **nna**, **ana**. Two actual premises (**aa**) are naturally impossible, unless the middle event has exceptionally the same polarity in the two premises. Also invalid, are **ap**, **pa** or **pp**.

- In addition to all the above, we could construct an equal number of valid moods, whose premises and/or conclusions involve a
, obviously. Such moods are easily validated by substituting the negation of a term for a term, in various ways. Some interesting results emerge, as the samples below show.*negative antecedent*

In figure one, all the primary moods can be reiterated, with a negative middle term (as in the sample below) and/or a negative minor term.

1/nnn

When this S is not M, it must be Q

When this S is P, it cannot be M

so, When this S is P, it must be Q.

In figure two, all the primary moods can be reiterated, with a negative major term (as in the sample below) and/or a negative minor term.

2/nnn

When this S is not Q, it must be M

When this S is P, it cannot be M

so, When this S is P, it must be Q.

In figure three, all the primary moods can be reiterated, with a negative minor term (as in the sample below) and/or a negative middle term.

3/npp

When this S is M, it must be Q

When this S is M, it can not-be P

so, When this S is not P, it can be Q.

In the fourth figure, we may switch the (mixed) polarities of the middle term, and/or of the major term, and/or insert a negative minor term. We thus have a total of 8 valid modes of polarity in each of the 4 figures.

These random examples demonstrate that the rules of polarity may seemingly be by-passed. Thus, for examples, we seem to process a negative minor premise in the first figure, or to obtain a positive conclusion in the second figure, or to draw a positive conclusion from a negative premise in the third figure. But of course, the rules of polarity are still essentially operative, the changes are illusory.

Still, such moods have practical significance. Without their clarification, we might miss out on possible inferences from data, or make errors. The reader is therefore advised to develop a full list of such syllogisms, as an exercise.

### 2. Summary and Quantities.

The following table neatly summarizes the results obtained in the previous section. Note the similarities and differences between the modes of modality here, and those for categorical syllogism.

Polarities | Valid | Subaltern | Invalid |

Figure One.

MQ | ++ | +- | -+ | – – | nnn | nnp | pnp |

PM | ++ | ++ | +- | +- | npp | naa | nna, ana |

PQ | ++ | +- | ++ | +- | aaa | ap, pa, pp |

plus 4 with negative minor term.

Polarities | Valid | Subaltern | Invalid |

Figure Two.

QM | ++ | +- | -+ | – – | nnn | nnp | pnp, (aaa) |

PM | +- | ++ | +- | ++ | npp | nna, ana | |

PQ | +- | +- | ++ | ++ | naa | ap, pa, pp | |

(aaa) |

plus 4 with negative minor term.

Figure Three.

MQ | ++ | +- | ++ | +- | npp | nnp | nnn |

MP | ++ | ++ | +- | +- | pnp | naa | nna |

PQ | ++ | +- | -+ | – – | aaa | ana | ap, pa, pp |

plus 4 with negative middle term.

Figure Four.

QM | +- | ++ | — | -+ | npp | nnp | nnn, pnp |

MP | ++ | -+ | ++ | -+ | naa | (aaa) | |

PQ | +- | +- | ++ | ++ | (aaa) | nna, ana | |

ap, pa, pp |

plus 4 with negative minor term.

In the first figure, 2 modal modes, and 1 actual mode, are valid (and these have 2 subalterns). For 8 polarity modes, this means a total of 24 (+16) valid moods. Similarly, in fig. 2, there are at least 24 (+8) valid moods, not counting the special cases of **aaa**. In fig. 3, the total is 24 (+24). In fig. 4, it is at least 16 (+8), not counting the special cases of **aaa**.

The grand total of primary moods is thus 88 (not counting specials alluded to in parentheses), of which 56 are modal and 32 are actual; plus 56 subalterns.

All the valid moods listed above are in the singular mode of quantity ‘**sss**’, but they may of course be ** quantified**. However, the rules of quantity are less stringent for conditional syllogism than with categorical syllogism.

This is due to **sss** being here valid throughout, because an individual instance of the subject, indicated by ‘this S’, effectively stands outside the syllogistic procedure as such, and remains recognizable independently of the three predicates, P, Q, and M which are being manipulated.

It follows that, *so long as one premise is universal*, a conclusion can be drawn, having the same quantity as the other premise; but no conclusion is possible from two particular premises, and the conclusion cannot be higher than the lower of the two premises.

In other words: **uuu**, **upp**, **pup**, **uss**, **sus**, are all valid, in all the figures, for all the moods established in **sss**. The only invalid inferences with regard to quantity, are therefore **upu**, **ups**, **puu**, **pus**, **ppp**, **ppu**, **pps**, **usu**, **suu**, obviously.

Below are the modes of quantity for each figure, with a minimum of examples, to illustrate some of the deviations from previous rules.

Thus, in the first and second figures, while **uuu**, **upp**, and **uss**, remain valid, we have additionally **pup** and **sus**. For examples,

1/sus

When this S is M, it must be Q

When any S is P, it must be M

so, When this S are P, it must be Q.

2/pup

When certain S are Q, they must be M

When any S is P, it can not-be M

so, When certain S are P, they can not-be Q.

In the third figure, in addition to **upp** and **pup**, the modes **uuu**, **uss** and **sus** are valid. For example,

3/uuu

When any S is M, it must be Q

When any S is M, it must be P

so, When any S is P, it can be Q.

In the fourth figure, for the significant mood listed above, instead of just **upp**, we also have **uuu**, **pup**, **uss**, **sus**. For example,

4/pup

When certain S are Q, they cannot be M

When any S is M, it can be P

so, When certain S are P, they can not-be Q.

The reader is invited to develop a full list of plural syllogisms, as an exercise.

### 3. Production.

Production of natural conditionals is their inference from categorical propositions. This shows us how to construct natural conditionals deductively, rather than empirically. The structure of the premises follows the model of categorical syllogism, while the conclusion encompasses all the original terms.

- The chief mood of such argument is in the
; it involves a necessary major, a potential minor, and a necessary conclusion, as follows:*first figure*

All P must be Q

This S can be P

therefore, When this S is P, it must be Q.

We manage, exceptionally, to reason in the **npn** mode, note, because the conclusion, though stronger than the minor premise, concerns a narrower set of circumstances (SP instead of just S).

This argument can be validated by exposition; for any circumstance in which this S is actually P, we know that it will also be Q according to the categorical syllogism **1/AnRR**. Note well that we are exceptionally drawing a necessary, though conditional, conclusion from a merely potential minor premise.

Alternatively, we can use reduction ad absurdum. Denying the conclusion means either that ‘this S cannot be P’, which contradicts the minor premise, or that ‘this S can be P and not Q’, which implies that, for this S at least, some P can not-be Q, in contradiction to the major premise. Thus, the conclusion is indubitable.

Note well that ‘When this S is P, it must be Q’ does not imply ‘All P must be Q’. Although natural conditionals may be inferred from categorical premises, it does not follow that that is the only way we can get to reach such conclusions. Natural conditionals can also be known by induction; so, they do not logically imply categoricals other than their bases and connections.

The negative version of the above mood is:

No P can be Q

This S can be P

therefore, When this S is P, it cannot be Q.

Note that if the major premise is necessary, and the minor premise is the actual or necessary ‘This S is or must be P’, then the conditional conclusion as such is unaffected; so these are subaltern moods of production.

If both premises are actual, concerning the same circumstances, the conclusion is a categorical conjunction of all three terms, which represents the actual form of natural conditional. The positive and negative versions of this **aaa** mode, still in the first figure, are:

All P are Q

This S is P

therefore, This S is P and Q.

No P is Q

This S is P

therefore, This S is P and not Q.

We may also have, with the same actual major, a necessary minor ‘This S must be P’, without change of conclusion (mode, **ana**).

Note that the **nnn** mode is also valid, by subalternation from **npn**. It is interesting to note, however, that given the premises ‘This S must be P and all P must be Q’ we would rather draw the categorical conclusion ‘This S must be Q’, than the inferior conditional ‘When this S is P, it must be Q’. It shows the essential continuity between categorical and conditional syllogism. Given that ‘Some S can be P’ (which is the base of the minor premise) the conditional conclusion is a subaltern of the categorical one.

Also, two necessary categorical premises, with adequate modality of subsumption, may also be used to draw an actual conjunctive conclusion (**nna**). All the above conclusions of course further imply that ‘When this S is P, it can be Q’ or ‘… nonQ’, respectively (as in the subaltern **aap** mode).

However, although **npn** and **nnn** are valid, the modes **npa** or **nna** are invalid, since a necessary conditional does not imply an actual conjunction. Also, the major premise could not be merely potential, since the middle term P would then not be distributive in respect of modality, even if the minor premise were necessary (**pnp**, or **ppp**). For the same reason, an actual major cannot be combined with a potential minor (**ap**), or vice versa (**pa**).

With regard to quantity, the rules of categorical syllogism remain applicable here, so that the major premise must be universal, while the minor may be universal or particular, as well as singular; the conclusion has the same quantity as the minor.

- So much for the first figure; the valid moods of the
follow from these, using the usual methods. Below is a quick overview, ignoring actuals and subalterns, which are obvious.*other figures*

In figure* two*, the model moods are in the **npn** mode:

No Q can be P

This S can be P

therefore, When this S is P, it cannot be Q.

All Q must be P

This S can not-be P

therefore, When this S is not P, it cannot be Q.

Observe, in the latter case, the production of a natural conditional with negative antecedent, exceptionally.

We can in both cases introduce different modalities and quantities, as we did in figure one. Note that only the minor premise may be potential or particular.

In figure *three*, the process seems rather contrived, though formally supportable, because of the change of position of the minor term. The model moods are:

This P must be Q

This P can be S

therefore, When certain S are P, they must be Q.

This P cannot be Q

This P can be S

therefore, When certain S are P, they cannot be Q.

This P can be Q

This P must be S

therefore, When certain S are P, they can be Q.

This P can not-be Q

This P must be S

therefore, When certain S are P, they can not-be Q.

Note the necessary conditional conclusion from a necessary major coupled with a merely potential minor, in contrast to the conclusion being no better than potential if the major is only potential, even though the minor is necessary. Thus, though modes **npn** and **pnp** are valid, the **pnn** mode is invalid.

With regard to other modalities and quantities, the rules of categorical syllogism apply here. Only the major premise may be negative; one of the premises must be necessary (or both actual); one of the premises must be particular (or both singular); and the conclusion is in any case particular.

For the *fourth* figure, again the impression of artificiality, but here is the significant model mood (mode, **npn**) for the record, anyway, without further comment:

No Q can be P

This P can be S

therefore, When certain S are P, they cannot be Q.

- Lastly, note that
allows us to form arguments involving four terms, in a categorical major premise and a natural conditional minor premise and conclusion. For example,*the combination of syllogism and production*

All M must be Q

When this S is P, it must be M

therefore, When this S is P, it must be Q.

Such argument need not be considered as a distinct process. We draw the proposition ‘This S can be M’ from the minor premise, and use this with the major premise to produce ‘When this S is M, it must be Q’, which is then coupled with the minor in a syllogism with the said conclusion.

- Some additional comments on production. Consider the first figure valid mood,

All P must be Q

All S can be P

therefore, When any S is P, it must be Q.

Note well the difference between this production of a natural conditional, and the production of a logical hypothetical from the same premises: in the latter case, the conclusion would be ‘If all P must be Q and all S can be P, then all S can be Q’, or even ‘If all P must be Q and all S can be P, then when any S is P, it must be Q’. The focus in natural production is on concrete actualities, whereas logical production is concerned with formal truths.

It is also well, in this context, to keep in mind the difference between a dispensive natural conditional, ‘When any S is P, it must be Q’, which implies a number of independent singulars; and a collectional one, ‘When all S are P, they are Q’, which refers to the conjunction of singulars as the required condition.

In the former case, we mean: ‘When this S is P, it is Q, and when that S is P, it is Q, and …’; whereas in the latter, ‘When this S is P and that S is P and …, they are Q’. The same can be said about particulars.

## 37. NATURAL APODOSIS AND DILEMMA.

### 1. Apodosis.

Natural apodosis is deductive argument mainly involving (i) a necessary natural conditional as major premise, and (ii) an actual categorical corresponding to the antecedent or to the negation of the consequent as minor premise, with (iii) an actual categorical corresponding to the consequent or to the negation of the antecedent, respectively, as conclusion. Other modalities are less typical, though derivable.

- Actual Moods.

The premier valid mood, from which all others may be derived, consists in ‘affirming the antecedent’ (*modus ponens*), as follows. Note well that the conclusion is not ‘This S must be Q’, in spite of our placing the necessary modality of the conditional proposition in the consequent; however, the conclusion ‘this S is P’, although not naturally necessary, is of course logically necessary given the premises; the mode is **naa**.

When this S is P, it must be Q

and This S is P,

hence, This S is Q.

The major premise informs us that, whatever the surrounding circumstances for this S, its being P is accompanied by its being Q. The apodosis merely takes it at its word, and applies it to the actual circumstance given by the minor premise, to obtain the conclusion.

The following mood follows from this by obversion:

When this S is P, it cannot be Q

and This S is P,

hence, This S is not Q.

The following mood, which consists in ‘denying the consequent’ (*modus tollens*), may be reduced to the primary one above, ad absurdum: deny the conclusion, while retaining the major premise, and the minor premise is contradicted.

When this S is P, it must be Q

and This S is not Q,

hence, This S is not P.

A complex contraposition underlies this argument, of course. The major premise does not by itself imply the contraposite ‘When this S is not Q, it cannot be P’; but when the major is combined with ‘This S can not-be Q’, as implied by the minor premise, the contraposite is inferable, as we saw in the chapter on eduction. With the contraposite, this mood becomes identical to the one before.

The next follows by obversion from this:

When this S is P, it cannot be Q

and This S is Q,

hence, This S is not P.

The actual moods draw an actual conclusion from a necessary major premise and an actual minor premise, in mode **naa**. The mode **nna** is accordingly valid, granting the actuality of the subject. Modes **aaa** or **ana** are valid for *modus ponens*, but their minor premise is redundant; in modus tollens, they are invalid, because the premises are incompatible.

- Modal Moods.

Modal moods are those with a modal conclusion from modal premises.

Moods with a necessary major and minor premise, affirming the antecedent, yield a necessary conclusion. These moods can be viewed as repetitive applications of the corresponding actual moods, since natural necessity means actuality in all circumstances, and they teach us that if the antecedent is naturally necessary, so must the consequent be. The following are **nnn** moods valid:

When this S is P, it must be Q

and This S must be P,

hence, This S must be Q.

When this S is P, it cannot be Q

and This S must be P,

hence, This S cannot be Q.

On the other hand, moods with a necessary major and minor premise, denying the consequent (and thus depending on our switching the positions of the events), are not valid, because their conclusion would contrary the major premise. Thus, the following **nnn** moods are invalid, note well:

When this S is P, it must be Q

and This S cannot be Q,

hence, This S cannot be P.

When this S is P, it cannot be Q

and This S must be Q,

hence, This S cannot be P.

Such apodosis is invalid, even with an unnecessary conclusion (as in **nnp**), since the major premise requires ‘this S can be P and Q (or nonQ)’ as its basis, and thus formally excludes the logical possibility of the attempted minor premise, let alone any conclusion.

As for mode **npp**, necessary major premise combined with a potential affirming minor and conclusion, form a valid mood, since as soon as the minor actualizes, so will the conclusion. For instances:

When this S is P, it must be Q

and This S can be P,

hence, This S can be Q.

When this S is P, it cannot be Q

and This S can be P,

hence, This S can not-be Q.

This mood teaches us that the connection together with the base of the antecedent suffice to define a natural conditional, since the base of the consequent (and also the compound basis) follow anyway. But we could also view this mood as redundant, granting that we already know that both the minor premise and conclusion are formally implicit in the major premise.

On the other hand, a necessary major premise combined with a potential denying minor and conclusion, form a more significant, as well as valid, mood of apodosis (mode **npp**). These arguments have already been encountered in the context of complex contraposition:

When this S is P, it must be Q

and This S can not-be Q,

hence, This S can not-be P.

When this S is P, it cannot be Q

and This S can be Q,

hence, This S can not-be P.

What of moods with a potential, instead of necessary, major premise? *Modus ponens* cases are redundant, and modus tollens cases are invalid, as shown below:

If the associated minor premise affirms the antecedent, whether necessarily or potentially, a necessary conclusion (mode **pnn** or **ppn**) is of course out of the question. Drawing a potential conclusion (in **pnp** or **ppp**) would teach us nothing new, since that is already implied in the basis of the major, anyway. For instance:

When this S is P, it can be Q

and This S can or must be P,

hence, This S can be Q.

If the associated minor premise denies the consequent necessarily, we cannot draw a conclusion, because the two premises are anyway contrary to each other. Thus, for instance, the following is invalid (mode **pnp**):

When this S is P, it can be Q

and This S cannot be Q,

hence, This S can not-be P.

If the associated minor premise denies the consequent potentially, we cannot draw a conclusion, because we have no guarantee that the some circumstances referred to by the major overlap with those referred to by the minor. Thus, for instance, the following is invalid (mode **ppp**):

When this S is P, it can be Q

and This S can not-be Q,

hence, This S can not-be P.

- We can construct further valid moods, analogous and equal in number to the above described, by substituting a
, ‘When this S is not P,…’ in all the majors. The polarity of the corresponding minor or conclusion must of course be changed to match, in every case.*negative antecedent*

- Also, we can
all the valid moods. One of the premises must be general, to guarantee overlap; the quantity of the conclusion then follows that of the other premise. Thus, we have two sets of quantified moods, with some overlapping cases (both premises general).*quantify*

Those with a general major premise, and any quantity in the minor, like:

When any S is P, it must be Q

and All/This/Some S is/are P,

hence, All/This/Some S is/are Q.

When any S is P, it must be Q

and All/This/Some S is/are not Q,

hence, All/This/Some S is/are not P.

And those with any quantity in the major premise, and a general minor, like:

When any/this/some S is/are P, it/they must be Q

and All S are P,

hence, All/This/Some S is/are Q.

When any/this/some S is/are P, it/they must be Q

and No S is Q,

hence, All/This/Some S is/are not P.

Similarly with the allowable changes in modality, and with negative consequents and/or antecedents, of course. Clearly, the rules of quantity here are less restrictive than those of modality; this is because the quantity of antecedent and consequent is one and the same, whereas the modality concerns their relationship.

Moods such as those below are, of course, not valid, because they go beyond the brief of the forms concerned. However, if we regard the minor premise as an adduction of evidence or counterevidence, we may view the suggested conclusion as tending to be confirmed.

When this S is P, it must be Q

and This S is Q (is given as evidence),

hence, This S is P (is somewhat confirmed).

When this S is P, it must be Q

and This S is not P (is given as counterevidence),

hence, This S is not Q (is somewhat confirmed).

Compare such natural adduction to logical adduction. Here, we are assuming that the actual set of circumstances surrounding the minor premise, is among the sets of natural circumstances in which the major premise holds, namely all the circumstances when this S is P or all the circumstances when this S is not Q.

### 2. Dilemma.

Natural *disjunctive* arguments are reducible to natural conditional processes, at least in the case of two alternatives. For example, the following apodosis could be validated by inferring ‘When this S is not P, it must be Q’ from the major premise.

This S must be P or Q

This S is not P

so, This S is Q.

Or again, the following sample of ‘syllogistic’ argument, admittedly somewhat forced and not likely to be used as such in practice, could be validated in a similar way.

This S must be M or Q

This S must be P or not M

so, This S must be P or Q

Likewise, we can develop arguments for production of natural disjunctives.

These are of course only the simplest samples. Other polarities, other modalities, other manners of disjunction, and multiple disjunction, would need be considered for full treatment of the field. But these topics will not be analyzed further, here.

*Natural dilemma*, however, deserves some attention, because of the improved insight into the meaning of natural necessity which it provides, and to stress its distinction from logical dilemma.

natural dilemma consists, as shown below, of premises and conclusion all of which are necessary; the major premise consists of a conditional whose antecedent is a natural disjunction (or, alternatively, of the equivalent conditionals in conjunction), the minor premise is disjunctive, and the conclusion is categorical.*Simple constructive*

When this S is M or N, it must be P,

but, This S must be M or N,

hence, This S must be P.

Whereas in apodosis to draw such a necessary conclusion, the minor premise had to be a categorical necessity, here we are taught that a necessary conclusion may still be drawn from a slightly less demanding minor premise, namely a disjunctive necessity — provided, of course, that the conditional major premise(s) is/are necessary.

We learn from this that if some event P is ‘bound to’ follow each of circumstances M, N, etc. (however many there be), and the set of circumstances M, N, etc. is exhaustive, then the event P is immovable and effectively independent of any circumstance. Thus, the dilemma as a whole tells us ‘Whether this S is M or N, it must be P’.

Note well that the minor premise and conclusion could not have been actual, as in apodosis. There is no actual form of natural disjunction; the proposition ‘This S is M or N’, taken literally, is a logical disjunctive, based on a doubt as to whether ‘This S is M’ or ‘This S is N’ is true, without implying that both these actualities are potential in the real world in the present circumstances.

- Note well the
*special case*of simple constructive natural dilemma:

‘When this S is M, it must be P, and when it is not M, it must be P’.

but, This S must be M or not M,

hence, This S must be P.

Or, more briefly, ‘Whether this S is or is not M, it must be P’. There is nothing in the structure of natural conditionals preventing contradictory antecedents from having the same necessary consequent. In such case, the consequent is absolutely, and not just relatively, necessary, so that the antecedents are redundant.

(This is the nearest thing to logical paradox, which we find in natural conditioning; there is of course no exact equivalent, since ‘When this S is not P, it must be P’ would imply that ‘This S both can not-be P, and must be P’, a natural impossibility.)

It is with this phenomenon in mind that we developed our original definition of natural necessity as ‘actuality *in every* circumstance, *whatever* the actual circumstance’. Strictly-speaking the concept of ‘in’ is more primitive than of ‘when’ or ‘or’; but the above dilemma serves as a clarification, anyway.

(Any seeming circularity is due to the fundamentality of the concepts involved; there is no inconsistency in that; nor is it redundant, because it aids our understanding, and the development of a formal logic of modality.)

- In contrast, the
natural dilemma has to consist of conditional major premise with a disjunctive consequent, which combined with an actual minor premise, yields a categorical actual conclusion, as shown below.*simple destructive*

When this S is P, it must be M or N

but, This S is not M and not N;

hence, This S is not P.

Why so? Because the other alternatives are meaningless. Had we formulated destructive dilemma as follows:

When this S is P, it must be M, and when this S is P, it must be N;

but, This S is not M and not N;

hence, This S is not P.

…we would be faced with two ordinary apodoses, each one of which would suffice to obtain the required conclusion.

If, on the other hand, we had formulated it as follows:

When this S is P, it must be M, and when this S is P, it must be N;

but, This S cannot be M or N

or even, This S must be not M or not N;

…we would be faced with incompatible premises, since the majors imply that ‘This S can be M and N’, and yet the minors deny that. Also, concluding that ‘This S cannot be P’ would deny the implication of the major that ‘This S can be P’.

(If the minor premise said ‘This S can no longer be M or N’, instead of ‘cannot’, then we might assume a similar loss of power for the antecedent, and conclude ‘This S can no longer be P’; however, that interpretation is far from certain: for it is conceivable that the major premise relationships are entirely different in such eventuality.)

- With regard to
natural dilemma, it takes the following constructive and destructive forms, for similar reasons.*complex*

When this S is M, it must be P, and when this S is N, it must be Q;

but, This S must be M or N,

hence, This S must be P or Q.

When this S is P, it must be M, and when this S is Q, it must be N;

but, This S must be not M and not N;

hence, This S must be not P or not Q.

Note that, in complex natural dilemma, there is a destructive form which is an exact analogue of the constructive, i.e. having a necessary disjunctive minor premise and conclusion. We can reduce the destructive to the constructive, by contraposing its major premise’s horns, on the basis of its minor premise.

- Lastly note,
of a natural dilemma, by a seemingly ‘equally cogent’ dilemma involving antithetical terms, is in no case logically possible, in view of the formal incompatibility between the needed minor premises. Try it and see.*rebuttal*

## 38. TEMPORAL CONDITIONALS.

### 1. Structure and Properties.

*Structure**.*The forms of conditional proposition of temporal modality, are very similar to those of natural modality. I will therefore analyze them only very briefly. They are presented below without quantifier, but of course should be used with a singular or plural quantifier.

When S is P, it is always Q

When S is P, it is never Q

S is P and Q

S is P and not Q

When S is P, it is sometimes Q

When S is P, it is sometimes not Q

(The symbolic notation for temporal conditionals could be similar to that used for naturals, except with the suffixes **c**,** t** instead of **n**,** p**; **m** and **a** are of course identical.)

Temporal conditional propositions have structures and properties very similar to their natural analogues. There is no need, therefore, to reiterate everything here, since only the modal type differs, while the categories of modality involved remain unchanged.

Temporal conditionals signify that at all, this given, or some time(s), within the bounds of any, the indicated, or certain S being P, it/each is also Q (or: nonQ), as the case may be. (Similarly, it goes without saying, with a negative antecedent, nonP.)

Here, ‘when’ means ‘*at such times as*’. The actuals (momentaries) exist ‘at the time tacitly or explicitly under consideration’, the modals (constants or temporaries) concern a plurality of (unspecified) times.

The antecedent and consequent events are actualities. The modal basis of their relationship is the temporal possibility: ‘this/those S is/are sometimes both P and Q (or: nonQ)’. The connection between them is expressed by a temporal modifier placed in the consequent; for constants, it is ‘this/those S is/are never both P and nonQ (or: Q)’, for temporaries, it is identical with the basis. The quantifier specifies the instances of S concerned.

The order of sequence of the events, though often left unsaid, should be understood. Each has a relative duration, as well as location in time. Expressions like ‘while’, ‘at the same time as’, ‘before’, ‘thereafter’, ‘whenever’, are used to specify such details.

*Properties**.*With regard to opposition, constant conditionals (like ‘Whenever S is P, it is Q’) do not formally imply the corresponding momentaries (‘S is now P and Q’, for example), although both the former and the latter do imply temporaries (their common basis, ‘S is sometimes P and Q’, here).

A constant like ‘When this S is P, it is always Q’, is contradicted by denial of either its basis or connection; that is, by saying ‘This S is never P’ or, ‘This S is sometimes both P and nonQ’. A temporary like ‘When this S is P, it is sometimes Q’, is contradicted by denying the base of either or both events; that is, by saying ‘This S is never both P and Q’.

Other oppositional relations follow from these automatically, and the same may be repeated for negative events. Momentaries are identical to, and behave like, actuals, of course.

The processes of translation, eduction, apodosis, syllogism, production, and dilemma, likewise all follow the same patterns for temporals as for naturals.

Temporal disjunction is also very similar to natural disjunction, and its logic can be derived from that of temporal subjunction.

### 2. Relationships to Naturals.

Although temporal and natural conditionals have analogous structure and properties, each within its own system, the continuity between the two systems is here somewhat more broken than it was in the context of categoricals.

In conditionals, natural necessity does not imply constancy. Compare, for instance, ‘When this S is P, it must be Q’ and ‘When this S is P, it is always Q’. Although the natural connection ‘This S cannot be P and nonQ’ implies the temporal connection ‘This S is never P and nonQ’ — the natural basis ‘this S can be P and Q’ does not imply (but is implied by) the temporal basis ‘this S is sometimes P and Q’.

Since the higher connection is coupled with an inferior basis, while the lower connection is coupled with a superior basis, the ‘must’ conditional as a whole is unable to subalternate the ‘always’ version. This is easy to understand, if we remember that even within natural conditioning, ‘must be’ does not imply ‘is’; it follows that ‘must be’ cannot imply ‘is always’, which is essentially a subcategory of ‘is’ (though it too does not imply ‘is’, as already mentioned).

This breach in modal continuity, in the context of conditionals, further justifies our regarding natural and temporal modal categories, as belonging to distinct systems of modality. In categorical relationships, these two types of modality differ merely in the frame of reference of their definitions (circumstances or times); but a more marked divergence between them takes shape when they are applied to conditioning.

For similar reasons, natural necessity does not even imply temporariness. On the other hand, temporariness does imply potentiality, since, for instance, ‘When this S is P, it is sometimes Q’ implies ‘When this S is P, it can be Q’. Here, the categorical continuity is still operative.

Also, the actualities for both types coincide: ‘in the present circumstances’ and ‘at the present time’ mean the same thing. ‘Circumstances’ refers to the existential layout of the world, how all the substantial causes are positioned in the dimensions of space; while ‘time’ focuses on the positioning of these various circumstances along the dimension of time; at any given present, these two aspects of a single happening are bound to correspond, like two sides of the same coin.

These first principles allow us to work out the valid processes which correlate natural and temporal conditionals in detail.

### 3. Mixed Modality Arguments.

I will not explore deductive arguments which mix natural and temporal modalities, in any great detail, but only enough to make the reader aware of their existence.

In syllogism, we should note valid arguments such as the following (which follow from **1/naa** by exposition):

1/ncc

When this S is M, it must be Q (or: cannot be Q)

When this S is P, it is always M

so, When this S is P, it is always Q (or: is never Q).

1/ntt

When this S is M, it must be Q (or: cannot be Q)

When this S is P, it is sometimes M

so, When this S is P, it is sometimes Q (or: nonQ).

However, an argument like the following would be invalid, because there is no guarantee that the circumstances for this S to be P are compatible with those for it to be Q (or, nonQ, as the case may be).

1/cnp

When this S is M, it is always Q (or: is never Q)

When this S is P, it must be M

so, When this S is P, it can be Q (or: nonQ).

This mode is invalid, note well. Although **1/ccc**, **1/cmm** and **1/ctt** are valid, the temporal conditionals **c**, **m**, or **t** are not subalterns of the natural conditional **n**.

In production, modes of mixed modal type are subalterns of modes of uniform type, in accordance with the rules of categorical syllogism. This may result in compound conclusions, as in the following case:

All P must be Q (implying, is always P)

This S is sometimes P (implying, can be P)

therefore, When this S is P, it must be Q (**1/npn**)

and, When this S is P, it is always Q (**1/ctc**)

(likewise with a negative major term.)

In apodosis, mixed-type ‘*modus ponens*’, like the following ones in **ncc** or **ntt**, are valid (since they can be reduced to a number of **naa** arguments):

When this S is P, it must be Q (or: nonQ)

and This S is sometimes, or always, P

hence, This S is sometimes or always Q (or: nonQ).

And also, note well, mixed-type ‘*modus tollens*’, like the following ones in **ncc** or **ntt**, are valid (since they can be reduced to a number of **naa** arguments):

When this S is P, it must be Q (or: nonQ)

and This S is sometimes not, or never, Q (or: nonQ)

hence, This S is sometimes not, or never, P.

This result is interesting, if we remember that the arguments below are not valid, since they involve inconsistent premises (the minor contradicts a base of the major):

When this S is P, it must be Q (or: nonQ)

and This S cannot be Q (or: nonQ)

hence, This S cannot be P.

When this S is P, it always be Q (or: nonQ)

and This S is never Q (or: nonQ)

hence, This S is never P.

Additionally, note, a constant major premise coupled with a naturally necessary minor premise, yield a conclusion, granting that for categoricals **n** implies **c**. Thus, **cnc** is valid, as a subaltern of **ccc**. But since **ccc** is invalid in cases of denial of the consequent, **cnc** only applies to cases of affirmation of the antecedent:

When this S is P, it is always Q (or: is never Q)

and This S must be P (implying, is always P)

hence, This S is always Q (or: is never Q).

We can similarly investigate disjunctive arguments of mixed modal type, and dilemma.

## 39. EXTENSIONALS: FEATURES, OPPOSITIONS, EDUCTIONS.

Very different from naturals and temporals, are the conditionals built on extensional modality. These are quite important, because they broaden the theory of classification, providing us with the formal means for more complex thinking processes.

### 1. Main Features.

*Actual forms**.*The following are prototypical forms of extensional conditional, those with three terms. The antecedent and consequent might in this context be called ‘occurrences’. We will first consider forms with actual occurrences, and thereafter deal with those with modal ones.

(The forms, if need be, could be symbolized like their categorical analogues, except for, say, an ampersand ‘**&**’ as prefix, to distinguish them also from natural or temporal conditionals.)

&**A**: Any S which is P, is Q

&**E**: No S which is P, is Q

&**R**: This S is P and Q

&**G**: This S is P and not Q

&**I**: Some S which are P, are Q

&**O**: Some S which are P are not Q

- Basis and Connection.

The basis of all these forms is a particular proposition of the form ‘Some S are P and Q (or nonQ)’, which incidentally implies that ‘some S are P’ and ‘Some S are Q (or nonQ)’. The basis is a particular conjunction of the same modality as the occurrences.

Note well, the difference between such extensional basis, and the basis ‘All/this/some S can be, or sometimes is/are, P and Q (or nonQ)’ of natural or temporal conditionals, which is a potential or temporary conjunction of the same quantity as the events. Contrast also to the basis of hypotheticals.

The connection implicit in ‘Any S which is P, is Q’ is the general proposition ‘No S are both P and not Q’; and that in ‘No S which is P, is Q’ (meaning, ‘Any S which is P, is not Q’) is ‘No S is P and Q’. Note that ‘Any S…’ can be expressed in many ways, like ‘In any case that S…’, or ‘Whatever S…’, or ‘Where S….’ In the forms ‘Some S which are P, are Q (or nonQ)’, the connection is identical with the basis.

Thus, to define the general forms of extensional conditional, we must mention both the connection and basis; the connection alone provides us only with a sort of logical conditional — an adequate basis is additionally required to form an extensional conditional. For the particular forms, the basis is all we need to define them. For singulars, we must present a specific case which fits the description; the basis follows incidentally.

The ** modal qualification** of the relation as a whole, here, is the quantity. Note that in practice we often say ‘In such case as S is P, it must or may be Q (or nonQ)’, with the intent to mean an extensional conditional; here, ‘must’ signifies generality, and ‘may’ particularity. What matters, is that we mean the relationship here discussed, however we choose to verbalize it.

In extensional conditionals, it is the (general, singular or particular) quantity which expresses the (extensional) necessity, existence or possibility of the relationship, so that it is essential to the relation. In contrast, in natural or temporal conditionals, the quantity is merely incidental, allowing us to summarize many individual events in one statement.

The forms ‘This S is P and Q (or nonQ)’ signify that we have found an instance of the subject-concept which displays the said conjunction. An ‘extensional possibility’ concerning the universal S, has been found ‘realized’, in this pointed-to instance of S. We could have written ‘In this case, S is P and Q (or nonQ)’.

The singular versions are also often expressed as ‘There is (or this is) an S which is P, which is Q (or nonQ)’, or ‘This S is a P, which is Q (or nonQ)’, to emphasize the mediative role played (which is more evident in plurals). These forms inform us, with reference to the sample of S, of the factual relationship between P and Q.

The expression *‘ which’* is interesting. It strings together two extreme terms, through the medium of a merely particular middle term. Because extensional conditionals have three terms, we do not need the distributive middle term of categorical syllogism to express the passage from minor to major term. The syllogism ‘This S is P and some P are Q, so this S is Q’ is invalid — unless we have inside information assuring us that the middle term is known to overlap in this case. That assurance is given us by the ‘which’.

Note lastly that the consequent may be positive or negative. Needless to say, the antecedent in the above forms may equally be negative: ‘In such case as S is not P,….’

- Function.

Extensional conditionals describe ‘cases’ of correspondences between the manifestations of distinct universals. Though their quantity is dispensive, as in categoricals, their focus is not so much the behavior of cases as that of universals.

Note that the antecedent and consequent occurrences may coincide in time, or be unequal or separate, like any two events. They may be transient, or permanent; they may be qualitative or concern action. But the message of such forms is not primarily these dynamic details, but the extensional relations between them.

It is as if the universal involved is regarded as an individual, something in itself, which changes over time. In fact, no actual, objective change needs be taking place. The time lapse involved may be subjective, relating merely to the observer, as he or she focuses on one instance after another of the unchanging universal. In extensional modality, opposites may happen simultaneously in objective time, because they happen in different instances.

Extensional conditional propositions differ from naturals and temporals, in that they study (record, report) the behaviors of universals, instead of individuals, *as if the various manifestations of a universal are like the various states of an individual*. Extensional contingency is diversity; incontingency is positive or negative universality.

Extensional modality is concerned with instances of the subject-concept; instances are its ‘modal units’, instead of surrounding circumstances or times. The effective subject of such a proposition is S-ness as such. The varying cases of S, signal varying hidden (extensional) conditions, and thus serve a function analogous to the various circumstances or times in the existence of an individual thing, which are natural or temporal conditions. This explains why all these modal types have many similar characteristics.

We see here an important underlying assumption concerning universals, that they are ruled by a kind of static and plural causality, similar to and yet distinct from the mobile causality relating individual events. For natural or temporal conditioning, real change is implied; for extensional conditioning, only real difference is implied.

Here, we are still concerned with real-world causality, but it is of a clearly different type. Natural and temporal causality essentially concern the changes within individual things stretching across time and the links between them (this is true for quantified forms as well as singulars, by subsumption). Whereas, extensional causality refers to the differences and ties affecting universals as such.

The logic of conditioning for this type of modality, investigates more intricate relationships, than those dealt with by Aristotle’s categorical propositions. These relationships have analogies to those found in natural and temporal conditioning, and even in logical conditioning, but they also have their own peculiar attributes and properties. We must therefore study them separately.

This research results in a better understanding of quantity and universals, and a powerful verbal and conceptual tool. The clarity of language it offers, will become apparent when we look into class-logic.

The main function of extensional conditionals is *classification, ordering of data*. These forms record the impacts of universals on each other, with reference to some or all of their instances. Extensionals are thus useful in explaining differences in structure or behavior patterns by reference to certain characteristics of the species.

For example, **in biology**. Suppose the species S1, S2, S3, display the attributes or properties {P1, Q1, R1}, {P2, Q2, R2}, {P3, Q3, R3}, respectively; we might infer that they stand in a hierarchy, proportional to the differences of degree between P1, P2, P3, or Q1, Q2, Q3, or R1, R2, R3. In this way, we conclude that, say, birds are related to reptiles, or men to monkeys. Although we have no film footage of natural and temporal transitions, we presume common ancestries (** theory of evolution**) with reference to character continuities.

But of course, strictly speaking, as our analysis of the definitional features of the various types of conditioning show, ** extensional comparisons are not proof of natural or temporal causation**. Awareness of the type of modality involved is therefore very important.

### 2. Modal and Other Forms.

- Modal Forms.

The antecedent and consequent of an extensional need not be both actual propositions (as above), but may involve any combination of natural and temporal modalities. I use the actuals as standard forms, because they suffice to analyze the main logical properties of extensional conditioning, but any natural or temporal category is a fitting occurrence.

To begin with, consider an extensional conditional of the form ‘This S can be P and can be Q’. Its intent is only to record that these two potentialities are each consistent with the subject-concept in the given case. The form does not insist that this S can be both P and Q at once. If we wanted to specify the latter, we would have to elaborate with a natural conditional of the form ‘When this S is P, it can be Q’. Note well the difference.

Thus, the said extensional is a wider, vaguer conjunction of two categoricals: ‘This S can be P, and this S can be Q’, whereas the corresponding natural presents the special case: ‘This S can be {P and Q}’. The natural form therefore subalternates the extensional form.

The purpose of the extensional is to specifically inform us of the identity of the indication ‘this S’ in the two potential occurrences, leaving open the issue as to whether or not their potentials can actualize in tandem. The purpose of the natural is to inform us of the concurrence of actual events, and not merely their potentialities, in the indicated instance, in some circumstances.

If the form ‘There is an S which can be P, which can be Q’ was taken to imply that that S, *as a P*, can be Q, then in cases where a P cannot be Q we would have to say ‘There is an S which can be P, which can *become* Q’. It follows that in cases of uncertainty about the compatibility of P and Q, we would say: ‘There is an S which can be P, which can be or become Q’.

It is therefore better to admit the extensional form in its widest sense, only implying that S can be Q, without determining whether SP can be or become Q. An extensional is concerned specifically with the extensional aspects of the relation (the coincidence of modal occurrences), and leaves the issue of circumstantial compatibility of the actual events to a natural proposition. Their functions are distinct.

The basis and connection of the corresponding general form ‘Any S which can be P, can be Q’ are: ‘Some S can be P, and (at least) these S can be Q’ and ‘No S both can be P and cannot be Q’, respectively. In every case, the implied basis is a positive conjunction of *particular* propositions (of equal extension), each of which has *the same* natural or temporal modality as the occurrence it underlies, note well. The connection, for general conditionals, is a general denial of the conjunction of the antecedent modality with the negation of the consequent modality. The basis and connection of the corresponding particular form ‘Some S which can be P, can be Q’, are one and the same proposition ‘Some S can be P, and these S can be Q’

It may be mentioned here, that the colloquialism ‘S can or can not be P’, does not disjoin ‘can’ and ‘can not’, but rather (redundantly) disjoins ‘P’ and ‘nonP’; it should more strictly be expressed as ‘S can and can not be P’ (the antinomy between P and nonP being given by the law of contradiction, anyway).

Modal extensionals, one or both of whose occurrences is/are of natural necessity, have different basis and connection. Thus, ‘an S which must be P, can be Q’ is based on ‘Some S must be P, and these can be Q’, whereas ‘an S which can be P, must be Q’ is based on ‘Some S can be P, and these must be Q’; and similarly with two natural necessities. Although such forms happen to imply that ‘these S can (or even must) be {P and Q}’ (and therefore that ‘some P can be Q’), that is not the primary message, and they are still very different from the natural conditionals with the same implications.

The reader is encouraged to always mentally compare, as we proceed with our study, the logical behavior of extensionals, with that of natural and temporal conditionals and hypotheticals of similar appearance. The evident differences in attributes and properties, serve to justify our making a distinction between these various forms.

We can similarly analyze other combinations of natural and/or temporal modalities, of whatever polarities and quantities. In all cases, the natural or temporal modality is effectively a part of the occurrence it appears in, and does not qualify the relation as a whole; it is the quantity which performs the task of modalizing the relation. (In that large sense, all plurals are ‘modal’, be their internal components actual or modal — in contrast to singulars which are ‘nonmodal’ with respect to extensional modality.)

Some random examples of occurrences of mixed modality are: ‘Any S which must be P, is Q’, or ‘Some S are sometimes P and always Q’, or ‘There are S which can be P, yet are never Q’.

In this text, we shall of course try to use a uniform terminology, at least in strictly formal presentations. But in practice, people are not always consistent in their choice of words to express the modal type of a conditional proposition. We may for example say ‘If or When S are P, they must be or are always Q’ and yet mean ‘All S which are P, are Q’.

To complicate matters further, we sometimes intend conditioning of mixed modal type — in structure, not just content. We may say ‘when any S is P, it must be Q’, and mean both that ‘All S can be both P and Q’ and that ‘Some S are both P and Q’; here, the extensional ‘Any S which is P, is Q’ is tacitly understood. Effectively, we are constructing a distinct type of conditioning, using *a compound type of modality*, which expresses a two-edged probability argument.

(Note, concerning symbolization: the seeming actuality of the symbols &**A**, &**E**, &**R**, &**G**, &**I**, &**O**, is irrelevant, what matters is that they specify the polarity and extensional modality concisely. If we insist on a symbolic notation to indicate the natural or temporal modalities in antecedent and consequent, we could insert two suffixes of modality, as in &**Anp** for example. But it is better to avoid complications; if we need to, we can always write a proposition in full.)

- Other Forms.

Extensional conditional propositions may also have more than three terms, which may be related in noncategorical ways.

The subject may remain the same in antecedent and consequent, while its predicates are more complex. For examples: ‘Any S which is P1 and P2, is Q’ has a conjunction of categoricals as antecedent; ‘Any S among those which ‘when they are P1, must be P2’, is Q’ has a natural conditional as antecedent. Likewise, the consequent may be more complex.

Also, the antecedent and consequent may conceivably concern different subjects. Since a ‘one for one’ correspondence is usually involved, though we can expect some common substratum to underlie them, and make possible their linkage somehow. For example, ‘For all S1 which are P, there is an S2 which is Q’ would occur if S1 and S2 are both, say, aspects of the same entity S, or are caused to occur together by some third thing S.

Extensional *disjunction* may be understood with reference to extensional conditionals. It is quite distinct in its implications from other modal types of disjunction.

With three terms and actual predications, the general form is ‘S are all P or Q’, meaning ‘Any S which is not P, is Q, and any which is not Q, is P’. This implies that ‘Some S are P and some not, and some S are Q, and some not’ (bases) and that ‘No S is {both nonP and nonQ}’ (connective). It does not imply that all S can be P, nor that all S can be Q, note well.

Here again, the different senses of ‘or’ would need to be considered, as well as the corresponding particular form, ‘S may be P or Q’, and the parallel negative forms, ‘No S is P or Q’ and ‘Some S are not P or Q’. More broadly, multiple disjunctions can be defined, with reference to the number of predicates which are found to occur together or apart, in any instances of the subject.

Disjunction of modal predications is also feasible, of course. For example, in ‘S all must be P or can not-be Q’, which means ‘Any S which can not-be P, must be Q, and any S which must be Q, can not-be P, though some S must be P, and some S can not-be Q’.

Note well that the natural modalities are parts of the occurrences, and have nothing to do with the conditioning as such, which is itself extensional. Also, do not confuse the above extensional interpretation, from that of a similarly worded logical disjunction, meaning ‘{All S must be P} or {All S can not-be Q}’.

Similarly, with any other internal polarities and modal categories and types, in any combinations. We can also construct forms with more than three terms, like ‘In all cases, an S1 is P or an S2 is Q’.

However, detailed analysis of these various forms will not be attempted here. Our treatment of the analogous forms in other types of modality, should serve as a model for further research in this area. The reader is invited to do the job.

### 3. Oppositions.

I shall only here sketch with a broad pen, the oppositions between extensional conditionals, among each other and in relation to categoricals. The reader should draw three-dimensional diagrams, to clarify all implications.

The singular form ‘This S is P and Q’ is contradicted by ‘This S is nonP and/or nonQ’, in the sense of a logical disjunction.

The general form ‘Any S which is P, is Q’ means ‘Some S are P, and these S are Q, and no S is both P and nonQ’; it may therefore be contradicted by saying ‘No S is both P and Q, or some S are both P and nonQ’. But each of these alternatives, whether denying the basis or denying the connection, taken by itself, is only contrary to the form as a whole.

The particular form ‘Some S which are P, are Q’ is contradicted by saying ‘No S is both P and Q’. This may arise because ‘No S is P’ or ‘No S is Q’, but it is also compatible with ‘Some S are P, and some (other) S are Q’. It follows that general denial of the antecedent or of the consequent, only contraries the basis.

Note that a proposition like ‘Any S which is P, is Q’, or its particular version, does not exclude the logical possibility that ‘All S are P’ and/or that ‘All S are Q’.

In extensional conditioning, a general proposition subalternates a particular one, since the latter is identical with the basis of the former, if they are alike in polarities and modalities. But (here, unlike in natural or temporal conditioning) *a general proposition does not subalternate a singular one*; saying that ‘any S which is P, is Q’ does not imply that this given S is among those which are P (and therefore Q). However, a singular proposition subalternates a particular one; saying that ‘this S is P and Q’ does imply that there are at least some cases of S (if only this one) which are P and Q.

Comparing forms with consequents of opposite polarity, the singulars ‘This S is P and Q’ and ‘This S is P and nonQ’ are merely contrary, since they may both be false, as in cases where ‘This S is not P’.

The generals ‘Any S which is P, is Q’ and ‘No S which is P, is Q’ (meaning, ‘Any S which is P, is not Q’) share the same partial basis ‘Some S are P’; but their connectives are respectively ‘No S is both P and nonQ’ and ‘No S is both P and Q’; thus, they disagree on whether the S which are P, are or are not Q, and are contrary.

Note well that ‘No S which is P, is Q’ means more than ‘No S is both P and Q’ (its connective); the former has as basis ‘Some S are P and nonQ’, whereas the latter does not have that implication, since it may be true *because* ‘No S is P’ and/or ‘No S is Q’.

As for ‘Some S which are P, are Q’ and ‘Some S which are P, are not Q’, they are compatible, but neither implies the other, since they may be referring to distinct cases of S. They are not subcontrary, since if ‘No S is P’ is true, both are false; they are therefore neutral to each other.

The parallel forms negating the antecedent can similarly be dealt with. Their antecedent is of course based on ‘Some S are not P’, instead of ‘Some S are P’, so they are bound to be compatible, with forms which imply the latter base. That is, for instance, ‘Any S which is P, is Q’ and ‘Any S which is not P, is Q’ may both be true, implying that ‘Some S are P, some not, but all S are Q’. Likewise for a negative consequent.

The four particular forms ‘Some S are P and Q’, ‘Some S are P and nonQ’, ‘Some S are nonP and Q’, ‘Some S are nonP and nonQ’, are together exhaustive: one of them must be true, though up to four of them may be true.

We can also find the oppositions between extensional conditionals whose occurrences have natural or temporal modalities other than actualities. The oppositions between categoricals obviously affect this issue. For example, *provided* ‘some S must be P’ is given, ‘Any S which is P, is Q’ implies ‘Any S which must be P, is Q’ (note well that the modality of Q is unaffected); but it may equally be of course that ‘only those S which are P and can not-be P, are Q’, in which case we must say so.

Still needing to be dealt with are the oppositions between extensional conditionals, and natural and temporal conditionals. Samples of such relationships have been hinted at throughout this chapter. A fuller picture is left to the reader to try and work out.

We will not go into further detail here. Once the similarities between extensional conditioning, and natural or temporal conditioning, are understood, all their attributes and properties can be predicted by analogy, if only we switch our focus to the appropriate modal type.

### 4. Translations and Eductions.

Extensional conditionals may be translated into the form of conjunctions of categoricals, by eliciting their defining basis and connection. One can also abridge them without error, by forming a narrowed subject out of the original subject and antecedent predication, as in ‘All/this/some SP is/are Q’, since it is given that ‘some S are P’.

With regard to eduction. For singulars, note the following: ‘This S is both P and Q’ is equivalent to ‘This S is both Q and P’, and implies ‘This S is neither {P and nonQ}, nor {nonP and Q}, nor {nonP and nonQ}’.

For plurals, obversion is always possible, i.e. ‘Any SP is Q’ implies ‘No SP is nonQ’, and ‘Some SP are Q’ implies ‘Some SP are not-nonQ’, obviously, and vice versa.

‘Any S which is P, is Q’, like ‘Some S which are P, are Q’, converts only to ‘Some S which are Q, are P’; ‘No S which is P, is Q’, like ‘Some S which are P, are not Q’, only convert by negation, to ‘Some S which are not Q, are P’.

The polarities may not be changed, without their extensional possibility being first given. Thus, only knowing that ‘Some S are not Q’ could we contrapose ‘Any S which is P, is Q’ to ‘Any S which is not Q, is not P’; likewise, only knowing that ‘Some S are Q’ could we contrapose ‘No S which is P, is Q’ to ‘No S which is Q, is P’.

Similarly, with more complex forms involving natural or temporal necessity or possibility. For example, ‘Any S which can be P, must be Q’ converts to ‘Some S which must be Q, can be P’ without proviso, but contraposes to ‘Any S which can not-be Q, cannot be P’ only if we are additionally given that ‘Some S can not-be Q’.

The subject S has remained the same throughout, note. Note well the differences between all these immediate inferences, and those applicable to similar looking natural or temporal conditionals.

## 40. EXTENSIONAL CONDITIONAL DEDUCTION.

### 1. Syllogism.

We can expect the valid quantity modes of extensional conditional syllogism to be analogous to the valid modality (not quantity) modes of natural or temporal conditional syllogism. The valid polarity modes are bound to be the same in all types of conditioning.

- Extensional conditional syllogism in the
*first*figure, has the valid plural modes**uuu**,**upp**. These may be validated by exposition, or we may reduce the particular version to the general ad absurdum (using the major premise). Negative moods may be derived from positive ones by obversion. The following moods are typical:

1/uuu.

Any S which is M, is Q

Any S which is P, is M

so, Any S which is P, is Q.

No S which is M, is Q

Any S which is P, is M

so, No S which is P, is Q.

1/upp.

Any S which is M, is Q

Some S which are P, are M

so, Some S which are P, are Q.

No S which is M, is Q

Some S which are P, are M

so, Some S which are P, are not Q.

Additionally, the mode **1/uss** is valid; that is, the minor premise and conclusion could equally well have been singular (though that would be closer to apodosis than syllogism). Providing the indicated instance of the subject is one and the same, the mode **1/sss** is also valid (though more to do with conjunction than conditioning), and indeed is the argument we appeal to repeatedly in exposition.

Subaltern modes are **uup**, **usp**, **ssp**. But the modes **uus**, **sus** are invalid, because here **u** does not imply **s** (unless we are additionally given that ‘This S is P’). Also, **pup**, **ppp**, **psp**, **spp**, are not valid: the major premise cannot be particular.

Also note, though the major premise consequent may be negative, the minor premise consequent has to be positive, unless the middle term is negative in both premises. We can further design an equal number of valid moods with a negative minor term, by substituting nonP for P.

Lastly, we could introduce other natural or temporal categories of modality in the premises. It goes without saying that the conclusion must be altered accordingly, in each case. For example:

Any S which can be M, is Q

Any S which must P, can be M

so, Any S which must be P, is Q.

Sorites can be formed with extensional conditionals, as with categoricals.

Note that *formal relations* are often left tacit in arguments. For instance, in the last example, if the minor premise consequent had been ‘must be M’, we could still draw the same conclusion, since ‘Any S which must be M, can be M’ is formally true (it being already given that ‘Some S must be M’).

- The remaining figures follow, using the appropriate methods of reduction. Typical examples of each are given below, without further ado. As in the first figure, many variations on these themes are workable:

In the *second* figure, the mode **2/uuu** is valid:

Any S which is Q, is M

No S which is P, is M

so, No S which is P, is Q.

No S which is Q, is M

Any S which is P, is M

so, No S which is P, is Q.

Similarly, with particular or singular minor premise and conclusion; that is, the modes **upp** and **uss** are valid. Subaltern modes are **uup**, **usp**; but the modes **uus**, **sus** do not work. The mode **sss** is valid, only if the middle term has the same polarity in both premises (contrary to the habitual configuration for this figure). The modes **pup**, **ppp**, **psp**, and **spp** are not valid, as before.

In the *third* figure, we have arguments like:

Any S which is M, is Q

Any S which is M, is P

so, Some S which are P, are Q.

No S which is M, is Q

Any S which is M, is P

so, Some S which are P, are not Q.

These **uup** moods (note the particular conclusion) are of course subaltern to those in **upp** or **pup**, in which one or the other premise is particular. Also valid, are moods with two singular premises, in **sss**; subaltern to this mode, are modes **uss** and **sus**, since the middle term is antecedent in both premises. But note that **uuu**, **uus**, and **ppp**, **psp**, **spp**, are all invalid modes.

In the *fourth* figure, we have (the significant mood):

No S which is Q, is M

Some S which is M, is P

so, Some S which are P, are not Q.

This argument is in mode **upp**; similarly valid is the mode **uss**, with a singular minor premise. Subaltern to these are **uup**, **usp**. All other modes are invalid, namely: **uuu**, **pup**, **sss**, **uus**, **sus**, **spp**, **psp**, **ppp**. Note that **sss** would require a contradictory middle term.

For all these figures, as in the first, other combinations of polarities may be introduced; see our treatment of this issue in the context of natural modality for full details. Likewise, as in the first figure, the occurrences may have any combinations of natural and/or temporal modalities.

- Note well, in all the figures, the analogies between the valid modes of extensional syllogism in quantitative issues (with
**u**,**s**,**p**symbols), and the valid modes of modality in natural (**n**,**a**,**p**symbols) or temporal (**c**,**m**,**t**symbols) conditional syllogism. These uniformities facilitate remembering.

However, note also, the differences between their respective treatments of quantity and modality. The valid quantity modes for extensionals differ from the valid quantity modes for naturals or temporals. And likewise, modality inferences differ. It is therefore important to be aware of the modal type of any conditional proposition.

### 2. Production.

The situations and results for extensional production are again clearly different from those concerning natural or temporal production.

- In the
*first*figure, production of extensional conditionals from categorical premises proceeds as in the following samples, mode**1/upu**:

All P are Q

Some S are P

therefore, Any S which is P, is Q.

No P is Q

Some S are P

therefore, No S which is P, is Q.

We thus are able to infer, given a universal major premise, a conditional universal from a particular minor; also of course inferable is the categorical ‘Some S are Q (or not Q)’. In thinking of the natural or temporal type, our conclusion would have been ‘Some S are P and Q (or nonQ)’, instead.

The above minor premise could equally be universal, with the same conditional conclusion, in the subaltern mode **uuu**; but here a better conclusion could be drawn, the categorical ‘All S are Q (or nonQ)’, which subalternates the extensional conditional. This again shows us the essential continuity between categorical and conditional argument.

With a singular minor premise, a singular conclusion can be drawn, the conjunction ‘This S is P and Q (or nonQ)’, so the mode **1/uss** is valid.

When the premises can have modalities besides actuality, the conclusion is also modal, but it must reflect a valid categorical syllogism, as in the following samples with natural modalities. Note how, in some cases, the conclusion retains similar occurrences, whereas in other cases the conclusion may alter modality and even copula, in accordance with earlier findings.

All P must be Q

Some S must be P

(whence, Some S must be Q)

so, Any S which must be P, must be Q.

All P must be Q

Some S can be P

(whence, Some S can be Q)

so, Any S which can be P, can be Q.

All P can be Q

Some S must be P

(whence, Some S can be or become Q)

so, Any S which must be P, can be or become Q.

All P can be Q

Some S can be P

(whence, Some S can be or become Q)

so, Any S which can be P, can be or become Q.

Contrast the extensional conditional conclusion in the second of these samples, to the natural conditional conclusion which could also be drawn from the same premises, namely ‘When certain S are P, they must be Q’). Their concerns are clearly distinct.

Similarly, for moods with a negative major term. And again similarly with temporal modalities, or with mixtures of modal and actual premises, or premises of mixed modal type. In every case, the rules of modal categorical must be respected, to produce a valid extensional conditional.

- The valid moods of the other figures follow from those of the first figure, as usual.

In figure *two*, we have mainly (mode** 2/upu**):

No Q is P

Some S are P,

therefore, No S which is P, is Q.

All Q are P

Some S are not P,

therefore, No S which is not P, is Q.

Note the polarity of the antecedent of the conclusion, in the latter case. With a singular minor premise, a singular conclusion could also be drawn (mode **2/uss**), of the form ‘This S is P (or nonP), and Q’.

In the *third* figure, we can draw a general extensional, if the major premise is general and the minor particular, singular or general (**3/upu**, **usu**, **uuu**); but we can draw only a particular extensional, if the major is particular or singular, and the minor premise is general (**3/pup**, **sup**). The main moods are thus:

All P are Q (or nonQ)

Some P are S,

therefore, Any S which is P, is Q (or nonQ).

Some P are Q (or nonQ)

All P are S,

therefore, Some S which are P, are Q (or nonQ).

In this figure, a singular premise does not yield a singular conclusion, because of the inappropriate positions of the terms.

In the *fourth* figure, we have:

No Q is P

Some P are S,

so, Any S which is P, is not Q.

Likewise with a singular or general minor premise. Again, a singular premise does not yield a singular conclusion, due to the position of terms.

Production of modal extensional conditionals in these figures, is also feasible — keeping in mind the rules of categorical syllogism, as well as the above models for each figure. The reader should explore some examples.

- Lastly, note that we can combine syllogism and production to form arguments involving a categorical major premise and a conditional minor premise and conclusion, as in the following example:

All M are Q

Any S which is P, is M,

so, Any S which is P, is Q.

The minor implies that ‘Some S are M’; this, together with the major premise, produces ‘Any S which is M, is Q’; which, in a syllogism with the original minor premise, in turn yields the required conclusion.

Similarly with modals, as for instance in:

All M must be Q

Any S which must be P, can be M,

so, Any S which must be P, can be Q.

Note also the following derivative argument, involving a categorical minor premise and a conditional major premise and conclusion. The fact that necessity implies possibility, because necessity is one of the species of possibility, gives us the hidden premise in parentheses, provided the categorical minor is true.

Any S which can be P, can be Q,

and Some S must be P

(whence, any S which must be P, can be P),

therefore, Any S which must be P, can be Q.

Here, we ‘produce’ a new, narrower, conditional from a given conditional, instead of the categorical ‘All P must be Q’; or this process could be viewed as ‘eduction’ complicated by a proviso.

Note that extensional conditionals can also be arrived at by inductive means (observation and generalization); they do not have to be deduced by syllogism or production.

### 3. Apodosis.

- Extensional apodosis follows the pattern set by the primary moods presented below. These are
(affirming the antecedent) arguments, in mode*modus ponens***uss**; they simply apply the principle expressed in the major premise to a singular case:

Any S which is P, is Q, No S which is P, is Q,

and This S is P, and This S is P,

hence, This S is Q. hence, This S is not Q.

The major premise cannot be particular. But the minor can be universal or particular, and the conclusion will have the same quantity. The plural moods are:

Any S which is P, is Q, No S which is P, is Q,

and All S are P, and All S are P,

hence, All S are Q. hence, No S is Q.

Note, with regard to the mode **uuu**, *modus ponens*, that the major premise is compatible with the minor and conclusion; general extensional conditionals only imply particular bases, and particularity means contingency or generality. Also:

Any S which is P, is Q, No S which is P, is Q,

and Some S are P, and Some S are P,

hence, Some S are Q. hence, Some S are not Q.

The mode **upp**, *modus ponens*, may be regarded in two ways: (i) it teaches us that all you need for definition of a general conditional is the base of the antecedent plus the connection, because the base of the consequent follows by such apodosis; or (ii) since we know that the two bases are formally implicit, such argument is in practice redundant.

However, the latter viewpoint is incorrect, because not all conditionals are formulated from knowledge of the basis and connection, but some are arrived at obliquely, as by syllogism, so that *modus ponens *in **upp** is informative, it aids understanding of the data in hand.

- The following moods are
(denying the consequent) arguments, in mode*modus tollens***uss**. These may be validated directly, by contraposition of the major premise on the basis of the minor; the conclusion is new information, emerging from the contraposite and the base of its antecedent in a*modus ponens*apodosis. Or we may validate them by reductio ad absurdum, contradicting the conclusion results in denial of the minor premise, by*modus ponens*.

Any S which is P, is Q, No S which is P, is Q,

and This S is not Q, and This S is Q,

hence, This S is not P. hence, This S is not P.

The major premise again cannot be particular. The minor and conclusion can be particular; but note well that they cannot be general, since they would contradict the bases of the major. The valid plural moods of *modus tollens *are, therefore, only the following:

Any S which is P, is Q, No S which is P, is Q,

and Some S are not Q, and Some S are Q,

hence, Some S are not P. hence, Some S are not P.

Thus, modes **uss** and **upp** are valid in both ponens and tollens extensional apodosis. But the mode **uuu** is only valid in ponens; in tollens, it is invalid, note well:

Any S which is P, is Q, No S which is P, is Q,

and No S is Q, and All S are Q,

hence, No S is P. hence, No S is P.

- We may of course introduce a negative antecedent into any of the arguments above or below; just replace P with nonP throughout. For examples:

Any S which is not P, is Q, No S which is not P, is Q,

and This S is not P, and This S is not P,

hence, This S is Q. hence, This S is not Q.

Also, any natural or temporal modality, or mixture of them, may be involved, provided we adhere to the set interpretations of extensional conditionals. The rules of quantity of the extensional apodosis process are the same with modals, as with actuals.

The following are some examples of modal *modus ponens*. Note the faithful transmission of natural modality from consequent to conclusion. If the antecedent is necessary, nothing less than a necessary minor will activate it.

Any S which can be P, can be Q,

and This S can be (or is or must be) P,

hence, This S can be Q.

Any S which must be P, can be Q,

and This S must be P

hence, This S can be Q.

Any S which can be P, must be Q,

and This S can be (or is or must be) P,

hence, This S must be Q.

Any S which must be P, must be Q,

and This S must be P,

hence, This S must be Q.

The following are some examples of modal *modus tollens*. It is interesting how, granting the premises, we are able to draw a conclusion of opposite natural modality, as well as polarity, to the antecedent. If the consequent is potential, nothing less than a necessary minor will activate it.

Any S which can be P, can be Q,

and This S cannot be Q,

hence, This S cannot be P.

Any S which must be P, can be Q,

and This S cannot be Q,

hence, This S can not-be P.

Any S which can be P, must be Q,

and This S can not-be (or is not or cannot be) Q,

hence, This S cannot be P.

Any S which must be P, must be Q,

and This S can not-be (or is not or cannot be) Q,

hence, This S can not-be P.

- Note well in all the above arguments, the differences between extensional apodosis, and natural or temporal such arguments.

Thus, in natural (or temporal) apodosis the major premise may be particular if the minor is general; but not here: in extensional apodosis the major must be general. On the other hand, in natural (or temporal) apodosis the consequent cannot be potential (or temporary), whereas here it can.

Such differences in process are due to the switched *roles* of the features of quantity and modality, from one type of conditioning to the next. In naturals or temporals, the conditioning is defined by the modality, and the quantity is incidental. In extensionals, the conditioning is defined by the quantity, and the modalities involved are incidental.

Lastly, note the existence here too of adductive arguments, which merely suggest a result, with some degree of probability, though not certainty:

Any S which is P, is Q

and This S is Q (is given as evidence)

hence, This S is P (is somewhat confirmed).

Any S which is P, is Q

and This S is not P (is given as counter-evidence)

hence, This S is not Q (is somewhat confirmed).

Compare extensional adduction, to logical, natural or temporal adduction. Here, we are expressing a likelihood that the indicated instance of the subject, is indeed one of the instances of the subject covered by the major premise. Note that though the conditional is general, it may be based on a very limited number of cases.

### 4. Extensional Dilemma.

Extensional *disjunctive* arguments are reducible to extensional conditional processes. It is important to always clarify just what we intend by the disjunction, because often different interpretations are feasible.

An example of an extensional disjunctive *apodosis*

All S are P or Q (implying Any S which is not P, is Q)

This S is not P

hence, This S is Q.

An example of disjunctive *syllogism* (reduced to two conditional syllogisms):

All S are M or Q (all S-nonM are Q, all S-nonQ are M)

All S are P or nonM (all S-nonP are nonM, all S-M are P)

hence, All S are P or Q (all S-nonP are Q, all S-nonQ are P)

Production of extensional disjunctives may likewise be achieved by production and reconstruction of extensional conditionals.

*Extensional dilemma* is more complicated, and worth exploring more deeply. The reader should compare it to logical, natural and temporal dilemma, to see the analogies and differences.

- The
*simple*dilemmas look as follows (taking ‘or’ to mean that one of the alternatives has to be applicable):

The simple *constructive* form:

Any S which is M or N, is P

but, All S are M or N

therefore, All S are P.

In this argument, the major premise tells us that those S which are not M, are N and P; and those S which are not N, are M and P; and none of all these S are both nonM and nonN. The minor premise tells us that all S fit the preconditions expressed in the major, yielding the conclusion that all S are also subsumed by the consequent, categorically.

The simple* destructive* form:

Any S which is P, is M or N

but, Some S are not M and not N

therefore, Some S are not P.

The major premise informs us that of all the S which are P, none is also both nonM and nonN. The minor premise presents us with some cases of S which are indeed both nonM and nonN. The conclusion is, therefore the latter S cannot be counted among the former, and there must be some S which are not P.

Note that the constructive version has a general minor premise and conclusion (mode, **uuu**), whereas the destructive version only works with a particular minor premise and conclusion (mode, **upp**). A constructive dilemma with a merely particular minor (**upp**) would yield a conclusion already known, since it is a base of the major; a destructive dilemma with a universal minor (**uuu**) would yield a conclusion contradictory to a base of the major.

In the singular, **uss** mode, the minor disjunction cannot be meant extensionally; where it happens in ordinary discourse, we intend a logical basis; the conclusion would still be valid on that basis, however. Logical basis disjunction is of course also sometimes intended within universals or particulars. More on this topic in the chapter on condensed propositions.

- If we look at the special case of antithetical antecedent predicates:

Any S which is M or not M, is P,

but All S are M or not M,

whence All S are P.

…we see this means that ‘*Whether* any S is M or not M, it is P’ implies ‘All S are P’. This reflects the compatibility of the propositions ‘Any S which is M, is P’ and ‘Any S which is not M, is P’, provided ‘All S are P’ to prevent their contraposability.

We can reword it as ‘Though some S are M and some others not M, all S are P’. The ‘though’ stresses the independence of the general consequent from the contingent antecedent: their ‘link’ is so strong, that it is effectively absent. This model allows us to understand universality as a type of necessity; what is found in all the cases of a subject is viewed as more ingrained in their nature, than attributes which differ from similar case to case.

- The
*complex*constructive and destructive extensional dilemmas, respectively, look as follows.

Any S which is M is P, and any S which is N is P,

but All S are M or N,

therefore, All S are P.

In this constructive version, the extensional disjunction in the minor premise ensures that all S fit the preconditions of one or the other of the horns of the major premise, and so make the antecedents exhaustive, and their common consequent general.

Any S which is P is M, and any S which is P is N,

but Some S are not M or not N,

therefore, Some S are not P.

In this destructive version, notice that the minor premise is disjunctive. It could have been, more narrowly, ‘Some S are not M and not N’; but since the conclusion may be drawn by apodosis from either of these negatives without the other, we can broaden the applicability of the argument by saying ‘or’. However, this disjunction may be intended as merely logical.

Note that the valid quantity modes here are **uuu** for the constructive, and **upp** for the destructive; as for simple dilemma, a constructive **upp** is uninformative, and a destructive **uuu** is logically impossible.

The singular **uss** mode is conceivable with a logical, rather than extensional, disjunctive minor, constructively or destructively; it is also conceivable in a destructive mood with ‘This S is not M and not N’ as the minor premise.

- All the above forms of dilemma may of course involve antecedents and consequents of other polarities, and of natural or temporal modalities other than actuality. The rules of modality here are similar to those of modal extensional apodosis. The reader should construct some examples of modal dilemma, to get acquainted with it.

Lastly note, there is no argument in extensional dilemma, equivalent to *rebuttal* of a logical dilemma by an ‘equally cogent’ dilemma. The minor premises required for that would be contradictory. The reader should experiment, and find out if this statement is correct.

## 41. MODALITIES OF SUBSUMPTION.

We need to analyze our presuppositions regarding the modalities of subsumption by the terms of categoricals, as distinct from the copulative modalities.

### 1. Formal Review.

In formulating the logic of modal categoricals so far, we have taken for granted certain ideal assumptions, which will now be reviewed.

. We granted that ‘All S are P’ implies ‘This S is P’. However, closer inspection suggests the truth of such subalternation, only on the proviso that we have directed our attention to something, which we designate by ‘this’, and have discerned that ‘this is S’.*Singular subsumption*

For, whereas ‘all S’ can be talked about without needing to be attentive any one S, the indicative ‘this’ requires a definite act of focusing on one thing, and judging whether or not it is S. This psychological requirement also means for logic that ‘this S is P’ and ‘this S is not P’ are both deniable at once, by saying ‘but this is not an S’.

Thus, **A** and **R**, or **E** and **O**, are only relatively subalternative, since this relation only works conditionally; absolutely speaking they are neutral to each other. Likewise, although **R** and **G** are relatively contradictory, they are absolutely only contrary. When the preliminary judgements regarding subsumption are settled, the relative opposition comes into effect; otherwise, the absolute opposition is operative. Similarly for modal singulars.

. We granted that ‘All S must be P’ implies ‘All S are P’. However, closer inspection suggests the truth of such subalternation, only on the proviso that there be Ss in the present actuality. We have to consider the two modalities of ‘all S’.*Actual subsumption*

Normally, we understand **An** to refer to ‘all S, ever’ (i.e. past, present, or future); although it could refer more restrictively to ‘all now S’. In the timeless (i.e. across time) case, there is no guarantee that any S exist in the present actuality, taken at random. In contrast, **A** is normally understood to refer to ‘all S now’, since any absent S are out of the present picture; although, if we view all the scattered actualities as one actuality, then we could say that the implication holds in the timeless case.

Thus, **An** and **A** may have distinct extensions. If they both mean the same ‘all S’, the subalternation holds. But if **An** means ‘all S at all times’ and **A** only means ‘all S at this time’, then **An** ceases to imply **A**, unless we have already established that ‘some S are actual’.

We can argue in the same way that ‘All S are P’ implies ‘All S can be P’, provided they have the same extension; if **A** means ‘all S now’ and **Ap** is understood to mean ‘all S ever’, the inference is illicit, and we can only accept that ‘Some S can be P’.

Thus, **An** implies **A**, and likewise for **En** and **E**, only conditionally. Also, **A** implies **Ap**, and likewise for **E** and **Ep**, only conditionally, though they still respectively imply **Ip** and **Op** unconditionally.

For the same reason, **A** and **O**, or **E **and **I**, may both be false, if it happens that ‘No S are actual’. Their contradictions apply at such times, but there are times when both can be denied. And likewise, the subcontrariety of **I** and **O** is only relative to there being actual Ss.

With regard to the interrelationships of modal propositions, since normally the subsumption of ‘all S’ has the same modality for all of them, such problems do not arise. They all imply, and presuppose, that Ss are potential. (It is true that if Ss do not exist even potentially, then modals behave like actuals without actual subjects; but this is another issue, dealt with later.)

Just as singulars like ‘This S is P’ presuppose ‘this is S’, so with any actual propositions we have to assume that ‘there are Ss at this time’: these are separate, preliminary judgements, which affect the logical properties of the propositions that conceal them.

. The above concerns subsumption by the subject. With regard to the predicate, it seems obvious that, if the subject of an affirmative, actual or necessary, proposition is actual, then so is the predicate, for the same extension. On this basis, we can convert ‘all or some S are P’ to ‘some P are S’. Also, since necessity implies actuality when the subject is actual,*Subsumption by the predicate***An**and**In**can be converted to**I**, under those conditions; otherwise, only to**Ip**.

In the case of the corresponding negatives, it would at first sight not be thought that the predicate needs be actual. However, if ‘No S is P’ is to be converted, there has to be actual Ps to support the actuality of the inference; if this precondition is not met the eduction is invalid. If only some P are actual, then **E** is convertible, but only to ‘Some P are not S’; if all P are actual, **E **is convertible fully to ‘No P are S’; if no P are actual, nothing actual about Ps may be denied or affirmed.

Also, since **En** implies **E**, conversion of ‘No S can be P’ to ‘No P is S’ is only conditionally feasible, even though that to ‘No P can be S’ is independent of actuality. (Note however, in passing, that the conversion of **En** to **En** does presuppose the potentiality of the predicate.) For **On** and **O**, such problem does not arise, since they are inconvertible in any case.

- All the above can be repeated with reference to temporal modality.

### 2. Impact.

Thus, the singular ‘this’ and the plural ‘all’ or ‘some’ are more weakly related than previously intimated. Also, actual copulae require at least actual subsumption by the terms, whereas modal copulae (whether necessary or possible) need only possible subsumption by the terms. The type of modality subsuming the terms corresponds to the type affecting the copula. In natural modal propositions, the subsumptions are potential; in temporals, they are temporary.

Thus, we have seen that many processes adopted as standard by both actual and modal logic, are only conditionally true. Some other logical processes, which depend on those considered above for their validity, may be expected to in turn become equally conditional. For example, if **E** is only conditionally convertible, then obverted conversion or inversion of **E** is likewise restricted.

Even syllogism may be affected. We have to look at the results of arguments, to make sure they are unconditional with reference to modal subsumption. For example, the mood **4/EIO** does convert both its premises unconditionally, because the middle term in the minor premise, allows conversion of the middle term in the major premise. In contrast, the mood **3/RRR** was rejected, essentially because the degree of specificity of the middle term could not be transferred to the minor term; but we could equally view this mood as conditionally valid, if we can indicate the subject.

We had made some ideal assumptions, to better emphasize the essential natures of the forms under consideration. These assumptions are reasonable — one would not normally formulate a proposition unless its subsumptive conditions seemed fulfilled; it is only in further ratiocination that an illicit process may occur, which yields a presumptive subsumption. However, we must be made aware of the exceptions and provisos, so that the system as a whole remain unassailable.

Thus, an avenue for further logical analysis is to check the unconditionality or conditionality of all our validations or rejections of logical processes. That investigation is left to the reader.

Note well that these theoretical requirements are not necessarily fulfilled in practice. There is a difference between ‘common parlance’, which is more flexible and approximate, and the ideal language of formal logic, which must needs have fixed and precise meanings.

For example, when in practice we say ‘All S are P’, we often mean **A**, but may also mean **An **or **Ac **or **Ap** or **At**, or even sometimes just **I** or **Ip** or **It**. Also, we may mean ‘all now’ or ‘all ever’. We may even misrepresent the terms. This is all harmless, if our thought is clear enough to oneself and successfully conveyed to others. One can reason logically with the rough sentences of everyday language, but there is less likelihood of error using formal language.

So long as the normative system is capable of verbalizing all situations encountered in practice, it is successful and sufficient. Thus, the science of Logic must extend its tentacles as far as necessary, enough to make possible the verbalization of any intention we may encounter in practice.

In that case, all casual statements must be carefully reformulated, to fit the standard forms provided by Logic, before they can be subjected to its rigid analysis. It is impossible to develop a system of Logic which parallels common practice exactly, because the variations in it are too arbitrary and too subjective.

Obviously, if the standard forms are not properly used, if the translation picks the wrong forms to express our pre-verbal intention, the results are likely to go awry. The process of forming a clarified thought is by no means automatic and guaranteed.

### 3. Primitives.

Completely categorical propositions may be called primitives. They vary in degree of *specificity*, but conceal no conditions.

** Indication** is the instrument of full specification. Only something which is precisely indicated — extensionally, naturally and temporally — is fully specified.

The indicative, singular and actual: ‘this thing, at this time, and in these circumstances, is so and so’, refers to an unnamed, pointed-to thing, existing in a pointed at time and set of circumstances. This form is specific extensionally, and temporally and naturally.

‘This’ (or that or these or those) is a sui generis term, which is meaningless without the presence in front of one of what is being referred to. One can say that ‘this is not so and so’ (to deny a statement starting with ‘this so and so is…’),* but one cannot say “this is not a ‘this’”*.

The next level of specificity is the indefinite, particular, actual: ‘there are, at this time and in these circumstances, some things which are so and so’, which informs us that, out-there somewhere in the world, ‘some things are so and so’. This form is unspecific extensionally, though still specific with regard to time or circumstance.

Further down the scale, the indicative singular modal ‘this thing is possibly so and so’, and the particular modal ‘some things are possibly so and so’, are indeed categorical, but unspecific. Note well that ‘this thing’ in singular modals is less specific than ‘this thing’ in singular actuals; because the latter concerns an actual relation, whereas the former concerns a modal one. The indicative is less demanding, here. Likewise for ‘some things’, the modality of subsumption depends on the modality of the copula.

The above mentioned primitive forms are the only absolutely categorical propositions. All other ‘categorical’ forms used by formal logic are more complex, and thus implicitly conditional. Their categorical format is somewhat conventional, artificial — hiding their compositeness.

The singular actual ‘This S is P (or not P)’ presupposes that ‘this thing is indeed S’, which may be said to specify the subject under discussion. As well, all actuals require and imply that the units subsumed by their terms be as actual as the copula between them (else, how would the relationship be viewed as actual?). Here, natural circumstances or times are being tacitly specified.

Plural actuals ‘All or Some S are P’ presuppose that ‘some things are indeed S’, which just means ‘there are actually unspecified Ss out there’. The specific actuality involved is supposed to be clearly understood.

Modals only require and imply that ‘this or some thing(s)’ — ‘are in some circumstances S’ (in the case of natural modality) or ‘are at some times S’ (in the case of temporal modality). Here, the circumstances or times for S remain unspecified, implying mere potentiality or temporariness of the subject, rather than a specified ‘this now’.

Similarly for the predicate, whatever the polarity of the copula, if conversion is accepted. We could alternatively, consistently, say that conversion of a universal negative is a valid process, only if the predicate is specific; in which case, the predicate of negative propositions does not need to be formally specific.

We cannot consistently say that all propositions are conditional, because then we would have no way to express categorically that the conditions have been met (as in apodosis). But it is logically permissible to regard the primitive statements ‘This thing is actually S’ and ‘There are actual Ss’ (= some things are S), as the only truly categorical forms, while all others as only relatively categorical.

### 4. Transformations.

Let us, therefore, reword the more complex categorical forms, in such a way that their implicit assumptions are brought out in the open, using primitives. We may call this ‘transformation’; it is done below, for actuals, then potentials, then naturally necessary propositions. A parallel listing can be made for temporal modality. We see that they all concern conjunctions involving the two terms, with varying degrees of specificity and complexity.

**R**: ‘This thing is now S and P’

**G**: ‘This thing is now S and not P’

**I**: ‘Some things are now S and P’

**O**: ‘Some things are now S and not P’

**A**: ‘Some things are now S and P, but nothing is now S and not P’

**E**: ‘Some things are now S and not P, but nothing is now S and P’

**Rp**: ‘This thing can be S and P’

**Gp**: ‘This thing can be S and not P’

**Ip**: ‘Some things can be S and P’

**Op**: ‘Some things can be S and not P’

**Ap**: ‘Some things can be S and P, and no other things can be S and P’.

**Ep**: ‘Some things can be S and not P, and no other things can be S and not P’.

**Rn**: ‘This thing can be S and P, but cannot be S and not P’

**Gn**: ‘This thing can be S and not P, but cannot be S and P’

**In**: ‘Some things can be S and P, but none of these things can be S and not P’

**On**: ‘Some things can be S and not P, but none of these things can be S and P’

**An**: ‘Some things can be S and P, but nothing can be S and not P’

**En**: ‘Some things can be S and not P, but nothing can be S and P’

Thus, the forms ‘All S are P’ or ‘Some S cannot be P’, and such, are really abbreviations, shorthand versions, of the above, more descriptive, forms. Their full definition shows many of them to be conjunctive, of two or more primitive categorical propositions.

Notice that the implicit conditionalities, may be a mix of extensional and natural, modal subjunctions. Plurals may be reworded in extensional conditional form, and modals in natural conditional form; so plural modals will involve both types of subjunction, one inside the other. Thus, we may have an extensional conditional, whose antecedent and consequent are two natural conditional propositions, involving different polarities.

For examples. **Ap **means: ‘For some things: in some circumstances, S and P coincide; but for other things: in no circumstances do S and P coincide’. **An** means ‘For some things: in some circumstances, S and P coincide; but for all things: in no circumstances do S and nonP coincide’.

Similarly with temporal modality, instead of natural, throughout.

I will not here analyze such forms further, although this is the obvious next step in the logical development of a complete system of modal logic. We would want to verify that the oppositions, eductions and syllogistic arguments, which were developed for complex categoricals, remain in force, when the later are transformed into their clearer, subjunctive versions. (If any inconsistencies in properties are uncovered, the transformations would have to be further perfected, until consistency is indeed achieved.)

### 5. Imaginary Terms.

Another issue relating to modality of subsumption is, how to view imaginary terms. This is a further complication, concerning logical modality.

An imaginary term may be built up out of certain suppositions and/or assumptions. ‘Supposition’ concerns what is already granted to be true in some cases, and/or in some circumstances or times, is singularized in an indicated instance and/or actualized in an indicated circumstance or time; whereas assumption concerns the granting of such particular, and/or potential or temporary subsumptions, to begin with.

Thus, supposition is based on given extensional, and/or natural or temporal possibility, and only presumes applicability to the specific instance or actuality; whereas assumption involves hypothetical constructs, it presumes the realization of what is merely logically conceivable. They differ in audacity, the former having more empirical grounds than the latter; but ultimately, they are both presumptive, bringing together certain events or characteristics in novel conjunctions, *with more specificity than contextually justified*.

Just as, with regard to extensional, natural or temporal modality, the modalities of the copula and terms affect each other — so, with regard to logical modality, the modalities of the copula and terms, are proportional. If a proposition involves some term of less than established status, then its truth is correspondingly no more than conceivable.

A concept which is believed to involve no presumptions may be viewed as realistic, while a concept is imaginary to the degree that it involves suppositions and assumptions. If we are at a stage where the projected parameters are still conceivable, then our concept tends towards realism with varying success. If we know already that the projections are not realizable, then our concept will remain imaginary.

In science, we construct imaginary concepts in the hope of eventually establishing them as realistic. But literature allows for pure imagination, whether it is in the form of a novel built on the suppositions that certain particulars, potentials or temporaries are in effect, or in the form of science fiction or fantasy built on unrealistic assumptions. The latter kind of imagination has no pretensions of literal truth, it is mere entertainment or example setting.

Thus, we can say that, apart from deliberate fictions, the difference between imaginary and realistic concepts is one of degree of contextual credibility. The degree is greatest, if no presumption was involved; intermediate, if only supposition was involved (the less supposition, the more realistic); and least, if assumption was involved (the more assumption, the more imaginary).

Our belief of a proposition is a function of our belief in its terms. If a term is imaginary, then we do not in the fullest sense accept the proposition as true, even if the formula makes internal sense. As the chances that our term be realistic increase, so accordingly does the proposition as a whole become closer to ‘true’ in the ultimate sense.

Thus, the hypotheticality of a term influences the degree of truth in the proposition. But such conditioning must be the exception, rather than the rule. We cannot consider knowledge as hypothetical ad infinitum: there has to be some definite knowledge.

Some propositions must be admitted as categorically true; the proof is that, if we claim all knowledge hypothetical, we thereby posit that claim as unconditionally true, and thus contradict ourselves. Because some propositions are unconditionally true, then these at least must involve realistic terms: ergo, some concepts must be admitted as realistic.

In practice, we commonly call even propositions with fictional terms ‘true’ — this is in the sense of internal consistency within a narrow framework, without regard to the unrealizability of the terms. For example: ‘Dragons are lizard-like’ has a mythical subject and yet is in a sense ‘true’. Here, ‘truth’ merely signifies an accurate description of a mental image known to be fictional; effectively, there is a tacit bracket saying ‘all this is imaginary’.

Closer scrutiny reveals that our example really means (should be rephrased as) ‘We have formed a fantasy, to be called a dragon, with an arbitrary description including the shape of a lizard’: so formulated, the proposition is factual. The format ‘Dragons are lizard-like’ is merely an abbreviation of that true statement; but taken literally, it is false (since there are no dragons).

In practice, we often have a fictional predicate in a negative proposition, as in ‘Lizards are not dragons’. This is formally more justifiable, since we can regard the convertibility of universal negatives as conditional on the factuality of the predicate. We could then demand that all subjects be factual, since nothing can really be said about nonexistents, without insisting on the same requirement for predicates. If so, ‘X does not exist’ would have to be worded ‘No existents are X’.

In conclusion, just as, with regard to the extensional, natural and temporal subsumption, we said that, whatever the polarity the copula, the terms must be specified in primitive form (indicatively for singulars or actuals, through the corresponding possibility for plurals or modals) — so, with regard to logical modality, subsumption in fully true propositions must be factual (or necessary), whereas subsumption in logically modal propositions need only be logical possibility (of varying degree: from mere notion, through relevant and consistent, up to logically necessary).

The rules of subsumption are essentially the same for all types of modality. In logical modality, a proposition has to be conceivable at some level, however low. The minimum requirement is that the words involved all mean something. That something may be any kind of appearance: one either rightly believed in, or realistic but disbelieved or unsure, or wrongly believed in, or unrealistic and disbelieved or unsure. But there must in any case be some kind of appearance, whether empirical, conceptually arrived at, or imaginary, which serves as the intent of the word.

These restrictions concern any proposition presented as having some possibility of truth. False propositions are not subject to law; they can even be meaningless or self-contradictory. Likewise, the antithesis, ‘nonX’, of a meaningful and consistent term, need not itself be so conceivable.

The various types of modality should not be viewed as making up a hierarchy along one line. Rather, each is like a dimension, at right-angles to the others, with analogous categories of modality. They thus are capable of combining together, while remaining mutually independent continua.

## 42. CONDENSED PROPOSITIONS.

Conditional propositions provide us with a powerful formal language, enabling us to elucidate a large variety of derivative forms we commonly use.

### 1. Forms with Complex Terms.

We are now in a position to consider ‘condensed’ propositions, which have a conjunctive or disjunctive subject and/or predicate. These propositions are made to appear like single categoricals, through this device of complex terms, but in fact conceal two or more standard propositions.

- The ‘subject’ may be a conjunction of subjects. Thus, ‘Every S1 and S2 is P’ normally means that whatever is {both S1 and S2}, is P, without implying (nor denying) that something S1 but not S2, or something S2 but not S1, satisfies the condition for being P.

Note in passing that complex propositions like ‘all ABC are XYZ’ are often used in practice, because we try to abbreviate a multiplicity of relations in a minimum of words. Such a form implies smaller statements like ‘some A are B’ or ‘some A are X’ or ‘some X are Y’. Such implicits may be the precise premises of categorical syllogisms, rather than the more complex form which is presented as a premise. This explains why theoretical logic may seem so much more bare than practical examples. For instance, ‘my computer sounds like a duck’ contains many smaller statements like: ‘I have a computer’ or ‘my computer makes sounds’.

- The ‘subject’ may be a disjunction of subjects. Thus, ‘Every S1 or S2 is P’ normally means that every S1 is P and every S2 is P, without telling us whether anything may or may not be S1 but not S2, or S2 but not S1.

- The ‘predicate’ may be a conjunction of predicates. Thus, ‘Every S is P1 and P2’ means that every S is P1 and every S is P2.

- The ‘predicate’ may be a disjunction of predicates. Thus, ‘Every S is P1 or P2’ normally means that some S are P1 and some S are P2, without telling us whether or not any S is both P1 and P2.

However, other interpretations of some of these forms are feasible. Firstly, our interpretation depends on whether the ‘or’ is understood as an ‘either-or’ or as an ‘and/or’ or as an ‘or-else’, and on how definite these disjunctions are. Secondly, our interpretation depends on the type of modality intended: is the ‘or’ intended to convey an extensional disjunction, or a natural or temporal one, or even a problemacy?

Form (a) admits of singular or particular versions ‘this/some {S1 and S2} is/are P’. Note that we often in practice intend the conjunction more loosely, so that we really mean the same as form (b).

Form (b) was above understood extensionally, so that the predicate was dispensed to both subjects generally. In that case, there are no corresponding singular or particular versions. However, we can say ‘this/some S is/are {P1 or P2}’, if we regard the disjunctive clause as a whole, rather than the disjuncts, as the subject. In this case, how the ‘or’ is understood, and the type of modality involved, becomes more variable.

Form (c) is the least ambiguous of them all, and readily admits of singular or particular versions ‘this/some S is/are P1 and P2’.

Form (d) was above understood extensionally, so that both predicates were dispensed to the subject particularly. In that case, there are no corresponding singular or particular versions. However, we can say ‘this/some S is/are {P1 or P2}’, if we regard the disjunctive clause as a whole, rather than the disjuncts, as the predicate. In this case, how the ‘or’ is understood, and the type of modality involved, becomes more variable.

To illustrate alternative interpretation, consider the form ‘Every S is P1 or P2’ again. If we wanted our ‘or’ to suggest that S may be split into two groups S1, S2, such that no S1 are S2, and all S1 are P1 and all S2 are P2, it would not suffice for us to say that ‘some S are P1, and all other S are P2’. We would have to make use of extensional conditionals, as follows: ‘any S which is P1, is not P2; and any S which is P2, is not P1’.

This last form is important because it introduces fractionating of a subject, which topic will be dealt with in more detail later.

Similarly, natural conditionals may be used to express other interpretations, such as ‘when any S is not P1, it is P2; and when any S is not P2, it is P1’. And likewise with temporal modality. We can even understand the ‘or’ in a logical sense, even as a mere problemacy; for instance, ‘if all S are P1, no S is P2; and if all S are P2, no S is P1’.

Thus, we see that the ambiguities of such condensed forms are dealt with through the instrument of our more precise conditional forms in each type of modality, and we do not need to develop a new logic for them (except as an exercise).

The condensed forms presented above were all affirmative and actual. We may similarly analyze negative forms, like ‘No S1 and S2 is P’, or modal forms, like ‘this S must be P1 or P2’. Note in passing that exceptive propositions, like ‘all S but S1 are P’, can be similarly analyzed. Note also that conditional propositions may also involve complex terms, and are similarly analyzable in a multitude of ways.

I will not go into such detail, however: I must move on; the job is left as an exercise for the reader, and for other logicians. It is clear, in any case, that the same principles apply.

In conclusion, it should have become obvious by now that the issue of complex terms, involving a conjunction or disjunction of subjects or predicates, is ultimately an enlargement of the issue of modalities of subsumption. Also, it shows clearly that the distinction between categorical and conditional forms, is ultimately somewhat arbitrary; there is a continuum of forms running into each other.

### 2. Making Possible or Necessary.

Another, unrelated, family of forms which condense conditionals, can be mentioned at this juncture: making possible or making necessary. These relate to causality. Here again, the concepts involved can be applied to any type of modality.

‘P makes Q possible’ signifies, in logical modality, that if nonP, then nonQ (nonP and nonQ are possible, and ‘nonP and Q’ is impossible), whereas if P, not-then nonQ (P and Q are possible). This commutes to ‘Q is made possible by P’. Clearly, only on the condition of P being true, does the possibility of Q being true have an effective chance of arising; P is thus said to be an *exclusive condition* for Q, a *sine-qua-non*.

‘P makes Q necessary’ signifies, in logical modality, that if P, then Q (P and Q are possible, and ‘P and nonQ’ is impossible), whereas if nonP, not-then Q (nonP and nonQ are possible). This commutes to ‘Q is made necessary by P’. Clearly, the truth of P alone would raise the mere possibility (within a contingency) of Q’s truth to an effective necessity (more precisely, it is Q’s realization, rather than Q itself, which becomes necessary); P is thus said to be a *sufficient condition* for Q.

These concepts ‘making possible’ and ‘making necessary’ are obviously correlative. If P makes Q possible, then nonP makes nonQ necessary; and if P makes Q necessary, then nonP makes nonQ possible. Also, if P makes Q possible, then Q makes P necessary; likewise, if P makes Q necessary, then Q makes P possible.

We have other concepts of a similar nature. Thus, ‘P is *necessary for* (or to) Q’ means that without P, Q would not be true; which is equivalent to ‘P makes Q possible’. Again ‘P *necessitates* Q’ means that in order for P to be true, Q would be required to be true; which is equivalent to ‘P makes Q necessary’.

We can similarly analyze the concepts of ‘making impossible’ (implying prevention or inhibition) and ‘making unnecessary’.

All this can be duplicated in other types of modality. Thus, in natural modality, ‘When this S is not P, it cannot be Q, and when this S is P, it can be Q’ implies that P makes Q potential in this S; also, ‘When this S is P, it must be Q, and when this S is not P, it can not-be Q’ implies that P makes Q (that is, Q’s actualization) a natural necessity in this S. Similarly with regard to temporal modality. In extensional modality, ‘Any S which is not P, is not Q, and some S which are P, are Q’ implies that P makes Q possible for S’s; also, ‘Any S which is P, is Q, and some S which are not P, are not Q’ implies that P makes Q necessary for S’s.

When issues of sequence arise, the possibility or necessity involved may be, if not simultaneous, precedent or subsequent in time.

These concepts obviously refer us to the various types and categories of *aetiology*. They allow us to begin a classification of causes. Within each modality, some causes are both exclusive (or necessary) and sufficient (or necessitating); some are only the one or the other; some are neither of these, but rather *occasional* (or contingent), meaning that they depend on additional partial conditions (a conjunctive antecedent) to effect the consequence. Many more subdivisions of causality are of course possible.

Clearly, a *formal logic of causality* can be derived from the logic of conditioning. Forms like ‘A being B causes C to be D’ are commonly used, and capable of precise analyses, by specifying the category and type of modality involved. I will not here develop this field of logic, since it is derivable; but it is important, and should be eventually done.

Many statements conceal this sort of form. In some cases, syllables like ‘en-’ or ‘-fy’, meaning ‘to make’, are used to signal causality, as in ‘he verified the statement’, meaning ‘he (did something which) caused the statement to be (accepted as) true’. In some cases, the verb entirely hides the causal aspect, as for instance in ‘water dissolves these crystals’, the verb ‘dissolves’ means ‘causes to dissolve’, and we can rephrase the whole more precisely as ‘when some water is mixed with these crystals, a solution is obtained’.

We enter here, also, into the subordinate realm of *teleology*, the study of needs in the context of given goals or purposes. For examples, with making possible. In logical pursuits: what must I prove first, before I can prove so and so? In natural or temporal causation: what should I do, in order to achieve so and so? In extensional choices: which of these things should I choose, to obtain so and so?

These forms play an important role in the formal logic of *ethical modality*. Granting certain standards of value, all the ways and means follow, with reference to objective aetiological (and teleological) relations. Something is permissible if compatible with *all* our ends, impermissible if incompatible with some of them; something is imperative if a *sine-qua-non* of some of our final causes, unimperative if not a *sine-qua-non* of any of them. A full study of ethical modality would have to analyze volition, and discuss the source of our ultimate norms. These issues are of course beyond the scope of the present treatise.

# PART V(a). CLASS LOGIC.

## 43. THE LOGIC OF CLASSES.

### 1. Subsumptive or Nominal.

We have to distinguish between the *subsumptive* use of a word, and its (say) *‘nominal’* use. In the former case, the word has an only incidental role, serving to direct our minds to the objects we label by it; in the latter case, the word itself is the object of our attention, while the things it refers to are incidental. Thus, for example, when we speak of dogs, we think of our tail-wagging and barking friends; but when we speak of “dogs”, we mean the word enclosed by inverted commas itself.

This should not be confused with the distinction between denotative and connotative terms, which we made in discussing permutation. We can take denotative terms subsumptively or nominally (as we did with above, comparing dogs and “dogs”), and we can take connotative terms likewise subsumptively or nominally (for example, caninity and “caninity”). That is not what is at issue here. What is at issue is, whether our focus is purely objective (as in, dogs and caninity), or we are focusing on the instrument (as in, “dogs” and “caninity”).

Thus, in subsumptive intent, we mean what the word refers to, and the instrument is transparent; whereas in nominal intent, the instrument itself is what we mean, and what the word refers to specifically is of lesser moment. Normally, our intent is subsumptive (let us symbolize this as an X); but sometimes, especially in epistemological discussions, our intent is nominal (let us symbolize this as “X”).

Nomenclatural propositions have the primary forms: “X” is the name of all X; or: all X are the referents of “X”.

We may extend the distinction between subsumptive and nominal intent to other aspects of our instruments of thought, not only to the verbal. An ‘idea’ or a ‘class’, or any similar construct, may like a word be considered ‘nominally’, in contrast to subsumptively. This does not mean to imply that words, ideas, classes, and such, are all one and the same thing, but only that they have in common the property we mentioned. There is no doubt that they are significantly different concepts, yet also somehow related.

The precise relation between these various concepts is not the topic of this chapter. Rather, we shall specifically explore some of the mechanics of classification, in an effort to better understand the logical relations between things and our concepts of them. This research into *the way knowledge is organized*, has been of great interest to logicians of this century, under the heading of ‘the logic of classes’.

### 2. Classes.

We think of a class and its members, as having a similar relation to that of a receptacle and the things it contains. The container, an elastic and permeable wrapping, is a figment of our imagination; yet, its shape and size are determined by the contents. This visual analogy is not perfect, but is a starting point.

The subsumptive outlook is directed at the contents, labeling each member as X; this is the only kind of classificatory relationship we have dealt with so far: it is the concern of Aristotelean logic. The nominal outlook is directed at the container, labeling the class as “X”; this gives rise to a new field of logic.

- Definitions.

For any thing X, we can invent a corresponding thing “X”, such that:

- whatever is X, is a member of “X”; and whatever is not X, is not a member of “X”.

Conversely, we say:

- “X” is the class of (all) X; and “X” is a class of anything which is X, and not a class of anything which is not X.

These two sets of statements mean the same thing, they are just two sides of the same coin, they commute each to the other; we call the whole relationship ‘class-membership’.

The above begin to formally define the difference between what we mean by X and “X”, relating them through a new pair of copulas, which are different from the copula ‘is’. In one direction, the copula is labeled ‘is a member of’, and has a subsumptive as subject and a nominal as predicate; in the other direction, the copula is labeled ‘is a class of’, and has a nominal as subject and a subsumptive as predicate.

Since in speech, unlike writing, we have no way to display inverted commas, we merge the two and say: this thing is a member of the class of X; the latter expression, class of X, is equivalent to “X”, in relation to X. We understand “X”, or the class of X, as a mental construct of some sort, which we intend to bear a certain relation to the things we have labeled as X. We assign the virtually same label to the construct as we did to the original things, except for a small distinguishing mark (“” or the class of) to keep their distinction in mind.

The plain name is subsumptive, referring directly to the things concerned, the marked name is nominal, referring rather to the invented correspondent of the things. Note that, although the member to class relation has some similarity to the relation of an individual to a group, they are not identical. The subsumptive versus nominal distinction, should not be confused with the dispensive versus collective (or even collectional) distinction, which we made earlier (in the discussion of quantity).

Thus far, what we have done is to point to a set of phenomena, which we commonly encounter in our current ways of thinking, and sorted them out somewhat, and named the various factors. But all we have achieved is at best a technical definition; a fuller definition requires some further understanding of the distinctive properties of these factors. That is what we will look into now.

Consider the following example, which accords with our normal manner of speaking. Dogs are dogs, and are members of “dogs” (or the class of dogs). But, dogs are not “dogs”, only members of “dogs”; and “dogs” is not a dog, and not a member of “dogs”. Note that there is no self-contradiction in saying that dogs are not “dogs”, or that “dogs” is not a dog, even though the statement that dogs are not dogs is of course absurd.

Such examples suggest the following features and processes. (Note that I concentrate mainly on the properties of ‘is a member of’; those of ‘is a class of’ follow obviously, I do not highlight them, to avoid repetitions.)

- Features.

Whereas in a proposition of the form ‘this thing is X’, the subject and predicate are both subsumptive — in a proposition of the form ‘this thing is a member of “X” (or the class of X)’, ** the predicate is nominal**. This principle is necessary, because the whole concept of membership was built with the intent to study that special kind of term we call nominal. Membership by definition relates any kind of thing to one kind of thing specifically: mental constructs.

With regard to the subject of membership, the above definition concerns only subsumptive subjects, but we shall presently consider nominal ones.

** What is X, is not “X”**, but only a member of “X”. The copula ‘is a member of’ positively relates two things, X and “X”, which the copula ‘is’ negatively relates, at least in examples like ours (dogs are not “dogs”, but only members of “dogs”).

** “X” is not an X, nor a member of “X”**. A class is not a member of itself: the relation of membership is not reversible, at least not in examples like ours (“dogs” is not a member of “dogs”, since it is not a dog).

With regard to the latter two principles, the examples only prove that they hold in some instances; however, we may generalize from such cases, if we find no examples to the contrary.

- Immediate inferences.

Obviously, by definition, since all X are X, all X are members of “X”; and since no nonX are X, no nonX are members of “X”. The class of X includes all things which are X, and excludes all things which are not X. Similar eductions apply for the class of nonX, or “nonX”. It follows that membership in “X” and membership in “nonX” are exact contradictories.

More broadly, we can infer from the above definition of membership that: ** if any X is Y, that X is a member of “Y”; and if any X is not Y, that X is not a member of “Y”**. Any thing which is X and also Y, is an X which is a member of “Y”; any thing which is X but not Y, is an X which is not a member of “Y”. That is, “X” is

*the*class of all X, but not

*the only*class for any X; there are normally other classes like “Y”, of which we can say that it is

*a*class of some or all X. For examples, retrievers are members of the class of dogs, but not members of the class of cats.

It follows that: if all X are Y, then all X are members of “Y”; if only some X are Y, then only some X are members of “Y”; if no X is Y, no X is a member of “Y”. (Note in passing, in the middle case, we regard the membership of some X in “Y”, as ‘accidental’, or ‘incidental’ to their being X, since not all X fall in this category; or we say that these X are Y, but not *‘qua’* X or not as X ‘*per se*’, not by virtue of being X.)

These statements are reversible: if all X are members of “Y”, then all X are Y; if some X are members of “Y”, then some X are Y; if some X are not members of “Y”, then some X are not Y; if no X are members of “Y”, then no X are Y.

- Deductive arguments.

We thus can construct the following syllogisms for the copulas ‘is (or is not) a member of’, on the basis of Aristotelean syllogisms for the copulas ‘is (or is not)’.

Figure 1.

All Y are members of “Z”,

all/this/some X is/are member(s) of “Y”,

so, all/this/some X is/are member(s) of “Z”.

Likewise, with negative major and conclusion.

Figure 2.

No Z are members of “Y”,

all/this/some X is/are member(s) of “Y”,

so, all/this/some X is/are not member(s) of “Z”.

Likewise, with positive major and negative minor.

Figure 3.

All/this/some Y are members of “Z”,

some/this/all Y is/are member(s) of “X”,

so, some X are members of “Z”.

Likewise, with negative major and conclusion.

Figure 4.

No Z are members of “Y”,

some Y are member(s) of “X”,

so, some X is/are not members of “Z”.

Such deductions are easily validated, by translating them into their customary forms. Note that a term may be subsumptive in one proposition and nominal in another, according to its position by virtue of the figure.

- Modal class-logic.

The definition of class-membership is easily modalized, if we wish to work out a more modal class logic.

Thus, for natural modalities: if something can be X, then it can be a member of “X”; and if something cannot be X, it cannot be a member of “X”; if something must be X, then it must be a member of “X”; and if something can not-be X, it can not-be a member of “X”. Similarly for temporal modalities. The quantification of these singular forms represents extensional modality.

Note that these definitions are in the form of extensional conditionals. The logical properties of their consequent forms are easily derived from the modal logic of their antecedent forms, which are ordinary categoricals. That includes: oppositions, eductions, and deductions.

### 3. Classes of Classes.

In the previous section, we defined and analyzed the membership of a non-class (subsumptive) in a class; now, we need to look into what we mean when we say of a class (nominal) that is a member of another class.

- Definitions.

We propose that, for any X and Y:

- if all X are Y, then “X” (or the class of X) is a class of Y, and therefore is a member of “classes of Y”, (or the class of classes of Y).

Conversely:

- if less than all or no X are Y, then “X” (or the class of X) is not a class of Y, and therefore not a member of “classes of Y” (or the class of classes of Y).

Note the variety in wording; we also often abbreviate ‘class(es) of Y’ to ‘*Y-class(es)*’.

This definition of so-called classes of classes reflects our common practice. For examples, since all dogs are animals, “dogs” is an animal-class, or a member of “animal-classes”; but since some dogs are not black animals, “dogs” is not a class of black animals, or a member of “classes of black animals”.

Now, this is an artifice. The reason why we construct this new concept is that we want to be able to talk about classes in the same way as we talk about things. We build up a parallel domain, in which classes bear relations to each other, somewhat similar to the relations between their ultimate referents. Thus far, the stratification of things had no equivalent in the realm of classes, since nominal terms were defined as predicates of the ‘is a member of’ copula. In order to place classes as subjects of similar propositions, we introduce appropriate special predicates: classes of classes. A class of classes is a subsumptive whose referents are specifically nominal.

Note that an ordinary class (that is, one which is not a class of classes) stands as subject of membership when the predicate is a class of classes; there are no grounds for assuming that an ordinary class can ever be a member of another ordinary class. We cannot, for instance, say “dogs” is a member of “animals”, but only, dogs are members of “animals”, or “dogs” is a member of “animal-classes”.

This was already suggested in the previous section, in the claim that “X” is not a member of “X”; now, we can generalize further, and say that “X” cannot be a member of any “Y”, granting that these terms are not classes of classes of anything. Other than the above defined case, *there are no known conditions regarding any X and Y, under which we could conclude that “X” is a member of “Y”**.*

Similarly, ** there are no known conditions under which propositions of the form: “X-classes” is a member of “Y”, may arise**. However, as we shall presently see, propositions of the form: all/some X-classes are (are not) members of “Y-classes”, do indeed arise, directly out of the definition of classes of classes. However, note that the subject is subsumptive here, not nominal.

Let us now investigate how successful our above definition of classes of classes is, some of the logical properties it implies.

- Features.

** “X” is an X-class, and a member of “X-classes”** (or the class of X-classes), since all X are X, and even though “X” is not an X, nor a member of “X”. This principle proceeds deductively from the definition, by substituting X where we find Y. It means that every class is a member of the class of classes bearing its name. It does not mean that it is a member of itself, however; we should not confuse a class with a class of classes; thus far, we have no cause to doubt the earlier postulate that classes cannot be members of themselves. For example, “dogs” is a dog-class, and a member of “dog-classes”.

However, ** no X is an X-class, nor a member of “X-classes”**, even though all X are members of “X”, and “X” is a member of “X-classes”. The definition of a class of classes refers to a nominal “X” as its subject, not a subsumptive X. The relationship of membership is not passed on all the way down the chain to the individuals subsumed by X; the only individuals subsumed by a class like “X-classes” are classes like “X”. For example, dogs are members of the class of dogs, but not of the class of dog-classes.

Similarly,** no X is a Y-class, nor a member of “Y-classes”**, even if all X are Y, and therefore members of “Y”. Contrast those statements to saying that “X” is a Y-class (or a class of Y), and therefore a member of “Y-classes” (or the class of Y-classes, or the class of classes of Y). Keep the distinctions clear.

We might strengthen these insights by calling ordinary classes, classes *‘of the first order’*, and classes of classes, classes *‘of the second order’*; then we can say: ** members of a class of the first order cannot be members of a class of the second order**; at best, they might be said to be members of a member of a class of the second order. This may be referred to as the

*principle of intransmissibility*of membership across orders of classification.

- Immediate Inferences.

Obviously, by definition, if “X” is a Y-class, then all X are Y; and if “X” is not a Y-class, at least some X are not Y. Likewise, with any of the alternative wordings.

The following theorems are important, because they construct propositions in which a class of classes is the subject, a novelty; thus far, classes of classes only appeared as predicates.

** If all X are Y, then all X-classes (including “X” itself) are Y-classes, or members of “Y-classes”, the class of Y-classes**. Proof is by exposition: consider any class “W” which fits the definition of an X-class, so that all W are X, then (since all X are Y) all W are Y, and it will follow that “W” is a Y-class; this can be repeated for any “W”, and even “X” fits in (since all X are X). For example, all dog-classes (such as “retrievers”) are animal-classes.

A corollary is: ** if “X” is a Y-class, then all (other) X-classes are (also) Y-classes**; the conclusion follows, since the premise implies that all X are Y.

*If some X are Y, then some X-classes are Y-classes**.* Proof: those things which are both X and Y can be said to be XY, and self-evidently all XY are X and all XY are Y; thus we have, in the case of “XY” at least, an X-class which is a Y-class.

*If some X are not Y, then some X-classes are not Y-classes**.* Proof: those things which are X but not Y can be said to be XnonY, and self-evidently all XnonY are X and no XnonY are Y; thus we have, in the case of “XnonY” at least, an X-class which is not a Y-class.

** If no X is Y, then no X-classes are Y-classes**. For if, say, “W” is an X-class, then all W are X; and since no X is Y, it follows that no W is Y, which means that “W” is not a Y-class.

Thus, note well, if some X are Y, it follows only that some X-classes are Y-classes, for we may find a class “W” (other than “X”) for which all W are X and yet no W is Y. Likewise, if some X are not Y, it follows only that some X-classes are not Y-classes, for we may find a class “W” (other than “X”) for which all W are X and also all W are Y.

Conversely, if all X-classes are Y-classes, then all X are Y; if some X-classes are Y-classes, then some X are Y; if some X-classes are not Y-classes, then some X are not Y; and if no X-classes are Y-classes, no X is Y.

- Deductive arguments.

It is important to note that syllogistic reasoning with the copula ‘is a member of’ depends for its validity on the manner of reference of its terms.

We saw that, if any X is a member of “Y”, and “Y” is a member of “Z”, it follows that that X is a member of “Z”. The proof being, since that X is Y, and all Y are Z, then that X is Z.

However, if even all X are members of “Y”, and “Y” is a member of “Z-classes”, it does not follow that any X is a member of “Z-classes”. For, even though it be implied that all X are Z, this only signifies, as already pointed out, that “X” is a member of “Z-classes”, not that any X is a Z-class.

Thus, we have the same arrangement of premises, with the copula ‘is a member of’ in both cases, yet the conclusions are of fundamentally different form. In the former case, subsumptives are members of an ordinary class; in the latter case, a nominal is member of a class of classes. This of course illustrates the earlier mentioned principle of intransmissibility of membership.

The following arguments may be validated with reference to the indicated Aristotelean syllogisms.

Figure 1 (from 1/AAA).

“Y” is a member of “Z-classes”,

and “X” is a member of “Y-classes”,

therefore, “X” is a member of “Z-classes”.

Figure 2 (from 2/AOO).

“Z” is a member of “Y-classes”,

and “X” is not a member of “Y-classes”,

therefore, “X” is not a member of “Z-classes”.

Figure 3 (from 3/OAO).

“Y” is not a member of “Z-classes”,

and “Y” is a member of “X-classes”,

therefore, “X” is not a member of “Z-classes”.

However, no other arguments of that sort are possible. In the first figure, a negative major premise, “Y” is not a member of “Z-classes”, would only imply that some Y are not Z, from which no conclusion can be drawn; and as for **1/AII**, it has no equivalent here, since “X” is a member of “Y-classes” would require that all X be Y. In the second figure, likewise with regard to a negative major premise; and as for **2/AEE**, it has no equivalent here, since “X” is not a member of “Y-classes” only implies that some X are not Y. We can similarly write off the remaining moods of the third figure. The fourth figure has no equivalent here, since the minor premise of **4/EIO** is not enough to imply membership of a class in a class of classes.

Thus, we only have three valid moods for propositions of this kind; no other moods are valid. The first is used for including a class in a class of classes, the other two for purposes of exclusion. These can be restated as follows, in accordance with the theorems of immediate inference:

Figure 1 (1/AAA)

“Y” is a Z-class (or, all Y-classes are Z-classes),

“X” is a Y-class (or, all X-classes are Y-classes),

so, “X” is a Z-class (or, all X-classes are Z-classes).

Figure 2 (2/AOO).

“Z” is a Y-class (or, all Z-classes are Y-classes),

“X” is not a Y-class (or, some X-classes are not Y-classes),

so, “X” is not a Z-class (or, some X-classes are not Z-classes).

Figure 3 (3/OAO)

“Y” is not a Z-class (or, some Y-classes are not Z-classes),

“Y” is an X-class (or, all Y-classes are X-classes),

so, “X” is not a Z-class (or, some X-classes are not Z-classes).

For examples. (i) The class of retrievers is a class of dogs, and the class of dogs is a class of animals, therefore “retrievers” is an animals-class. (ii) “Roses” is a class of plants, but “dogs” is not a class of plants, therefore “dogs” is not a member of “classes of roses”. (iii) “roses” is not a class of animals, but “roses” is a class of plants, therefore “plants” is not a member of the class of classes of animals.

Although the subsumptive relation between classes and classes of classes allows for only these three valid moods, it is clear that the subsumptive relation of classes of classes with each other allows for a fuller range of syllogistic argument. The three arguments indicated in brackets are obviously not all the valid moods for such terms, but any valid Aristotelean syllogism might be applied here. For example: some X-classes are Y-classes, no Y-classes are Z-classes, therefore some X-classes are not Z-classes. The explanation is simply that first order classes are effectively singular, whereas second order class subsume many such singulars.

- Modal class-of-classes logic.

To modalize the concept of classes of classes, we would have to appeal to a *collectional* proposition, of the form ‘*all* X can be Y. This, you may recall, signifies, not only that for each X there are some circumstances in which it is Y, but also that there is at least one set of circumstances in which all X *at the same time* are Y.

The modal definitions are: for any X and Y, if *all* X simultaneously can be Y, then “X” can be a class of Y; but if some X cannot be Y, or all X can be Y, but not *all* at once, then “X” cannot be a class of Y; and if all X must be Y, then “X” must be a class of Y; but if some X can not-be Y, then “X” can not-be a class of Y.

From these definitions, the entire modal logic of classes of classes is easily derived, with reference to the logic of ordinary modal categoricals and collectionals. Note that the defining propositions are all intended as extensional conditionals, but two of them are special in that they contain a collectional antecedent.

## 44. HIERARCHIES AND ORDERS.

### 1. First Order Hierarchies.

- Reconsider the definition: if all X are Y, then “X” is a class of Y (or member of “Y-classes”). The condition only implies that some Y are X.

In the case where all Y are X, they are coextensive and their relation is reciprocal; then “Y” is also a class of X (or member of “X-classes”), and “X” and “Y” are members of each other’s group of classes (which does not mean that they are members of each other, note well); such classes may be called equal. “X” is an equal-class of “Y”, signifies that X-ness and Y-ness are two ‘aspects’ of the same ultimate referents.

But in the case where some Y are not X, they cover a different extension and their relation is uneven. “X” is a member of Y-classes, but “Y” is not a member of X-classes. In such case we say that “X” is a ** subclass** of “Y” and that “Y” is an

**of “X”. Alternatively, we say that “X” is a lower class than “Y”, and “Y” is a higher class than “X”; or again, we speak of species and genus.**

*overclass*Note in passing, we often define a species by stating its genus (or one of its genera) together with a differentia; the latter is that character in the ultimate referents of a species, which distinguishes them from the ultimate referents of other species of same genus; the referents of all the species have in common the generic character.

Thus, we here introduce three new copulas, one of which is reversible, and two of which are relative to each other. These of course may be denied, making six altogether. These copulas differ from those previously defined, in that *the subject and predicate are both nominal*. Their function is to establish, or more precisely express, the *hierarchies* among classes. These various relations have their own logic, which can be analyzed in detail as we did for previous ones; I will not get into that here, however (the reader is invited to do the job).

- We call
*‘*, listing the subclasses of a class; If the subclasses of the latter are in turn listed, we call the process ‘subdivision’. We represent these relations on paper by means of (upside down)**division**’*‘trees’*, in which the highest class (or summum genus) is placed at the top, and successively divided into lower classes, like a downward branching.

Since all classes ultimately fall under the heading of “things”, there is only one big tree; however, we may speak of branch systems as trees, too. Note that we must have at least one general positive proposition ‘all X are Y’ and/or ‘all Y are X’, to be able to say that “X” and “Y” are in the same tree, or branch of a tree. Otherwise, they are neither equal, nor lower, nor higher classes, in relation to each other, but are in separate trees, or branches of a tree.

If we stand back and consider all possible classes, we see that, though they form a single tree, it is not flat. We have a multitude of hierarchies, all stemming down from “things”, in three dimensions. Hierarchies with entirely different referents, have no intersecting branch lines; hierarchies with some but not all referents in common, have some intersecting branch lines; hierarchies with all the same referents have the same branch lines.

The latter occurs when we have two sets of equal classes: they run along the same branch lines, but they signify different ‘principles of division’, different aspects of the same referents. Thus, for example, humans can be divided into those with male sex-organs and those with female sex-organs, or alternatively, into those without bosoms and those with bosoms: yet these two divisions overlap exactly (ignoring exceptions).

The ultimate referents of all these classes are at the very bottom, in a ‘horizontal’ plane (representing the space-time continuum). There is, as it were, a fanning-out below the lowest classes, to cover the ultimate referents. The relation of referents to lower or higher classes is the same (membership), but it is not the same as the relation of lower classes to higher classes (hierarchy), note well.

### 2. Second Order Hierarchies.

- With all this in mind, we see that what a class of classes does is refer us to all the subclasses of a class, plus the class itself. Thus, we should not confuse a class of classes with a first-order overclass, which stands higher up in the continuum of classes.

Whereas an upper first-order class is nominal, and bears certain hierarchical relations to others like it — a class of classes subsumes a class and its subclasses, without thereby becoming part of the same hierarchy, and thus constitutes a second order. Thus, ‘hierarchy’ and ‘order’ are two distinct ways we can stratify classes, and should not be confused.

The two orders of class, “X” and “X-classes”, for any X, are *not comparable*. The former refers to all things which are X as its members, the latter refers to all (mental) groupings of things which are X as its members. The one concerns numerous individual things, the other untold collectives (in every which way) of these very same things. Their world of reference is one and the same in size, so it is hard to say which is ‘bigger’. The number of referents each has is different, but (in most cases) incalculable.

- If we apply the definition of classes of classes to classes of classes, we obtain the following result:
*if all X-classes are Y-classes, then “X-classes” is a class of Y-classes, or a member of “classes of Y-classes”*. Here, now, we have classes of classes of classes. We can repeat the process, and obtain an infinity of levels upon levels. But it does not seem to mean anything more than “Y-classes”, to me at least.

The basis on which we form various classes about anything, is in the things they concern. For example, the different kinds of dogs differ in sizes, colors, and so on. Beyond that, the ‘containers’ as such are uniform, there is nothing to distinguish them from each other, other than the differences observed in their ‘contents’. Thus, to pile up level upon level, over and above classes of things and classes of classes of things, is a meaningless redundancy. We may reasonably conclude that *there is no order of classification above the two already considered*.

*We may, however, organize second order classes into hierarchies among themselves*, on the basis of statements like ‘all X-classes are Y-classes’. In that case, “X-classes” is an equal-class or subclass of “Y-classes”; and similarly in other cases, in accord with the above definitions of hierarchical relations.

Obviously, the hierarchies in the second order *reflect* those in the first order, on the basis of inferences like ‘if all X are Y, then all X-classes are Y-classes’. This just signifies that formal eductions are feasible from one system to the other.

However, the relationship of second-order to first-order classes is not hierarchical, but simply subsumptive. It is like the relation of first-order classes to their ultimate referents — namely, a relationship of inclusion as members; it is not like the relation of higher classes to lower classes of one and the same order.

For first-order classes, as we pointed out, the theater of reference is the space-time continuum, represented as a horizontal plane. For second-order classes, the theater of reference is the vertical dimension in which the tree of first-order classes evolves. However, the tree of second order classes need not be viewed as implying yet another dimension; we can view it as a distinct branch system within the same vertical dimension. The two orders of classes are layered in neat harmony with each other.

What distinguishes the second-order classes is that their members are first-order classes, but not the members of first order-classes. Thus, the lowest second-order classes ‘fan-out’ to first-order classes, but stop short of similarly relating to the members of first-order classes.

- In conclusion, it is important to keep in mind that the concept of ‘inclusion’ has many meanings. It can mean inclusion of things in a first order class, or inclusion of first order classes in a second order class, or inclusion of a subclass in an overclass. These relations are not one and the same, though we call them all ‘inclusion’.

In practice, we are not always clear about the exact distinctions between subsumptives and nominals; first order classes (or simply, classes) and second order classes (or, classes of classes); equal-classes, subclasses and overclasses. But we have to be careful, because as we saw, their logical properties vary considerably.

### 3. Extreme Cases.

It is important to understand that the concepts of classes, or classes of classes, are *purely relational*. Although we colloquially use these expressions as if they were terms, there is no such thing as a ‘class’ which is not a class *of* certain things, or a class of classes *of* certain things. The word ‘of’ is operative here, and should not be ignored. It follows that we cannot say that classes are classes, or that “classes” is a class, or make similar statements, except very loosely speaking; we can only strictly say that such and such are classes *of* so and so.

Our habit of speaking of ‘classes’ or ‘classes of classes’ without awareness of the subtext, causes us to think that ‘classes’ is a collection of all classes, supposedly including all ‘classes of classes’ together with all ‘classes not of classes’, and even ‘classes’ itself and also ‘non-classes’. Similar ambiguity is generated by ‘classes of classes’. It is all very confusing, and due to the above mentioned imprecision.

If we want to think at once of *the events *of class-relating-to-its-members, we of course may do so. This is a class of all the ‘lines’ joining classes to their members (whether first-order classes to ultimate referents, or second-order classes to first-order classes). The resulting concept is, however, what we call ‘subsumption’ (or ‘membership’, in the reverse direction). If we want to think of hierarchical relations, we again may do so; but the resulting concept is again a copula.

If we want to speak of *the terms* of such relations, say, all classes *indefinitely* — that is, without having to specify what they are classes of — we strictly should say ‘the classes of anything’, where ‘anything’ is understood like a variable ‘X’, standing for any kind of thing we choose to substitute in its place. Likewise, for classes of classes (of anything), or with reference to hierarchies.

The *largest* possible class, is the class *of* all things (including real and illusory things), or simply “things”; it is not just ‘classes’. From our definition, since every thing is a thing, every thing is a member of “things” or the class of things. The largest possible class of classes, is the class of all classes *of* things, or simply the “things-class”; it is not just ‘classes of classes’. This means, again by definition, since all things are things, “things” (or the class of things) is a class of things, or a member of the “things-class”, or the class of classes of things.

Since a nominal (the class of anything) is itself a thing, it follows that the classes “things” and “things-classes” are both things, and so members of “things”. Additionally, since for any X, “X” is an X-class, it follows that “things” is a member of “things-classes”. Thus, exceptionally, the classes “things” and “things-classes” seem to be equal to each other and, somehow, members of themselves. They are (it is) the summum genus of all hierarchies.

When this summum genus branches out into species like “dogs”, “machines”, and such, it is preferably called “things”; when we focus on its subsumption, not of the ultimate referents, but of the ideational instruments standing between it and them, we call it “things-classes”; alternatively, we may embrace both these categories.

## 45. ILLICIT PROCESSES IN CLASS LOGIC.

### 1. Self-membership.

With regard to the issue of self-membership, more needs to be said. Intuitively, to me at least, the suggestion that something can be both container and contained is hard to swallow.

Now, self-membership signifies that a nominal is a member of an exactly identical nominal. Thus, that all X are X, and therefore members of “X”, does not constitute self-membership; this is merely the definition of membership in a first order class by a non-class.

We saw that, empirically, at least with ordinary examples, “X” (or the class of X) is never itself an X, nor therefore a member of “X”. For example, “dogs” is not a dog, nor therefore a member of “dogs”.

I suggested that this could be generalized into an inductive postulate, if no examples to the contrary were forthcoming. My purpose here is to show that all apparent cases of self-membership are illusory, due only to imprecision of language.

That “X” is an X-class, and so a member of “X-classes”, is not self-membership in a literal sense, but is merely the definition of membership in a second order class by a first order class. For example, “dogs” is a class of dogs, or a member of “classes of dogs”, or member of the class of classes of dogs.

Nor does the formal inference, from all X are X, that all X-classes are X-classes, and so members of “X-classes” (or the class of classes of X), give us an instance of what we strictly mean by self-membership; it is just tautology. For example, all dog-classes are members of “classes of dogs”.

Claiming that an X-class may be X, and therefore a member of “X”, is simply a wider statement than claiming that “X” may be X, and not only seems equally silly and without empirical ground, but would in any case not formally constitute self-membership. For example, claiming “retrievers” is a dog.

As for saying of any X that it *is* “X”, rather than a member of “X”; or saying that it *is* some other X-class, and therefore a member of “X-classes” — such statements simply do not seem to be in accord with the intents of the definitions of classes and classes of classes, and in any case are not self-membership.

The question then arises, is “X-classes” itself a member of “X-classes”? The answer is, no, even here there is no self-membership. The impression that “X-classes” might be a member of itself is due to the fact that it concerns X, albeit less directly so than “X” does. For example, dog-classes refers to “retrievers”, “terriers”, and even “dogs”; and thus, though only indirectly, concerns dogs.

However, more formally, “X-classes” does not satisfy the defining condition for being a member of “X-classes”, which would be that ‘all X-classes are X’ — just as: “X” is a member of “X-classes”, is founded on ‘all X are X’. As will now be shown, this means that the above impression cannot be upheld as a formal generality, but only at best as a contingent truth in some cases; as a result, all its force and credibility disappears.

If we say that *for any and every* X, all X-classes are X, we imply that for all X, “X” (which is one X-class) is X; but we have already adduced empirical cases to the contrary; so the connection cannot be general and formal. Thus, we can only claim that perhaps *for some* X, all X-classes are X; but with regard to that eventuality, no examples have been adduced.

Since we have no solid grounds (specific examples) for assuming that “X” or “X-classes” is ever a member of itself, and the suggestion is fraught with difficulty; and we only found credible examples where they were not members of themselves — we are justified in presuming, by generalization, that: *no class of anything, or class of classes of anything, is ever a member of itself.*

I can only think of one possible exception to this postulate, namely: “things” (or “things-classes”). But I suspect that, in this case, rather than saying that the class is a member of itself, we should regard the definition of membership as failing. That is, though this summum genus *is* a thing, it is not ‘a member of’ anything.

### 2. The Russell Paradox.

The Russell Paradox is modern example of double paradox, discovered by British logician Bertrand Russell.

He asked whether the class of “all classes which are not members of themselves” is or not a member of itself. If “classes not members of themselves” is not a member of “classes not members of themselves”, then it is indeed a member of “classes not members of themselves”; and if “classes not members of themselves” is a member of “classes not members of themselves”, then it is also a member of “classes which are members of themselves”. Thus, we face a contradiction either way.

In contrast, the class of “all classes which are members of themselves” does not yield a similar difficulty. If “self-member classes” is not a member of “self-member classes”, then it is a member of “classes not members of themselves”; but if “self-member classes” is a member of “self-member classes”, no antinomy follows. Hence, here we have a single paradox coupled with a consistent position, and a definite conclusion can be drawn: “self-member classes” is a member of itself.

Now, every absurdity which arises in knowledge should be regarded as an opportunity for advancement, a spur to research and discovery of some previously unknown detail. So what is the hidden lesson of this puzzle?

As I will show, the Russell Paradox proceeds essentially from an equivocation; it is more akin to the sophism of the Barber paradox, than to that of the Liar paradox. For *whether self-membership is possible or not, is not the issue*. Russell believed that some classes, like “classes” include themselves; though I disagree with that, my disagreement is not my basis for dissolving the Russell paradox. For it is not the concept of self-membership which results in a two-way inconsistency. It is the concept of non-self-membership which does so; and everyone agrees that at least some (if not all, as I believe) classes do not include themselves: for instance, “dogs” is not a dog.

What has stumped so many logicians with regard to the Russell paradox, was the assumption that we can form concepts at will, if we but formulate a verbal definition. But this viewpoint is without justification. The words must have a demonstrable meaning; in most cases, they do; but in some cases, they are isolated or pieced together without attention to their intrinsic structural requirements. We cannot, for instance, use the word ‘greater’ without specifying ‘than what?’; many words are attached, and cannot be reshuffled at random. The fact that we commonly, in everyday discourse, use words loosely, to avoid boring constructions, does not give logicians the same license.

### 3. Impermutability.

The solution to the problem is so easy, it is funny, though I must admit I was quite perplexed for a while. It is simply that: *propositions of the form ‘X (or “X”) is (or is not) a member of “Y” (or “Y-classes”)’ cannot be permuted**.* *The process of permutation is applicable to some forms, but not to all forms.*

- In some cases, where we are dealing with relatively simple relations, the relation can be attached to the original predicate, to make up a new predicate, in an ‘S is P’ form of proposition, in which ‘is’ has a strictly classificatory meaning. Thus, ‘X is-not Y’ is permutable to ‘X is nonY’, or ‘X is something which is not Y’; ‘X has (or lacks) Y-ness’ is permutable to ‘X is a Y-ness having (or lacking) thing’; ‘X does (or does not do) Y’ is permutable to ‘X is a Y-doing (or Y-not-doing) thing’. In such cases, no error arises from this artifice.

But in other cases, permutation is not feasible, because it falsifies the logical properties of the relation involved. We saw clear and indubitable examples of this in the study of modalities.

For instance, the form ‘X can be Y’ is not permutable to ‘X is something capable of being Y’, for the reason that we thereby change the subject of the relation ‘can be’ from ‘X’ to ‘something’, and also we change a potential ‘can be’ into an actual ‘is (capable of being)’. As a result of such verbal shenanigans, formal errors arise. Thus, ‘X is Y, and all Y are capable of being Z’ is thought to conclude ‘X is capable of being Z’, whereas in fact the premises are quite compatible with the contradictory ‘X cannot be Z’, since ‘X can become Z’ is a valid alternative conclusion, as we saw earlier.

It can likewise be demonstrated that ‘X can become Y’ is not permutable to ‘X is something which can become Y’, because then the syllogism ‘X is Y, all Y are things which can become Z, therefore X is something which can become Z’ would seem valid, whereas its correct conclusion is ‘X can be or become Z’, as earlier seen. Thus, modality is one kind of relational factor which is not permutable. Even though we commonly say ‘X is capable or incapable of Y’, that ‘is’ does not have the same logical properties as the ‘is’ in a normal ‘S is P’ proposition.

- The Russell Paradox reveals to us the valuable information that the copula ‘is a member (or not a member) of’ is likewise not open to permutation to ‘is something which is a member (or not a member) of’.

The original ‘is’ is an integral part of the relation, and does not have the same meaning as a solitary ‘is’. The relation ‘is or is not a member of’ is an indivisible whole; you cannot just cut it off where you please. The fact that it consists of a string of words, instead of a single word, is an accident of language; just because you can separate its verbal constituents does not mean that the objective relation itself can similarly be split up.

Permutation is a process we use, when possible, to bypass the difficulties inherent in a special relation; in this case, however, we cannot get around the peculiar demands of the membership relations by this artifice. The Russell paradox locks us into the inferential processes previously outlined; it tells us that there are no other legitimate ones, it forbids conceptual short-cuts.

The impermutability of ‘is (or is not) a member of’ signifies that you cannot form a class of ‘self-member classes’ or a class of ‘non-self-member classes’. These are not terms, they are relations. Thus, the Russell paradox is fully dissolved by denying the conceptual legitimacy of its terms. There is no way for us to form such concepts; they involve an illicit permutation. The connections between the terms are therefore purely verbal and illusory.

The definition of membership is ‘if something *is* X, then it is a member of “X”‘ or ‘if all X *are* Y, then “X” is a member of “Y-classes”‘. The Russell paradox makes us aware that the ‘is’ in the condition has to be a normal, solitary ‘is’, it cannot be an ‘is’ isolated from a string of words like ‘is (or is not) a member of’. If this antecedent condition is not met, the consequent rule cannot be applied. In our case, the condition *is not met*, and so the rule does not apply.

- Here, then, is how the Russell paradox formally arises, step by step. We will signal permutations by brackets like this: {}.

Let “X” signify any class, of any order:

(i) If “X” is a member of “X”, then “X” is {a member of itself}. Call the enclosed portion Y; then “X” is Y, defines self-membership.

(ii) If “X” is not a member of “X”, then “X” is {not a member of itself}. Call the enclosed portion nonY; then “X” is nonY, defines non-self-membership.

Next, apply the general definitions of membership and non-membership to the concepts of Y and nonY we just formed:

(iii) whatever is not Y, is nonY, and so is a member of “nonY”.

(iv) whatever is Y, is not a member of “nonY”, since only things which are nonY, are members of “nonY”.

Now, the double paradox:

(v) if “nonY” is not a member of “nonY”:

— then, by putting “nonY” in place of “X” in (ii), “nonY” is {not a member of itself}, which means it is nonY;

— then, by (iii), “nonY” is a member of “nonY”, which contradicts the starting premise.

(vi) if “nonY” is a member of “nonY”:

— then, by putting “nonY” in place of “X” in (i), “nonY” is {a member of itself}, which means it is Y;

— then, by (iv), “nonY” is not a member of “nonY”, which contradicts the starting premise.

Of all the processes used in developing these arguments, only one is of uncertain (unestablished) validity: namely, permutation of ‘is a member of itself’ to ‘is {a member of itself}’, or of ‘is not a member of itself’ to ‘is {not a member of itself}’. Since all the other processes are valid, the source of antinomy has to be such permutation. Q.E.D.

- The existence of impermutable relations suggests that we cannot regard all relations as somehow residing
*within*the things related, as an indwelling component of their identities. We are pushed to regard some relations, like modality or membership, as bonds standing outside the terms, which are not actual parts of their being.

Thus, for example, that ‘this S can be P’ does not have an ontological implication that there is some actual ‘mark’ programmed in the actual identity of this S, which records that it ‘can be P’. For this reason, the verbal clause {can be P} cannot be presumed to be a unit; there is nothing corresponding to it in the actuality of this S, the potential relation does not cast an actual shadow.

Thus, there must be a reality to ‘potential existence’, outside of ‘actual existence’. When we say that ‘this S can be P’, we consider this potentiality to be P as somehow part of the ‘nature’ of this S. But the S we mean, itself stretches in time, past, ‘present’, and future; it also has ‘potential’ existence, and is wider than the actual S.

The same can be argued for can not, or must or cannot. Thus, natural (and likewise temporal) modalities refer to different degrees, or levels, of existence.

Similarly, the impermutability of membership relations, signifies that they stand external to their terms, leaving no mark on them, even when actual.

It seems like a reasonable position, because if every relation of something to everything else, implied some corresponding trait inside that thing, then each thing in the world would have to contain an infinite number of messages, one message for its relations to each other thing. Much simpler, is to regard relations (at least, those which are impermutable) as having a separate existence from their terms, as other contents of the universe.

# PART V(b). ADDUCTION.

## 46. ADDUCTION.

### 1. Logical Probability.

Induction, in the widest sense, is concerned with finding the probable implications of theses. Deduction may then be viewed as the ideal or limiting case of induction, when the probability is maximal or 100%, so that the conclusion is necessary. In a narrower sense, induction concerns all probabilities below necessity, when a deductive inference is not feasible.

- All this refers to logical probability. A thesis is logically possible if there is some chance, any chance, of it being found true, rather than false. ‘Probability’ signifies more defined possibility, to degrees of possibility, as it were.

Thus, we understand that low probability means fewer chances of truth as against falsehood; high probability signifies greater chances of such outcome; even probability implies that the chances are equal. High and low probability are also called probability (in a narrower sense) and improbability (with the im- prefix suggesting ‘not very’), respectively. Necessity and impossibility are then the utter extremes of probability and improbability, respectively.

There are levels of possibility, delimited by the context, the logical environment. This can be said even with regard even to formal propositions. Taken by itself, any proposition of (say) the form ‘S is P’, is possible. But, for instance, in the given context ‘S is M and M is P’, that proposition becomes (relatively) necessary: its level of possibility has been formally raised. Alternatively, in the given context ‘S is M and M is not P’, that proposition becomes (relatively) impossible: its level of possibility has been formally lowered.

The same applies with specific contents. At first sight, every statement about anything seems logically ‘possible’. This just means that the form is acceptable, there exist other contents for it of known value — a well-guarded stamp of approval.

As we analyze it further, however, we find the statement tending either toward truth or toward falsehood. We express this judgement by introducing a modality of probability into the statement. We place the statement in a logical continuum from nil to total credibility.

In any case, we know from experience that such probabilities are rarely permanent. They may increase or decrease; they may first rise, then decline, then rise again. They vary with context changes. Keeping track of these probabilities is the function of induction. For example, when a contradiction arises between two or more propositions, they are all put in doubt somewhat, and their negations are all raised in our esteem to some extent, until we can pinpoint the fault more precisely.

- In the chapter on credibility, we described degrees of credibility as impressions seemingly immediately apparent in any phenomenon. Thus, credibility is a point-blank, intuitive notion. In the chapter on logical modality, on the other hand, we showed that the definitions of unspecific plural modalities coerced us into the definition of logical probabilities with reference to a majority or minority of contexts. Thus, knowledge of logical probability presupposes a certain effort and sophistication of thought, a greater awareness of context.

Here, we must inquire into the relation between credibility and logical probability.

Every proposition has, *ab-initio*, *some* credibility, if only by virtue of our being able to formulate it with any meaning. This intuitive credibility is undifferentiated, in the sense that, so long as it is unchallenged, it is virtually, effectively, total. But at the same time, this credibility is not very informative or decisive, because the opposite thesis may have been ignored or may be found to have equal credibility.

As we begin to consider the proposition in its immediate context, and we find contradictions (or even sense some unspecified cause for doubt), the credibility becomes more comparative, and it is certified or annulled, or seen as more or less than extreme one way or the other, or as problematic (equally balanced).

As our perspective is broadened, and we project changes in context, the problematic credibilities become more qualified — that is, they are quantified by some specific logical probability, so that they shift more decidedly in either direction. Thus, problemacy (median credibility) may be viewed as the very minimum, the beginning, of probability.

In this way, all the plural logical modalities may be viewed as ‘filtering down’ to the single-context level of truth or falsehood. This transmission of modality, from the high level of many-contexts to the low level of the present context, may be immediately apparent (as in the case of necessities and impossibilities), or may gradually develop over time (as with all contingent probabilities).

As probabilities vary, through new inputs of raw data into the actual context, so that more alternative contexts are imaginable, and through closer scrutiny of available data — the credibilities under their influence also and proportionally change.

Logical probability, as formally defined, is impossible to know with finality. The exception is in the extreme cases of logical necessity or impossibility, which can be known even without access to all conceivable contexts, through the one-time discovery of self-evidence or self-contradiction (in paradoxical propositions); these modalities are permanent.

But in all cases of logical probability based on contingency, there is no way to make a sure statement of the form ‘In most contexts,….’ All we can refer to are: most of the contexts *considered so far*; these may in reality be a minority of all possible contexts, for all we know. Such modal statements are therefore not static, never entirely final.

We have shifted the concept of logical probability from its rigid formal definition as ‘true in most contexts’, to a more practical version: ‘true in most *known* contexts’. It thus is no longer implied to be static; but it is now flexible, and suggests comparison of credibilities with a reasonable degree of purpose.

Thus, the concepts of (comparative) credibility and logical probability ultimately blur, and can to some extent be used interchangeably. However, if we understand logical probability in its strictest sense, as *based on and implying* logical possibility, then it should not be confused with credibility, which is even applicable to logically impossible propositions (until their self-contradiction is discovered). Here, I use ‘probability’ in an indeterminate sense, so as to avoid the issue.

The main purpose of induction is to lead us to facts, to hopefully true specific contents. How we know their logical probabilities is not a separate or additional goal for inductive research; it is one and the same issue with that of knowing their truths. In the process of pursuit of facts, by evaluating our current distance from the establishment of truth, *we are incidentally also finding their logical probabilities*.

Ultimately, we would like to construct a clear, step-by-step, *model* of human knowledge, showing precisely how each proposition in it is arrived at; but in the meantime, the processes involved can be broadly defined. How exactly do we get to know these logical gradations? They are not arbitrary, not expressions of subjective preference, not intuitive guesses; there is a system to such evaluations.

### 2. Providing Evidence.

The investigation of this problem in general terms, that is, without reference to specific forms, may be called *‘adduction’*. Adduction provides us with the rules of evidence and counterevidence, which allow us to weight the varying probabilities of theses.

The more evidence we adduce for our proposed thesis, the more it is *confirmed* (strengthened); the more evidence we adduce to a contrary thesis, the more is ours *undermined* (weakened). These valuations should not be confused with proof and refutation, which refer to the ideal, extreme powers of evidence.

Adduction is performed by means of the logical relations described by hypothetical and disjunctive propositions. These, we saw, are normally based on the separate logical possibility of two theses, and inform us about the logical modalities of their conjunctions, together or with each other’s antitheses. They establish connections of varying degree, direction, and polarity.

Now, ‘If P, then Q’ represents necessary connection, the highest level; it could be stated as ‘if P, necessarily Q’. Accordingly, ‘if P, then nonQ’, incompatibility, could be stated ‘if P, impossibly Q’. The contradictories of these would be ‘if P, possibly Q’ (= ‘if P, not-then nonQ’) and ‘if P, possibly not Q’ (= ‘if P, not-then Q’). We can, following this pattern, think in terms of probabilities of connection.

- Adductive argument evolves out of apodosis. It most typically takes the forms:

If P, then Q | If P, then Q |

and Q | but not P |

hence, probably P. | hence, probably not Q. |

These conclusions, so far, do not express the precise degree of probability; they do indicate that the possible result has *increased in probability*. The possibility of the result is already implicit in the major premise to some extent. A deductive, necessary, conclusion would not be justified. But we are one step ahead, in that it is conceivable that the minor premise is true *because* the proposed conclusion was true.

We argue backwards, from the consequent to the antecedent, or from the denial of the antecedent to the denial of the consequent. As apodosis, this is of course invalid; but here we view the minor premise as an *index to*, rather than proof of, the conclusion.

The more hypotheses suggest a conclusion, the more probably will it turn out to be true. The less hypotheses suggest a conclusion, the more probably will it turn out to be false. Thus, *‘ evidence’* may be defined as whatever increases the logical probability of a thesis by any amount, and

*‘*refers to sources of decrease.

**counterevidence**’Through adduction, we mentally shift from incipient credibility and problemacy, to a more pondered logical probability.

Note that the first mood, the affirmative one, is strictly more correct than the second, negative, mood. For, in the negative case, we presuppose the major premise not to be complemented by ‘if nonP, then Q’, even though the latter is a formally conceivable adjunct. That is, we are presuming that ‘nonQ’ is logically possible, without prior justification, since this is not always part of the basis of the major premise. Whereas, in the positive case, if ‘if nonP, then Q’ were also given, the additional conclusion ‘probably not P’ would balance but not strictly contradict ‘probably P’, and also allow Q to be logically necessary.

It follows that the conclusion of the negative mood is more precisely, ‘if nonQ is at all possible, then it is now more probable’. But since, as earlier pointed out, every proposition is at first encounter logically possible, this is not a very significant distinction. The issue of basis is more serious for natural, temporal or extensional conditionals than for logical conditionals.

We can simply say that if ‘nonQ’ turns out to be logically impossible for other reasons, then of course the initial possibility is thenceforth annulled. Such an eventuality is not excluded by the negative adductive argument, just as the positive version allows for the eventual denial of P, anyway.

Note then that the loose sense of logical probability here intended does not imply that ‘P is logically possible’ (in the first mood) or that ‘nonQ is logically possible’ (in the second mood), unless these possibilities were part of the tacit basis of the major premise. Logical possibility must still be strictly understood as signifying an established necessity or contingency.

- Other moods of adduction follow by changing the polarities of theses. These represent other valuable approaches to provision of evidence or counterevidence, confirmation or undermining.

If P, then nonQ | If P, then nonQ |

and not Q | but not P |

hence, probably P. | hence, probably Q. |

If nonP, then Q | If nonP, then Q |

and Q | but P |

hence, probably not P. | hence, probably not Q. |

If nonP, then nonQ | If nonP, then nonQ |

and not Q | but P |

hence, probably not P. | hence, probably Q. |

Note that if the major premise is contraposed, the conclusion remains the same. This shows that the listed moods constitute a consistent system.

We can also form disjunctive adductive arguments, like the following, with any number of theses:

P or else Q | P and/or Q |

but not P | but P |

hence, probably Q | hence, probably not Q |

- It is clear that if the major and/or minor premise in all these arguments were probabilistic, instead of fully necessary or factual, some probability would still be transmitted down to the conclusion, albeit a proportionately more tenuous one.

This principle of ‘transmissibility’ of credibility, let us call it, is very important to logic, because it means that, although deductive logic was designed with absolutely true premises in mind, its results are still applicable to premises of only relative truth. Thus, deductive processes also have some inductive utility.

We previously made a clear distinction between the ‘uppercase’ forms of hypothetical, like ‘if P, then nonQ’, which involve a logically necessary connection, with the lowercase forms, like ‘if P, not-then Q’, which merely establish a compatibility. This distinction is especially important in deductive argument, such as apodosis.

We can conceive of less than necessary major premises, having forms like ‘if P, possibly or probably Q’. Some probability is still transmitted down to the conclusion, though of course again much more tentatively and insignificantly. We can regard thus arguments like the following as also adductive; in fact, they are the most comprehensive formats of adductive argument.

If P, probably Q, | If P, probably Q, |

and probably P, | and probably Q, |

hence, probably Q. | hence, probably P. |

If P, probably Q, | If P, probably Q, |

and probably not P, | and probably not Q, |

hence, probably not Q. | hence, probably not P. |

In such argument, the probabilities involved may have any degree. Also, the premises may have very different probabilities; and the probability of the conclusion depends on the overlap, if any, of the conditions for realization of the premises, so that it is generally far inferior. It is normally very difficult to quantify such probabilities precisely; but, when we can estimate the degrees of the premises, we can accordingly calculate the degree of the conclusion (which may be zero, if there is no overlap).

We could thus expand our definitions of apodosis and adduction, so that they are equivocal. In that case apodosis and adduction (in the narrow senses we adopted) would respectively be: forward and backward apodosis (in the larger sense), or necessary/deductive and merely-probable/inductive adduction (in the larger sense). This is mentioned only to show the continuity of the two processes.

Note that when we formulate hypothetical propositions, we often order the theses according to their probabilities. ‘If P, then Q’ may intend to implicitly suggest, that P is so far more probable than Q, and may be used deductively to improve the probability of Q; or that Q is so far more probable, and may be used to inductively to raise the probability of P. Tacitly, this signifies an argument with a necessary major premise, and a probabilistic minor premise and conclusion.

Similarly, by the way, for disjunctive argument. Premises and conclusion may have any degrees of logical probability. Also, the minor premise may be implicit in the major, by virtue of our ordering the alternatives, from the most likely (mentioned first to attract our attention) to the least (relegated to the periphery of our attention); or from the least likely (because easiest to eliminate) to the most (the leftover alternative, when we reach the end of the sentence).

### 3. Weighting Evidence.

We have thus far described adductive argument, but have not yet validated it. We have to explain why the probable conclusion is justified, and clarify by how much the logical probability is increased. The answer to this question is found in the hidden structure of such argument, the pattern of thought which underlies it.

- Let us suppose that P1, P2,… Pn are the full list of all the conceivable theses, each of which is separately capable of implying Q, so that the denial of all of them at once results in denial of Q. This means:

If P1, then Q; and if P2, then Q; etc.

or, more succinctly,

If P1 or P2 or… Pn, then Q.

And, since the list is exhaustive,

If not-P1 and not-P2 …and not-Pn, then notQ.

(i) In that ideal situation, we can say that if Q is found true, then each of P1, P2,… Pn has prima facie an equal chance of having anteceded that truth. We know at least one of them must be true (since otherwise Q would be false), but not precisely which. Each carries an nth part of the total probability which this necessity embraces. Thus, the degree of probability is in principle knowable, and the process justifiable.

If one of the alternative antecedents is thereafter found false, the number of alternatives is decreased, and so the probability of each of the remainder is proportionately increased. Where only one alternative remains it becomes maximally probable, that is, necessary; and the conclusion is deductive rather than adductive or inductive (in the narrow sense).

In practice, we do not always know or consider all the alternatives; even when we think we are aware of them all, it may only be an assumption, a generalization. Still, the principle remains, even if the degree of probability we assign to the conclusion turns out to be inexact. This is because we are here dealing with logical probability, which is intrinsically tentative and open to change. That is just the function and raison-d’être of logical probability, to monitor the current status of propositions in an evolving body of knowledge.

(ii) If, not yet knowing whether Q is true or false, we find one of the alternatives, say P1, false, we can say that we are one step closer to the eventuality that all are false, from which the falsehood of Q would follow. In that case, the probability of Q being false has increased by an increment of 1/nth.

If thereafter say P2 is also found false, the chances of Q being false are further increased. When all the conditions of that event are fulfilled, the probability becomes maximal — a necessity.

- In formal terms, what the above means is that ‘If P, necessarily Q’ is convertible to ‘If Q,
*(a bit more)*probably P’. Similarly, ‘If P, necessarily Q’ is invertible to ‘If not P,*(a bit more)*probably not Q’. Even if we do not know what, and how many, are the other shareholders of the overall probability, these inferences retain their value.

In aetiological terms, we thus have two sources of probability increase. A thesis (here, P1 for instance) may be rendered more probable by the truth of another (viz., here, Q), of which it is an alternative contingent cause. Or a thesis (here, nonQ) may be rendered more probable by the truth of another (viz., here, not-P1 for instance), which is a component of a necessary cause of it.

Thus, more broadly, probability is transmitted across the logical relationship signified by hypotheticals: in both directions, from antecedents to consequents and vice versa, and to varying degrees, reflecting the intensity of the link.

Each such probability change is relative: it applies within that limited environment which we projected. In practice, the degree of probability we assign to a thesis is a complex result of innumerable such incremental changes. Needless to say, when a thesis is strengthened, its contraries are proportionately weakened; and vice versa.

A thesis may be increasingly confirmed for a variety of reasons, and at the same time increasingly undermined for a variety of other reasons. What matters is its resultant probability, its overall rating, the sum and average of all the affirming and denying forces impinging upon it, at the present stage of knowledge development.

If follows that, though the alternative theses are, to begin with, of equal weight, they may, in a broader context, be found of unequal weight. In that case, we select the relatively most weighty, the logically most probable, as our preferred thesis at any stage of the proceedings.

All the above can be repeated with respect to disjunctions. Consider two or more theses, each with some degree of credibility from other sources. If they are found to be contrary, their credibilities are all proportionately lowered, since we know they cannot all be true. If they are found to be subcontrary, their credibilities are all proportionately raised, since we know they cannot all be false. However, in the case of exact contradictories, their independent credibilities are unaffected, since their mutual exclusion and exhaustiveness offset each other.

- Lastly, note that we have to clearly discriminate between: exhausting the known possibilities, on the one hand, and open-mindedness to the eventual possibility that new alternatives be found one day, on the other hand.

At any given stage in the development of knowledge we have to bow to all the apparent finalities; this does not prevent us from accepting the principle that some correction might later be called upon. On the other hand, that attitude of receptiveness to change should not be allowed to belittle our trust in acquired certainties.

When all but one of the known theories concerning some phenomena have been eliminated, or one theory is shown to be their only conceivable explanation, we must accept our conclusion as final and unassailable, provided no inconsistency or specific cause for doubt remains. The truth that some such certainties have in the past been overturned, does not logically imply that this particular certainty will ever be overturned.

There is a formal difference between the status of logical possibility within a context, and the general admission that context does change, which stands outside of any context. They are not identical in power: the former affects contextual reasoning, the latter plays no active part in deliberations, being only an open-ended philosophical truth without specific applicability.

We ordinarily think assertorically, in terms of statements like ‘if P, then Q’, meaning ‘if P is established, then Q may be claimed to be known’. But sometimes we remain dubious, and say ‘if perhaps P, then perhaps Q’. Some people reason in this manner more often than others, hanging on to uncertainties so insistently that they inhibit the forward motion of their knowledge.

But such reasoning, which may be called ‘problematic logic’, is essentially no different from assertoric logic. Its inferences are exactly parallel, the only difference is the explicit emphasis it puts on the probabilities of the theses.

Perhaps the legitimate context for such statements would be whenever we inquire into eventual developments of knowledge. Right now, say, P is to all appearances true; but there is always an off-chance that it might turn out not to be true, after all; in that case, we ask, *what would happen if* P was not true. We look ahead, even though we are without strict justification, in order *to be prepared* for eventual alternatives to ‘established fact’.

### 4. Other Types of Probability.

As we saw in the discussion of *de-re *conditioning, adduction is also feasible using natural, temporal or extensional conditionals, but it must be stressed that the emergent probability is essentially in logical modality. We might call it para-logical probability, meaning not *purely* logical, if we wish to underline the faint difference, which relates to source of judgment.

- A categorical proposition always has adductive implications. ‘Most (or Few) S are P’ is taken to imply ‘This S is probably (or improbably) P’; that is, for any random S, the logical probability is high (or low) that it will be P, in proportion to the quantity. We consider the likelihood that the given case of S happens to be one of those which are P.

Likewise, ‘This S is P in most (or few) circumstances’ implies ‘This S is probably (or improbably) P’ that, for any randomly chosen circumstance, there is a logical probability that this S will be P in it, commensurate with the number of natural circumstances favoring such event. We consider the likelihood that the given circumstance surrounding this S happens to be one of those in which this S is P. Similarly with temporal modality.

When two or more of the extensional, and natural or temporal, modalities are involved in a proposition, the logical effect is compounded. The logical probability is increased (or decreased) to some extent by each of the *de-re *modalities, and the resultant is whatever it happens to be.

- Such transmission of logical probability, from a plural
*de-re*proposition down*to a single-unit case*for the type of modality concerned, on the ground of a majority or minority of instances, circumstances or times — is also to be found with conditionals. The following are some typifying examples:

- In
**extensional**adduction:

Any S which is P, is Q,

and this S is Q — therefore, this S is probably P;

or: and this S is not P — so, this S is probably not Q.

- In
**natural**adduction:

When this S is P, it must be Q,

and this S is Q — therefore, it is probably P;

or: and this S is not P — so, it is probably not Q.

- In
**temporal**adduction:

When this S is P, it is always Q,

and this S is Q — therefore, it is probably P;

or: and this S is not P — so, it is probably not Q.

These concepts can be further broadened by reference to majoritive or minoritive conditionals, in arguments like the *de-re *adductions here shown, and likewise for corresponding apodoses. Some logical probability is still transmitted down from premises to conclusion.

Thus, if the major premises in such arguments had been the extensional ‘Most (or few) S which are P, are Q’, or the natural ‘When this S is P, it is in most (or few) circumstances Q’, or the equivalent temporal conditional — the conclusion would still have some degree of logical probability, proportionately to the numbers of instances, circumstances or times involved. Likewise, in cases of compound modal type.

If the minor premises were respectively of the form ‘Most S are Q’ (or ‘Most S aren’t P’), or ‘This S is in most circumstances Q’ (or ‘This S is in most circumstances not P’), or the equivalent temporal categorical — a probable conclusion can likewise be drawn. Note, however, that if the minor premise is of low *de-re *probability, it does not follow that the conclusion is likewise of low probability; all we can say is that the conclusion has very slightly increased in probability. Likewise, in cases of compound modal type.

A probabilistic major premise, of any modal type or combination of modal types, together with a probabilistic minor premise, of any modal type or combination of modal types, yield a conclusion of some, though much diminished, degree of logical probability.

More broadly still, such conditional major premises, and indeed the minor premises, may have varying degrees of purely logical probabilities as propositions in a knowledge context, quite apart from the inherent ‘para-logical’ (*de-re*) probabilities just discussed. In that case, the resultant logical probability is still further diminished.

We can similarly adduce evidence through *de-re *disjunctive adduction, in each or any combination of these types of modality.

## 47. THEORY FORMATION.

### 1. Theorizing.

Every theory involves an act of imagination. We go beyond the given data, and try to mentally construct a new image of reality capable of embracing the empirical facts. The more nimble our imagination, the greater our chances of reaching truth. Think how many people were stumped by the constancy of the velocity of light discovered by the Michelson-Morley experiment, until an Einstein was able to conceive a solution!

Without creativity our understanding would be very limited. We need it both to construct hypotheses, and to uncover their implications. Neither of these achievements is automatic. Conceiving alternatives and prevision both involve work of imagination.

In practice, no theory is devoid of hidden assumptions, besides its stated postulates. We may try to be as explicit as possible, but often later discover new dependencies. Thus, with Newton’s assumption of Euclidean geometry, which was much later discarded in the General Relativity theory.

Thus, our theorizing is always to some extent limited by our ability to make mental projections, and the depth and breadth of our conceptual insight.

These faculties of course depend very much on the mind being fed by new empirical input. Creativity depends on the ideas provided us by new experience, and revision of fundamentals depends on the stimulus of discovered difficulties.

Each individual has his own limits. People often remain attached to preconceptions, and are unable or refuse to consider alternatives. This can be a weakness or vice, but it is also a normal part of the way the mind works.

We have to hold something steady while considering the impact of new perspectives. We cannot re-invent the wheel all the time, without justification. We review our presuppositions, only when the need arises, when some empirical problem presents itself.

This does not exclude ‘art for art’s sake’. The pursuit of theoretical improvements is always permissible. But it is anyway serial. We are mentally unable to change all our knowledge at once, but are forced to proceed in an orderly, structured manner, gradually focusing on this or that proposed change while the rest is taken for granted.

Logical and mathematical skills also count for much in the development of theories. Many a wild speculation is built on unsound reasoning. These skills include, among many others: clarifying inter-relationships, finding analogies and implications, distinctions and contradictions, ordering information.

A good grasp of the methodology of adduction is very important. It opens minds to the ever-present possibility of alternative explanations and further testing. Adduction is essentially a process of trial and error.

The tentative, and often transient, nature of theories, as well as their ability to make impressive predictions, has been exemplified in some stunning scientific revolutions in the past few centuries. Even seemingly unshakable theories have been known to fall, and some of the discoveries occasioned by the new perspectives would have seemed unthinkable previously.

There is much to learn by observing the ‘life’ of theories, their historical courses, the ways they have augmented or displaced, complemented or contested, each other, their dynamics.

### 2. Structure of Theories.

Any one general proposition can of course be viewed as in itself a theory, and the processes of generalization and particularization are samples of adduction. The relation of a general proposition to particular observations, is logically one of antecedent and consequent, though the chronological order may be the reverse.

However, we normally use the term ‘theory’ in larger, more complex, situations. We think of a rational system for understanding some subject-matter. The sciences of course consist of theories, which attempt to explain the empirical phenomena facing them. But we also build small personal theories about events in our lives of concern only to ourselves.

Let us examine the structure of theories. A ** theory** (say,

**T**) consists of a number of conceptual and/or mathematical propositions. Among these propositions, some cannot be derived from the others: they may be called primary; the others, being of a derivative nature may be called secondary. The derivation, of course, is supposed to be logically or mathematically flawless.

Among the primary propositions, some are distinctive to that theory: they are called its ** postulates** (label these

**p1**,

**p2**,

**p3**, etc.). Postulates should be as limited in number, as simple in conception and broad-based, as we can make them. Though postulates may be particular (as for instance in a theory concerning historical events), the postulates of sciences are normally general propositions. These are usually obtained by generalization from directly observable particulars, but not always (consider, for instance, the idea of curved space).

If a primary proposition is not distinctive to that theory, but found in all other theories of the subject under investigation, then it is not essentially part of that specific theory, but stands outside it to some extent. Such external primaries may be transcendent axioms, or they may be borrowed from some adjacent or wider field of investigation, taken for granted so long as that other theory holds.

The secondary propositions are called the theory’s ** predictions** (label these

**q1**,

**q2**,

**q3**, etc.), even if not distinctive to that theory.

Some predictions are testable, open to empirical observation, perhaps through experiment; some predictions are intrinsically difficult to test. To the extent that a theory offers untestable predictions, it tends to be viewed as speculative. Among the testable predictions, some are normally already tested: they provided the raw data around which the theory was built; others may be novel items, which anticipate yet unobserved phenomena, providing us with opportunity for further testing.

Predictions are derived from the postulates by a process of production, mediated by the relatively external primaries. We regard the external primaries as categorical, as far as our theory is concerned, so that they may remain tacit, though they underlie the connections between our postulates and predictions.

Thus, postulates are hypothetically linked to predictions, in the way of antecedent to consequent. The antecedent need not include the external primaries, since the latter are considered as affirmed anyway, and were used to establish the connection. For example, Newton’s laws of motion were the postulates distinguishing his mechanics, while his epistemological, ontological, algebraic and geometrical assumptions lay outside the scope of his theory as such.

Theories often draw on findings in other domains outside their direct concern, and may have powerful repercussions in other domains. Thus, Newton had to develop calculus for his mechanics; this mathematical tool might well have been researched independently, as indeed it was by Liebnitz, but it was also stimulated or given added meaning when its value to physics became apparent.

A theory, then, may be described as follows, formally:

T = If p1 and p2 and… , then q1 and q2 and…

Note that this overall relation may in some cases be supplemented by narrower ones. It may be that all the postulates are required to make all the predictions; or it may be that some of the postulates are alone sufficient to make some of the predictions.

### 3. Criteria.

Theories serve both to explain (unify, systematize, interpret) known data, and to foresee the yet unknown, and thus guide us in further research, and in action. The criteria for upholding a theory are many and complex; they fall under three headings:

- Criteria of
. A theory may be upheld as possibly true, so long as it is meaningful, internally consistent, applicable to (i.e. indeed implying) the phenomena under investigation, and consistent with all other observation to date.*relevance*

This possibility of truth signifies no more than that the theory is conceivable, and has some initial degree of probability. This may be called relevance.

- Criteria of
. But the work of induction is not complete until the theory has been compared to others, which may be equally thinkable and defensible in the given context. Induction depends on critically pitting theories against each other.*competitiveness*

Two or more theories may each fulfill the conditions of relevance, and yet be incompatible with each other. They might converge in some respects, having some postulates and/or predictions in common, but found divergent in other respects.

It might be possible to reconcile them, finding postulates which succeed in encompassing the ones in conflict, while retaining the same uniform predictions. Or we may have to find exclusive predictions for each, which can be tested empirically to help us make a choice between postulates.

This is where adduction comes into play. It is the process used to evaluate, compare, and select theories through their predictions. It is the main tool for the induction of theories, commonly known as ‘the scientific method’.

criteria. Although utility is a relatively ‘subjective’ standard for evaluating theories, being man-centered, it plays a considerable role. For us, knowledge is not a purely theoretical enterprise, but a practical necessity for survival. We use it to support and improve our lives.*Utilitarian*

We judge a theory to some extent by how accessible it is to our minds, by virtue of its simplicity, or the elegance of its ordering of information. All other things being equal, we would choose the theory which approaches this ideal most closely, on the general grounds that the world is somehow simple and beautiful. The onus of proof is on the more complex, the more *‘far-fetched’*, theories: avoidable complications need additional justification.

However, simplicity should not be confused with *superficiality*. People often opt for overly simplistic viewpoints, which only take the most obvious data into consideration, and ignore deeper issues. A theory should preferably be simple, but not at the expense of accuracy; it must cover more known phenomena and answer more questions, than any other, to be credible. The easy solution often has a limited data base, and reveals a *naive* outlook.

Apart from such rationalistic and esthetic bias, we also look at the implementation value of a theory. Even if a theory or group of theories is/are known to contain some contradictions, we may hang on to them, in the absence of a viable substitute. We assume that the problem will eventually be resolved; meanwhile, we need a tool for prediction, decision-making and action, however flawed. Thus, for example, with the particle-wave dichotomy in physics.

We will look at some of the dynamics of theory selection in more formal terms, in the next chapter.

### 4. Control.

It must be stressed that the primary problem in theorizing is producing a theory in the first place. It is all very well to know in general how a theory is structured, but that does not guarantee we are able to even think of an interpretation of the facts. All too often, we lack a hypothesis capable of embracing all the available data.

Very often, theories regarded as being ‘in conflict’, are in fact not strictly so. One may address itself to part of the data, while the other manages to deal with another segment of the data; but neither of them faces all the data. Their apparent conflict is due to their implicit ambition to fit all the facts and problems, but in reality we have no all-embracing theories before us.

However, quite often, we do easily think up a number of alternative theories. In that case, we are wise to resort to *structured theorizing and testing*, to more clearly pose the problems and more speedily arrive at their solutions.

This is known to scientists as ‘controlled experiment’, which consists in changing (by small alterations or thorough replacement) one of the variables involved, while ‘keeping all other things equal’. The method is applicable equally to forming theories and to testing them (by simple observation or experiment).

Structuring consists in ordering one’s ideas in a hierarchy, so as to systematically try them out, and narrow down the alternatives.

- List the independent
*issues*. A subject-matter may raise several questions, which do not seemingly affect each other; these various domains of concern must first be identified. For example, in geometry, whether or not space is continuous, and whether or not parallels meet, seem to be two separate issues.

- For each issue, list the alternative postulates, which might provide an answer. Combine the various postulates of each issue, with the various postulates of all other issues involved, to yield a number of theories (equal to the product of the numbers of postulates in the various issues). Some of these combinations may be logically inconsistent, and eliminatable immediately; in other words, there may be some partial or conditional dependencies between the issues.

- Within each issue, distinguish between alternative postulates which are radically different, and between postulates which may be viewed as minor alterations of one common assumption. In the former case, we may expect to eventually find some radically different predictions from the alternative postulates. In the latter case, varying the main postulate may merely cause small variations in the predictions, and the work involved is more one of fine tuning our theory.

- The best way to test ideas is to organize them in terms of successive specific theses and antitheses, as follows:

Starting with the seemingly broadest, most independent issue, focus on one postulate p1, and find for it a prediction q1, which is denied by the denial of that postulate, thus:

If p1, then q1, but if not p1, then not q1.

Next, suppose that p1 wins that contest, and concentrate on the next issue; within that issue, consider one postulate p2, and again look for some exclusive prediction q2 for it:

If p2, then q2, but if not p2, then not q2.

Proceeding in this manner, we can gradually foresee the course of all possible events, and eventually of course test our results experientially. This is an ideal pattern, in that it is not always easy to find such distinctive implications; but it often works.

The trick, throughout the process of theorizing and testing is to structure one’s thoughts, so as to advance efficiently to the solutions of problems. A purposeful, constructive, orderly approach, is obviously preferable to a hesitant, vague, muddled one. It often helps to use paper and pencil, or computer, and draw flow-charts; it generates new ideas. Sometimes, of course, it is wise not to insist, and to let the mind find its way intuitively.

I would like to here praise the inventors and developers of the modern personal computer, and all software. Imagination and verbal memory greatly improve the mind’s ability to formulate and test thoughts. The invention of the written word, and pen and paper to draw and write with, provided us with an enormous expansion in these capabilities.

The word-processing and other computer applications increase our mental powers still further, by an enormous amount. A patient person can keep improving ideas on a screen, again and again, to degrees which were previously beyond reach. This has and will make possible tremendous advances in human thinking.

## 48. THEORY SELECTION.

### 1. The Scientific Method.

The ‘scientific method’ consists in trying out every conceivable imaginary construct, and seeing which of them keep fitting all new facts, and which do not. Those which cease to fit, must be eliminated (or at least corrected). Those which continue to fit, are to that extent increasingly probable, until they in turn cease to fit. Whatever theory alone survives this eliminative process, is effectively proved, since all the shares of probability have been inherited by it.

In practice, the construction of alternative postulates, and the discovery of the full implications of each, are both gradual processes. We do not know these things immediately. Also, the given context is not static, but itself grows and changes as we go along. This feeds our imagination and insight, helping theory developments, and stimulating further research.

We may start with one or two partially developed theories, and slowly find additional alternatives and make further predictions, as events unfold and the need arises. The extent of our creative and rational powers affects the exhaustiveness of our treatment.

Several theories concerning some group of phenomena may, at any stage in the development of knowledge, simultaneously equally fulfill the criteria of relevance; namely, conceptual meaningfulness, internal consistency, ability to explain the phenomena in question, and compatibility with all other empirical givens so far.

In formal terms, this simply means that competing theories T1, T2, T3,… may, while being contrary to each other, each still logically imply the already experienced phenomena Q. That is, the hypotheticals ‘if T1, then Q’, ‘if T2, then Q’, etc., are formally compatible, even though ‘T1 or else T2 or else T3…’ is true.

The statement that our list of theories for Q is exhaustive, has the form ‘If T1 or T2 or T3… , then Q’, plus ‘one of T1, T2, T3… must be true’. Although it may be hard to prove that our list is exhaustive, we may contextually assume it to be so, if every effort has been expended in finding the alternative explanations.

Each theory contains a number of postulates: T1 = p11 + p12 + p13 +…, T2 = p21 + p22 + p23 +…, and so on. Some of these postulates might well be found in more than one theory; it may be, for instance, that p13 = p29 = p36. But each theory must have at least one distinctive postulate or a distinctive combination of postulates, which makes it differentiable from all the others.

Also, the phenomenon or group of phenomena labeled Q are already known empirically, and supposed to be equally embraced by the various theories put forward. But each theory may have other implications, if we can determine them through reason, open to empirical testing, though not yet tested.

Each theory has a set of predictions: T1 = q11 + q12 + q13 +…, T2 = q21 + q22 + q23 +…, and so on. Some of these must be in common, constituting the given phenomena Q which gave rise to our theorizing in the first place. That is, say, Q = q15 = q27 = q31.

The rest may likewise be all identical, one for one; or some overlaps may occur here and there, while some predictions found here are missing there; or, additionally, some conflicting predictions may occur, so that one or more theories affirm some prediction that certain other(s) deny.

In principle, it is conceivable that the various theories all make only the same predictions, in which case they are factually indistinguishable, and we cannot choose between them on an empirical basis, though we may still refer to utilitarian criteria.

Most often, however, we may eventually find distinctive further predictions for each theory, or at least some which are not common to all. A difference in postulates usually signifies a difference in predictions. Here, we must be careful to differentiate between:

- a prediction implied by, say, T1, but neither implied nor excluded by T2, T3, etc. — if such a prediction passes the test of experience, T1 is confirmed, but T2, T3,… are neither confirmed nor rejected, though their probabilities are diminished by the increased probability of T1; whereas if such a prediction fails the test of experience, T1 is rejected, while T2, T3,… become more probable by virtue of being less numerous than before; and:
- a prediction implied by, say, T1, and logically excluded by T2, T3, etc. — if such a prediction turns out empirically successful, T2, T3… are rejected, and (if only T1 is leftover) T1 is proved; whereas if such a prediction turns out empirically unsuccessful, T2, T3,… are confirmed by their anticipation of the negative event, while T1 is rejected.

Thus, theory selection depends on finding distinctive predictions, which can be used in adductive argument or apodosis. These should be empirically testable predictions, of course.

If one or more theories have an implication which the others lack, though are compatible with, or if one or more theories have an implication which the others are incompatible with — we have at least an eventual source of divergent probabilities, allowing us to prefer some theories over others, even if we cannot eliminate any of them; and in some cases, we may be able to eliminate some of them, and maybe ultimately all but one of them.

These methods are of course well known to scientists today. But all this concerns not only scientists at work, but the development of opinions by individuals in every domain. It is the ‘trial and error’ process through which we all learn and improve our knowledge.

Even if at a later stage we might manage to validate some of our beliefs more deductively and systematically, this is the method we usually use to initially feel our way to them and develop them. Knowing the ‘scientific method’ explicitly and clearly can help individuals to make their personal thinking on topics remote from abstract science more scientific.

### 2. Compromises.

We have described the ideal pattern of scientific evaluation of theories; but, in practice things are not always so neat, and we often have to make do with less than perfect intellectual situations.

- For a start, the coexistence of conflicting theories may be viewed less generously as a source of doubt for all of them; they may each be corroborated by the delimited data they explain, but their mutual incompatibility is a significant inconsistency in itself.

We may remain for years with equally cogent, yet irreconcilable theories, which we are unable to decide between. Our minds are often forced to function with a baggage of unresolved contradictions.

In such case, we suspend judgment, and make use of each theory for pragmatic purposes, without considering any as ultimately true as a theoretical image of reality.

Even as we may give more credence to one theory as the more all-embracing and most-confirmed, or as the simplest and most-elegant, we may still withhold final judgment, and not regard that theory as our definite choice, because the evidence does not seem to carry enough conviction.

- Sometimes the available theories only partially explain the given data. They may embrace some details in common, with comparable credibility, but one may be more useful than the others in some areas, while another is more thorough in other respects.

Although this suggests that the theories have distinct implications, they are each supportable on different grounds, perhaps with the same overall probabilities. We may not find a way to choose between them empirically, or to unify them somehow.

In such case, narrowing the field by elimination of alternatives is hardly our main concern; rather, we are still at a stage where we need a unifying principle, we effectively do not have a theory in the full sense of the term. An example of this is the particle-wave dichotomy, and the search for a unified field theory to resolve it.

Sometimes, we know our list of available theories is faulty, because their connections to the data are not entirely satisfactory and convincing. In that case, our ‘if-then-’ statements are themselves probabilistic, rather than necessary. Our ideas then had better be called notions or speculations.

- Sometimes, no theory at all can be found for the phenomena at hand, for years. There may be seemingly insurmountable antinomies. We are forced to wait for an inspiration, a new idea, a new insight, a new observation, which might lead us to a satisfactory solution.

Because it is in some domains very difficult to develop a meaningful and consistent conceptual framework, we may be forced to accept one which is conceptually or logically flawed, as a working hypothesis.

Sometimes, the problem may be shelved, because its impact lies elsewhere, creating doubts and questions in distant disciplines. For example, Heisenberg’s Uncertainty Principle seems to assault our common-sense conceptions of determinism for inanimate matter: this might later be resolved by Physics itself, or might remain an issue for Philosophy to deal with.

In practice, an imperfect tool of knowledge is often better than none at all. We prefer to have a theory formulated in terms of vague or seemingly contradictory concepts, with practical value, than to remain paralyzed by a dogmatic insistence on an elusive ideal.

- Thus, sometimes, although a theory may apparently be strictly speaking felled by hard evidence, and we are unable to pinpoint its mistakes, we may nonetheless pragmatically hang on to it, if there is no other to replace it. We simply mentally attach a reservation to it, retain an awareness of its limitations, and move on cautiously to practical applications.

This is especially justifiable when the reason for its empirical rejection was an extreme situation, or ‘boundary case’, not encountered in the normal course of events. We then recognize the need to specify some limiting conditions to the theory, without being able to fulfill this need more precisely at the present stage.

### 3. Theory Changes.

Even when a theory is found empirically wrong, yet has alternatives, we may avoid outright rejection, and rather first seek to rectify it somehow, limiting it in scope or shifting some of its postulates slightly. This is feasible on the ground that there must have been some grain of truth in the original insight, and we may be able to tailor our assumptions to fit the new data.

Even if we cannot immediately conceive a correction, we may still choose to hang on to the original idea in the hope of its eventual redemption. We all carry a baggage of beliefs through life, which we know lead to contradictions or have been apparently disproved or rendered very improbable; we keep them in mind for further verification, anyway. This attitude taken to an extreme is of course contrary to logic, but within reasonable bounds it has some utility.

The pursuit of truth is not cold and vengeful, as it were, towards flawed theories, intent on rarefying the alternatives at all costs. Rather, it is a process of flexible adaptation to changing logical conditions. Our goal is, after all, to indeed arrive at truth, and not merely to give the impression that we did.

If we manage to modify a theory well enough to fit the new facts, then effectively we have developed a new theory. It may be a new version of the old, but still merits consideration as a theory in its own right.

We defined a theory as a number of distinctive postulates together implying a number of predictions. More loosely, the range of applicability of a theory might be varied, without radically affecting the substance of its proposals or its details.

Also, we may distinguish between essential postulates and postulates open to change. The former may be generic proposals, the latter specifics within them which we have not yet resolved — postulates within postulates, as it were. Likewise, we might distinguish between generic predictions, which are necessary consequences, and their specifics, which may be less firmly bound to the postulates.

With these thoughts in mind, we can talk of a theory ‘changing’, while remaining essentially the same theory. This may refer to changes in scope or changes in detail which do not affect the main thrust of a hypothesis. In other words, a theory may involve logical conditional propositions, as well as categoricals, leaving room for variations.

Denial of a postulate may mean: either denial of the broadness of the postulate, without excluding the possibility that a more moderate formulation is acceptable, or denial of a specific position, which can be replaced by another specific position with the same generic impact, or radical denial of a generic position, in the sense that all its possible embodiments are consequently denied.

Denial of a prediction may accordingly either merely cause us to regard the theory as having a more limited applicability than originally thought, or to make relatively small corrections in our assumptions, or force us to formulate a completely new theory.

Thus denial of a postulate or prediction does not necessarily mean rejection of the whole theory as such, it may be only partly discredited, requiring a less ambitious or a slightly altered formulation.

Accordingly, a new theory may totally replace an old one, or it may embrace it as a special case. For example, Einstein’s Relativity resulted in our particularization of Newtonian mechanics to commonplace physical levels; it was thenceforth seen as inapplicable to more extreme astronomical or sub-atomic situations, but retained much of its usefulness.

### 4. Exclusive Relationships.

We know from apodosis that affirmation of a postulate implies acceptance of all its necessary predictions (even those untestable empirically), and denial of a prediction obliges us to reject (or at least change) the postulates which necessitate it.

Denial of a postulate does not engender denial of its still untested predictions; it only diminishes their probability. However, empirically untestable predictions can still be discarded, if we can show them to be logically exclusive to some empirically rejected postulate(s). The argument is a valid apodosis:

Only if postulates p, then predictions q

(implying: if notp, then notq),

but not p,

hence, not q.

Doubt may remain, depending on how sure we are of the postulate’s denial, and especially on the strength of the exclusiveness. Also, what has been said does not prevent the possibility that a slightly different version of the predictions still hold.

Likewise, affirmation of a prediction does not in itself prove any of the postulates giving rise to it, but only confirms them. However, theoretical postulates can still be established, if we can show them to make some logically exclusive empirically tested prediction(s).

Only if postulates p, then predictions q

(implying: if notp, then notq),

but q,

hence, p.

This too is a valid apodotic argument. Again, such exclusiveness may often be hard to determine indubitably, but the principle remains valid.

It is not always easy or even possible to find such exclusive relationships. In such case, we are of course limited to the adductive approach. Note that, just as necessity is the extreme of probability, so apodosis is the limiting case of adduction: they differ in degree, not in essence.

Thus, it is not permissible to regard, as some philosophers seem to have intimated, science as incapable of certitude in disproof of empirical matters, or of certitude in proof of theoretical constructs. Admittedly, a good deal of theory selection is based on the processes of adduction and elimination; but this is only one arrow in the arsenal of the scientific method.

If we regard science as capable of establishing logical (or mathematical) connections for the purposes of mere confirmation or undermining of theories, then it is equally capable in principle of establishing exclusive connections which can be used for the above described demonstration purposes.

All the hypothetical forms are structurally identical, irrespective of the polarities of their theses. If any one of them is recognized as accessible to science, then they are all equally so. If we can rely on the ‘if p, then q’ of adduction, then we can just as well rely on the ‘if notp, then notq’ of exclusive apodoses.

There is no intent, here, to underrate the importance of competitive induction, only to point out that other, more certain, means are *sometimes* available to us, though not always. What is at issue here is the suggestion that we only have a choice of a-priori, axiomatic knowledge versus a posteriori, probabilistic knowledge.

There is an in-between alternative: knowledge which is at once theoretical, and certifiable, and empirical. It is arrived at through the logical discovery of exclusive relationships between postulates and predictions. This methodology has the stamp of approval of logical science, and is perfectly reliable.

Indeed, all our so-called mind-set concepts, even the axioms of logic, have such exclusive-empirical grounding, as well as self-evidence (i.e. self-contradiction of their contradictories). Every particular proposition, for example, appeals to this reasoning. More generally, any concept which appears as sole available interpretation or explanation of the experienced phenomena is justifiable on that basis.

## 49. SYNTHETIC LOGIC.[4]

### 1. Synthesis.

Knowledge requires inquisitiveness and creativity. It cannot advance far inertially. The role of the knower is to actively ask questions and look for answers, not to sit back passively and assume all is well. Knowledge is a *constructive* activity.

In forming one’s opinions, one has to *think things through*, and not unfocus one’s stare and avoid the effort. One should not rely excessively on generally-held opinion, though of course its general acceptance is in most cases well-earned. One is duty-bound to verify, repair, and contribute, if one can.

Knowing is not mere maintenance work, ‘when something goes wrong, fix it’, but involves searching for flaws or improvements even without apparent cause. Speculation, the attitude of ‘what if things are otherwise than they now seem or are said to be?’, has considerable value in the pursuit of truth.

In forming our world-view, we all make use of some prejudicial ideas, or preconceptions. We take for granted many basic assumptions, often unconsciously, without awareness of having made them, without ever having analyzed them to any great extent, without having tried the alternative assumptions.

Some such assumptions become deeply ingrained in a sub-culture, a culture, a period of history, or all human thinking. If such a philosophical prejudice is institutionalized, it is called a dogma. But our concern here is also with unconscious dogmas. My purpose in this chapter is to show informally how such ideas can be brought out into the open and evaluated.

The first thing is always a willingness to face the issue explicitly, and confront the possibly unpleasant results. Next, try to reconcile the apparent opposites, find a *synthesis* of some sort. Look for the ultimate premises, and even if speculatively, consider alternative conceptions which are capable of fitting the known facts.

The synthesis of knowledge is an attempt to ‘wrap it all up’, or at least take stock of the situation as a whole thus far. You lay out the data you have, and you firmly evaluate their significance on your current opinions:

- Where are you at?
- What do you know, what don’t you know?
- What do you need to know?
- What can you know, what can’t you know?

An inventory and a summation, to the best of one’s ability.

### 2. Self-Criticism.

Thus far, one’s logic may have been lenient. One perhaps wanted to get ahead, to cover ground. There was no time for scrupulous analysis of the degrees of logical probability in one’s information and inferences. Now, the whole must be reviewed, each part considered in the light of all the others. One must disengage oneself, and become a neutral referee between contending ideas.

One must challenge one’s previous viewpoints. One must look at things *more critically*, less intent on the object than on the process which led us to our viewpoint. It is time to linger on detail, digress a little, consider the full impact of what one is saying.

This may mean taking-off in all directions, even to the point of looking into metaphysical implications. One should not limit one’s vision to one field, but range as far and wide as necessary to prove a point. One may appeal to epistemological reasons, or consider ontological outcomes.

Initially, we accept our deductions and inductions with fair-minded tolerance. But, in the final analysis, the limits of one’s certainties must be emphasized. There are different degrees of strictness of outlook; different modalities of implication. There is a ‘take it for granted’, working level; and there is a more severe, philosophical level.

Within philosophy, ‘anything goes’, and even doubts about logic, about the laws of thought or the trustworthiness of experience, have some legitimacy. At this strict level, it is healthy to give skepticism some rein, to enable us to judge with honest detachment (though total skepticism remains invalid, since paradoxical).

For instance, an adductive argument is ordinarily allowed; it is acknowledged to increase the probability of the conclusion. But viewed deductively, its inference is worthless. Synthetic logic probes into theories by considering, not only their internal consistency and continuing confirmation, but more fully and deeply:

- What are the ultimate assumptions?
- What are the implied conclusions?
- Are there alternative premises or inferences?
- How do they compare and contrast, how much do they agree or disagree?
- How reliable are the apparent consistencies and how serious are the seeming inconsistencies?
- How solid are the logical connections between postulates and predictions, and what are they based on?
- What is the data, and how empirical is it?

The enterprise of science is an open pursuit of knowledge. If it is objective, as it wants to be, then it should have no prejudice as to what the object presented to it is, or how it got there. The process of adduction, we saw, has the form:

If Theory, then Predictions:

Yes to any of these predictions,

therefore, possibly yes to the theory.

(but if No to any prediction, no to the theory.)

This may be countered by the equally valid adduction:

If Other Theory, then Same or Other Predictions:

Yes to any of those predictions,

therefore, possibly yes to the other theory.

(but if No to any prediction, no to the theory.)

Now, note the following methodological implications, according to strict logic. Here, the emphasis is more on the criteria of relevance and competitiveness. Utilitarian or esthetic criteria are not granted much weight, so that a far-fetched theory may be as respectable as a more obvious one.

- If the two theories make predictions which coincide exactly, or if none of their predictions logically impinge on each other, there is no way to choose between them. They are effectively undifferentiated, or irrelevant to each other.
- If the two theories have some different prediction(s), but these differences are in practice or in principle untestable, again there is no ground for preferring the one to the other. But we may not regard untestable predictions as strictly logically equivalent to non-predictions.
- If the two theories have been confirmed by adduction to an equal degree of logical probability — that is, as many times, by equally firmly-implied and credible phenomena, whether these phenomena be the same or different — no conclusion is permissible. The logical modality is the same.

All this applies as well to theories with mutually exclusive postulates, and to theories with postulates which are independent of each other.

### 3. Fairness.

Clearly, the mere fact that someone takes up a theory of his own, and keeps testing it, and finds it repeatedly confirmed, does *not* in itself make his work fully scientific, and in accord with the neutral demands of logic.

The scientific approach, under the terms set by epistemology (*not* ontology, mind you), is to consider all other available theories, and busy oneself to an equal extent in testing and confirming *them too*. If difficulties arise, we are duty-bound to try to repair *all* the known theories with equal zeal, and not just the one we hope will win, for whatever personal reasons.

The same methodological demands should be made for one’s own pet theory, as one makes for others’; and the same leniency should be granted to others’ theories, as one grants to one’s own.

Similarly, one should refrain from negative pronouncements on sectors of human inquiry about which one is not adequately informed. In other words, one may regard oneself as a specialist, advancing a limited domain of the inquiry, without laying claim to any authority beyond those limits.

To be professional in the pursuit of knowledge, completely objective and neutral, without prejudice, one must proceed in accord with the rules of argument set by logic. The scientist who merely works on one theory at a time, without regard to the inadequacy of his methodology, is kidding himself and everyone else; he has ignored the alternatives, his conclusions are strictly invalid.

Of course, one can only do one thing at a time; but one must always keep the global perspective in mind, or refrain from comment.[5]

# PART VI. FACTORIAL INDUCTION.

## 50. ACTUAL INDUCTION.

### 1. The Problem.

Induction is the branch of Logic concerned with determining how general propositions — and, more broadly, how necessary propositions — are established as true, from particular or potential data.

By ‘actual induction’, I mean induction of actual propositions; by ‘modal induction’, I mean induction of modal propositions (referring to *de-re *modality).

We saw, in the analysis of Deductive processes, that although we can infer a general or particular proposition from other general propositions, through opposition, eduction or syllogism, it seems impossible to deductively infer general truths from particular ones only.

Indeed, it is even, according to the rules of syllogism, just about impossible to deduce a particular proposition from particular premises only: there has to be a general premise; the only exceptions to this rule are found in eduction, and in a limited number of third figure syllogisms, which allow us to obtain particular conclusions without use of a general premise: but these are too special to be claimed as important sources.

If, then, virtually all deduction presupposes the prior possession of general premises, where do these first general premises originate, or more precisely, how are they themselves shown to be true? Obviously, if such first premises, whatever their content, are open to doubt and of little credibility, then all subsequent deduction from them, however formally trustworthy, may be looked upon with healthy skepticism. As computer programmers say, “Garbage in, garbage out”. Conclusions drawn from spurious premises could nonetheless be true, but it would be mere chance, not proof.

Furthermore, these ‘first general premises’ we mentioned are not few in number. We are not talking here of a few First Principles, like the axioms of logic, from which exclusively all knowledge is to be derived. We require an extremely large number of first general premises, with all sorts of contents, to be able to develop a faithful image of our actual knowledge base. While mathematical sciences, like arithmetic, algebra or geometry, can seemingly be reduced to a very limited number of axioms, this is a feat not easy to duplicate in sciences like physics or psychology, or in everyday thinking.

If, now, we introspect, and observe our actual thinking processes as individuals, and analyze the actual historical development of Science, the accumulation of knowledge by humankind as a whole, we see clearly that, although deduction plays a large and important role, it is not our only source of knowledge. Even axioms in mathematics have been identified over time, and been subject to improvement or change. In practice, however faultless our deductions, our knowledge is clearly an evolving, flexible, thing. Ideas previously ignored, eventually make their appearance in our body of knowledge; thoughts once considered certain, turn out to be incorrect, and are modified or abandoned.

The primary source of knowledge is not deduction, but *observation*. This term is to be understood here is its broadest, and most neutral, sense, including both passive experiences and those experimentally generated.

Observation is to be understood as in itself a neutral event. It is consciousness, awareness, of appearances, phenomena, such as they present themselves, without judgement as to their ultimate meaning or value in the full scheme of things. Observation concerns the given, in its most brutal, unordered, unprocessed form.

Any *interpretation* that we attach to an observation, is to be regarded as a separate phenomenon; the distinction between these two is not always easy to make, nevertheless. Interpretation, in contrast to observation, attempts to relate phenomena, to place them in a supposed order of things, to evaluate their credibility and real significance in the widest possible context. It is a relatively complex mental process, and more subject to error. Its purpose is to tell us whether, all things considered, an experience was illusory or real.

### 2. Induction of Particulars.

In this treatise, I will evolve an original theory of induction, in considerable detail, with reference to categorical propositions: first for actuals, then more broadly for modals. I will not here deal with natural, temporal, or extensional conditionals, at all, but it will become obvious that the same methods and principles can be extended to those forms as well, though the formulas involved are bound to be enormously more complex; I leave the task to future logicians with my compliments!

The first step in induction is formulation of particular propositions on the basis of observation. This is a more complicated process than we might at first sight suppose. It does not merely consist in observation of a perceptible phenomenon, but includes the conceptual factor of abstraction of ‘universals’, the similarities on which we base our verbalization of terms, copula, and particular quantity. Pure observation forms no judgement; it is meditation on, simple consciousness of, the object at hand. The moment a thought is expressed, even a particular proposition, we have interpretation, conceptual correlation. The question of truth or falsehood is yet a separate judgement.

It follows, in passing, that a particular proposition based on observation of concrete phenomena, cannot be viewed as extremely superior in value to one based on observation of abstract phenomena. Both involve abstraction of sorts and verbalization. Their difference is only in the qualitative character of object involved, in the relative accessibility of the evidence.

Now, all observation concerns primarily individual instances. We have seen that singular propositions point to a single specific individual under consideration (referred to by ‘this’), whereas particular propositions are quantitatively indefinite and need not specify the individuals they concern (we just say ‘some’). A plural but specific proposition, involving the quantity ‘these’, is essentially singular in nature, or a conjunction of singulars; it differs from a genuine particular, which is more broadly intended. We have seen, too, that singulars imply particulars, by formal opposition.

Normally, unless the subject is a namable individual person or animal, a uniquely complex entity we deal with on a regular basis, our singular propositions are only temporary furniture in our knowledge base. I may say to you “look, this rose, unlike the others in my garden, is blue’ or “this particle swerved to the left in our experiment”, but ultimately, the individual is ignored or forgotten, and only an indefinite particular proposition is retained in the record. Furthermore, although a particular can be inferred from one singular, it is more often based on a pl