The Logic of CAUSATION
Definition, Induction and Deduction of Deterministic Causality
Phases I, II and III
Avi Sion, Ph. D.
(C) Copyright Avi Sion, 1999, 2000, 2003, 2005, 2008, 2010.
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Previously published, 1999 (rev. 2000), 2003 (rev. 2005), 2008 (internet only).
Third edition, 2010.
By Avi Sion, Geneva, Switzerland.
Abstract
The Logic of Causation is a treatise of formal logic and of aetiology. It is an original and wide-ranging investigation of the definition of causation (deterministic causality) in all its forms, and of the deduction and induction of such forms. The work was carried out in three phases over a dozen years (1998-2010), each phase introducing more sophisticated methods than the previous to solve outstanding problems. This study was intended as part of a larger work on causal logic, which additionally treats volition and allied cause-effect relations (2004).
The Logic of Causation deals with the main technicalities relating to reasoning about causation. Once all the deductive characteristics of causation in all its forms have been treated, and we have gained an understanding as to how it is induced, we are able to discuss more intelligently its epistemological and ontological status. In this context, past theories of causation are reviewed and evaluated (although some of the issues involved here can only be fully dealt with in a larger perspective, taking volition and other aspects of causality into consideration, as done in Volition and Allied Causal Concepts).
Phase I: Macroanalysis. Starting with the paradigm of causation, its most obvious and strongest form, we can by abstraction of its defining components distinguish four genera of causation, or generic determinations, namely: complete, partial, necessary and contingent causation. When these genera and their negations are combined together in every which way, and tested for consistency, it is found that only four species of causation, or specific determinations, remain conceivable. The concept of causation thus gives rise to a number of positive and negative propositional forms, which can be studied in detail with relative ease because they are compounds of conjunctive and conditional propositions whose properties are already well known to logicians.
The logical relations (oppositions) between the various determinations (and their negations) are investigated, as well as their respective implications (eductions). Thereafter, their interactions (in syllogistic reasoning) are treated in the most rigorous manner. The main question we try to answer here is: is (or when is) the cause of a cause of something itself a cause of that thing, and if so to what degree? The figures and moods of positive causative syllogism are listed exhaustively; and the resulting arguments validated or invalidated, as the case may be. In this context, a general and sure method of evaluation called ‘matricial analysis’ (macroanalysis) is introduced. Because this (initial) method is cumbersome, it is used as little as possible – the remaining cases being evaluated by means of reduction.
Phase II: Microanalysis. Seeing various difficulties encountered in the first phase, and the fact that some issues were left unresolved in it, a more precise method is developed in the second phase, capable of systematically answering most outstanding questions. This improved matricial analysis (microanalysis) is based on tabular prediction of all logically conceivable combinations and permutations of conjunctions between two or more items and their negations (grand matrices). Each such possible combination is called a ‘modus’ and is assigned a permanent number within the framework concerned (for 2, 3, or more items). This allows us to identify each distinct (causative or other, positive or negative) propositional form with a number of alternative moduses.
This technique greatly facilitates all work with causative and related forms, allowing us to systematically consider their eductions, oppositions, and syllogistic combinations. In fact, it constitutes a most radical approach not only to causative propositions and their derivatives, but perhaps more importantly to their constituent conditional propositions. Moreover, it is not limited to logical conditioning and causation, but is equally applicable to other modes of modality, including extensional, natural, temporal and spatial conditioning and causation. From the results obtained, we are able to settle with formal certainty most of the historically controversial issues relating to causation.
Phase III: Software Assisted Analysis. The approach in the second phase was very ‘manual’ and time consuming; the third phase is intended to ‘mechanize’ much of the work involved by means of spreadsheets (to begin with). This increases reliability of calculations (though no errors were found, in fact) – and also allows for a wider scope. Indeed, we are now able to produce a larger, 4-item grand matrix, and on its basis find the moduses of causative and other forms needed to investigate 4-item syllogism. As well, now each modus can be interpreted with greater precision and causation can be more precisely defined and treated.
In this latest phase, the research is brought to a successful finish! Its main ambition, to obtain a complete and reliable listing of all 3-item and 4-item causative syllogisms, being truly fulfilled. This was made technically feasible, in spite of limitations in computer software and hardware, by cutting up problems into smaller pieces. For every mood of the syllogism, it was thus possible to scan for conclusions ‘mechanically’ (using spreadsheets), testing all forms of causative and preventive conclusions. Until now, this job could only be done ‘manually’, and therefore not exhaustively and with certainty. It took over 72’000 pages of spreadsheets to generate the sought for conclusions.
This is a historic breakthrough for causal logic and logic in general. Of course, not all conceivable issues are resolved. There is still some work that needs doing, notably with regard to 5-item causative syllogism. But what has been achieved solves the core problem. The method for the resolution of all outstanding issues has definitely now been found and proven. The only obstacle to solving most of them is the amount of labor needed to produce the remaining (less important) tables. As for 5-item syllogism, bigger computer resources are also needed.
Contents
Phase One: Macroanalysis.
Chapter 1. The Paradigm of Causation. 11
Chapter 2. The Generic Determinations. 16
- Strong Determinations. 16
- Parallelism of Strongs. 19
- Weak Determinations. 21
- Parallelism of Weaks. 28
- The Four Genera of Causation. 29
- Negations of Causation. 31
Chapter 3. The Specific Determinations. 34
Chapter 4. Immediate Inferences. 46
Chapter 5. Causative Syllogism.. 58
Chapter 6. List of Positive Moods. 69
Chapter 7. Reduction of Positive Moods. 85
Chapter 8. Matricial Analyses. 109
- Matricial Analysis. 109
- Crucial Matricial Analyses in Figure 1. 112
- Crucial Matricial Analyses in Figure 2. 127
- Crucial Matricial Analyses in Figure 3. 133
Chapter 9. Squeezing Out More Information. 141
- The Interactions of Determinations. 141
- Negative Moods. 145
- Negative Conclusions from Positive Moods. 148
- Imperfect Moods. 152
Chapter 10. Wrapping Up Phase One. 155
Appendix 1: J. S. Mill’s Methods: A Critical Analysis. 161
- The Joint Method of Agreement and Difference. 162
- The Method of Agreement 165
- The Method of Difference. 167
- The Method of Residues. 170
- The Method of Concomitant Variations. 172
Phase Two: Microanalysis.
Chapter 11. Piecemeal Microanalysis. 179
- Binary Coding and Unraveling. 179
- The Generic Determinations. 181
- Contraction and Expansion. 183
- Intersection, Nullification and Merger. 187
- Negation. 190
Chapter 12. Systematic Microanalysis. 193
- Grand Matrices. 193
- Moduses in a Two-Item Framework. 195
- Catalogue of Moduses, for Three Items. 199
- Enumeration of Moduses, for Three Items. 204
- Comparing Frameworks. 206
Chapter 13. Some More Microanalyses. 209
- Relative Weaks. 209
- Items of Negative Polarity in Two-Item Framework. 212
- Items of Negative Polarity in Three-Item Framework. 216
- Categoricals and Conditionals. 221
Chapter 14. Main Three-Item Syllogisms. 231
- Applying Microanalysis to Syllogism. 231
- The Moduses of Premises. 233
- The Moduses of Conclusions. 236
- Dealing with Vaguer Propositions. 243
Chapter 15. Some More Three-Item Syllogisms. 246
Chapter 16. Outstanding Issues. 261
- Four-Item Syllogism. 261
- On Laws of Causation. 262
- Interdependence. 267
- Other Features of Causation Worthy of Study. 270
- To Be Continued. 271
Phase Three: Software Assisted Analysis.
Chapter 17. Resuming the Research. 275
Chapter 18. Moduses of the Forms. 280
- 2-Item Framework Moduses. 280
- 3-Item Framework Moduses. 281
- 4-Item Moduses of the Forms. 282
- Interpretation of the Moduses. 283
Chapter 19. Defining Causation. 294
- Back to the Beginning. 294
- The Puzzle of No Non-connection. 297
- The Definition of Causation. 301
- Oppositions and Other Inferences. 303
Chapter 20. Concerning Complements. 306
- Reducing Numerous Complements to Just Two. 306
- Dependence Between Complements. 307
- Exclusive Weak Causation. 308
- The Need for an Additional Item (or Two). 311
Chapter 21. Causative Syllogisms. 313
Chapter 22. Scanning for Conclusions. 328
- Methodology. 328
- Forms Studied and their Oppositions. 330
- 3-Item Syllogisms. 331
- 4-Item Syllogisms. 334
Chapter 23. Exploring Further Afield. 351
- Possible Forms of Premises. 351
- Dealing with Negative Items. 352
- Preventive Syllogisms and their Derivatives. 354
- Syllogisms with Negative Premise(s). 356
- Causal Logic Perspective. 358
Chapter 24. A Practical Guide to Causative Logic. 361
- What is Causation?. 361
- How is Causation Known?. 363
- A List of the Main Causative Arguments. 367
- Closing Remark. 369
Avi Sion
The Logic of Causation:
Phase One:
Macroanalysis
Phase I: Macroanalysis. Starting with the paradigm of causation, its most obvious and strongest form, we can by abstraction of its defining components distinguish four genera of causation, or generic determinations, namely: complete, partial, necessary and contingent causation. When these genera and their negations are combined together in every which way, and tested for consistency, it is found that only four species of causation, or specific determinations, remain conceivable. The concept of causation thus gives rise to a number of positive and negative propositional forms, which can be studied in detail with relative ease because they are compounds of conjunctive and conditional propositions whose properties are already well known to logicians.
The logical relations (oppositions) between the various determinations (and their negations) are investigated, as well as their respective implications (eductions). Thereafter, their interactions (in syllogistic reasoning) are treated in the most rigorous manner. The main question we try to answer here is: is (or when is) the cause of a cause of something itself a cause of that thing, and if so to what degree? The figures and moods of positive causative syllogism are listed exhaustively; and the resulting arguments validated or invalidated, as the case may be. In this context, a general and sure method of evaluation called ‘matricial analysis’ (macroanalysis) is introduced. Because this (initial) method is cumbersome, it is used as little as possible – the remaining cases being evaluated by means of reduction.
Chapter 1. The Paradigm of Causation
1. Causation.
Causality refers to causal relations, i.e. the relations between causes and effects. This generic term has various, more specific meanings. It may refer to Causation, which is deterministic causality; or to Volition, which is (roughly put) indeterministic causality; or to Influence, which concerns the interactions between causation and volition or between different volitions.
The term ‘causality’ may also be used to refer to causal issues: i.e. to negative as well as positive answers to the question “are these things causally related?” In the latter sense, negations of causality (in the positive sense) are also causality (in the broad sense). This allows us to consider Spontaneity (i.e. causelessness, the lack of any causation or volition) as among the ‘causal’ explanations of things.
A study of the field of causality must also include an investigation of non-causality in all its forms. For, as we shall see, even if we were to consider spontaneity impossible, the existence of causality in one form or other between things in general does not imply that any two things taken at random are necessarily causally related or causally related in a certain way. We need both positive and negative causal propositions to describe the relations between things.
In the present work, The Logic of Causation, we shall concentrate on causation, ignoring for now other forms of causality. Causative logic, or the logic of causative propositions, has three major goals, as does the study of any other type of human discourse.
- To define what we mean by causation (or its absence) and identify and classify the various forms it might take.
- To work out the deductive properties of causative propositions, i.e. how they are opposed to each other (whether or not they contradict each other, and so forth), what else can be immediately inferred from them individually (eduction), and what can be inferred from them collectively in pairs or larger numbers (syllogism).
- To explain how causative propositions are, to start with, induced from experience, or constructed from simpler propositions induced from experience.
Once these goals are fulfilled, in a credible manner (i.e. under strict logical supervision), we shall have a clearer perspective on wider issues, such as (d) whether there is a universal law of causation (as some philosophers affirm) or spontaneity is conceivable (as others claim), and (e) whether other forms of causality (notably volition, and its derivative influence) are conceivable.
Note well, we shall to begin with theoretically define and interrelate the various possible forms of causation, leaving aside for now the epistemological issue as to how they are to be identified and established in practice, as well as discussions of ontological status.
We shall thus in the present volume primarily deal with the main technicalities relating to reasoning about causation, and only later turn our attention to some larger epistemological and ontological issues (insofar as they can be treated prior to further analysis of the other forms of causality). The technical aspect may at times seem tedious, but it is impossible to properly understand causation and its implications without it. Most endless debates about causation (and more generally, causality) in the history of philosophy have arisen due to failure to first deal with technical issues.
2. The Paradigmic Determination.
Causation, or deterministic causality, varies in strength, according to the precise combinations of conditioning found to hold between the predications concerned. We may call the different forms thus identified the determinations of causation.
The paradigm, or basic pattern, of causation is its strongest determination. This has the form:
If the cause is present, the effect is invariably present;
if the cause is absent, the effect is invariably absent.
Our use, here, of the definite article, as in the cause or the effect, is only intended to pinpoint the predication under consideration, without meaning to imply that there is only one such cause or effect in the context concerned. Use of an indefinite article, as in a cause or an effect, becomes more appropriate when discussing a multiplicity of causes or effects, which as we shall later see may take various forms.
We may rewrite the above static formula in the following more dynamic expression:
If the cause shifts from absent to present, the effect invariably shifts from absent to present;
if the cause shifts from present to absent, the effect invariably shifts from present to absent;
We shall presently see how this model is variously reproduced in lesser determinations. For now, it is important to grasp the underlying principle it reflects.
The essence of causation (or ‘effectuation’) is that when some change is invariably accompanied by another, we say that the first phenomenon that has changed has “caused” (or “effected”) the second phenomenon that has changed. In the above model, the changes involved are respectively from the absence to the presence of the first phenomenon (called the cause) and from the absence to the presence of the second phenomenon (called the effect); or vice versa. We may, incidentally, commute this statement and say that the effect has been caused (or effected) by the cause.
Now, some comments about our terminology here:
The term “change”, here, must be understood in a very broad sense, as referring to any event of difference, whatever its modality.
- Its primary meaning is, of course, natural change, with reference to time or more to the point with respect to broader changes in surrounding circumstances[1]. Here, the meaning is that some object or characteristic of an object which initially existed or appeared, later did not exist or disappeared (ceasing to be), or vice-versa (coming to be); or something existed or appeared at one place and time and recurred or reappeared at another place, at another time (mutation, alteration or movement). This gives rise to temporal and natural modalities of causation.
- Another, secondary sense is diversity in individuals or groups. This signifies that an individual object has different properties in different parts of its being[2]; or that a kind of object has some characteristic in some of its instances and lacks that characteristic (and possibly has another characteristic, instead) in some other of its instances. This gives rise to spatial and extensional modalities of causation.
- Tertiary senses are epistemic or logical change, which focus respectively on the underlying acts of consciousness or the status granted them: something is at first noticed and later ignored, or believed and later doubted, or vice-versa, by someone. This gives rise to epistemic and logical modalities of causation.
Regarding the terms “present” and “absent” (i.e. not present), they may be understood variously, with reference to the situations just mentioned. They may signify existence or appearance or instancing (i.e. occurrence in some indicated cases) or being seen or being accredited true – or the negations of these.
The term “phenomenon” is here, likewise, intended very broadly, to include physical, mental or spiritual phenomena (things, appearances, objects), concrete or abstract. Also, a phenomenon may be static or dynamic: that is, the changing cause and effect need not be a quality or quantity or state or position, though some such static phenomena are always ultimately involved; the cause and effect may themselves be changes or events or movements. For instance, motion is change of place, acceleration is change in the speed or direction of motion. What matters is the switch from presence to absence, or vice-versa, of that thing, whatever its nature (be it static or dynamic). The cause and effect need not even be of similar nature; for example, a change of quality may cause a change of quantity.
Another term to clarify in the above principle is “accompanied”. Here again, our intent is very large. The cause and effect may be in or of the same object or different objects, adjacent or apart in space, contemporaneous or in a temporal sequence. The definition of causation contains no prejudice in these respects, though we may eventually find fit to postulate relatively non-formal rules, such as that in natural causation the effect cannot precede the cause in time or that all causation at a distance implies intermediate contiguous causations[3].
Indeed, it is in some cases difficult for us, if not impossible, to say which of the two phenomena is the cause and which is the effect. And this often is not only an epistemological issue, but more deeply an ontological one. For, though there is sometimes a direction of causation to specify, there is often in fact no basis for such a specification. The phenomena named ‘cause’ and ‘effect’ are in a reciprocal relation of causation; the terms cause and effect are in such cases merely verbal distinctions. All that we can say is that the phenomena are bound together, and either can be accessed through the other; the labels applied to them become a matter of convenience for purposes of discourse.
Finally, the term “invariably” has to be stressed. How such constancy is established is not the issue here; we shall consider that elsewhere. In the paradigm of causation given above, it would not do for the conjunction of the cause and effect, or the conjunction of their negations, to be merely occasional. We would not regard such varying conjunctions as signifying genuine causation, but quite the opposite as signs of mere coincidence, happenstance of togetherness. Post hoc ergo propter hoc. The problem is complicated in lesser determinations of causation; but as we shall see it can be overcome, a constancy of conjunction or of non-conjunction is always ultimately involved.
In this context, a warning is in order. When something is invariably accompanied by another, we say that the first (the presence or absence of the cause) “is followed by” the second (the presence or absence of the effect). This refers to causal sequence and should not be confused with temporal sequence; the term “followed” is ambivalent (indeed, it is also used in relation to spatial or numerical series). Even though causal and temporal sequence are often both involved (which is why the term “to follow” is equivocal), causal sequence may occur without temporal sequence (even in natural causation) or in a direction opposite to temporal sequence (though supposedly not in natural causation, certainly in logical causation, and by abstraction of the time factor also in extensional causation). The context usually makes the intent clear, of course.
Now, for some formal analysis:
In our present treatment of causation, we shall focus principally on the logical ‘mode’ of causation, note well. There are (as we shall later discuss) other modes, notably the natural, the temporal, the spatial and the extensional, whose definitions differ with respect to the type of modality considered. Having investigated modality and conditioning in detail in a previous treatise (Future Logic, 1990), I can predict that most of the behavior patterns of logical causation are likely to be found again in the other modes of causation; but also, that some significant differences are bound to arise.
Returning now to the paradigm of causation, it may be expressed more symbolically as follows, using the language of logical conditioning (as developed in my Future Logic, Part III):
If C, then E; and
if notC, then notE.
A sentence of the form “If P, then Q” means “the conjunction of P and the negation of Q is impossible”, i.e. there are no knowledge-contexts where this conjunction (P + notQ) credibly occurs. Such a proposition can be recast in the contraposite form “If notQ, then notP”, which means “the conjunction of notQ and the negation of notP is impossible” – the same thing in other words.
Such a proposition, note, does not formally imply that P is possible or that notQ is possible. Normally, we do take it for granted that such a proposition may be realized, i.e. that P is possible, and therefore (by apodosis) Q is possible and the conjunction “P and Q” is possible; and likewise that notQ is possible, and therefore (by apodosis) notP is possible and the conjunction “notQ and notP” is possible.
However, in some cases such assumption is unjustified. It may happen that, though “If P, then Q” is true, P is impossible, in which case “If P, then notQ” must also be true; or it may happen that, though “if P, then Q” is true, notQ is impossible, in which case “If notP, then Q” must also be true. These results are paradoxical, yet quite logical. I will not go into this matter in detail here, having dealt with it elsewhere (see Future Logic, ch. 31). It is not directly relevant to the topic under discussion, except that it must be mentioned to stress that such paradox cannot occur in the context of causation (except to deny causation, of course).
Therefore, when discussing causation, it is tacitly understood that:
C is contingent and E is contingent[4].
That is, each of C, E is possible but unnecessary; likewise, by obversion, for their negations, each of notC, notE is possible but unnecessary. If any of these positive or negative terms is by itself necessary or impossible, it is an antecedent or consequent in valid (and possibly true) propositions, but it is not a cause or effect within the causation specified. This is, by the way, one difference in meaning between the expressions cause/effect, and the expressions antecedent/consequent. We shall see, as we deal with lesser determinations of causation, that their meanings diverge further. All the more so, when the terms cause/effect are used in other forms of causality.
Furthermore, as above shown with reference to “P” and “Q”, granting the contingencies of C and E, each of the propositions “If C, then E” and “If notC, then notE” implies the following possibilities:
The conjunction (C + E) is possible; and
the conjunction (notC + notE) is possible.
All this is hopefully clear to the reader. But we must eventually consider its implications with reference to statements dealing with lesser determinations of causation or statements denying causation.
Chapter 2. The Generic Determinations
1. Strong Determinations.
The strongest determination of causation, which we identified as the paradigm of causation, may be called complete and necessary causation. We shall now repeat the three constituent propositions of this form and their implications, all of which must be true to qualify:
- If C, then E;
- if notC, then notE;
- where: C is contingent and E is contingent.
As we saw, these propositions together imply the following:
The conjunction (C + E) is possible;
the conjunction (notC + notE) is possible.
Clauses (i) and (iii) signify complete causation. With reference to this positive component, we may call C a complete cause of E and E a necessary effect of C. Where there is complete causation, the cause is said to make necessary (or necessitate) the effect[5]. This signifies that the presence of C is sufficient (or enough) for the presence E.
Clause (ii) and (iii) signify necessary causation. With reference to this negative component, we may call C a necessary cause of E and E a dependent effect of C. Where there is necessary causation, the cause is said to make possible (or be necessitated by) the effect. This signifies that the presence of C is requisite (or indispensable) for the presence E[6].
Clause (iii) is commonly left tacit, though as we saw it is essential to ensure that the first two clauses do not lead to paradox. Strictly speaking, it would suffice, given (i), to stipulate that C is possible (in which case so is E) and E is unnecessary (in which case so is C). Or equally well, given (ii), that C is unnecessary (in which case so is E) and E is possible (in which case so is C). The possibilities of the conjunctions (C + E) and (notC + notE), logically follow, and so need not be included in the definition.
Looking at the paradigm, we can identify two distinct lesser determinations of causation, which as it were split the paradigm in two components, each of which by itself conforms to the paradigm through an ingenuous nuance, as shown below.
Also below, I list the various clauses of each definition, renumbering them for purposes of reference. Then a table is built up, including all the causal and effectual items involved (positive and negative) and all their conceivable combinations[7]. The modus of each item or combination, i.e. whether it is defined or implied as possible or impossible, or left open, is then identified. In each case, the source of such modus is noted, i.e. whether it is given or derivable from given(s).
Complete causation:
- If C, then E;
- if notC, not-then E;
- where: C is possible.
No. | Element/compound | Modus | Source/relationship | |
1 | C | possible | (iii) | |
2 | notC | possible | implied by (ii) | |
3 | E | possible | implied by (i) + (iii) | |
4 | notE | possible | implied by (ii) | |
5 | C | E | possible | implied by (i) + (iii) |
6 | C | notE | impossible | (i) |
7 | notC | E | open | |
8 | notC | notE | possible | (ii) |
Complete causation conforms to the paradigm of causation by means of the same main clause (i); whereas its clause (ii), note well, concerning what happens in the absence of C, substitutes for the invariable absence of E (i.e. “then notE”), the not-invariable presence of E (i.e. “not-then E”). However, remember, contraposition of (i) implies that “If notE, then notC”, meaning that in the absence of E we can be sure that C is also absent[8].
Clause (ii) means that (notC + notE) is possible, so we are sure from it that C is unnecessary and E is unnecessary; also it teaches us that C and E cannot be exhaustive. Technically, it would suffice for us to know that notE is possible, for we could then infer clause (ii) from (i); but it is best to specify clause (ii) to fit the paradigm of causation. As for clause (iii), we need only specify that C is possible; it follows from this and clause (i) that (C + E) is possible and so that E is also possible.
Note well the nuance that, to establish such causation, the effect has to be found invariably present in the presence of the cause, otherwise we would commit the fallacy of post hoc ergo propter hoc; but the effect need not be invariably absent in the absence of the cause: it suffices for the effect not to be invariably present.
The segment of the above table numbered 5-8 (shaded) may be referred to as the matrix of complete causation. It considers the possibility or impossibility of all conceivable conjunctions of all the items involved in the defining clauses or the negations of these items.
Necessary causation:
- If notC, then notE;
- if C, not-then notE;
- where: C is unnecessary.
No. | Element/compound | Modus | Source/relationship | |
1 | C | possible | implied by (ii) | |
2 | notC | possible | (iii) | |
3 | E | possible | implied by (ii) | |
4 | notE | possible | implied by (i) + (iii) | |
5 | C | E | possible | (ii) |
6 | C | notE | open | |
7 | notC | E | impossible | (i) |
8 | notC | notE | possible | implied by (i) + (iii) |
Necessary causation conforms to the paradigm of causation by means of the same main clause (i)[9]; whereas its clause (ii), note well, concerning what happens in the presence of C, substitutes for the invariable presence of E (i.e. “then E”), the not-invariable absence of E (i.e. “not-then notE”). However, remember, contraposition of (i) implies that “If E, then C”, meaning that in the presence of E we can be sure that C is also present[10].
Clause (ii) means that (C + E) is possible, so we are sure from it that C is possible and E is possible; also it teaches us that C and E cannot be incompatible. Technically, it would suffice for us to know that E is possible, for we could then infer clause (ii) from (i); but it is best to specify clause (ii) to fit the paradigm of causation. As for clause (iii), we need only specify that C is unnecessary; it follows from this and clause (i) that (notC + notE) is possible and so that E is also unnecessary.
Note well the nuance that, to establish such causation, the effect has to be found invariably absent in the absence of the cause, otherwise we would commit the fallacy of post hoc ergo propter hoc; but the effect need not be invariably present in the presence of the cause: it suffices for the effect not to be invariably absent.
Note the matrix of necessary causation, i.e. the segment of the above table numbered 5-8 (shaded).
Lastly, notice that complete and necessary causation are ‘mirror images’ of each other. All their characteristics are identical, except that the polarities of their respective cause and effect opposite: C is replaced by notC, and E by notE, or vice-versa. The one represents the positive aspect of strong causation; the other, the negative aspect. Accordingly, their logical properties correspond, mutatis mutandis (i.e. if we make all the appropriate changes).
Following the preceding analysis of necessary and complete causation into two distinct components each of which independently conforms to the paradigm, we can conceive of complete causation without necessary causation and necessary causation without complete causation. These two additional determinations of causation are conceivable, note well, only because they do not infringe logical laws; that is, we already know that the various propositions that define them are individually and collectively logically compatible.
2. Parallelism of Strongs.
Before looking into weaker determinations of causation, we must deal with the phenomenon of parallelism.
The definition of complete causation does not exclude that there be some cause(s) other than C – such as say C_{1} – having the same relation to E. In such case, C and C_{1} may be called parallel complete causes of E. The minimal relation between such causes is given by the following normally valid 2nd figure syllogism (see Future Logic, p.162):
If C, then E (and if notC, not-then E / and C is possible);
and if C_{1}, then E (and if notC_{1}, not-then E / and C_{1} is possible);
therefore, if notC_{1} not-then C (= if notC, not-then C_{1} – by contraposition).
The possibility of parallel complete causes is clear from the logical compatibility of these premises, which together merely imply that in the absence of E both C and C_{1} are absent. The main clauses of the premises can be merged in a compound proposition of the form “If notE, then neither C nor C_{1}”, which by contraposition yields “If C or C_{1}, then E”. Thus, such parallel causes may be referred to as ‘alternative’ complete causes (in a large sense of the term ‘alternative’).
Since the conclusion of the above syllogism is subaltern to each of the propositions “if notC_{1}, then notC” and “if notC, then notC_{1}”, it may happen that C implies C_{1} and/or C_{1} implies C – but they need not do so. Likewise, since the conclusion is compatible with the proposition “if C_{1}, then notC” or “if C, then notC_{1}”, it may happen that C and C_{1} are incompatible with each other – but they do not have to be. The conclusion merely specifies that C and C_{1} not be exhaustive (i.e. be neither contradictory nor subcontrary; this is the sole formal specification of the disjunction in “If C or C_{1}, then E”).
Similarly, still in complete causation, E need not be the exclusive necessary effect of C; there may be some other thing(s) – such as say E_{1} – which invariably follow C, too. In such case, E and E_{1} may be called parallel necessary effects of C. The minimal relation between such effects is given by the following normally valid 3rd figure syllogism (see Future Logic, pp. 162-164):
If C, then E (and if notC, not-then E_{1} / and C is possible);
and if C, then E_{1} (and if notC, not-then E_{1} / and C is possible);
therefore, if E_{1}, not-then notE (= if E, not-then notE_{1} – by contraposition).
The possibility of parallel necessary effects is clear from the logical compatibility of these premises, which together merely imply that in the presence of C both E and E_{1} are present. The main clauses of the premises can be merged in a compound proposition of the form “If C, then both E and E_{1}”. Thus, such parallel effects may be said to be ‘composite’ necessary effects.
Since the conclusion of the above syllogism is subaltern to each of the propositions “if E_{1}, then E” and “if E, then E_{1}”, it may happen that E_{1} implies E and/or E implies E_{1} – but they need not do so. Likewise, since the conclusion is compatible with the proposition “if notE_{1}, then E” or “if notE, then E_{1}”, it may happen that E and E_{1} are exhaustive – but they do not have to be. The conclusion merely specifies that E and E_{1} not be incompatible (i.e. be neither contradictory nor contrary).
Again, mutatis mutandis, the definition of necessary causation does not exclude that there be some cause(s) other than C – such as say C_{1} – having the same relation to E. In such case, C and C_{1} may be called parallel necessary causes of E. The minimal relation between such causes is given by the following normally valid 2nd figure syllogism (see Future Logic, p. 162):
If notC, then notE (and if C, not-then notE / and notC is possible);
and if notC_{1}, then notE (and if C_{1}, not-then notE / and notC_{1} is possible);
therefore, if C_{1}, not-then notC (= if C, not-then notC_{1} by contraposition).
The possibility of parallel necessary causes is clear from the logical compatibility of these premises, which together merely imply that in the presence of E both C and C_{1} are present. The main clauses of the two premises can be merged in a compound proposition of the form “If E, then both C and C_{1}”, which by contraposition yields “If notC or notC_{1}, then notE”. Thus, such parallel causes may be referred to as ‘alternative’ necessary causes (in a large sense of the term ‘alternative’).
Since the conclusion of the above syllogism is subaltern to each of the propositions “if C_{1}, then C” and “if C, then C_{1}”, it may happen that C_{1} implies C and/or C implies C_{1} – but they need not do so. Likewise, since the conclusion is compatible with the proposition “if notC_{1}, then C” or “if notC, then C_{1}”, it may happen that C and C_{1} are exhaustive – but they do not have to be. The conclusion merely specifies that C and C_{1} not be incompatible (i.e. be neither contradictory nor contrary; this is the sole formal specification of the disjunction in “If notC or notC_{1}, then notE”).
Similarly, still in necessary causation, E need not be the exclusive dependent effect of C; there may be some other thing(s) – such as say E_{1} – which are invariably preceded by C, too. In such case, E and E_{1} may be called parallel dependent effects of C. The minimal relation between such effects is given by the following normally valid 3rd figure syllogism (see Future Logic, p. 162-164):
If notC, then notE (and if C, not-then notE / and notC is possible);
and if notC, then notE_{1 }(and if C, not-then notE_{1} / and notC is possible);
therefore, if notE_{1}, not-then E (= if notE, not-then E_{1} by contraposition).
The possibility of parallel dependent effects is clear from the logical compatibility of these premises, which together merely imply that in the absence of C both E and E_{1} are absent. The main clauses of the premises can be merged in a compound proposition of the form “If notC, then neither E nor E_{1}”. Thus, such parallel effects may be said to be ‘composite’ dependent effects.
Since the conclusion of the above syllogism is subaltern to each of the propositions “if notE_{1}, then notE” and “if notE, then notE_{1}”, it may happen that E implies E_{1} and/or E_{1} implies E – but they need not do so. Likewise, since the conclusion is compatible with the proposition “if E_{1}, then notE” or “if E, then notE_{1}”, it may happen that E and E_{1} are incompatible with each other – but they do not have to be. The conclusion merely specifies that E and E_{1} not be exhaustive (i.e. be neither contradictory nor subcontrary).
It happens that parallel causes or parallel effects are themselves causally related. That this is possible, is implied by what we have seen above. Since each of the following pairs of items may have any formal relation with one exception, namely:
- parallel complete causes cannot be exhaustive (since “if notC, not-then C_{1}” is true for them); and parallel necessary effects cannot be incompatible (since “if E, not-then notE_{1}” is true for them);
- parallel necessary causes cannot be incompatible (since “if C, not-then notC_{1}” is true for them); and parallel dependent effects cannot be exhaustive (since “if notE, not-then E_{1}” is true for them);
… it follows that either one of parallel causes C and C_{1 }may be a complete or necessary cause of the other; and likewise, either one of parallel effects E and E_{1} may be a complete or necessary cause of the other.
In certain situations, as we shall see in a later chapter, it is possible to infer such causal relations between parallels. But, it must be stressed, the mere fact of parallelism does not in itself imply such causal relations.
In sum, complete and/or necessary causation should not be taken to imply exclusiveness (i.e. that a unique cause and a unique effect are involved); such relation(s) allow for plurality of causes or effects in the sense of parallelism as just elucidated.
Indeed, it is very improbable that we come across exclusive relations in practice, since every existent has many facets, each of which might be selected as cause or effect. Our focusing on this or that aspect as most significant or essential, is often arbitrary, a matter of convenience; though often, too, it is guided by broader considerations, which may be based on intuition of priorities or complicated reasoning.
In any case, it is important to distinguish plurality arising in strong causation, which signifies alternation of causes or composition of effects, as above, from plurality arising in weak causation, which signifies composition of causes or alternation of effects, which we shall consider in the next section.
3. Weak Determinations.
Having clarified the complete and necessary forms of causation, as well as parallelism, we are now in a position to deal with lesser determinations of causation. Let us first examine partial causation; contingent causation will be dealt with further on.
Partial causation:
- If (C1 + C2), then E;
- if (notC1 + C2), not-then E;
- if (C1 + notC2), not-then E;
- where: (C1 + C2) is possible.
No. | Element/compound | Modus | Source/relationship | ||
1 | C1 | possible | implied by (iii) or (iv) | ||
2 | notC1 | possible | implied by (ii) | ||
3 | C2 | possible | implied by (ii) or (iv) | ||
4 | notC2 | possible | implied by (iii) | ||
5 | E | possible | implied by (i) + (iv) | ||
6 | notE | possible | implied by (ii) or (iii) | ||
7 | C1 | E | possible | implied by (i) + (iv) | |
8 | C1 | notE | possible | implied by (iii) | |
9 | notC1 | E | open | ||
10 | notC1 | notE | possible | implied by (ii) | |
11 | C2 | E | possible | implied by (i) + (iv) | |
12 | C2 | notE | possible | implied by (ii) | |
13 | notC2 | E | open | ||
14 | notC2 | notE | possible | implied by (iii) | |
15 | C1 | C2 | possible | (iv) | |
16 | C1 | notC2 | possible | implied by (iii) | |
17 | notC1 | C2 | possible | implied by (ii) | |
18 | notC1 | notC2 | open | ||
19 | C1 | C2 | E | possible | implied by (i) + (iv) |
20 | C1 | C2 | notE | impossible | (i) |
21 | C1 | notC2 | E | open | |
22 | C1 | notC2 | notE | possible | (iii) |
23 | notC1 | C2 | E | open | |
24 | notC1 | C2 | notE | possible | (ii) |
25 | notC1 | notC2 | E | open | |
26 | notC1 | notC2 | notE | open |
Two phenomena C1, C2 may be called partial causes of some other phenomenon E, only if all the above conditions (i.e. the four defining clauses) are satisfied. In such case, we may call E a contingent effect of each of C1, C2. Of course, the compound (C1 + C2) is a complete cause of E, since in its presence, E follows (as given in clause (i)); and in its absence, i.e. if not(C1 + C2), E does not invariably follow (as evidenced by clauses (ii) and (iii)). Rows 19-26 of the above table (shaded) constitute the matrix of partial causation.
We may thus speak of this phenomenon as a composition of partial causes; and stress that C1 and C2 belong in that particular causation of E by calling them complementary partial causes of it. Indeed, instead of saying “C1 and C2 are complementary partial causes of E”, we may equally well formulate our sentence as “C1 (complemented by C2) is a partial cause of E” or as “C2 (complemented by C1) is a partial cause of E”. These three forms are identical, except for that the first treats C1 and C2 with equal attention, whereas the latter two lay stress on one or the other cause. Such reformatting, as will be seen, is useful in some contexts.
We may make a distinction between absolute and relative partial causation, as follows. The ‘absolute’ form specifies one partial cause without mentioning the complement(s) concerned; it just says: “C1 is a partial cause of E”, meaning “C1 (with some unspecified complement) is a partial cause of E”. This is in contrast to the ‘relative’ form, which does specify a complement, as in the above example of “C1 (complemented by C2) is a partial cause of E”. This distinction reflects common discourse. Its importance will become evident when we consider negations of such forms.
One way to see the appropriateness of our definition of partial causation, its conformity to the paradigm of causation, is by resorting to nesting (see Future Logic, p. 148). We may rewrite it as follows:
From (i) | if C2, then (if C1, then E); |
from (ii) | if C2, then (if notC1, not-then E); |
from (iii) | if notC2, not-then (if C1, then E).[11] |
Clause (i) tells us that given C2, C1 implies E. Clause (ii) tells us that given C2, notC1 does not imply E. Thus, under condition C2, C1 behaves like a complete cause of E. Moreover, clause (iii) shows that under condition notC2, C1 ceases to so behave. Similarly, mutatis mutandis, C2 behaves conditionally like a complete cause of E.[12]
Let us now examine the definition of partial causation more closely. The terminology adopted for it is obviously intended to contrast with that for complete causation.
Clause (i) informs us that in the presence of the two elements C1 and C2 together, the effect is invariably also present. However, that clause alone would not ensure that both C1 and C2 are relevant to E, participants in its causation. We need clause (ii) to establish that without C1, C2 would not by itself have the same result. And, likewise, we need clause (ii) to establish that without C2, C1 would not by itself have the same result.
Suppose, for instance, clause (ii) were false; then, combining it with (i), we would obtain the following simple dilemma:
If (C1 + C2), then E – and – if (notC1 + C2), then E;
therefore, if C2, then E.
That is, C2 would be a complete cause of E, without need of C1, which would in such case be an accident in the relation “If (C1 + C2), then E”, note well. Similarly, if clause (iii) were false, it would follow that C1 is sufficient by itself for E, irrespective of C2. In the special case where both (ii) and (iii) are denied, C1 and C2 would be parallel complete causes of E (compatible ones, since they are conjoined in the antecedent of clause (i)). Therefore, as well as clause (i), clauses (ii) and (iii) have to specified for partial causation.
Furthermore, our definition of partial causation thus mentions three combinations of C1, C2 and their respective negations, namely:
- C1 + C2
- notC1 + C2
- C1 + not C2
And it tells us what happens in relation to E in each of these situations: in the first, E follows; in the next two, it does not. One might reasonably ask, what about the fourth combination, namely:
- notC1 + notC2?[13]
Well, for that, there are only two possibilities: either E follows or it does not. Note first that both these possibilities are logically compatible with clauses (i), (ii) and (iii).
Suppose that “If (notC1 + notC2), then E” is true. In that case, notC1 and notC2 would each have the same relation to E that C1 and C2 have by virtue of clauses (i), (ii), (iii). For if we combine this supposed additional clause with clauses (ii) and (iii), we see that, whereas E follows the conjunction of notC1 and notC2, E does not follow the conjunction of not(notC1) with notC2 or that of notC1 with not(notC2). In that case, we would simply have two, instead of just one, compound causes of E, namely (C1 + C2) and (notC1 + notC2), sharing the same clauses (ii) and (iii) which establish the relevance of each of the elements. Though at first sight surprising, such a state of affairs is quite conceivable, being but a special case of parallel causation! Thus, the proposition “If (notC1 + notC2), then E” may well be true. But may it be false? Suppose that its contradictory “If (notC1 + notC2), not-then E” is true, instead. Here again, the causal significance of the first three clauses remains unaffected. We can thus conclude that what happens in the situation “notC1 + notC2”, i.e. whether E follows or not, is irrelevant to the roles played by C1 and C2. Our definition of partial causation through the said three clauses is thus satisfactory.
Lastly the following should be noted. If we replaced clauses (ii) and (iii) by “If not(C1 + C2), not-then E”, to conform with clause (i) to the definition of complete causation, we would only be sure that the compound (C1 + C2) causes E. It does not suffice to establish that both its elements are involved in that causation, since it could be adequately realized by the eventuality that “If (notC1 + notC2), not-then E”. For this reason, too, clauses (ii) and (iii) are unavoidable.
Regarding clause (iv), which serves to ensure that the first three clauses do not lead to paradox, it is easy to show that the possibility of the conjunction (C1 + C2) is the minimal requirement. For this through clause (i) implies that E is possible and (C1 + C2 + E) is possible. Additionally, clause (ii) means that (notC1 + C2 + notE) is possible, and therefore implies that (notC1 + C2) is possible and each of notC1, C2, notE is possible. Similarly, clause (iii) means that (C1 + notC2 + notE) is possible, and therefore implies that (C1 + notC2) is possible and each of C1, notC2, notE is possible. It is thus redundant to specify these various contingencies.
The methodological principle underlying the definition of partial causation is well known to scientists and oft-used. It is that to establish the causal role of any element such as C1, of a compound (C1 + C2…) in whose presence a phenomenon E is invariably present, we must find out what happens to E when the element C1 is absent while all other elements like C2 remain present. That is, we observe how the putative effect is affected by removal of the putative cause while keeping all other things equal[14]. Only if a change in status occurs (minimally from “then E” to “not-then E”), may the element be considered as participating in the causation, i.e. as a relevant factor.
Once this is understood, it is easy to generalize our definition of partial causation from two factors (C1, C2) to any number of them (C1, C2, C3…), as follows:
- If (C1 + C2 + C3…), then E;
- if (notC1 + C2 + C3…), not-then E;
- if (C1 + notC2 + C3…), not-then E;
- if (C1 + C2 + notC3…), not-then E;
…etc. (if more than three factors);
and (C1 + C2 + C3…) is possible.
Clause (i) establishes the complete causation of the effect E by the compound (C1 + C2 + C3…). But additionally there has to be for each element proof that its absence would be felt: this is the role of clauses (ii), (iii), (iv)…, each of which negates one and only one of the elements concerned. Thus, the number of additional clauses is equal to the number of factors involved.
Whatever the relation to E of other possible combinations of the elements and their negations, the partial causation of E by elements C1, C2, C3… is settled by the minimum number of clauses specified in our definition. As we saw, with two factors the combination “notC1 + notC2” is not significant. Similarly, we can show that with three factors the following combinations are not significant:
- notC1 + notC2 +C3
- notC1 + C2 + notC3
- C1 + notC2 + notC3
- notC1 + notC2 + notC3
And so forth. Generally put, if the number of elements is n, the number of insignificant combinations will be is 2^{n} – (1 + n). Whether any of these further combinations implies or does not imply E does not affect the role of partial causation signified by the defining clauses for the factors C1, C2, C3… per se. Other causations may be involved in certain cases, but they do not disqualify or diminish those so established.
The very last clause, that (C1 + C2 + C3…) is possible, is required and sufficient, for reasons already seen.
Clearly, we can say that the more factors are involved, the weaker the causal bond. If C is a complete cause of E, it plays a big role in the causation of E. If C1 is a partial cause of E, with one complement C2, it obviously plays a lesser role than C. Similarly, the more complements C1 has, like C2, C3…, the less part it plays in the whole causation of E. We may thus view the degree of determination involved as inversely proportional to the number of causes involved, though we may (note well) be able to assign different weights to the various partial causes[15].
Note finally that we can facilitate mental assimilation of multiple (i.e. more than two) partial causes through successive reductions to pairs of partial causes, one of which is compound. Thus, (C1 + C2 + C3 + …) may be viewed as (C1 + (C2 + C3 +…)), provided all the above mentioned conditions are entirely satisfied.
Let us now turn our attention to contingent causation.
Contingent causation:
- If (notC1 + notC2), then notE;
- if (C1 + notC2), not-then notE;
- if (notC1 + C2), not-then notE;
- where: (notC1 + notC2) is possible.
No | Element/compound | Modus | Source/relationship | ||
1 | C1 | possible | implied by (ii) | ||
2 | notC1 | possible | implied by (iii) or (iv) | ||
3 | C2 | possible | implied by (iii) | ||
4 | notC2 | possible | implied by (ii) or (iv) | ||
5 | E | possible | implied by (ii) or (iii) | ||
6 | notE | possible | implied by (i) + (iv) | ||
7 | C1 | E | possible | implied by (ii) | |
8 | C1 | notE | open | ||
9 | notC1 | E | possible | implied by (iii) | |
10 | notC1 | notE | possible | implied by (i) + (iv) | |
11 | C2 | E | possible | implied by (iii) | |
12 | C2 | notE | open | ||
13 | notC2 | E | possible | implied by (ii) | |
14 | notC2 | notE | possible | implied by (i) + (iv) | |
15 | C1 | C2 | open | ||
16 | C1 | notC2 | possible | implied by (ii) | |
17 | notC1 | C2 | possible | implied by (iii) | |
18 | notC1 | notC2 | possible | (iv) | |
19 | C1 | C2 | E | open | |
20 | C1 | C2 | notE | open | |
21 | C1 | notC2 | E | possible | (ii) |
22 | C1 | notC2 | notE | open | |
23 | notC1 | C2 | E | possible | (iii) |
24 | notC1 | C2 | notE | open | |
25 | notC1 | notC2 | E | impossible | (i) |
26 | notC1 | notC2 | notE | possible | implied by (i) + (iv) |
Two phenomena C1, C2 may be called contingent causes of some other phenomenon E, only if all the above conditions (i.e. the four defining clauses) are satisfied. In such case, we may call E a tenuous effect[16] of each of C1, C2. Of course, the compound (notC1 + notC2) is a necessary cause of E, since in its presence, notE follows (as given in clause (i)); and in its absence, i.e. if not(notC1 + notC2), notE does not invariably follow (as evidenced by clauses (ii) and (iii)). Rows 19-26 of the above table (shaded) constitute the matrix of contingent causation.
We may thus speak of this phenomenon as a composition of contingent causes; and stress that that C1 and C2 belong in that particular causation of E by calling them complementary contingent causes of it. Indeed, instead of saying “C1 and C2 are complementary contingent causes of E”, we may equally well formulate our sentence as “C1 (complemented by C2) is a contingent cause of E” or as “C2 (complemented by C1) is a contingent cause of E”. These three forms are identical, except for that the first treats C1 and C2 with equal attention, whereas the latter two lay stress on one or the other cause. Such reformatting, as will be seen, is useful in some contexts.
We may make a distinction between absolute and relative contingent causation, as follows. The ‘absolute’ form specifies one contingent cause without mentioning the complement(s) concerned; it just says: “C1 is a contingent cause of E”, meaning “C1 (with some unspecified complement) is a contingent cause of E”. This is in contrast to the ‘relative’ form, which does specify a complement, as in the above example of “C1 (complemented by C2) is a contingent cause of E”. This distinction reflects common discourse. Its importance will become evident when we consider negations of such forms.
Here again, we can demonstrate that our definition of contingent causation conforms to the paradigm of causation through nesting. We may rewrite it as follows:
From (i) | if notC2, then (if notC1, then notE); |
from (ii) | if notC2, then (if C1, not-then notE); |
from (iii) | if C2, not-then (if notC1, then notE). |
Clause (i) tells us that given notC2, notC1 implies notE. Clause (ii) tells us that given notC2, C1 does not imply notE. Thus, under condition notC2, C1 behaves like a necessary cause of E. Moreover, clause (iii) shows that under condition C2, C1 ceases to so behave. Similarly, mutatis mutandis, C2 behaves conditionally like a necessary cause of E.
Note well that the main clause of contingent causation is not “If not(C1 + C2), then notE”[17], but more specifically “If (notC1 + notC2), then notE”. Considering that in partial causation the antecedent is (C1 + C2) and that this compound behaves as a complete cause, one might think that in contingent causation the antecedent would be a negation of the same compound, i.e. not(C1 + C2), which would symmetrically behave as a necessary cause. But the above demonstration of conformity to paradigm shows us that this is not the case. The explanation is simply that two of the alternative expressions of “If not(C1 + C2), then notE”, namely “If (C1 + notC2), then notE” and “If (notC1 + C2), then notE” are contradictory to clauses (ii) and (iii), respectively. Therefore, only “If (notC1 + notC2), then notE” is a formally appropriate expression in this context. Our definition of contingent causation is thus correct.
We need not repeat our further analysis of partial causation for contingent causation; all that has been said for the former can be restated, mutatis mutandis, for the latter. For partial and contingent causation are ‘mirror images’ of each other. The one represents the positive aspect of weak causation; the other, the negative aspect. All their characteristics are identical, except that the polarities of their respective causes and effect are opposite: C1 is replaced by notC1, C2 by notC2, and E by notE, or vice-versa.
Note that partial and contingent causation each involves a plurality of causes, though in a different sense from that found in parallelism.
We should also mention that partial causation often underlies alternation or plurality of effects.
Consider the form “If C, then (E or E_{1})”, which may be interpreted as “the conjunction (C + notE + notE1) is impossible”, and therefore implies “If (C + notE), then E_{1}” and “If (C + notE_{1}), then E”. Take the latter, for instance, and you have a type (i) clause. If additionally it is true that (notC + notE_{1} + notE), (C + E_{1} + notE), (C + notE_{1}) are possible conjunctions, you have clauses of types (ii), (iii) and (iv), respectively. In such case[18], C is a partial cause of E (the other partial cause being notE_{1} or, more precisely, some complete and necessary cause of notE_{1}).
Just as we may have plurality of effects in partial causation, so we may have it in contingent causation.
Note, concerning the term ‘occasional’. When parallel complete causes may occur separately (i.e. neither implies the other), they are often called occasional causes; however, note well, the same term is often used to refer to partial causes, in the sense that each of them is effective only when the other(s) is/are present. The term occasional effect is used with reference to alternation of effects; i.e. when a cause has alternative effects, each of the latter is occasional; but the term is also applicable more generally, to any effect of a partial cause as such, i.e. to contingent effects.
Partial and contingent causation may conceivably occur in tandem or separately; i.e. no formal inconsistency arises in such cases.
4. Parallelism of Weaks.
Before going further let us here deal with parallelism in relation to the weaker determinations of causation.
In partial causation, this would mean, that there are two (or more) sets of two (or more) partial causes, viz. C1, C2… and C3, C4… (and so forth), with the same effect E:
If (C1 + C2…), then E; etc.
If (C3 + C4…), then E; etc.
…
Clearly, we have ‘plurality of causes’ in both senses of the term at once, here. By “etc.”, I refer to the further clauses involved in partial causation, such as “if (C1 + notC2), not-then E” and so on, here left unsaid to avoid repetitions. Such statements may be merged; thus, the above two become a single statement in which each bracketed conjunction constitutes an alternative complete cause:
If (C1 + C2…) or (C3 + C4…) or…, then E; etc.
The bracketed conjunctions, as we have seen when dealing with parallel complete causes, may be interrelated in various ways except be exhaustive. These interrelations would be expressed in additional statements. The resulting information, including the above statement where all the conjunctions are disjoined in a single antecedent and all statements not explicitated[19] here, can then be analyzed in great detail by tabulating all the items and their negations, and considering the modus of each combination. We can, in this way, have a clear picture of all eventualities, and avoid all ambiguity.
Similarly, mutatis mutandis, for contingent causation:
If (notC1 + notC2…), then notE; etc.
If (notC3 + notC4…), then notE; etc.
…
We may merge these complex causal statements, consider additional specifications regarding the opposition of alternatives, and analyze the mass of information through a table.
Note the following special cases of the above parallelisms.
A partial cause may be found common to two (or more) such causations with the same effect; if say C3 is identical with C1, C1 would have C2… as complement(s) in the first relation and C4… as complement(s) in the second, without problem. But may something (say C1) be a partial cause in one relation and its negation (say, notC1 = C3) a partial cause in the other? Yes, since the negation of E would imply both not(C1 + C2…) and not(notC1 + C4…), which is consistent; except that in such case the two compounds could not occur together.
Similarly, a contingent cause may be found common to two (or more) such causations with the same effect; if say C3 is identical with C1, notC1 would have notC2… as complement(s) in the first relation and notC4… as complement(s) in the second, without problem. But may something (say C1) be a contingent cause in one relation and its negation (say, notC1 = C3) a contingent cause in the other? Yes, since the negation of notE would imply both not(notC1 + notC2…) and not(C1 + notC4…), which is consistent; except that in such case the two compounds could not occur together.
5. The Four Genera of Causation.
We have found the minimal formal definitions of, respectively, complete, necessary, partial and contingent causation. We are now in a position to begin synthesizing our accumulated findings concerning these determinations of causation. Remember how we developed these four concepts….
We started with the paradigm of causation (later named complete and necessary causation). From this we abstracted two constituent forms, or (strong) determinations, which we called complete causation and necessary causation. Then we derived by means of an analogy two additional forms, or (weak) determinations, which we called by way of contrast partial causation and contingent causation.
These four constructs apparently exhaust what we mean by causation, in view of their respective conceptual derivations from the paradigm of causation, and of their symmetry in relation to each other and the whole. No further expressions of the concept of causation, direct or indirect, seem conceivable.
The four forms thus identified can thus be referred to as the genera of causation, or as its generic determinations. And we can safely postulate that:
Nothing can be said to be a cause or effect of something else (in the causative sense), if it is not related to it in the way of at least one of these four genera of causation.
We shall need symbols for these four genera, to facilitate their discussion. I propose (remember them well) the following letters, simply:
n for Necessary causation,
m for coMplete causation (to rhyme with n),
p for Partial causation, and
q for ‘Qontingent’ causation (to rhyme with p)[20].
This notation will be found particularly useful when we deal with causative syllogism. We will also occasionally distinguish between absolute and relative partial or contingent causation, by means of the symbols: p_{abs} and q_{abs }for absolutes (i.e. those not mentioning any complement) and p_{rel} and q_{rel }for relatives (i.e. those specifying some complement). Unless specified as relative, p and q may always be considered absolute.
It follows from what we have just said that we may interpret the causative proposition “P is a cause of Q” as “P is a complete or necessary or partial or contingent cause of Q (or a consistent combination of these alternatives)”.
It is easy to demonstrate that any compounds of the four genera involving both m and p, and/or both n and q, are inconsistent, i.e. formally excluded. That is, one and the same thing cannot be both a complete and partial cause of the same effect; for if clause (i) of m, namely “if C1, then E”, is true, then clause (iii) of p, namely “if (C1 + notC2), not-then E”, cannot be true, and vice-versa. Similarly, necessary and contingent causation, i.e. n and q, are incompatible. We shall see at a later stage that certain other combinations are also formally impossible.
We shall consider the remaining, consistent compounds involving the four generic determinations, which we shall call the specific determinations, in the next chapter.
We may, as already suggested, refer to something as a strong cause, if it is a complete and/or necessary cause; and to something as a weak cause, if it is a partial and/or contingent cause. Conversely, a necessary and/or dependent effect may be said to be a strong effect; and a contingent and/or tenuous effect, it may be said to be a weak effect. Mixtures of these characters are conceivable, as we shall see.
Another classification based on common characters: if something is known to be a complete or partial cause, it may be called a ‘contributing cause’[21]; and if something is known to be a necessary or contingent cause, it may be called a ‘possible cause’. Likewise, if something is known to be a necessary or contingent effect, it may be called a ‘possible effect’; and if something is known to be a dependent or tenuous effect, it may be called (say) a ‘subject effect’.
Moreover: we have characterized complete and partial causation as positive aspects of causation; and necessary and contingent causation as its negative aspects, comparatively. We may in this sense, relative to a given set of items, speak of ‘positive’ or ‘negative’ causation. The latter, of course, should not be confused with negations of causation. Accordingly, we may refer to positive or negative causes or effects.
The reader is referred to the Appendix on J. S. Mill’s Methods, for comparison of our treatment of causation in this chapter (and the next).
6. Negations of Causation.
So far, we have only considered in detail positive causative propositions, i.e. statements affirming causation of some determination. We must now look at negative causative propositions, i.e. statements denying causation of some determination or any causation whatever. For this purpose, to avoid the causal connotations implied by use of symbols like C and E for the items involved, we shall rather use neutral symbols like P and Q.
Statements denying causation may be better understood by studying the negations of conditional propositions.
A ‘positive hypothetical’ proposition has the form “If X, then Y” (which may be read as X implies Y, or X is logically followed by Y); it means by definition “the conjunction (X + notY) is impossible”. Its contradictory is a ‘negative hypothetical’ proposition of the form “If X, not-then Y”[22] (which may be read as X does not imply Y, or X is not logically followed by Y); it means by definition “the conjunction (X + notY) is possible”.
In the positive form, though X and notY are together impossible, they are not implied (or denied) to be individually impossible. In the negative form, since X and notY are possible together, each of X, notY is also formally implied as possible. In either form, there is no formal implication that notX be possible or impossible, or that Y be possible or impossible. As for the remaining conjunctions (X + Y), (notX + Y), (notX + notY) – nothing can be inferred concerning them, either.
However, as we have seen, when such statements appear as implicit clauses of causation, the interactions between clauses will inevitably further specify the situation for many of the items concerned.
The negation of complete causation or necessary causation, through statements like “P is not a complete cause of Q” or “P is not a necessary cause of Q”, is feasible if any one or more of the three constituent clauses of such causation is deniable. That is, such negation consists of a disjunctive proposition saying “not(i) and/or not(ii) and/or not(iii)”, which may signify non-causation or another determination of causation (necessary instead of complete, or vice-versa, or a weaker form of causation).
To give an example: the denial of “P is a complete cause of Q” means “if P, not-then Q” and/or “if notP, then Q” and/or “P is impossible”. These alternatives may give rise to different outcomes; in particular note that if “P is impossible” is true, then P cannot be a cause at all, and if “if P, then Q” and “if notP, then Q” are both true, then Q is necessary, in which case Q cannot be an effect at all.
The negation of strong causation as such means the negation of both complete and necessary causation.
With regard to negation of partial or contingent causation, we must distinguish two degrees, according as a given complement is intended or any complement whatever.
The more restricted form of negation of partial causation or contingent causation mentions a complement, as in statements like “P1 (complemented by P2) is not a partial cause of Q” or “P1 (complemented by P2) is not a contingent cause of Q”. Such negation is feasible if any one or more of the four constituent clauses of such causation is deniable. That is, such negation consists of a disjunctive proposition saying “not(i) and/or not(ii) and/or not(iii) and/or not(iv)”.
In contrast, note well, the negation of partial causation or contingent causation through statements like “P1 is not a partial cause of Q” or “P1 is not a contingent cause of Q”, is more radical. “P1 is not a partial cause of Q” means “P1 (with whatever complement) is not a partial cause of Q” – it may thus be viewed as a conjunction of an infinite number of more restricted statements, viz. “P1 (complemented by P2) is not a partial cause of Q, and P1 (complemented by P3) is not a partial cause of Q, and… etc.”, where P2, P3, etc. are all conceivable complements. Similarly with regard to “P1 is not a contingent cause of Q”.
A restricted negative statement is very broad in its possible outcomes: it may signify that P1 is not a cause of Q at all, or that P1 is instead a complete or necessary cause of Q, or that P1 is a weak cause of Q but a contingent rather than partial one or a partial rather than contingent one, or that P1 is a partial or contingent cause (as the case may be) of Q but with some complement other than P2.
A radical negative statement comprises many restricted ones, and is therefore less broad in its possible outcomes, specifically excluding that P1 be involved in a partial or contingent causation (as the case may be) with any complement(s) whatsoever. A restricted negation is relative to a complement (say, P2); a radical negation is a generality comprising all similar restricted negations for the items concerned (P1, Q), and is therefore relative to no complement (neither P2, nor P3, etc.).
The negation of weak causation as such means the negation of both partial and contingent causation, either in a restricted sense (i.e. relative to some complement) or in a radical sense (i.e. irrespective of complement).
This brings us to the relation of non-causation, which is also very complex.
As we saw, the positive causative proposition “P is a cause of Q” may be interpreted as “P is a complete or necessary or partial or contingent cause of Q”. Accordingly, we may interpret the negative causative proposition “P is not a cause of Q” as “P is not a complete and not a necessary and not a partial and not a contingent cause of Q”, i.e. as a denial of all four genera of causation in relation to P and Q (with whatever complement).
It is noteworthy that we cannot theoretically define non-causation except through negation of all the concepts of causation, which have to be defined first[23]. In contrast, on a practical level, we proceed in the opposite direction: in accord with general rules of induction, we presume any two items P and Q to be without causative relation, until if ever we can establish inductively or deductively that a causative relation obtains between them.[24]
Nevertheless, ‘non-causation’ refers to denial of causation, and is not to be confused with ignorance of causation; it is an ontological, not an epistemological concept.
Note well that non-causation is not defined by the propositions “if P, not-then Q, and if notP, not-then notQ”. Such a statement, though suggestive of non-causation, is equally compatible with partial and/or contingent causation; so it cannot suffice to distinguish non-causation. To specify a relation of non-causation, we have to deny every determination of causation.
Furthermore, “P is not a cause of Q” refers to relative non-causation – it is relative to the items P and Q specifically, and does not exclude that Q may have some other cause P_{1}, or that P may have some other effect Q_{1}. Two items, say P and Q, taken at random, need not be causatively related at all (even in cases where they happen to be respectively causatively related to some third item, as will be seen when we study syllogism in later chapters). In such case, P and Q are called accidents of each other; their eventual conjunction is called a coincidence.
Relative non-causation is an integral part of the formal system of deterministic causality. We have to acknowledge the possibility, indeed inevitability, of such a relation. If I say “the position of stars does not affect[25] people’s destinies”, I mean that there is no causal relation specifically between stars and people; yet I may go on to say that stars affect other things or that people are affected by other things, without contradicting myself.
Relative non-causation should not be confused with absolute non-causation. The causelessness of some item A would be expressed as “nothing causes A”, a proposition summarizing innumerable statements of the form “B does not cause A; C does not cause A;…etc.”, where B, C,… are all existents other than A. Similarly, the effectlessness of some item A would be expressed as “nothing is caused by A”, a proposition summarizing innumerable statements of the form “B is not caused by A; C is not caused by A;… etc.”, where B, C,… are all existents other than A.
We thus see that whereas positive causative propositions are defined by conjunctions of clauses, negative ones are far more complex in view of their involving disjunctions.
The negations of determinations, or the negation altogether of causation, should not themselves be regarded as further determinations, since they by their breadth allow for non-causation (between the items concerned), note.
Chapter 3. The Specific Determinations
1. The Species of Causation.
We shall now look into the consistent combinations of the four genera of causation, symbolized as m, n, p, q, with each other or their negations. Implicit in our gradual development of these concepts of causation from a common paradigm, was the idea that they are abstractions, indefinite concepts that are eventually concretized in the more specific and definite compounds.
We have already found some of their combinations, namely mp and nq to be inconsistent. This was due to incompatibilities between clauses of their definitions, or in other words, certain rows of their matrices. Thus, row 6 of m (C + notE is impossible) is in conflict with row 22 (C1 + notE is possible) of p; similarly, row 7 of n (notC + E is impossible) is in conflict with row 23 (notC1 + E is possible) of q.
It is also possible to prove certain other combinations to be logically impossible. This can be done formally, but not at the present stage of development, because we do not yet have the technical means at this stage to treat negations of generic determinations. To define notm, notn, notp, notq in verbal terms would be extremely arduous and confusing. I will therefore for now merely affirm to you that combinations of any one positive generic determination with the negations of the three other generic determinations, for the very same terms, are inconsistent.
By elimination, we are left with only four consistent compounds, i.e. remaining combinations give rise to no inconsistency, i.e. whose respective clauses do not contradict each other. This means that, from the logical point of view, they are conceivable, and therefore worthy of further formal treatment. We may refer to them as the specific determinations, or species of causation.
The following table (where + and – signify, respectively, affirmation and denial of a determination) lists all combinations of the generics and identifies the logically possible specifics among them:
No. of genera | Compound | m | n | p | q | Modus |
Four | mnpq | + | + | + | + | mp, nq impossible |
Three | mnp | + | + | + | – | mp impossible |
mnq | + | + | – | + | nq impossible | |
mpq | + | – | + | + | mp impossible | |
npq | – | + | + | + | nq impossible | |
Two | mp | + | – | + | – | mp impossible |
nq | – | + | – | + | nq impossible | |
mn | + | + | – | – | possible | |
mq | + | – | – | + | possible | |
np | – | + | + | – | possible | |
pq | – | – | + | + | possible | |
Only one | m-alone | + | – | – | – | will be proved impossible |
n-alone | – | + | – | – | will be proved impossible | |
p-alone | – | – | + | – | will be proved impossible | |
q-alone | – | – | – | + | will be proved impossible | |
None | non causation | – | – | – | – | possible |
The formulae given in the above table for each specific determination is as brief as possible. For instance, since m implies the negation of p and n implies the negation of q, ‘mn’ (meaning both complete and necessary causation) tacitly implies ‘notp and notq’ (neither partial nor contingent causation, with whatever complement); the latter negations need not therefore be mentioned. Similarly, an expression like m-alone signifies the affirmation of one generic determination (here, m) and the denial of all three others (i.e. notn and notq, as well as notp). This notation is far from ideal, but suffices for our current needs, since many combinations are eliminated at the outset.
We see that four specific determinations, namely mn, mq, np, pq, are formed by conjunction of positive causative propositions; these we shall call (following J. S. Mill’s nomenclature) joint determinations. It follows from the above table that each generic determination has only two species. Each generic determination may therefore be interpreted as a disjunction of its two possible embodiments; thus, m means mn or mq; n means mn or np; p means np or pq; and q means mq or pq. Also note, we could refer to mn as ‘only-strong causation’ and to pq ‘only-weak causation’, while mq and np are ‘mixtures of strong and weak’.
The four specific determinations formed by composing positive causative propositions with negative ones, namely m-alone, n-alone, p-alone, q-alone, will be called lone determinations. This expression is introduced at this stage to contrast it with generic and joint determinations. Clearly, one should not confuse an isolated generic symbol such as m with the corresponding specific symbol m-alone; I use this heavy notation to ensure no confusion arises. Moreover, nota bene: In the above table, these forms are eliminated at the outset, because they concern absolute partial or contingent causation, i.e. they are irrespective of complement and mean m-alone_{abs} etc. But as we shall later see, when they involve relative partial or contingent causation, i.e. when some complement is specified (in p_{rel} or q_{rel} or their negations), so that they mean m-alone_{rel} etc., they remain possible forms. This need not concern us at the moment, but is said to explain why these forms need to be named.
We would label as, simply, causation (or ‘any causation’), the disjunctive proposition “m or n or p or q”, or the more specific “mn or mq or np or pq”. Such positive propositions merely imply causation, if they involve less disjuncts or an isolated generic or joint determination. The contradictory of causation, non-causation, is the only remaining allowable combination, our table being exhaustive. This last possible combination involves negation of all four generic or joint determinations, note well. That is, it means “neither m nor n nor p nor q” or equally “neither mn nor mq nor np nor pq”.
The above table also allows us to somewhat interpret complex negations. The negation of any compound is equivalent to the disjunction of all remaining four compounds (three of causation and one of non-causation). For instance “not(mn)” means mq, np, pq, or non-causation. Similarly with any other formula.
Note that where one of the weak determinations is denied by reason of the affirmation of the contrary strong determination (m in the case of p, or n in the case of q), any and all proposed complements are denied. Where one of the weaks is affirmed (even if the other is radically denied), at least one complement is implied; and of course, the contrary strong determination is denied. In all other cases, we must remember to be careful and distinguish between restricted and radical negations of p or q, as already explained in the previous chapter.
2. The Joint Determinations.
We shall now examine in detail the four joint determinations, symbolized by mn, mq, np, and pq, each of which is obtained by consistent conjunction of two generic determinations. Each is thus a species shared by the two genera constituting it. Thus, mn is a specific case of m and a specific case of n; and so forth.
We have already encountered one of these joint determinations, viz. complete and necessary causation, the paradigm of causation. We shall now examine it in further detail, and also treat the other three joint determinations.
Complete and Necessary causation by C of E:
- If C, then E;
- if notC, not-then E (may be left tacit);
- where: C is possible.
And:
- if notC, then notE;
- if C, not-then notE (may be left tacit);
- where: C is unnecessary.
No. | Element/compound | Modus | Source/relationship | |
1 | C | Possible | (iii) | |
2 | notC | Possible | (vi) | |
3 | E | Possible | implied by (v) | |
4 | notE | possible | implied by (ii) | |
5 | C | E | possible | (v) or implied by (i) + (iii) |
6 | C | notE | impossible | (i) |
7 | notC | E | impossible | (iv) |
8 | notC | notE | possible | (ii) or implied by (iv) + (vi) |
Notice how the merger of clauses (i), (ii) and (iii) with (iv), (v) and (vi) renders clauses (ii) and (v) redundant (though still implicit). Rows 5-8 of the above table (shaded) constitute the matrix of complete-necessary causation.
Complete but Contingent causation by C1 of E:
- If C1, then E;
- if notC1, not-then E (may be left tacit);
- where: C1 is possible (may be left tacit).
And:
- if (notC1 + notC2), then notE;
- if (C1 + notC2), not-then notE;
- if (notC1 + C2), not-then notE;
- where: (notC1 + notC2) is possible.
No. | Element/compound | Modus | Source/relationship | ||
1 | C1 | possible | (iii) or implied by (v) | ||
2 | notC1 | possible | implied by (vi) or (vii) | ||
3 | C2 | possible | implied by (vi) | ||
4 | notC2 | possible | implied by (v) or (vii) | ||
5 | E | possible | implied by (v) or (vi) | ||
6 | notE | possible | implied by (iv) + (vii) | ||
7 | C1 | E | possible | implied by (v) | |
8 | C1 | notE | impossible | (i) | |
9 | notC1 | E | possible | implied by (vi) | |
10 | notC1 | notE | possible | (ii) or implied by (iv) + (vii) | |
11 | C2 | E | possible | implied by (vi) | |
12 | C2 | notE | open | if #12 is impossible, so is #24; and in view of (i): if #12 is possible, so is #24 | |
13 | notC2 | E | possible | implied by (v) | |
14 | notC2 | notE | possible | implied by (iv) + (vii) | |
15 | C1 | C2 | open | if #15 is impossible, so is #19; and in view of (i): if #15 is possible, so is #19 | |
16 | C1 | notC2 | possible | implied by (v) | |
17 | notC1 | C2 | possible | implied by (vi) | |
18 | notC1 | notC2 | possible | (vii) | |
19 | C1 | C2 | E | open | if #19 is possible, so is #15; and in view of (i): if #19 is impossible, so is #15 |
20 | C1 | C2 | notE | impossible | implied by (i) |
21 | C1 | notC2 | E | possible | (v) |
22 | C1 | notC2 | notE | impossible | implied by (i) |
23 | notC1 | C2 | E | possible | (vi) |
24 | notC1 | C2 | notE | open | if #24 is possible, so is #12; and in view of (i): if #24 is impossible, so is #12 |
25 | notC1 | notC2 | E | impossible | (iv) |
26 | notC1 | notC2 | notE | possible | implied by (iv) + (vii) |
Notice how the merger of clauses (i), (ii) and (iii) with (iv), (v), (vi) and (vii) renders clauses (ii) and (iii) redundant (though still implicit). Rows 19-26 of the above table constitute the matrix of complete-contingent causation.
Concerning the four positions labeled open in the above table, note that the moduses of Nos. 12 and 24 are tied and likewise those of Nos. 15 and 19. Proof for the first two: if #12 (C2 + notE) is impossible, #24 (notC1 + C2 + notE) must also be impossible; if #24 (notC1 + C2 + notE) is impossible, then knowing #20 (C1 + C2 + notE) to be impossible, #12 (C2 + notE) must also be impossible; the rest follows by contraposition. Proof for the other two: if #15 (C1 + C2) is impossible, #19 (C1 + C2 + E) must also be impossible; if #19 (C1 + C2 + E) is impossible, then knowing from (i) that #20 (C1 + C2 + notE) is impossible, #15 (C1 + C2) must also be impossible; the rest follows by contraposition. The interpretation of these open cases is as follows.
(a) Suppose #12 is impossible; this means that “If C2, then E”. We know from #14 that “If notC2, not-then E”; and from #3 that “C2 is possible”. Whence, C2 satisfies the definition for being a complete cause of E, just like C1. Thus, in such case, C1 and C2 are simply parallel complete (and contingent) causes of E. This is quite conceivable, and as we have seen in an earlier section such causes may be compatible or incompatible. If #15 is possible, they are compatible; and if #15 is impossible, they are incompatible.
(b) Suppose #12 is possible; this means that “If C2, not-then E”, in which case C2 is not a complete cause of E. This is quite conceivable, covering situations where one of the contingent causes (namely, C1) is also complete, while the other (C2) is not complete. Additionally, we can say: if #15 is possible, they are compatible; and if #15 is impossible, they are incompatible; there is no problem of consistency either way.
However, a very interesting question arises in such case: is a contingent but not complete cause (like C2, here) bound to be a partial cause? C2 is certainly not a partial cause of E in conjunction with C1, since C1 is a complete cause of E. Therefore, if C2 is a partial cause of E, it will be so in conjunction with some other partial cause of E, say C3. But since C3 is unmentioned in our original givens, its existence is not formally demonstrable. We thus have no certainty that an incomplete contingent cause is implicitly a partial contingent cause! We will return to this issue later.
Partial yet Necessary causation by C1 of E:
- If notC1, then notE;
- if C1, not-then notE (may be left tacit);
- where: C1 is unnecessary (may be left tacit).
And:
- if (C1 + C2), then E;
- if (notC1 + C2), not-then E;
- if (C1 + notC2), not-then E;
- where: (C1 + C2) is possible.
No. | Element/compound | Modus | Source/relationship | ||
1 | C1 | possible | implied by (vi) or (vii) | ||
2 | notC1 | possible | (iii) or implied by (v) | ||
3 | C2 | possible | implied by (v) or (vii) | ||
4 | notC2 | possible | implied by (vi) | ||
5 | E | possible | implied by (iv) + (vii) | ||
6 | notE | possible | implied by (v) or (vi) | ||
7 | C1 | E | possible | (ii) or implied by (iv) + (vii) | |
8 | C1 | notE | possible | implied by (vi) | |
9 | notC1 | E | impossible | (i) | |
10 | notC1 | notE | possible | implied by (v) | |
11 | C2 | E | possible | implied by (iv) + (vii) | |
12 | C2 | notE | possible | implied by (v) | |
13 | notC2 | E | open | if #13 is impossible, so is #21; and in view of (i): if #13 is possible, so is #21 | |
14 | notC2 | notE | possible | implied by (vi) | |
15 | C1 | C2 | possible | (vii) | |
16 | C1 | notC2 | possible | implied by (vi) | |
17 | notC1 | C2 | possible | implied by (v) | |
18 | notC1 | notC2 | open | if #18 is impossible, so is #26; and in view of (i): if #18 is possible, so is #26 | |
19 | C1 | C2 | E | possible | implied by (iv) + (vii) |
20 | C1 | C2 | notE | impossible | (iv) |
21 | C1 | notC2 | E | open | if #21 is possible, so is #13; and in view of (i): if #21 is impossible, so is #13 |
22 | C1 | notC2 | notE | possible | (vi) |
23 | notC1 | C2 | E | impossible | implied by (i) |
24 | notC1 | C2 | notE | possible | (v) |
25 | notC1 | notC2 | E | impossible | implied by (i) |
26 | notC1 | notC2 | notE | open | if #26 is possible, so is #18; and in view of (i): if #26 is impossible, so is #18 |
Notice here again how the merger of clauses (i), (ii) and (iii) with (iv), (v), (vi) and (vii) renders clauses (ii) and (iii) redundant (though still implicit). Rows 19-26 of the above table (shaded) constitute the matrix of partial-necessary causation.
Concerning the four positions labeled open in the above table, note that the moduses of Nos. 13 and 21 are tied and likewise those of Nos. 18 and 21. These statements may be proved in the same manner as done for the preceding table; this is left to the reader as an exercise. We can also interpret these situations in similar ways. If #13 is impossible, C2 is a partial and necessary cause of E, parallel to C1; and notC2 is either compatible or incompatible with notC1 according to whether #18 is possible or impossible. If #13 is possible, C2 is a partial but not necessary cause of E, and notC2 is either compatible or not with notC1, according to whether #18 is possible or not.
However, it is not formally demonstrable that an unnecessary partial cause is implicitly a contingent partial cause; and the implications of this finding (or absence of finding) will have to be considered later.
Partial and Contingent causation by C1 of E:
- If (C1 + C2), then E;
- if (notC1 + C2), not-then E;
- if (C1 + notC2), not-then E;
- where: (C1 + C2) is possible.
And:
- if (notC1 + notC2), then notE;
- if (C1 + notC2), not-then notE;
- if (notC1 + C2), not-then notE;
- where: (notC1 + notC2) is possible.
No. | Element/compound | Modus | Source/relationship | ||
1 | C1 | possible | implied by (iii) or (iv) or (vi) | ||
2 | notC1 | possible | implied by (ii) or (vii) or (viii) | ||
3 | C2 | possible | implied by (ii) or (iv) or (vii) | ||
4 | notC2 | possible | implied by (iii) or (vi) or (viii) | ||
5 | E | possible | implied by (vi) or (vii) | ||
6 | notE | possible | implied by (ii) or (iii) | ||
7 | C1 | E | possible | implied by (vi) | |
8 | C1 | notE | possible | implied by (iii) | |
9 | notC1 | E | possible | implied by (vii) | |
10 | notC1 | notE | possible | implied by (ii) | |
11 | C2 | E | possible | implied by (vii) | |
12 | C2 | notE | possible | implied by (ii) | |
13 | notC2 | E | possible | implied by (vi) | |
14 | notC2 | notE | possible | implied by (iii) | |
15 | C1 | C2 | possible | (iv) | |
16 | C1 | notC2 | possible | implied by (iii) or (vi) | |
17 | notC1 | C2 | possible | implied by (ii) or (vii) | |
18 | notC1 | notC2 | possible | (viii) | |
19 | C1 | C2 | E | possible | implied by (i) + (iv) |
20 | C1 | C2 | notE | impossible | (i) |
21 | C1 | notC2 | E | possible | (vi) |
22 | C1 | notC2 | notE | possible | (iii) |
23 | notC1 | C2 | E | possible | (vii) |
24 | notC1 | C2 | notE | possible | (ii) |
25 | notC1 | notC2 | E | impossible | (v) |
26 | notC1 | notC2 | notE | possible | implied by (v) + (viii) |
Rows 19-26 of the above table (shaded) constitute the matrix of partial-contingent causation. We note that here none of the original clauses are made redundant by the combination of partial and contingent causation. Furthermore, no position in the above table is left open, with regard to the possibility or impossibility of the item or combination concerned.
Additionally we can say that if C1 and C2 are, as here, complementary partial contingent causes of E, then they have the same set of relations to each other and to E. But this does not mean that if C1 and C2 are complementary partial causes of E, they are bound to be complementary contingent causes of E, since as we have seen both or just one of them may be necessary cause(s) of E. Similarly, we cannot say that if C1 and C2 are complementary contingent causes of E, they are bound to be complementary partial causes of E, since as we have seen both or just one of them may be complete cause(s) of E.
There may, of course, be more than one complement to C1 (i.e. complements C3, C4…, in addition to C2) in the last three joint determinations, mq, np or pq. Such cases may be similarly treated, as we have explained when considering the weaker generic determinations separately.
It is with reference to the joint determinations mq and np that the utility of reformatting sentences about partial or contingent causation becomes apparent. An mq proposition is best stated as “C1 is a complete and (complemented by C2) a contingent cause of E”, and an np proposition is best stated as “C1 is a necessary and (complemented by C2) a partial cause of E”.
We must now consider the hierarchy between the above four forms, since there are clearly differences in degree in the ‘bond’ between cause(s) and effect. Causation is obviously at its strongest when both complete and necessary (mn). It is difficult to say which of the next two forms (mq or np) is the stronger and which the weaker, they are not really comparable to each other; all we can say is that they are both less determining than the first and more determining than the last; let us call them middling determinations. Causation is weakest for each factor involved in partial and contingent causation (pq).
With regard to parallelism, we can infer that it is conditionally possible with reference to our previous findings in the matter.
Two complete-necessary causes, C, C_{1}, of the same effect E, may be parallel, provided they are neither exhaustive nor incompatible with each other, i.e. provided “if C, not-then notC_{1} and if notC, not-then C_{1}” is true.
For complete-contingent causation, it is conceivable that C1, C2 have this relation to E and C3, C4 have this same relation to E, provided the complete causes C1 and C3 are not exhaustive and the compounds (notC1 + notC2) and (notC3 + notC4) are not exhaustive. An interesting special case is when C2 = C4, i.e. when the two complete causes have the same complement in the contingent causation of E.
For partial-necessary causation, it is conceivable that C1, C2 have this relation to E and C3, C4 have this same relation to E, provided the necessary causes C1 and C3 are not incompatible and the compounds (C1 + C2) and (C3 + C4) are not exhaustive. An interesting special case is when C2 = C4, i.e. when the two necessary causes have the same complement in the partial causation of E.
For partial-contingent causation, the same condition of non-exhaustiveness between the parallel compounds involved applies. And here, too, note the special case when C2 = C4 as interesting.
Tables involving all the items concerned and their negations in all combinations may be constructed to analyze the implications of such parallelisms in detail.
The negations of the four joint determinations may be reduced to the denial of one or both of their constituent generic determinations. That is, not(mn) means ‘not-m and/or not-n’; not(mq) means ‘not-m and/or not-q’; not(np) means ‘not-p and/or not-n’; and not(pq) means ‘not-p and/or not-q’. Each of these alternative denials in turn implies denial of one or more of the constituent clauses, obviously.
3. The Significance of Certain Findings.
Let us review how we have proceeded so far. We started with the paradigm of causation, namely, complete necessary causation. We then abstracted its constituent “determinations”, the complete and the necessary aspects of it, and by negation formulated another two generic determinations, namely partial and contingent causation. We then recombined these abstractions, to obtain all initially conceivable formulas. Some of these formulas (mp, nq) could be eliminated as logically impossible by inspecting their definitions and finding contradictory elements in them. Others (the lone determinations, obtained by conjunction of only one generic determination and the negations of all three others) were eliminated on the basis of later findings not yet presented here. This left us with only five logically tenable specific causative relations between any two items, namely the four joint determinations (the consistent conjunctions generic determinations) and non-causation (the negation of all four generic determinations).
When I personally first engaged in the present research, I was not sure whether or not the (absolute) lone determinations were consistent or not. Because each lone determination involves three negative causative propositions in conjunction, and each of these is defined by disjunction of the negations of the defining clauses of the corresponding positive form, it seemed very difficult to reliably develop matrixes for them. I therefore, as a logician[26], had to assume as a working hypothesis that they were logically possible. It is only in a later phase, when I developed “matricial microanalysis” that I discovered that they can be formally eliminated. Take my word on this for now. This discovery was very instructive and important, because it signified that causation is more “deterministic” than would otherwise have been the case.
If lone determinations had been logically possible, causation would have been moderately deterministic. For two items might be causatively related on the positive side, but not on the negative side, or vice-versa. Something could be only a complete cause (or only a partial cause) of another without having to also be a necessary or contingent one; or it could be only a necessary cause (or only a contingent cause) of another without having to also be a complete or partial one. But as it turned out there is logically no such degree of freedom in the causative realm.
If two things are causatively related at all, they have to be ultimately related in one (and indeed only one) of the four ways described as the joint determinations[27], i.e. in the way of mn, mq, np, or pq. The concepts m, n, p, q are common aspects of these four relations and no others. There is no “softer” causative relation. Causation is “full” or it is not at all; no “holes” are allowed in it. We can formulate the following “laws of causation” in consequence:
- If something is a complete or partial cause of something, it must also be either a necessary or (with some complement or other) a contingent cause of it.
- If something is a necessary or contingent cause of something, it must also be either a complete or (with some complement or other) a partial cause of it.
- In short, since a lone determination is impossible, if something is at all a causative of anything, it must be related in the way of a joint determination with it.
These laws have the following corollaries:
- If something is neither a necessary nor contingent cause of something, it must also be neither a complete nor (with whatever complement) a partial cause of it.
- If something is neither a complete nor partial cause of something, it must also be either neither a necessary nor (with whatever complement) a contingent cause of it.
- In short, since a lone determination is impossible, if two things are known not to be related in the way of either pair of contrary generic determinations (i.e. m and p, or n and q), they can be inferred to be not causatively related at all.
Also:
- The complement of a partial cause of something, being also itself a partial cause of that thing, must either be a necessary or (with some complement or other) a contingent cause of that thing.
- The complement of a contingent cause of something, being also itself a contingent cause of that thing, must either be a complete or (with some complement or other) a partial cause of that thing.
With regard to the epistemological question, as to how these causative relations are to be established, we may say that they are ultimately based on induction (including deduction from induced propositions): we have no other credible way to knowledge. Causative propositions may of course be built up gradually, clause by clause (see definitions in the previous chapter).
As I showed in my work Future Logic, the positive hypothetical (i.e. if/then) forms, from which causatives are constructed, result from generalizations from experience of conjunctions between the items concerned (which generalizations are of course revised by particularization, when and if they lead to inconsistency with new information). The negative hypothetical (i.e. if/not-then) forms are assumed true if no positive forms have been thus established, or are derived by the demands of consistency from positive forms thus established. In their case, an epistemological quandary may be translated into an ontological fait accompli (at least until if ever reason is found to prefer a positive conclusion).
We may first, by such induction (or deduction thereafter), propose one of the four generic determinations in isolation. The proposed generic determination is effectively treated as a joint determination “in-waiting”, a convenient abstraction that does not really occur separately, but only within conjunctions. We are of course encouraged by methodology to subsequently vigorously research which of the four joint determinations can be affirmed between the items concerned. In cases where all such research efforts prove fruitless, we are simply left with a problematic statement, such as (to give an instance) “P is a complete cause, and either a necessary or a contingent cause, of Q”.
But, since lone determination does not exist, we can never opt for a negative conclusion, like “P is a complete cause, but neither a necessary nor a contingent cause, of Q”. We may not in this context effectively generalize from “I did not find” to “there is not” (a further causative relation). We may not interpret a structural doubt as a negative structure, an uncertainty as an indeterminacy.
In the history of Western philosophy, until recent times, the dominant hypothesis concerning causation has been that it is applicable universally. Some philosophers mitigated this principle, reserving it for ‘purely physical’ objects, excepting beings with volition (humans, presumably G-d, and even perhaps higher animals). A few, notably David Hume, denied any such “law of causation” as it has been called.
But in the 20th Century, the idea that there might, even in Nature (i.e. among entities without volition), be ‘spontaneous’ events gained credence, due to unexpected developments in Physics. That idea tended to be supported by the Uncertainty Principle of Werner Heisenberg for quantum phenomena, interpreted by Niels Bohr as an ontological (and not merely epistemological) principle of indeterminacy, and the Big-Bang theory of the beginning of the universe, which Stephen Hawking considered as possibly implying an ex nihilo and non-creationist beginning.
We shall not here try to debate the matter. All I want to do at this stage is stress the following nuances, which are now brought to the fore. The primary thesis of determinism is that there is causation in the world; i.e. that causal relations of the kind identified in the previous chapter (the four generic determinations) do occur in it. Our above-mentioned discovery that such causation has to fit in one of the four specific determinations may be viewed as a corollary of this thesis, or a logically consistent definition of it.
This is distinct from various universal causation theses, such as that nothing can occur except through causation (implying that causation is the only existing form of causality), or that at least nothing in Nature can do so (though for conscious beings other forms of causality may apply, notably volition), among others.
We shall analyze such so-called laws of causation in a later chapter; suffices for now to realize that they are extensions, attempted generalizations, of the apparent fact of causation, and not identical with it. Many philosophers seem to be unaware of this nuance, effectively regarding the issue as either ‘causation everywhere’ or ‘no causation anywhere’.
The idea that causation is present somewhere in this world is logically quite compatible with the idea that there may be pockets or borders where it is absent, a thesis we may call ‘particular (i.e. non-universal) causation’. We may even, more extremely, consider that causation is poorly scattered, in a world moved principally by spontaneity and/or volition.
The existence of causation thus does not in itself exclude the spontaneity envisaged by physicists (in the subatomic or astronomical domains); and it does not conflict with the psychological theory of volition or the creationist theory of matter[28].
Apparently, then, though determinism may be the major relation between things in this world, it leaves some room, however minor (in the midst or at the edges of the universe), for indeterminism.
We will give further consideration to these issues later, for we cannot deal with them adequately until we have clarified the different modes of causation.
Chapter 4. Immediate Inferences
1. Oppositions.
The logical interrelations between the truths and falsehoods of propositions involving the same items are referred to as their ‘oppositions’. This expression is unfortunate, because in everyday speech (and often in logical discourse) it connotes more specifically a ‘conflict’ between propositions; whereas in the science of logic, the term is intended more broadly as a ‘face-off’. Thus, the possible oppositions between two propositions are:
- contradiction (provided they cannot be both true and they cannot be both false);
- contrariety (provided they cannot be both true);
- subcontrariety (provided they cannot be both false);
- subalternation (provided one, called the subalternant, cannot be true if the other, called the subaltern, is false – though the latter may be true if the former is false);
- equivalence (provided neither can be true if the other is false, and neither can be false if the other is true);
- neutrality (provided either can be true or false without the other being true or false).
Contradictory or contrary propositions are incompatible or mutually exclusive; propositions otherwise related are compatible or conjoinable. Equivalents mutually imply each other; among subalternatives, the subalternant implies but is not implied by the subaltern. Propositions are neutral to each other if they are not related in any of the other ways above listed.
Let us now consider the oppositions between the four generic determinations. We can show, with reference to the definitions in the preceding chapter, that:
- If “P is a complete cause of Q”, then “(whatever P is complemented by) P is not a partial cause of Q”;
- if “P (with whatever complement) is a partial cause of Q”, then “P is not a complete cause of Q”.
- If “P is a necessary cause of Q”, then (whatever P is complemented by) “P is not a contingent cause of Q”;
- if “P (with whatever complement) is a contingent cause of Q”, then “P is not a necessary cause of Q”.
For clause (i) of complete causation, viz. “if P, then Q”, implies both “if (P + R), then Q” and “if (P + notR), then Q”, for any item R whatsoever; whereas clause (iii) of partial causation implies that there is an item R such that “if (P + R), then Q”. Similarly, necessary and contingent causation have conflicting implications, and therefore cannot both be true.
More briefly put, m and p are incompatible and n and q are incompatible. Other than that, no incompatibilities or implications exist between the four generic determinations. P may have no causative relation to Q at all, without any inconsistency ensuing. Thus, m and p are contrary (but not contradictory) and n and q are contrary (but not contradictory). As for the pairs m and n, m and q, n and p, p and q – they are all neutral to each other.
With regard to the negations of generic determinations, it follows that they are all neutral to each other. Of course, by definition, not-m is the contradictory of m, not-n is the contradictory of n, not-p is the contradictory of p, and not-q is the contradictory of q. Also, as above seen, m implies not-p, and n implies not-q. But between the four negations themselves, no incompatibilities or implications exist.
It should be stressed that partial causation is not to be considered as identical with the negation of complete causation, but only as one of the possible outcomes of such negation. That is, it would be illogical to infer from “P is not a complete cause of Q” that “P is a partial cause of Q”, or from “P is not a partial cause of Q” that “P is complete a cause of Q”. The labels ‘complete’ and ‘partial’ could be misleading, connoting a relation of inclusion between whole and part; here, note well, ‘complete’ excludes ‘partial’, and vice-versa. Similarly, of course, contingent causation is not equivalent to the negation of necessary causation.
Most importantly, keep in mind the inferences already mentioned in the last section of the preceding chapter, namely:
- If m is true, then n or q must be true.
- If n is true, then m or p must be true.
- If p is true, then n or q must be true.
- If q is true, then m or p must be true.
- If neither m nor p is true, then neither n nor q can be true.
- If neither n nor q is true, then neither m nor p can be true.
With regard to the oppositions between the four joint determinations.
Each of the four joint determinations obviously implies but is not implied by its constituent generic determinations. That is, m and n are subalterns of mn, m and q are subalterns of mq, and so forth. It follows that each joint determination is contrary to the negations of its constituent generic determinations. That is, mn is contrary to not-m and to not-n; and so forth. Or in other words, if either or both of its constituent generic determinations is/are denied, the joint determination as a whole must be denied.
Furthermore, the four joint determinations are all mutually exclusive. That is, if any one of them is true, the three others have to be false. For if mn is true, mq cannot be true (since n and q are incompatible), and np cannot be true (since m and p are incompatible), and pq cannot be true (for both reasons). Similarly, if we affirm mq, we must deny the combinations mn, np, pq; and so forth. On the other hand, the negation of any joint determination has no consequence on the others; they may all be false without resulting inconsistency.
2. Eductions.
Immediate inference is inference of a conclusion from one premise, in contrast to syllogistic (or mediate) inference. ‘Opposition’ is one form of it, in which the items concerned retain the same position and polarity. ‘Eduction’ is another form of it, involving some change in position and/or polarity of the items occurs.
Let us now look into the feasibility of eductions from causative propositions, with reference to their definitions. We shall for now ignore the issue of direction of causation, dealt with further on. All the usual eductive processes[29], namely inversion, conversion, and contraposition, obversion, obverted-inversion, obverted-conversion, and obverted-contraposition, can be used in the ways shown below. First however, we must consider eduction by negation of the complement, a process applicable to the weak determinations.
- Negations of the complement (from R to notR) for the same items (P-Q). This concerns the weak determinations, and results in a negative conclusion.
- “P (complemented by R) is a partial cause of Q” implies “P (complemented by notR) is not a partial cause of Q”.
Proof: Clause (i) of “P (complemented by R) is a partial cause of Q” and clause (iii) of “P (complemented by notR) is a partial cause of Q” contradict each other; therefore they are incompatible propositions.
In contrast, note, “P (complemented by R) is a partial cause of Q” is compatible with “P (complemented by notR) is a contingent cause of Q”.
- “P (complemented by R) is a contingent cause of Q” implies “P (complemented by notR) is not a contingent cause of Q”.
Proof: In a similar manner, mutatis mutandis.
In contrast, note, “P (complemented by R) is a contingent cause of Q” is compatible with “P (complemented by notR) is a partial cause of Q”.
Negation of the complement for the joint determination pq follows by conjunction:
- If P (complemented by R) is a partial and contingent cause of Q, then P (complemented by notR) is neither a partial nor a contingent cause of Q.
- Inversions (changes from P-Q to notP-notQ); the conclusion is called the inverse of the premise.
All four generic determinations are invertible to a positive causative proposition, simply by substituting not{notP} for P, not{notQ} for Q. In the case of weak determinations, additionally, not{notR} replaces R; and moreover, eduction by negation of the complement of the positive conclusion yields a further negative conclusion. Thus,
- “P is a complete cause of Q” implies “notP is a necessary cause of notQ”.
And vice-versa. In contrast, note, “P is a complete cause of Q” and “notP is a complete cause of notQ” are merely compatible.
- “P is a necessary cause of Q” implies “notP is a complete cause of notQ”.
And vice-versa. In contrast, note, “P is a necessary cause of Q” and “notP is a necessary cause of notQ” are merely compatible.
- “P (complemented by R) is a partial cause of Q” implies “notP (complemented by notR) is a contingent cause of notQ”.
And vice-versa. It follows by negation of the complement that:
- “P (complemented by R) is a partial cause of Q” implies “notP (complemented by R) is not a contingent cause of notQ”.
In contrast, note, “P (complemented by R) is a partial cause of Q” is compatible with “notP (complemented by R) is a partial cause of notQ” and with “notP (complemented by notR) is a partial cause of notQ”.
- “P (complemented by R) is a contingent cause of Q” implies “notP (complemented by notR) is a partial cause of notQ”.
And vice-versa. It follows by negation of the complement that:
- “P (complemented by R) is a contingent cause of Q” implies “notP (complemented by R) is not a partial cause of notQ”.
In contrast, note, “P (complemented by R) is a contingent cause of Q” is compatible with “notP (complemented by R) is a contingent cause of notQ” and with “notP (complemented by notR) is a contingent cause of notQ”.
Notice, with regard to the positive implications of the weak determinations, that P, Q, and R all change polarity. Evidently, inversions involve a change of determination from positive (complete or partial) to negative (necessary or contingent, respectively), or vice-versa.
With regard to the joint determinations, their inversions follow from those relative to the generic determinations.
Inversion of mn or pq is possible, without change of determination (i.e. to mn or pq, respectively), since the changes for each constituent determination balance each other out; and all items change polarity. Thus:
- If P is a complete and necessary cause of Q, then notP is a complete and necessary cause of notQ.
- If P (complemented by R) is a partial and contingent cause of Q, then notP (complemented by notR) is a partial and contingent cause of notQ; also, notP (complemented by R) is not a partial or contingent cause of notQ.
Inversion of mq or np is possible, though with changes of determination (i.e. to np or mq, respectively); and all items change polarity. Thus:
- If P is a complete and (complemented by R) a contingent cause of Q, then notP is a necessary and (complemented by notR) a partial cause of notQ; also, notP (complemented by R) is not a partial cause of notQ.
- if P is a necessary and (complemented by R) a partial cause of Q, then notP is a complete and (complemented by notR) a contingent cause of notQ; also, notP (complemented by R) is not a contingent cause of notQ.
With regard to negative causative propositions, we can easily derive analogous inversions on the basis of[30] the above findings:
- “P is not a complete cause of Q” implies “notP is not a necessary cause of notQ”.
- “P is not a necessary cause of Q” implies “notP is not a complete cause of notQ”.
- “P (complemented by R) is not a partial cause of Q” implies “notP (complemented by notR) is not a contingent cause of notQ”.
- “P (complemented by R) is not a contingent cause of Q” implies “notP (complemented by notR) is not a partial cause of notQ”.
Similarly for the negations of joint determinations.
- Conversions (changes from P-Q to Q-P); the conclusion is called the converse of the premise.
The strong generic determinations are convertible, as follows:
- “P is a complete cause of Q” implies “Q is a necessary cause of P”.
Proof: Clause (i) of the given proposition may be contraposed to “if notQ, then notP”; clauses (i) and (iii) together imply that (P + Q) is possible, which means that “if Q, not-then notP”; and clause (ii) implies “notQ is possible”. Thus, the conditions for the said conclusion are satisfied, and conversion is feasible.
And vice-versa. In contrast, note, “P is a complete cause of Q” and “Q is a complete cause of P” are merely compatible.
- “P is a necessary cause of Q” implies “Q is a complete cause of P”.
Proof: In a similar manner, mutatis mutandis.
And vice-versa. In contrast, note, “P is a necessary cause of Q” and “Q is a necessary cause of P” are merely compatible.
The weak generic determinations are also convertible, as follows:
- “P (complemented by R) is a partial cause of Q” implies “Q (complemented by notR) is a contingent cause of P”.
Proof: Clause (i) of the given proposition means that (P + R + notQ) is impossible, which may be restated as “if (notQ + R), then notP”; clauses (i) and (iv) together imply that (P + R + Q) is possible, which means that “if (Q + R), not-then notP”; clause (iii) means that (P + notR + notQ) is possible, which may be restated as “if (notQ + notR), not-then notP”; and clause (ii) implies “(notQ + R) is possible”. Thus, the conditions for the said conclusion are satisfied (reading not{notR} instead of R), and conversion is feasible.
And vice-versa. It follows by negation of the complement that:
- “P (complemented by R) is a partial cause of Q” implies “Q (complemented by R) is not a contingent cause of P”.
In contrast, note, “P (complemented by R) is a partial cause of Q” is compatible with “Q (complemented by R) is a partial cause of P” and with “Q (complemented by notR) is a partial cause of P”.
- “P (complemented by R) is a contingent cause of Q” implies “Q (complemented by notR) is a partial cause of P”.
Proof: In a similar manner, mutatis mutandis.
And vice-versa. It follows by negation of the complement that:
- “P (complemented by R) is a contingent cause of Q” implies “Q (complemented by R) is not a partial cause of P”.
In contrast, note, “P (complemented by R) is a contingent cause of Q” is compatible with “Q (complemented by R) is a contingent cause of P” and with “Q (complemented by notR) is a contingent cause of P”.
Note well, with regard to the positive implications of the weak determinations, that R changes polarity, while P, Q do not; in this sense, their conversion may be qualified as imperfect. Evidently, conversions involve a change of determination from positive (complete or partial) to negative (necessary or contingent, respectively), or vice-versa.
With regard to the joint determinations, their conversions follow from those relative to the generic determinations.
Conversion of mn is possible, without change of determination (i.e. to mn), since the changes for each constituent determination balance each other out. Thus:
- If P is a complete and necessary cause of Q, then Q is a complete and necessary cause of P.
Conversion of pq is possible, without change of determination (i.e. to pq), for the same reason; but the subsidiary item (R) changes polarity in the positive implication. Thus:
- If P (complemented by R) is a partial and contingent cause of Q, then Q (complemented by notR) is a partial and contingent cause of P; also, Q (complemented by R) is not a partial or contingent cause of P.
Conversion of mq or np is possible, though with changes of determination (i.e. to np or mq, respectively); also, the subsidiary item (R) changes polarity in the positive implication. Thus:
- If P is a complete and (complemented by R) a contingent cause of Q, then Q is a necessary and (complemented by notR) a partial cause of P; also, Q (complemented by R) is not a partial cause of P.
- If P is a necessary and (complemented by R) a partial cause of Q, then Q is a complete and (complemented by notR) a contingent cause of P; also, Q (complemented by R) is not a contingent cause of P.
With regard to negative causative propositions, we can easily derive analogous conversions on the basis of the above findings:
- “P is not a complete cause of Q” implies “Q is not a necessary cause of P”.
- “P is not a necessary cause of Q” implies “Q is not a complete cause of P”.
- “P (complemented by R) is not a partial cause of Q” implies “Q (complemented by notR) is not a contingent cause of P”.
- “P (complemented by R) is not a contingent cause of Q” implies “Q (complemented by notR) is not a partial cause of P”.
Similarly for the negations of joint determinations.
- Contrapositions (changes from P-Q to notQ-notP); the conclusion is called the contraposite of the premise.
All four generic determinations are contraposable, simply by conversion of their inverses:
- “P is a complete cause of Q” implies “notQ is a complete cause of notP”.
And vice-versa. In contrast, note, “P is a complete cause of Q” and “notQ is a necessary cause of notP” are merely compatible.
- “P is a necessary cause of Q” implies “notQ is a necessary cause of notP”.
And vice-versa. In contrast, note, “P is a necessary cause of Q” and “notQ is a complete cause of notP” are merely compatible.
- “P (complemented by R) is a partial cause of Q” implies “notQ (complemented by R) is a partial cause of notP”.
And vice-versa. It follows by negation of the complement that:
- “P (complemented by R) is a partial cause of Q” implies “notQ (complemented by notR) is not a partial cause of notP”.
In contrast, note, “P (complemented by R) is a partial cause of Q” is compatible with “notQ (complemented by R) is a contingent cause of notP” and with “notQ (complemented by notR) is a contingent cause of notP”.
- “P (complemented by R) is a contingent cause of Q” implies “notQ (complemented by R) is a contingent cause of notP”.
And vice-versa. It follows by negation of the complement that:
- “P (complemented by R) is a contingent cause of Q” implies “notQ (complemented by notR) is not a contingent cause of notP”.
In contrast, note, “P (complemented by R) is a contingent cause of Q” is compatible with “notQ (complemented by R) is a partial cause of notP” and with “notQ (complemented by notR) is a partial cause of notP”.
Notice, with regard to the positive implications of the weak determinations, that while P, Q change polarity, R does not; in this sense, their contraposition may be qualified as imperfect. Evidently, contrapositions distinctively do not involve changes of determination.
With regard to the joint determinations, their contrapositions follow from those relative to the generic determinations.
Contraposition of mn is possible, without change of determination (i.e. to mn). Thus:
- If P is a complete and necessary cause of Q, then notQ is a complete and necessary cause of notP.
Contraposition of pq, mq or np is possible, without change of determination (i.e. to pq, mq or np, respectively); and the subsidiary item (R) does not change polarity in the positive implication. Thus:
- If P (complemented by R) is a partial and contingent cause of Q, then notQ (complemented by R) is a partial and contingent cause of notP; also, notQ (complemented by notR) is not a partial or contingent cause of notP.
- If P is a complete and (complemented by R) a contingent cause of Q, then notQ is a necessary and (complemented by R) a partial cause of notP; also, notQ (complemented by notR) is not a partial cause of notP.
- If P is a necessary and (complemented by R) a partial cause of Q, then notQ is a complete and (complemented by R) a contingent cause of notP; also, notQ (complemented by notR) is not a contingent cause of notP.
With regard to negative causative propositions, we can easily derive analogous contrapositions on the basis of the above findings:
- “P is not a complete cause of Q” implies “notQ is not a complete cause of notP”.
- “P is not a necessary cause of Q” implies “notQ is not a necessary cause of notP”.
- “P (complemented by R) is not a partial cause of Q” implies “notQ (complemented by R) is not a partial cause of notP”.
- “P (complemented by R) is not a contingent cause of Q” implies “notQ (complemented by R) is not a contingent cause of notP”.
Similarly for the negations of joint determinations.
- Obversions (changes from P-Q to P-notQ); the conclusions are called obverses of the premise.[31]
All four generic determinations are obvertible in various ways, though the obverses are negative causative propositions.
- “P is a complete cause of Q” implies “P is not a complete cause of notQ” and “P is not a necessary cause of notQ”.
Proof: Clauses (i) and (iii) of “P is a complete cause of Q” together imply that (P + Q) is possible, whereas clause (i) of “P is a complete cause of notQ” implies that conjunction impossible; therefore they are incompatible propositions. Also, clause (i) of “P is a complete cause of Q” and clause (ii) of “P is a necessary cause of notQ” contradict each other; therefore they are incompatible.
- “P is a necessary cause of Q” implies “P is not a necessary cause of notQ” and “P is not a complete cause of notQ”.
Proof: In a similar manner, mutatis mutandis.
- “P (complemented by R) is a partial cause of Q” implies “P (complemented by R) is not a partial cause of notQ” and “P (complemented by notR) is not a contingent cause of notQ”.
Proof: Clauses (i) and (iv) of “P (complemented by R) is a partial cause of Q” together imply that (P + R + Q) is possible, whereas clause (i) of “P (complemented by R) is a partial cause of notQ” implies that conjunction impossible; therefore they are incompatible propositions. Also, clause (ii) of “P (complemented by R) is a partial cause of Q” and clause (i) of “P (complemented by notR) is a contingent cause of notQ” contradict each other; therefore they are incompatible. Notice in the latter case, the change in polarity of the complement (from R to notR), as well as the change in determination (from p to q).
In contrast, note well, “P (complemented by R) is a partial cause of Q” is compatible with “P (complemented by R) is a contingent cause of notQ”, and with “P (complemented by notR) is a partial cause of notQ”.
- “P (complemented by R) is a contingent cause of Q” implies “P (complemented by R) is not a contingent cause of notQ” and “P (complemented by notR) is not a partial cause of notQ”.
Proof: In a similar manner, mutatis mutandis. Notice in the latter case, the change in polarity of the complement (from R to notR), as well as the change in determination (from q to p).
In contrast, note well, “P (complemented by R) is a contingent cause of Q” is compatible with “P (complemented by R) is a partial cause of notQ”, and with “P (complemented by notR) is a contingent cause of notQ”.
With regard to the joint determinations, their obversions follow from those relative to the generic determinations.
- If P is a complete and necessary cause of Q, then P is neither a complete nor a necessary cause of notQ.
- If P is a complete and contingent cause of Q, then P is neither a complete, nor (complemented by R) a contingent, cause of notQ, and P is neither a necessary, nor (complemented by notR) a partial, cause of notQ.
- If P is a necessary and partial cause of Q, then P is neither a necessary, nor (complemented by R) a partial, cause of notQ, and P is neither a complete, nor (complemented by notR) a contingent, cause of notQ.
- If P (complemented by R) is a partial and contingent cause of Q, then P (whether complemented by R or notR) is neither a partial nor a contingent cause of notQ.
- Obverted inversions (changes from P-Q to notP-Q); the conclusions are called obverted-inverses of the premise.
All four generic determinations may be subjected to obverted-inversion, by successive inversion then obversion. The conclusions are therefore negative causative propositions.
- “P is a complete cause of Q” implies “notP is not a complete cause of Q” and “notP is not a necessary cause of Q”.
- “P is a necessary cause of Q” implies “notP is not a necessary cause of Q” and “notP is not a complete cause of Q”.
- “P (complemented by R) is a partial cause of Q” implies “notP (complemented by R) is not a partial cause of Q” and “notP (complemented by notR) is not a contingent cause of Q”.
In contrast, note well, “P (complemented by R) is a partial cause of Q” is compatible with “notP (complemented by notR) is a partial cause of Q”, and with “notP (complemented by R) is a contingent cause of Q”.
- “P (complemented by R) is a contingent cause of Q” implies “notP (complemented by R) is not a contingent cause of Q” and “notP (complemented by notR) is not a partial cause of Q”.
In contrast, note well, “P (complemented by R) is a contingent cause of Q” is compatible with “notP (complemented by notR) is a contingent cause of Q”, and with “notP (complemented by R) is a partial cause of Q”.
With regard to the joint determinations, their obverted-inversions follow from those relative to the generic determinations, as usual.
- Obverted conversions (changes from P-Q to Q-notP); the conclusions are called obverted-converses of the premise.
All four generic determinations may be subjected to obverted-conversion, by successive conversion then obversion. The conclusions are therefore negative causative propositions.
- “P is a complete cause of Q” implies “Q is not a complete cause of notP” and “Q is not a necessary cause of notP”.
- “P is a necessary cause of Q” implies “Q is not a necessary cause of notP” and “Q is not a complete cause of notP”.
- “P (complemented by R) is a partial cause of Q” implies “Q (complemented by R) is not a partial cause of notP” and “Q (complemented by notR) is not a contingent cause of notP”.
In contrast, note well, “P (complemented by R) is a partial cause of Q” is compatible with “Q (complemented by notR) is a partial cause of notP”, and with “Q (complemented by R) is a contingent cause of notP”.
- “P (complemented by R) is a contingent cause of Q” implies “Q (complemented by R) is not a contingent cause of notP” and “Q (complemented by notR) is not a partial cause of notP”.
In contrast, note well, “P (complemented by R) is a contingent cause of Q” is compatible with “Q (complemented by notR) is a contingent cause of notP”, and with “Q (complemented by R) is a partial cause of notP”.
With regard to the joint determinations, their obverted-conversions follow from those relative to the generic determinations, as usual.
- Obverted contrapositions, also known as conversions by negation (changes from P-Q to notQ-P); the conclusions are called obverted-contraposites of the premise.
All four generic determinations may be subjected to obverted-contraposition, by successive contraposition then obversion. The conclusions are therefore negative causative propositions.
- “P is a complete cause of Q” implies “notQ is not a complete cause of P” and “notQ is not a necessary cause of P”.
- “P is a necessary cause of Q” implies “notQ is not a necessary cause of P” and “notQ is not a complete cause of P”.
- “P (complemented by R) is a partial cause of Q” implies “notQ (complemented by R) is not a partial cause of P” and “notQ (complemented by notR) is not a contingent cause of P”.
In contrast, note well, “P (complemented by R) is a partial cause of Q” is compatible with “notQ (complemented by notR) is a partial cause of P”, and with “notQ (complemented by R) is a contingent cause of P”.
- “P (complemented by R) is a contingent cause of Q” implies “notQ (complemented by R) is not a contingent cause of P” and “notQ (complemented by notR) is not a partial cause of P”.
In contrast, note well, “P (complemented by R) is a contingent cause of Q” is compatible with “notQ (complemented by notR) is a contingent cause of P”, and with “notQ (complemented by R) is a partial cause of P”.
With regard to the joint determinations, their obverted-contrapositions follow from those relative to the generic determinations, as usual.
We may finally note the following derivative eductions, though they are virtually useless except that they partly summarize the preceding findings:
- If P is a strong cause of Q, then notP is a strong cause of notQ (inversion), and Q is a strong cause of P (conversion), and notQ is a strong cause of notP (contraposition).
- If P (complemented by R) is a weak cause of Q, then notP (complemented by notR) is a weak cause of notQ (inversion), and Q (complemented by notR) is a weak cause of P (conversion), and notQ (complemented by R) is a weak cause of notP (contraposition).
- If P is a cause of Q, then notP is a cause of notQ (inversion), and Q is a cause of P (conversion), and notQ is a cause of notP (contraposition).
Moreover, we can say:
- If P is a strong cause of Q, then P is not a strong cause of notQ (obversion), and notP is not a strong cause of Q (obverted inversion), and Q is not a strong cause of notP (obverted conversion), and notQ is not a strong cause of P (obverted contraposition).
But similar negative implications are not possible for “P (complemented by R) is a weak cause of Q”, in view of variations in the complement in such cases. It follows that similar negative implications are not possible for “P is a cause of Q”.
Finally, concerning the weak determinations, it should be noted that wherever the inference results in no change of complement, i.e. wherever the premise and conclusion concern the same complement, the complement need not be mentioned at all. That is, we can in some cases simply say: if “P is a partial (or contingent) cause of Q”, then “(the new cause) is (or is not) a partial (or contingent) cause of (the new effect)” (as the case may be), on the tacit understanding that the complement, whatever it happens to be, has not been altered.
More broadly, whether or not the complement changes polarity, it is clear that we do not need to specify or even remember its precise content, in order to perform the inference. When the complement is unchanged, we need not mention it at all (or, to be sure, we can say in the conclusion “with the same complement, whatever it be”); and when it is changed, we can add in the conclusion “with the negation of the initial complement, whatever it be, as complement”. It is good to know this, because it allows us to proceed with inferences without immediately having to or being able to pin-point the complement involved.
Note lastly, that all immediate inferences could also be validated or invalidated, as the case may be, by means of matricial analysis (see later). I have here preferred the less systematic, but also less voluminous, method of reduction to conditional arguments.
All these inferences add to our knowledge and understanding of causative propositions, of course. Some of them will prove useful for validations or invalidations of causative syllogisms by direct reduction to others.
3. The Directions of Causation.
Now, the implications between different forms of causative propositions identified above, such as that “P is a complete cause of Q” implies “Q is a necessary cause of P”, demonstrate that our definitions of causation were incomplete. For we well know that causation has a direction! However, bear with me – we deal with this issue.
Strictly-speaking, when we utter a statement of the form “P is a (complete, necessary, partial, contingent) cause of Q”, we imply a tacit clause specifying the direction (or ordering of items), in addition to the various clauses (treated in the preceding chapter) concerning determination. This means that denial of the tacit clause on direction would suffice to deny the causation concerned, even if all the other clauses are affirmed. However, there are good reasons why in our formal treatment we are wise to keep the issue of direction separate.
First of these is the epistemological fact that the direction of causation is not always known. We may by inductive or deductive means arrive at knowledge of all the other clauses, and yet be hard put to immediately specify the direction. If we wished to summarize our position in such case, and were not permitted to use the language of causation, we would have to introduce a relational expression other than “is a … cause of” (say, “is a … determinant of”) to allow us to verbalize the situation. Causation would then be defined as the combination of this relation (“determination”) with a directional clause. This is feasible, but in my view redundant; we can manage without such an artifice.
Secondly, we have to consider the ontological fact that causation does not always occur in only one direction: it may occur in both. Sometimes, the direction is exclusively from P to Q, or from Q to P; but sometimes, the causal relation is two-way or reversible. Moreover, reversible causation is not always reciprocal: there may be one determination in one direction, and another in the opposite sense; or there may, in the case of weak causations, be different complements in each direction. For this reason, too, we are wise to handle the issue of direction flexibly, considering it expressed in an additional clause, but left ‘hidden’ or ignored until specifically dealt with. This is the course adopted in the present work.
The directional clause for a causative proposition can be a phrase qualifying the sentence “P is a … cause of Q”, a phrase of the form “in the direction from P to Q” (which is identical with ‘notP to notQ’) and/or “in the direction from Q to P” (which is identical with ‘notQ to notP’). We must additionally allow for (hopefully temporary) ignorance with the phrase “direction unknown”.
We allow for only two directions, not four, note well. “P to Q” and its inverse “notP to notQ” are one and the same direction; likewise “Q to P” and “notQ to notP” are identical in direction. In this manner, causative statements remain always or formally invertible – but strictly-speaking only sometimes or conditionally convertible or contraposable, specifically when the causation is known to be reversible. That is, whereas inversion is ontologically universal, conversion and contraposition have the status of formal artifices until and unless their ontological applicability is established in a given case. The latter two eductions, of course, go together; if either is applicable, so is the other (since the contraposite is the inverse of the converse).
As we shall see, consideration of direction of causation affects other deductive processes in a similar manner, i.e. making them conditional instead of universal. Thus, in causative syllogism, arguments in the first figure guarantee the direction implicit in the conclusion (given the directions implied by the premises), whereas arguments in the other two figures cannot do so.
However, this is not a great difficulty, because we know that wherever a causative conclusion is drawn, the direction of causation has to be either as implied by that conclusion or as implied by its formal converse or both. Thus, the issue of direction is relatively minor. It is without impact on the inferred ‘bond’, on the fact that there is a certain (strong or weak) causative relationship between the items concerned; the only problem it sets for us is in which form this relationship is expressed, as ‘P-Q’ or its converse ‘Q-P’.
The real problem with direction of causation is identifying how it is to be induced in the first place. We shall try to solve this problem later, in the chapter on induction of causation. For now, suffices to say the following. In de dicta (logical) causation, theses are hierarchized by their epistemological roles (an axiom causes but is not caused by a resulting theorem, even if the latter implies the former, for instance); in de re (natural, temporal, spatial or extensional) causation, the order of things is often dictated by temporal or spatial sequences, for instances (logical issues also come into play).
There are cases in practice where deciding which item is the cause and which is the effect is virtually a matter of convention. This may occur in reciprocal causation, as well as in causation with permanently unknown (i.e. practically unknowable) direction. In such cases, the expressions “the cause” and “the effect” merge into one, becoming mere verbal differentiations. This is often true in the logical mode, and in the spatial and extensional modes; it occurs more rarely in the temporal and natural modes. The reason being that the only really absolute rule of direction we know is temporal sequence; other rules, though credible, are open to debate.
It should be stressed that the concept of direction (or orientation) concerns not only causation, but more broadly space and time in a variety of guises. It is therefore an issue in a wider and deeper ontological and epistemological context, not one reserved to causation. It might be viewed as one of the fundamental building-blocks of knowledge, and therefore not entirely definable with reference to other concepts.
It may be exemplified concretely by drawing a line on paper (this expresses its spatial component), and running a finger along it first one way, then the other (this expresses its temporal component, since the movement takes time to cover space); and saying “though the path covered is the same in both instances, the first movement is to be distinguished from the second – and this difference will be called one of direction”. In this manner, the words ‘from’ and ‘to’, though very abstract[32], are shown to be meaningful, i.e. to symbolize a communicable distinction, which can by analogy be applied in other contexts.
Such visual and mechanical demonstration merely aids the intuition[33] in focusing upon the intention of verbal expressions of direction. It does not, of course, by itself suffice to clearly define directionality in the context of causation, or to establish the direction of causation in particular cases. We must search for more precise means to achieve these ends. But we at least have a sort of ostensive-procedural definition of directionality in general, which gives some meaning to clauses like “from P to Q” and “from Q to P”.
The propositions “P causes Q” and “Q causes P” are simply declared unequal. Causation in general is symbolized by a string of words, namely “P”-”causation”-”Q”, analogous to a line; this line of relation is, however, to be taken as two-fold, i.e. as occasionally different in the senses P-Q and Q-P. What this difference signifies more deeply in formal terms, we cannot yet say; but we do believe that it exists, and wish to prepare for its linguistic expression by such declaration.
Chapter 5. Causative Syllogism
1. Causal or Effectual Chains.
The topic of concatenation of causations is an important field of research, though a tedious one. It is important not only to the natural sciences, which need to monitor or trace causal or effectual chains, but also to law and ethics.
To grasp its practical value in legal or ethical discourse, consider this example[34]: a motorist overruns a pedestrian, who in the hospital where he is rushed is additionally the victim of some medical mishap – can the motorist be blamed for the poor pedestrian’s subsequent misfortunes? Such questions can only be convincingly answered through a systematic and wide-ranging reflection on causal logic.
The concept of concatenation refers primarily to ‘chain reactions’: P causes Q, which causes R, and so on; or conversely, R is effected by Q, which is effected by P, and so forth.
Clearly, the concepts of cause and effect here are relative to each other. In the context of deterministic causality, nothing is absolutely a cause or absolutely an effect; it is always the cause or effect of something.
All we wish to point out here is the obvious: that a phenomenon Q which is a cause in relation to another phenomenon R may itself stand as effect in relation to yet a third phenomenon P. Similarly, a phenomenon Q which is an effect in relation to another phenomenon P may itself stand as cause in relation to yet a third phenomenon R.
When we speak in terms of chains like P-Q-R, we stand back from the underlying bipolar relations of cause and effect and focus on the wider picture. The items P, Q, R may then be referred to, more indifferently, as successive links in the chain.
Needless to say, concatenation of events implies but is not implied by the seriality of events (in whatever appropriate sense of the term ‘series’). Furthermore, even knowing that P causes Q and that Q causes R, we cannot presume concatenation. A series P-Q-R may be said to really form a chain, only if we can demonstrate that P, through the intermediary of Q, indeed causes R. This is not always feasible, for as we have seen the verb “causes” has a large variety of meaning.
You cannot just say “P causes Q and Q causes R, therefore P causes R” indiscriminately. This is one reason why a theoretical treatment of causal logic is essential to scientific thinking.
The search for concatenations varies in motive. Sometimes we are looking for the cause(s) of a cause, sometimes for the effect(s) of an effect, sometimes for some intermediary between a cause and an effect. We need not assume at the outset that all phenomena are bound to have causes and effects ad infinitum, nor that there has to be an infinity of intermediaries between any two given items.
A cause without apparent prior cause would be called a primary cause; an effect without apparent posterior effect would be called an ultimate effect. A cause and effect without apparent intermediary would be referred to as immediate or contiguous; if they have an intermediary, they would be referred to as mediated.
If we speculate that Existence as a whole has a Beginning and/or an End, then of course we may speak of that as a First Cause and/or a Last Effect. Likewise, we need not ab initio prejudice the issue concerning specific events within Existence, be it infinite or finite, and at least to start with make allowances for (in some sense) causeless or effectless phenomena.
We have so far mentioned what may be called orderly concatenation. We also search for chains in the context of parallelism of causes, or of effects. We may need to know whether parallel causes or parallel effects are themselves causally related, and thus order them in relation to the initial cause or terminal effect concerned. In such case, we are identifying one of the two causes or two effects (as the case may be) as an intermediary between the other two items.
It should be stressed, however, that the arguments about parallelism considered here cannot strictly-speaking tell us which one of the two causes (or two effects) causes the other; for as we have mentioned in the preceding chapters, sometimes there is a hidden issue of direction of causation to consider. This issue has to be resolved separately, with reference to spatial, temporal, or other conceptual or logical considerations[35]. We shall simply ignore this problem of ordering for now, and regard the tacit condition as always satisfied.
We should, additionally, in passing, mention the phenomenon of spiraling causation, which we commonly refer to as vicious circles. This phenomenon is a special case of concomitant variation[36]. It occurs when an increase or decrease in a cause C (C± x_{1}) causes an increase or decrease in an effect E (E± y_{1}), which in turn causes another increase or decrease in C (C± x_{1}± x_{2}), which in turn causes another increase or decrease in E (E± y_{1}± y_{2}), and so forth.
The spiral need not constitute an infinite chain, even if complete causation is involved at each step, because each of the causations involved is independent of (i.e. not formally implied by) its predecessors, note well. Even so, a spiral may come to a halt because it is in fact implicitly conditional, i.e. partial causation is involved at each step. But we can also conceive of infinite spirals, in the case of ongoing processes continuing as long as the universe lasts.
The problem of causal or effectual chains is, as we shall see, essentially syllogistic. We need to identify which syllogisms involving causative propositions as premises yield such propositions as conclusion. In this research, it is as important to expose the invalidity of certain syllogisms as to identify the valid syllogisms, for inappropriate reasoning is common[37].
2. Some Instructive Examples.
Before undertaking a systematic presentation and evaluation of causative syllogisms, I will propose some formal examples to acquaint the reader with some of the issues involved.
Consider, to begin with, the two causative syllogisms listed below (on the left):
Q is a complete cause of R; | => | Given that if Q then R |
P is a complete cause of Q; | => | and that if P then Q, |
so, P is a complete cause of R. | <= | it follows that if P then R. |
Q is a necessary cause of R; | => | Given that if notQ then notR |
P is a necessary cause of Q; | => | and that if notP then notQ, |
so, P is a necessary cause of R. | <= | it follows that if notP then notR. |
These typify orderly concatenation. Here, Q may be viewed as an intermediate cause of R after P, or as an intermediate effect of P before R. This arrangement of items is known to logicians as a ‘first figure’ syllogism. The first sentence in each case is called the ‘major premise’; the second one, the ‘minor premise’; the third, the ‘conclusion’.
In each case, notice, the premises and conclusion involve the same strong determination. We know that the conclusion may legitimately be drawn from the premises, because we can readily ‘reduce’ the argument to one previously known to logical science (shown on the left). That is, each premise given in the former implies a premise of the latter, whose conclusion in turn (granting certain provisions) implies that of the former.
The minimal provisions, as we have seen when defining these determinations, is as follows: in the case of complete causation, they are that P be possible and R be unnecessary; and in the case of necessary causation, they are that P be unnecessary and R be possible. We know these provisos are indeed met, in each case, being implied by the minor and major premises, respectively.
Ergo, these syllogisms are ‘valid’, they can be freely used, irrespective of what the items P, Q, R symbolize.
In contrast, consider the following two causative syllogisms:
If Q is a complete cause of R | => | That if Q then R |
and P is a necessary cause of Q, | => | and if notP then notQ, |
how are P and R then related? | … | yield only “if P, not-then notR”. |
If Q is a necessary cause of R | => | That if notQ then notR |
and P is a complete cause of Q, | => | and if P then Q, |
how are P and R then related? | … | yield only “if notP, not-then R”. |
These examples differ from the preceding two in that the premises are of different (though equally strong) determination, note. If we attempt to ‘reduce’ these arguments as before, we find no way to do so. We must thus admit that, in their case, we cannot demonstrably conclude either complete or necessary causation, and it would be misleading to think of the series P-Q-R as a chain. These combinations of premises are therefore ‘invalid’ arguments; we cannot reason with them without risking errors.
To be precise, these two arguments merely teach us that it would be wrong to deduce complete and/or necessary causation; but they do not exclude the possibility of such strong relations between P and R occurring independently of the intermediate item Q. The conclusion “if P, not-then notR” only precludes that “if P, then notR” (i.e. that P and R be incompatible), but not for instance that “if P, then R”. Similarly, the conclusion “if notP, not-then R” only precludes that “if notP, then R” (i.e. that P and R be exhaustive), but not for instance that “if notP, then notR”. Alternatively, for all we know, weaker forms of causation may apply or no causation at all.
Now, consider the following two causative syllogisms:
R is a complete cause of Q;
P is a necessary cause of Q;
therefore, P is a necessary cause of R.
R is a necessary cause of Q;
P is a complete cause of Q;
therefore, P is a complete cause of R.
These typify parallelism of causes. Notice the positions of the items involved, here: Q is an effect in common to P and R (whereas in orderly concatenation it was an effect of P and a cause of R); this is known to logicians as ‘second figure’ argument. In this case, as we shall later show, the syllogisms are valid[38], i.e. logically acceptable, albeit their having premises of different (though equally strong) determinations. The conclusion, notice, has the same determination as the minor premise. On the other hand, as we will show later, if the premises (with P, Q, R in a similar arrangement) have the same determination, i.e. both concern complete causation or both necessary causation, we are not permitted to draw any causative conclusion.
Finally, consider the following two causative syllogisms:
Q is a complete cause of R;
Q is a necessary cause of P;
therefore, P is a complete cause of R.
Q is a necessary cause of R;
Q is a complete cause of P;
therefore, P is a necessary cause of R.
These typify parallelism of effects. Notice the positions of the items involved, here: Q is a cause in common to P and R (whereas in orderly concatenation it was an effect of P and a cause of R, or in parallelism of causes it was an effect in common to P and R); this is known to logicians as ‘third figure’ argument. In this case, as we shall later show, the syllogisms are valid[39], i.e. logically acceptable, albeit their having premises of different (though equally strong) determinations. The conclusion, notice, has the same determination as the major premise. On the other hand, as we will show later, if the premises (with P, Q, R in a similar arrangement) have the same determination, i.e. both concern complete causation or both necessary causation, we are not permitted to draw any causative conclusion.
These examples reveal some of the complexities of causative argument.
We see from them that the ordering of the items involved in the premises affects the logical possibility of drawing a conclusion. In the first figure, two identical strong determinations yield a valid conclusion (of the same determination), whereas a mixture of such determinations is fruitless. In the second and third figures, the opposite is true; and furthermore, these differ from each other, in that a valid conclusion in the second figure follows the determination of the minor premise, whereas one in the third figure follows that of the major premise.
The problem becomes even more complicated when we investigate weak causations, which involve at least three items each (instead of two, as with strong causations). We discover, to give an extreme example, that whatever the figure considered, no conclusion can be drawn from two premises each of which concerns partial or contingent causation only. We then wonder what combinations of premises may be used to draw a conclusion about weak causation.
More broadly, considering that we have to deal with three figures, and eight possible determinations of causation for each premise, we have to examine 3*64=192 combinations, or ‘moods’ (as logicians say). What conclusion, if any, can be drawn from each one of those arguments; and how do we go about demonstrating it? Furthermore, we have so far mentioned syllogisms with only affirmative causative propositions; what of syllogisms involving propositions denying causation or a particular determination of causation?
Clearly, we cannot hope to reason correctly about causation without first dealing with causative syllogism in a thorough and systematic manner, so that we know precisely when an argument is valid and when it is not. If we limit our research to a few frequently used arguments, like those above shown, we will miss many opportunities for valid inference and risk making some invalid inferences. And in view of the volume of the problem, it has to be treated in as global a manner as possible.
This is our task in the next few chapters.
The research is tedious, because causative propositions are, as we have seen, very complex; they are each composed of two or more clauses, and most of these clauses are positive or negative conditional propositions, i.e. themselves complex.
In the simplest syllogisms, those involving strong determinations of causation only, and therefore the minimum number of (i.e. three) items, we can readily reduce causative reasoning to syllogism involving conditional propositions. The latter are reasonably well-known to logicians and to the public at large; a full treatment of them may be found in my work Future Logic.
But soon we find such simple methods inadequate. Syllogisms involving weak determinations or mixtures of strong and weak determinations are too complicated for us to feel secure with the results obtained by means of reduction. For certainty, I have had to develop a more complex method, called matricial analysis.
3. Figures and Moods.
A syllogism, we know thanks to Aristotle, consists of at least two premises and a conclusion. The premises together contain at least three items (terms or theses), at least one of which they have in common, and the conclusion contains at least two items, each of which was contained in a premise not containing the other.
Our job in syllogistic reasoning is to obtain from the premises, i.e. the given data, the information we need to construct the putative conclusion. If the premises, together and without reference to unstated assumptions, justify the conclusion, the syllogism proposed is valid deduction; otherwise it is invalid.
Validation (i.e. showing valid) justifies a form of reasoning; removing any uncertainty we may have about it or teaching us a new way of inference. Invalidation (i.e. showing invalid) is just as important, to contrast valid with invalid moods and thus set the limits of validity, and most of all to prevent us making mistakes in our thinking.
A putative conclusion may be invalid in the way of a non-sequitur, meaning that the conclusion does not conflict with anything in the premises, but just does not logically follow from them. Or, worse, it may be invalid in the way of antinomy, meaning that the conclusion is inconsistent (contradictory or contrary) with something in the premises[40].
In the case of a non-sequitur, we may be able to save the situation by stipulating some condition(s) under which the conclusion would follow; in that event, we may call the conclusion conditionally valid, or add the condition(s) to be satisfied to our premises as an additional premise to obtain an unconditionally valid conclusion, or again consider that we have a disjunctive conclusion whose alternatives include the satisfaction or non-satisfaction of the said condition(s).
In the case of an antinomy, we can redeem things by proposing the contradictory or a contrary of our invalid conclusion as a valid conclusion; if the invalid conclusion is a compound, we may be able to obtain a valid conclusion of the kind desired by negating some element(s) in it.
We are usually able to infer some information from the premises; but if this information does not add up to a causative proposition of some kind, we here consider the conjunction of the premises as a failure. For our task, in the present context, is not an investigation of deduction in the broadest sense, but specifically deduction of causative propositions from other causative propositions. Thus, do not be surprised if a syllogism is declared invalid even though some elements of a putative compound conclusion were inferable.
The evaluations of some moods may seem immediately or intuitively obvious; but some moods are too complicated for that and require careful examination. Some causative syllogisms, as already mentioned, can be validated by direct reduction to already established, non-causative syllogisms. Others are too complex for that, and can only be validated through matricial analysis, i.e. with painstaking reference to their corresponding matrix; this method will be described in detail later. Still others, though complex, can be validated by direct and/or indirect reduction (also known as reduction ad absurdum) to causative syllogisms already validated by other means (namely by matricial analysis).
Aristotle taught us that a syllogism may have one of three figures, according to the placement of the three items (terms or theses) in its premises and conclusion, as follows:
Figure 1 | Figure 2 | Figure 3 | |
Major premise: | Q – R | R – Q | Q – R |
Minor premise: | P – Q | P – Q | Q – P |
Conclusion: | P – R | P – R | P – R |
Notice, in each of the figures, the positions of the item found in both premises but not in the conclusion (namely Q; this is known as the middle item). Notice also the various positions of the other two items, one of which (R, the major item) is found only in the major premise (traditionally stated first) and conclusion (traditionally stated last), and the other of which (P, the minor item) is found only in the minor premise (traditionally stated second) and conclusion. The positions of the items tell us which ‘figure’ the reasoning is in.[41]
Each Aristotelian figure refers to three items (P, Q, R). But in the present context we are also dealing with some four-item (P, Q, R, S) arguments, which as we shall see can be combined in three different ways (and many more, which we shall deal with in a later chapter). Thus, we shall have to refer to subfigures. We can call Aristotle’s primary arrangement subfigure (a), and the three additional arrangements subfigures (b), (c), (d).
Subfigures | a | b | c | d |
Definitions | both premisesstrong only | major premisestrong only | minor premisestrong only | neither premisestrong only |
Figure 1 | QR | QR | Q(S)R | Q(P)R |
PQ | P(S)Q | PQ | P(S)Q | |
PR | P(S)R | P(S)R | P(S)R | |
Figure 2 | RQ | RQ | R(S)Q | R(P)Q |
PQ | P(S)Q | PQ | P(S)Q | |
PR | P(S)R | P(S)R | P(S)R | |
Figure 3 | QR | QR | Q(S)R | Q(P)R |
QP | Q(S)P | QP | Q(S)P | |
PR | P(S)R | P(S)R | P(S)R |
In (a), both premises involve strong determinations only; that is why there are only two items per premise (and in the conclusion). In (b) and (c), one premise (the major or minor, respectively) has only two items (implying the presence of only strong determination) and the other premise (and conclusion) has three items (implying the presence of joint strong and weak, or of only weak, determination). In (d), each premise (and the conclusion) involves three items (implying the absence of only-strong determination).
It is seen that the three-item symbolism (P, Q, R for the minor, middle and major items, respectively) is retained in four-item figures, except that we have an additional item (call it the subsidiary item, symbol S) appearing in a premise and the conclusion: S represents ‘outside interference’, as it were, in relation to the triad P-Q-R.
The important thing to note about this subsidiary item is that though it has to be mentioned in theoretical exposition and evaluation, as here, to place it and judge its impact, it need not be mentioned[42] in practice, because the conclusion follows the premises whatever its content happen to be. That is, the premise concerned and the conclusion need not specify “(complemented by S)”.
On the other hand, the clause “(complemented by P)” in the major premise of subfigures 1d, 2d and 3d, cannot be ignored in practice, since the middle item Q might well cause the major item R with some complement(s) other than the minor item P, rather than with P. Even though P is mentioned in association with Q in the minor premise, that in itself does not imply the causation in the major premise to be true with it: this knowledge must be obtained by other means to enable the inference of the conclusion.
If, in our present context, we specify as well as the figure the precise determination and polarity involved in each of the premises and in a putative conclusion, we have pinpointed the precise mood under discussion. This expression refers to the formal aspects of a syllogism, which distinguish it from all others. Thus, for each figure of syllogism, there are many conceivable moods.
The mood determinations (numbered 1-9 for reference) found in each subfigure are given in the table below. These tell us the determinations of the premises involved, which may be ‘strong only’ (abbreviation, so), a ‘mix of strong and weak’ (sw/ws), or ‘weak only’ (wo). Due to the numbers of items allowed for a premise in each subfigure, the number of determinations found in each subfigure varies.
Subfig. | a | b | c | d | |||||
Determ. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Major | so | so | so | sw/ws | wo | sw/ws | sw/ws | wo | wo |
Minor | so | sw/ws | wo | so | so | sw/ws | wo | sw/ws | wo |
There are 64 positive moods per figure, a total of 192 moods in all three of them. The system proposed now is to use three-digit identification numbers, or mood numbers. The hundreds will identify the figure 1, 2 or 3. The tens (#s 1-8, no 0 or 9) will tell us the major premise’s determination. The units (#s 1-8, no 0 or 9) will specify the minor premise’s determination. The subfigures (above labeled a-d) and modes (above labeled 1-9), shown in the preceding tables, are not explicitly mentioned in the mood number, but are tacitly implied by it.
Subfig. | Determ. | Premises | Moods Nos. | Qty | |
a | 1 | majorminor | strong onlystrong only | tens 1, 4, 5units 1, 4, 5 | 9 |
b | 2 | majorminor | strong onlysw, ws | tens 1, 4, 5units 2, 3 | 6 |
3 | majorminor | strong onlyweak only | tens 1, 4, 5units 6, 7, 8 | 9 | |
c | 4 | majorminor | sw, wsstrong only | tens 2, 3units 1, 4, 5 | 6 |
5 | majorminor | weak onlystrong only | tens 6, 7, 8units 1, 4, 5 | 9 | |
d | 6 | majorminor | sw, wssw, ws | tens 2, 3units 2, 3 | 4 |
7 | majorminor | sw, wsweak only | tens 2, 3units 6, 7, 8 | 6 | |
8 | majorminor | weak onlysw, ws | tens 6, 7, 8units 2, 3 | 6 | |
9 | majorminor | weak onlyweak only | tens 6, 7, 8units 6, 7, 8 | 9 |
I could of course have used letters instead of numbers to symbolize the different moods, but fearing to confuse the reader with yet more letter-symbols (the science of logic abounds with them) I have preferred number-symbols. Note that there are no moods numbered 01-10, 19-20, 29-30, 39-40, 49-50, 59-60, 69-70, 79-80, or 89+. The table below clarifies the meaning of each mood number within any given figure.
Minor | Major premise | |||||||
premise | mn=1 | mq=2 | np=3 | pq=4 | m=5 | n=6 | p=7 | q=8 |
mn=1 | 11 | 21 | 31 | 41 | 51 | 61 | 71 | 81 |
mq=2 | 12 | 22 | 32 | 42 | 52 | 62 | 72 | 82 |
np=3 | 13 | 23 | 33 | 43 | 53 | 63 | 73 | 83 |
pq=4 | 14 | 24 | 34 | 44 | 54 | 64 | 74 | 84 |
m=5 | 15 | 25 | 35 | 45 | 55 | 65 | 75 | 85 |
n=6 | 16 | 26 | 36 | 46 | 56 | 66 | 76 | 86 |
p=7 | 17 | 27 | 37 | 47 | 57 | 67 | 77 | 87 |
n=8 | 18 | 28 | 38 | 48 | 58 | 68 | 78 | 88 |
It is useful to expand the above table as done below, to show precisely what combination of determinations in the premises each mood number refers to.
Minor | Major premise | |||||||
premise | mn=1 | mq=2 | np=3 | pq=4 | m=5 | n=6 | p=7 | q=8 |
mn=1 | 11 | 21 | 31 | 41 | 51 | 61 | 71 | 81 |
major | mn | mq | np | pq | m | n | p | q |
minor | mn | mn | mn | mn | mn | mn | mn | mn |
mq=2 | 12 | 22 | 32 | 42 | 52 | 62 | 72 | 82 |
major | mn | mq | np | pq | m | n | p | q |
minor | mq | mq | mq | mq | mq | mq | mq | mq |
np=3 | 13 | 23 | 33 | 43 | 53 | 63 | 73 | 83 |
major | mn | mq | np | pq | m | n | p | q |
minor | np | np | np | np | np | np | np | np |
pq=4 | 14 | 24 | 34 | 44 | 54 | 64 | 74 | 84 |
major | mn | mq | np | pq | m | n | p | q |
minor | pq | pq | pq | pq | pq | pq | pq | pq |
m=5 | 15 | 25 | 35 | 45 | 55 | 65 | 75 | 85 |
major | mn | mq | np | pq | m | n | p | q |
minor | m | m | m | m | m | m | m | m |
n=6 | 16 | 26 | 36 | 46 | 56 | 66 | 76 | 86 |
major | mn | mq | np | pq | m | n | p | q |
minor | n | n | n | n | n | n | n | n |
p=7 | 17 | 27 | 37 | 47 | 57 | 67 | 77 | 87 |
major | mn | mq | np | pq | m | n | p | q |
minor | p | p | p | p | p | p | p | p |
q=8 | 18 | 28 | 38 | 48 | 58 | 68 | 78 | 88 |
major | mn | mq | np | pq | m | n | p | q |
minor | q | q | q | q | q | q | q | q |
Note that if you divide the above table in four equal squares, the top left square involves premises with only joint determinations, the bottom left one a joint major premise with a generic minor premise, the top right square involves a generic major premise with a joint minor premise, and finally the bottom right square premises with only generic determinations.
In my listing of moods in the next chapter, I do not follow their numerical order. Rather, I present the moods in a diagonal order with reference to the above table, starting with the strongest (top left hand corner) and ending with the weakest (bottom right hand corner).
Four moods, involving only strong determinations (namely, Nos. 11, 14, 41 and 44), have no ‘mirror images’; the remaining sixty moods may be treated in pairs, for each has a mirror image (thus, 12 and 13 are essentially the same, as are 21 and 31, and so forth). I present explicitly the more positive mood of each pair (e.g. 12), and only mention its mirror image (e.g. 13).
Moods with a stronger major premise are listed before moods with a stronger minor premise (e.g. 12, 13 before 21, 31). Moods with premises of uniform determination are listed before moods of mixed determination (e.g. 22, 33 before 23, 32). And so forth, the goal being to present all moods in a natural order.
Chapter 6. List of Positive Moods
1. Valid and Invalid Moods.
As we have seen in the preceding chapter, causative syllogism with both premises affirmative has 64 conceivable moods in each of three figures. In the present chapter, we shall list all these moods, and for each mood specify whether it is valid or invalid, and briefly the basis of this evaluation.
For any positive mood, there are four initially conceivable, putative conclusions, corresponding to the four generic determinations, which we have symbolized as m (for complete causation), n (for necessary causation), p (for partial causation) and q (for contingent causation). However, at most two such conclusions may be valid for any given mood, since the determinations m and p are contrary and n and q are contrary. Thus, there are eight logically possible conclusions for any positive causative syllogism, namely:
mn, mq, np, pq, m, n, p, q.
A putative conclusion is valid – if it logically follows from the given premises, i.e. if its contradictory is logically incompatible with them or any of their implications. A putative conclusion is declared invalid – if it is not valid, for whatever reason; the reason may be that the premises themselves are inconsistent, or that the contradictory of the putative conclusion is compatible with them (in which case the putative conclusion is a non-sequitur), or that the putative conclusion is incompatible with the premises (in which case the putative conclusion is an antinomy and its contradictory is valid).
If one of the eight joint or generic determinations is demonstrably inferable from the premises concerned, the mood is valid. If none of them can be legitimately drawn from the premises, the mood is invalid. Additionally, some moods are invalid at the outset because the premises concerned are in fact incompatible in some respect(s); i.e. at least one clause of each is implicitly denied by at least one clause of the other.
We shall, to repeat, in the present chapter only list the moods and their valid conclusion(s) if any, and state succinctly the basis of these results. In the next two chapters, we will show how these results were obtained, systematically and in detail; i.e. we will justify our claims.
Note that, in accord with the tradition in logic, if a mood is valid, only the correct conclusion(s) is/are mentioned in the listing; other conclusions, not mentioned, are tacitly implied to be incorrect. But it is well to keep both the valid and invalid conclusions in mind; for the purpose of the whole exercise is not only to instruct us in proper reasoning, but also to save us from improper reasoning!
As will be seen, some conclusions have to be validated or invalidated by matricial analysis; moods with at least one conclusion treated by matricial analysis may be called primary. The remaining conclusions may be validated or invalidated by reduction to the primary moods; moods all of whose conceivable conclusions have been treated by reduction may be called secondary or derived.
As for moods invalid due to inconsistency between the premises, they need not of course be subjected to matricial analysis or reduction. Note that it may be possible to affirm or deny some conclusion(s) from some of their clauses, if the inherent contradiction is disregarded; but that would be nonsensical, for if all the clauses are taken into consideration, we have to admit that the premises in question cannot in fact come together to yield such conclusion(s).
All evaluations could be performed by matricial analysis; but this process is long-winded, so we try and avoid it as much as possible. Such avoidance is anyway not sheer laziness on our part, for it is instructive to be aware of the interrelations between moods which reduction reveals. We learn, in this way, that causative syllogisms together constitute a close-knit totality, a system.
It should be stressed that the issue of direction of causation is ignored throughout the present formal treatment. In figure 1, this is no problem; i.e. given the directions of causation implied in the premises (namely, from P to Q and from Q to R), the direction of causation implied in an eventual valid conclusion (viz. from P to R) follows necessarily. But in figures 2 and 3, any eventual valid conclusions must be regarded as conditionally valid, i.e. on the proviso that the implied direction of causation (viz. from P to R) is established by other means.
However, if it turns out that a figure 2 or 3 conclusion is found not to satisfy this condition, the underlying implications between the items concerned (P and R) may still in certain cases result in a causative conclusion in the reverse direction. Such cases are formally predictable, simply by transposition of the premises concerned. If such transposition has some causative conclusion, then the direction of causation implied by that conclusion (i.e. from R to P) will be unconditionally valid. For if there is causation between P and R, it is bound to be in one direction or the other.
- Strong determinations. If two premises yield the conclusion ‘P is a complete cause of R’, then their transposition will yield the converse conclusion ‘R is a necessary cause of P’. If we do not know the direction of causation, we cannot know which of these conclusions is the correct one, but we do know that at least one of them must be. If we know that it is not this one, then we know it must be that one. Similarly, with the eventual conclusions ‘P is a necessary cause of R’ and ‘R is a complete cause of P’.[43]
- Weak determinations. If two premises yield the conclusion ‘P (complemented by S) is a partial cause of R’, and this conclusion is found unjustified with regard to the issue of direction of causation, then its converse has to be admitted as valid, viz. ‘R (complemented by notS) is a contingent cause of P’ (note well the change of polarity of the complement). Similarly, if we know that an eventual conclusion of the form ‘P (complemented by S) is a contingent cause of R’ is inapplicable with respect to the issue of direction of causation, then we may affirm ‘R (complemented by notS) is a partial cause of P’ instead.
The following statistics, based on the listings below, are of interest:
- In figure 1, out of 64 conceivable positive moods, 30 are valid and 34 are invalid (of which 10, due to inconsistency in the premises).
- In figure 2, out of 64 conceivable positive moods, 18 are valid and 46 are invalid (of which 6, due to inconsistency in the premises).
- In figure 1, out of 64 conceivable positive moods, 18 are valid and 46 are invalid (of which 10, due to inconsistency in the premises).
Thus, out of the 192 positive moods considered, 66 (34%) are valid and 126 (66%) are invalid. Obviously, in view of this validity rate, such reasoning cannot be left to chance!
2. Moods in Figure 1.
§1. Mood No. 111 = mn/mn/mn. | VALID |
Q is a complete and necessary cause of R;P is a complete and necessary cause of Q;so, P is a complete and necessary cause of R. | by reduction to moods 155, 166. |
No mirror mood. |
§2. Mood No. 112 = mn/mq/mq. | VALID |
Q is a complete and necessary cause of R;P (complemented by S) is a complete and contingent cause of Q;so, P (complemented by S) is a complete and contingent cause of R. | by reduction to moods 118, 155. |
No. 113 = mn/np/np (similarly, through 117, 166). |
§3. Mood No. 121 = mq/mn/mq. | VALID |
Q (complemented by S) is a complete and contingent cause of R;P is a complete and necessary cause of Q;so, P (complemented by S) is a complete and contingent cause of R. | by reduction to moods 155, 181. |
No. 131 = np/mn/np (similarly, through 166, 171). |
§4. Mood No. 122 = mq/mq. | INVALID |
Q (complemented by P) is a complete and contingent cause of R;P (complemented by S) is a complete and contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | due to inconsistency of premises. |
No. 133 = np/np (similarly). |
§5. Mood No. 123 = mq/np. | INVALID |
Q (complemented by P) is a complete and contingent cause of R;P (complemented by S) is a partial and necessary cause of Q;does it follow that P is (complemented by S) a cause of R? No! | due to inconsistency of premises. |
No. 132 = np/mq (similarly). |
§6. Mood No. 114 = mn/pq/pq. | VALID |
Q is a complete and necessary cause of R;P (complemented by S) is a partial and contingent cause of Q;so, P (complemented by S) is a partial and contingent cause of R. | by reduction to moods 117, 118. |
No mirror mood. |
§7. Mood No. 141 = pq/mn/pq. | VALID |
Q (complemented by S) is a partial and contingent cause of R;P is a complete and necessary cause of Q;so, P (complemented by S) is a partial and contingent cause of R. | by reduction to moods 171, 181. |
No mirror mood. |
§8. Mood No. 124 = mq/pq/q. | VALID |
Q (complemented by P) is a complete and contingent cause of R;P (complemented by S) is a partial and contingent cause of Q;so, P (complemented by S) is a contingent cause of R. | by reduction to 128 or 184 and by matricial analysis. |
No. 134 = np/pq/p (similarly, through 137 or 174 and MA). |
§9. Mood No. 142. pq/mq. | INVALID |
Q (complemented by P) is a partial and contingent cause of R;P (complemented by S) is a complete and contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | due to inconsistency of premises. |
No. 143 = pq/np (similarly). |
§10. Mood No. 144 = pq/pq/pq. | VALID |
Q (complemented by P) is a partial and contingent cause of R;P (complemented by S) is a partial and contingent cause of Q;so, P (complemented by S) is a partial and contingent cause of R. | by reduction to moods 147+148, or 174+184. |
No mirror mood. |
§11. Mood No. 115 = mn/m/m. | VALID |
Q is a complete and necessary cause of R;P is a complete cause of Q;so, P is a complete cause of R. | by reduction to moods 111, 112, 155. |
No. 116 = mn/n/n (similarly, through 111, 113, 166). |
§12. Mood No. 151 = m/mn/m. | VALID |
Q is a complete cause of R;P is a complete and necessary cause of Q;so, P is a complete cause of R. | by reduction to moods 111, 121, 155. |
No. 161 = n/mn/n (similarly, through 111, 131, 166). |
§13. Mood No. 125 = mq/m/m. | VALID |
Q (complemented by S) is a complete and contingent cause of R;P is a complete cause of Q;so, P is a complete cause of R. | by reduction to 121, 155 and by matricial analysis. |
No. 136 = np/n/n (similarly, through 131, 166 and MA). |
§14. Mood No. 126 = mq/n. | INVALID |
Q (complemented by S) is a complete and contingent cause of R;P is a necessary cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 121 and by matricial analysis. |
No. 135 = np/m (similarly, through 131 and MA). |
§15. Mood No. 152 = m/mq/m. | VALID |
Q is a complete cause of R;P (complemented by S) is a complete and contingent cause of Q;so, P is a complete cause of R. | by reduction to 112, 155 and by matricial analysis. |
No. 163 = n/np/n (similarly, through 113, 166 and MA). |
§16. Mood No. 153 = m/np. | INVALID |
Q is a complete cause of R;P (complemented by S) is a partial and necessary cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 113 and by matricial analysis. |
No. 162 = n/mq (similarly, through 112 and MA). |
§17. Mood No. 117 = mn/p/p. | VALID |
Q is a complete and necessary cause of R;P (complemented by S) is a partial cause of Q;so, P (complemented by S) is a partial cause of R. | by reduction to 113, 114 and by matricial analysis. |
No. 118 = mn/q/q (similarly, through 112, 114 and MA). |
§18. Mood No. 171 = p/mn/p. | VALID |
Q (complemented by S) is a partial cause of R;P is a complete and necessary cause of Q;so, P (complemented by S) is a partial cause of R. | by reduction to 131, 141 and by matricial analysis. |
No. 181 = q/mn/q (similarly, through 121, 141 and MA). |
§19. Mood No. 127 = mq/p. | INVALID |
Q (complemented by P) is a complete and contingent cause of R;P (complemented by S) is a partial cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 124 and by matricial analysis. |
No. 138 = np/q (similarly, through 134 and MA). |
§20. Mood No. 128 = mq/q/q. | VALID |
Q (complemented by P) is a complete and contingent cause of R;P (complemented by S) is a contingent cause of Q;so, P (complemented by S) is a contingent cause of R. | by reduction to 122, 124 and by matricial analysis. |
No. 137 = np/p/p (similarly, through 133, 134 and MA). |
§21. Mood No. 172 = p/mq. | INVALID |
Q (complemented by P) is a partial cause of R;P (complemented by S) is a complete and contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | due to inconsistency of premises. |
No. 183 = q/np (similarly). |
§22. Mood No. 173 = p/np. | INVALID |
Q (complemented by P) is a partial cause of R;P (complemented by S) is a partial and necessary cause of Q;does it follow that P is (complemented by S) a cause of R? No! | due to inconsistency of premises. |
No. 182 = q/mq (similarly). |
§23. Mood No. 145 = pq/m. | INVALID |
Q (complemented by S) is a partial and contingent cause of R;P is a complete cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 141 and by matricial analysis. |
No. 146 = pq/n (similarly, through 141 and MA). |
§24. Mood No. 154 = m/pq. | INVALID |
Q is a complete cause of R;P (complemented by S) is a partial and contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to 114, 124 and by matricial analysis. |
No. 164 = n/pq (similarly, through 114, 134 and MA). |
§25. Mood No. 147 = pq/p/p. | VALID |
Q (complemented by P) is a partial and contingent cause of R;P (complemented by S) is a partial cause of Q;so, P (complemented by S) is a partial cause of R. | by reduction to mood 144 and by matricial analysis. |
No. 148 = pq/q/q (similarly, through 144 and MA). |
§26. Mood No. 174 = p/pq/p. | VALID |
Q (complemented by P) is a partial cause of R;P (complemented by S) is a partial and contingent cause of Q;so, P (complemented by S) is a partial cause of R. | by reduction to mood 134 and by matricial analysis. |
No. 184 = q/pq/q (similarly, through 124 and MA). |
§27. Mood No. 155 = m/m/m. | VALID |
Q is a complete cause of R;P is a complete cause of Q;so, P is a complete cause of R. | by reduction to 111, 112 and by matricial analysis. |
No. 166 = n/n/n (similarly, through 111, 113 and MA). |
§28. Mood No. 156 = m/n. | INVALID |
Q is a complete cause of R;P is a necessary cause of Q;does it follow that P is a cause of R? No! | by reduction to moods 111, 113, 121. |
No. 165 = n/m (similarly, through 111, 112, 131). |
§29. Mood No. 157 = m/p. | INVALID |
Q is a complete cause of R;P (complemented by S) is a partial cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to moods 113, 114, 124. |
No. 168 = n/q (similarly, through 112, 114, 134). |
§30. Mood No. 158 = m/q. | INVALID |
Q is a complete cause of R;P (complemented by S) is a contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to moods 112, 114, 152. |
No. 167 = n/p (similarly, through 113, 114, 163). |
§31. Mood No. 175 = p/m. | INVALID |
Q (complemented by S) is a partial cause of R;P is a complete cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to moods 131, 135. |
No. 186 = q/n (similarly, through 121, 126). |
§32. Mood No. 176 = p/n. | INVALID |
Q (complemented by S) is a partial cause of R;P is a necessary cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to moods 131, 136, 141. |
No. 185 = q/m (similarly, through 121, 125, 141). |
§33. Mood No. 177 = p/p. | INVALID |
Q (complemented by P) is a partial cause of R;P (complemented by S) is a partial cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to 133, 134 and by matricial analysis. |
No. 188 = q/q (similarly, through 122, 124 and MA). |
§34. Mood No. 178 = p/q. | INVALID |
Q (complemented by P) is a partial cause of R;P (complemented by S) is a contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to moods 134, 138. |
No. 187 = q/p (similarly, through 124, 127). |
3. Moods in Figure 2.
§1. Mood No. 211 = mn/mn/mn. | VALID |
R is a complete and necessary cause of Q;P is a complete and necessary cause of Q;so, P is a complete and necessary cause of R. | by reduction to mood 111. |
No mirror mood. |
§2. Mood No. 212 = mn/mq/mq. | VALID |
R is a complete and necessary cause of Q;P (complemented by S) is a complete and contingent cause of Q;so, P (complemented by S) is a complete and contingent cause of R. | by reduction to mood 112. |
No. 213 = mn/np/np (similarly, through 113). |
§3. Mood No. 221 = mq/mn/n. | VALID |
R (complemented by S) is a complete and contingent cause of Q;P is a complete and necessary cause of Q;so, P is a necessary cause of R. | by reduction to mood 256 and by matricial analysis. |
No. 231 = np/mn/m (similarly, through 265 and MA). |
§4. Mood No. 222 = mq/mq. | INVALID |
R (complemented by P) is a complete and contingent cause of Q;P (complemented by S) is a complete and contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by matricial analysis. |
No. 233 = np/np (similarly, through MA). |
§5. Mood No. 223 = mq/np. | INVALID |
R (complemented by P) is a complete and contingent cause of Q;P (complemented by S) is a partial and necessary cause of Q;does it follow that P is (complemented by S) a cause of R? No! | due to inconsistency of premises. |
No. 232 = np/mq (similarly). |
§6. Mood No. 214 = mn/pq/pq. | VALID |
R is a complete and necessary cause of Q;P (complemented by S) is a partial and contingent cause of Q;so, P (complemented by S) is a partial and contingent cause of R. | by reduction to mood 114. |
No mirror mood. |
§7. Mood No. 241 = pq/mn. | INVALID |
R (complemented by S) is a partial and contingent cause of Q;P is a complete and necessary cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by matricial analysis. |
No mirror mood. |
§8. Mood No. 224 = mq/pq. | INVALID |
R (complemented by P) is a complete and contingent cause of Q;P (complemented by S) is a partial and contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by matricial analysis. |
No. 234 = np/pq (similarly, through MA). |
§9. Mood No. 242. pq/mq. | INVALID |
R (complemented by P) is a partial and contingent cause of Q;P (complemented by S) is a complete and contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | due to inconsistency of premises. |
No. 243 = pq/np (similarly). |
§10. Mood No. 244 = pq/pq. | INVALID |
R (complemented by P) is a partial and contingent cause of Q;P (complemented by S) is a partial and contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by matricial analysis. |
No mirror mood. |
§11. Mood No. 215 = mn/m/m. | VALID |
R is a complete and necessary cause of Q;P is a complete cause of Q;so, P is a complete cause of R. | by reduction to mood 115. |
No. 216 = mn/n/n (similarly, through 116). |
§12. Mood No. 251 = m/mn/n. | VALID |
R is a complete cause of Q;P is a complete and necessary cause of Q;so, P is a necessary cause of R. | by reduction to mood 161. |
No. 261 = n/mn/m (similarly, through 151). |
§13. Mood No. 225 = mq/m. | INVALID |
R (complemented by S) is a complete and contingent cause of Q;P is a complete cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to moods 221, 222. |
No. 236 = np/n (similarly, through 231, 233). |
§14. Mood No. 226 = mq/n/n. | VALID |
R (complemented by S) is a complete and contingent cause of Q;P is a necessary cause of Q;so, P is a necessary cause of R. | by reduction to moods 221, 256. |
No. 235 = np/m/m (similarly, through 231, 265). |
§15. Mood No. 252 = m/mq. | INVALID |
R is a complete cause of Q;P (complemented by S) is a complete and contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 162. |
No. 263 = n/np (similarly, through 153). |
§16. Mood No. 253 = m/np/n. | VALID |
R is a complete cause of Q;P (complemented by S) is a partial and necessary cause of Q;so, P is a necessary cause of R. | by reduction to mood 163. |
No. 262 = n/mq/m (similarly, through 152). |
§17. Mood No. 217 = mn/p/p. | VALID |
R is a complete and necessary cause of Q;P (complemented by S) is a partial cause of Q;so, P (complemented by S) is a partial cause of R. | by reduction to mood 117. |
No. 218 = mn/q/q (similarly, through 118). |
§18. Mood No. 271 = p/mn. | INVALID |
R (complemented by S) is a partial cause of Q;P is a complete and necessary cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to moods 231, 241. |
No. 281 = q/mn (similarly, through 221, 241). |
§19. Mood No. 227 = mq/p. | INVALID |
R (complemented by P) is a complete and contingent cause of Q;P (complemented by S) is a partial cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 224. |
No. 238 = np/q (similarly, through 234). |
§20. Mood No. 228 = mq/q. | INVALID |
R (complemented by P) is a complete and contingent cause of Q;P (complemented by S) is a contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 222. |
No. 237 = np/p (similarly, through 233). |
§21. Mood No. 272 = p/mq. | INVALID |
R (complemented by P) is a partial cause of Q;P (complemented by S) is a complete and contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | due to inconsistency of premises. |
No. 283 = q/np (similarly). |
§22. Mood No. 273 = p/np. | INVALID |
R (complemented by P) is a partial cause of Q;P (complemented by S) is a partial and necessary cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 233. |
No. 282 = q/mq (similarly, through 222). |
§23. Mood No. 245 = pq/m. | INVALID |
R (complemented by S) is a partial and contingent cause of Q;P is a complete cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 241. |
No. 246 = pq/n (similarly, through 241). |
§24. Mood No. 254 = m/pq. | INVALID |
R is a complete cause of Q;P (complemented by S) is a partial and contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 164. |
No. 264 = n/pq (similarly, through 154). |
§25. Mood No. 247 = pq/p. | INVALID |
R (complemented by P) is a partial and contingent cause of Q;P (complemented by S) is a partial cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 241. |
No. 248 = pq/q (similarly, through 244). |
§26. Mood No. 274 = p/pq. | INVALID |
R (complemented by P) is a partial cause of Q;P (complemented by S) is a partial and contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 234. |
No. 284 = q/pq (similarly, through 224). |
§27. Mood No. 255 = m/m. | INVALID |
R is a complete cause of Q;P is a complete cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 165. |
No. 266 = n/n (similarly, through 156). |
§28. Mood No. 256 = m/n/n. | VALID |
R is a complete cause of Q;P is a necessary cause of Q;so, P is a necessary cause of R. | by reduction to mood 166. |
No. 265 = n/m/m (similarly, through 155). |
§29. Mood No. 257 = m/p. | INVALID |
R is a complete cause of Q;P (complemented by S) is a partial cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 167. |
No. 268 = n/q (similarly, through 158). |
§30. Mood No. 258 = m/q. | INVALID |
R is a complete cause of Q;P (complemented by S) is a contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 168. |
No. 267 = n/p (similarly, through 157). |
§31. Mood No. 275 = p/m. | INVALID |
R (complemented by S) is a partial cause of Q;P is a complete cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to moods 231, 241. |
No. 286 = q/n (similarly, through 221, 241). |
§32. Mood No. 276 = p/n. | INVALID |
R (complemented by S) is a partial cause of Q;P is a necessary cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to moods 231, 233. |
No. 285 = q/m (similarly, through 221, 222). |
§33. Mood No. 277 = p/p. | INVALID |
R (complemented by P) is a partial cause of Q;P (complemented by S) is a partial cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 233. |
No. 288 = q/q (similarly, through 222). |
§34. Mood No. 278 = p/q. | INVALID |
R (complemented by P) is a partial cause of Q;P (complemented by S) is a contingent cause of Q;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 234. |
No. 287 = q/p (similarly, through 224). |
4. Moods in Figure 3.
§1. Mood No. 311 = mn/mn/mn. | VALID |
Q is a complete and necessary cause of R;Q is a complete and necessary cause of P;so, P is a complete and necessary cause of R. | by reduction to mood 111. |
No mirror mood. |
§2. Mood No. 312 = mn/mq/n. | VALID |
Q is a complete and necessary cause of R;Q (complemented by S) is a complete and contingent cause of P;so, P is a necessary cause of R. | by reduction to mood 365 and matricial analysis. |
No. 313 = mn/np/m (similarly, through 356 and MA). |
§3. Mood No. 321 = mq/mn/mq. | VALID |
Q (complemented by S) is a complete and contingent cause of R;Q is a complete and necessary cause of P;so, P (complemented by S) is a complete and contingent cause of R. | by reduction to mood 121. |
No. 331 = np/mn/np (similarly, through 131). |
§4. Mood No. 322 = mq/mq. | INVALID |
Q (complemented by P) is a complete and contingent cause of R;Q (complemented by S) is a complete and contingent cause of P;does it follow that P is (complemented by S) a cause of R? No! | due to inconsistency of premises. |
No. 333 = np/np (similarly). |
§5. Mood No. 323 = mq/np. | INVALID |
Q (complemented by P) is a complete and contingent cause of R;Q (complemented by S) is a partial and necessary cause of P;does it follow that P is (complemented by S) a cause of R? No! | due to inconsistency of premises. |
No. 332 = np/mq (similarly). |
§6. Mood No. 314 = mn/pq. | INVALID |
Q is a complete and necessary cause of R;Q (complemented by S) is a partial and contingent cause of P;does it follow that P is (complemented by S) a cause of R? No! | by matricial analysis. |
No mirror mood. |
§7. Mood No. 341 = pq/mn/pq. | VALID |
Q (complemented by S) is a partial and contingent cause of R;Q is a complete and necessary cause of P;so, P (complemented by S) is a partial and contingent cause of R. | by reduction to mood 141. |
No mirror mood. |
§8. Mood No. 324 = mq/pq. | INVALID |
Q (complemented by P) is a complete and contingent cause of R;Q (complemented by S) is a partial and contingent cause of P;does it follow that P is (complemented by S) a cause of R? No! | by matricial analysis. |
No. 334 = np/pq (similarly, through MA). |
§9. Mood No. 342. pq/mq. | INVALID |
Q (complemented by P) is a partial and contingent cause of R;Q (complemented by S) is a complete and contingent cause of P;does it follow that P is (complemented by S) a cause of R? No! | due to inconsistency of premises. |
No. 343 = pq/np (similarly). |
§10. Mood No. 344 = pq/pq. | INVALID |
Q (complemented by P) is a partial and contingent cause of R;Q (complemented by S) is a partial and contingent cause of P;does it follow that P is (complemented by S) a cause of R? No! | by matricial analysis. |
No mirror mood. |
§11. Mood No. 315 = mn/m/n. | VALID |
Q is a complete and necessary cause of R;Q is a complete cause of P;so, P is a necessary cause of R. | by reduction to mood 116. |
No. 316 = mn/n/m (similarly, through 115). |
§12. Mood No. 351 = m/mn/m. | VALID |
Q is a complete cause of R;Q is a complete and necessary cause of P;so, P is a complete cause of R. | by reduction to moods 151. |
No. 361 = n/mn/n (similarly, through 161). |
§13. Mood No. 325 = mq/m. | INVALID |
Q (complemented by S) is a complete and contingent cause of R;Q is a complete cause of P;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 126. |
No. 336 = np/n (similarly, through 135). |
§14. Mood No. 326 = mq/n/m. | VALID |
Q (complemented by S) is a complete and contingent cause of R;Q is a necessary cause of P;so, P is a complete cause of R. | by reduction to mood 125. |
No. 335 = np/m/n (similarly, through 136). |
§15. Mood No. 352 = m/mq. | INVALID |
Q is a complete cause of R;Q (complemented by S) is a complete and contingent cause of P;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 312 and by matricial analysis. |
No. 363 = n/np (similarly, through 313 and MA). |
§16. Mood No. 353 = m/np/m. | VALID |
Q is a complete cause of R;Q (complemented by S) is a partial and necessary cause of P;so, P is a complete cause of R. | by reduction to moods 313, 356. |
No. 362 = n/mq/n (similarly, through 312, 365). |
§17. Mood No. 317 = mn/p. | INVALID |
Q is a complete and necessary cause of R;Q (complemented by S) is a partial cause of P;does it follow that P is (complemented by S) a cause of R? No! | by reduction to moods 313, 314. |
No. 318 = mn/q (similarly, through 312, 314). |
§18. Mood No. 371 = p/mn/p. | VALID |
Q (complemented by S) is a partial cause of R;Q is a complete and necessary cause of P;so, P (complemented by S) is a partial cause of R. | by reduction to mood 171. |
No. 381 = q/mn/q (similarly, through 181). |
§19. Mood No. 327 = mq/p. | INVALID |
Q (complemented by P) is a complete and contingent cause of R;Q (complemented by S) is a partial cause of P;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 324. |
No. 338 = np/q (similarly, through 334). |
§20. Mood No. 328 = mq/q. | INVALID |
Q (complemented by P) is a complete and contingent cause of R;Q (complemented by S) is a contingent cause of P;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 324. |
No. 337 = np/p (similarly, through 334). |
§21. Mood No. 372 = p/mq. | INVALID |
Q (complemented by P) is a partial cause of R;Q (complemented by S) is a complete and contingent cause of P;does it follow that P is (complemented by S) a cause of R? No! | due to inconsistency of premises. |
No. 383 = q/np (similarly). |
§22. Mood No. 373 = p/np. | INVALID |
Q (complemented by P) is a partial cause of R;Q (complemented by S) is a partial and necessary cause of P;does it follow that P is (complemented by S) a cause of R? No! | due to inconsistency of premises. |
No. 382 = q/mq (similarly). |
§23. Mood No. 345 = pq/m. | INVALID |
Q (complemented by S) is a partial and contingent cause of R;Q is a complete cause of P;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 146. |
No. 346 = pq/n (similarly, through 145). |
§24. Mood No. 354 = m/pq. | INVALID |
Q is a complete cause of R;Q (complemented by S) is a partial and contingent cause of P;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 314. |
No. 364 = n/pq (similarly, through 314). |
§25. Mood No. 347 = pq/p. | INVALID |
Q (complemented by P) is a partial and contingent cause of R;Q (complemented by S) is a partial cause of P;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 344. |
No. 348 = pq/q (similarly, through 344). |
§26. Mood No. 374 = p/pq. | INVALID |
Q (complemented by P) is a partial cause of R;Q (complemented by S) is a partial and contingent cause of P;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 334. |
No. 384 = q/pq (similarly, through 324). |
§27. Mood No. 355 = m/m. | INVALID |
Q is a complete cause of R;Q is a complete cause of P;does it follow that P is a cause of R? No! | by reduction to mood 156. |
No. 366 = n/n (similarly, through 165). |
§28. Mood No. 356 = m/n/m. | VALID |
Q is a complete cause of R;Q is a necessary cause of P;so, P is a complete cause of R. | by reduction to mood 155. |
No. 365 = n/m/n (similarly, through 166). |
§29. Mood No. 357 = m/p. | INVALID |
Q is a complete cause of R;Q (complemented by S) is a partial cause of P;does it follow that P is (complemented by S) a cause of R? No! | by reduction to moods 313, 314. |
No. 368 = n/q (similarly, through 312, 314). |
§30. Mood No. 358 = m/q. | INVALID |
Q is a complete cause of R;Q (complemented by S) is a contingent cause of P;does it follow that P is (complemented by S) a cause of R? No! | by reduction to moods 312, 314. |
No. 367 = n/p (similarly, through 313, 314). |
§31. Mood No. 375 = p/m. | INVALID |
Q (complemented by S) is a partial cause of R;Q is a complete cause of P;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 176. |
No. 386 = q/n (similarly, through 185). |
§32. Mood No. 376 = p/n. | INVALID |
Q (complemented by S) is a partial cause of R;Q is a necessary cause of P;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 175. |
No. 385 = q/m (similarly, through 186). |
§33. Mood No. 377 = p/p. | INVALID |
Q (complemented by P) is a partial cause of R;Q (complemented by S) is a partial cause of P;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 334. |
No. 388 = q/q (similarly, through 324). |
§34. Mood No. 378 = p/q. | INVALID |
Q (complemented by P) is a partial cause of R;Q (complemented by S) is a contingent cause of P;does it follow that P is (complemented by S) a cause of R? No! | by reduction to mood 334. |
No. 387 = q/p (similarly, through 324). |
Chapter 7. Reduction of Positive Moods
1. Reduction.
The method of reduction was first theoretically identified by Aristotle, though of course it had been practically used by human beings long before. Reduction, in its broadest sense, consists in showing that an argument is valid or invalid because another argument is valid or invalid. Thus, reduction is not a primary process of evaluation but a method for transmitting validity or invalidity, and therefore presupposes that we have some other means for establishing certain fundamental validities or invalidities.
The ‘other means’, in our case, is matricial analysis; we shall use this method for a number of validations and invalidations, but before we do so we want to find out the minimum number of moods of causative syllogism which have to be so treated. For as already said, matricial analysis is a cumbersome, though essential and certain, method; and we wish to facilitate our task. Furthermore, while this method treats each mood as ‘an island onto itself’, reduction reveals the precise interrelations between moods, which we ought to be aware of.
Reduction is a short-cut. In the field of causative syllogism, reduction has many guises. The broad Aristotelian distinction between direct reduction and indirect (or ad absurdum) reduction is of course applicable here; but we may find fit to subdivide the concept of direct reduction.
Within the domain of positive moods of any figure, the validity of conclusions is transmitted by direct reduction and the invalidity of conclusions is transmitted by indirect reduction, within the same figure. The main implication which concerns us here is subalternation by joint determinations of generic determinations (thus, mn implies m and n, mq implies m and q, np implies p and n, and pq implies p and q). Since subalternation is one-way implication, different implications are used for validations and invalidations.
But there is also reduction from one figure to another, for which the eductive process of conversion is appropriate. Some second figure moods may be directly reduced to first figure moods, by conversion of the major premise; and some third figure moods may be directly reduced to first figure moods, by conversion of the minor premise. Since conversion works both ways, such reductions serve for both validation and invalidation.
We can thus distinguish between two sorts of direct reduction of positive moods, with reference to the precise sort of implication appealed to, i.e. subalternation (within the same figure) or conversion (across figures).[44]
It should be noted that the validity of any conclusion implies the invalidity of conflicting putative conclusions (thus if m is true, p cannot be, and vice versa; and if n is true, q cannot be, and vice versa); though note well that the invalidity of a putative conclusion does not imply the validity of its opposite, i.e. both m and p or both n and q may be invalid). We can on this basis save ourselves some work; this also might be viewed as a sort of indirect reduction.
Before going further, let us point out that some moods are composed of incompatible premises. Such moods may be declared invalid without further ado. This occurs specifically in subfigure (d) of each figure, where the premises have two items in common (namely, P and Q). Here, if the minor premise has a strong component, it may conflict with the weak component(s) of the major premise.
We shall now identify the implications between moods within any of the figures, due to inclusion within compound forms (joint determinations) of their constituent forms (generic determinations). The following table lists all implications between premises (note well) of moods; it is based on information given in Table 5.6, developed in the chapter on causative syllogism, listing the 64 moods conceivable in each figure.
The mood numbers on the left imply the adjacent mood numbers on the right. | |||||||||||||||
Along rows of table listing all moods: | |||||||||||||||
11 | 51 | 12 | 52 | 13 | 53 | 14 | 54 | 15 | 55 | 16 | 56 | 17 | 57 | 18 | 58 |
11 | 61 | 12 | 62 | 13 | 63 | 14 | 64 | 15 | 65 | 16 | 66 | 17 | 67 | 18 | 68 |
21 | 51 | 22 | 52 | 23 | 53 | 24 | 54 | 25 | 55 | 26 | 56 | 27 | 57 | 28 | 58 |
21 | 81 | 22 | 82 | 23 | 83 | 24 | 84 | 25 | 85 | 26 | 86 | 27 | 87 | 28 | 88 |
31 | 61 | 32 | 62 | 33 | 63 | 34 | 64 | 35 | 65 | 36 | 66 | 37 | 67 | 38 | 68 |
31 | 71 | 32 | 72 | 33 | 73 | 34 | 74 | 35 | 75 | 36 | 76 | 37 | 77 | 38 | 78 |
41 | 71 | 42 | 72 | 43 | 73 | 44 | 74 | 45 | 75 | 46 | 76 | 47 | 77 | 48 | 78 |
41 | 81 | 42 | 82 | 43 | 83 | 44 | 84 | 45 | 85 | 46 | 86 | 47 | 87 | 48 | 88 |
Down columns of table listing all moods: | |||||||||||||||
11 | 15 | 21 | 25 | 31 | 35 | 41 | 45 | 51 | 55 | 61 | 65 | 71 | 75 | 81 | 85 |
11 | 16 | 21 | 26 | 31 | 36 | 41 | 46 | 51 | 56 | 61 | 66 | 71 | 76 | 81 | 86 |
12 | 15 | 22 | 25 | 32 | 35 | 42 | 45 | 52 | 55 | 62 | 65 | 72 | 75 | 82 | 85 |
12 | 18 | 22 | 28 | 32 | 38 | 42 | 48 | 52 | 58 | 62 | 68 | 72 | 78 | 82 | 88 |
13 | 16 | 23 | 26 | 33 | 36 | 43 | 46 | 53 | 56 | 63 | 66 | 73 | 76 | 83 | 86 |
13 | 17 | 23 | 27 | 33 | 37 | 43 | 47 | 53 | 57 | 63 | 67 | 73 | 77 | 83 | 87 |
14 | 17 | 24 | 27 | 34 | 37 | 44 | 47 | 54 | 57 | 64 | 67 | 74 | 77 | 84 | 87 |
14 | 18 | 24 | 28 | 34 | 38 | 44 | 48 | 54 | 58 | 64 | 68 | 74 | 78 | 84 | 88 |
Note well that each implication may in turn imply others, i.e. one must follow up implications of implications. For instance, 11 implies 15 and 16, and 51 and 61; in turn, 15 implies 55 and 65, and 16 implies 56 and 66; also, 51 implies 55 and 56, and 61 implies 65 and 66. Similarly for other premises, as shown in the above table.
Also note, some of the implications shown in the above table may be useless in practice for a given figure: this occurs when a mood referred to has inconsistent premises.
The following are the principles for inference of validity or invalidity. Note well the condition that the validating or invalidating mood be internally consistent; as we explained, it can happen, in a given figure, that they are not so.
- If the premises of one of the above moods, say Y, are consistent and imply those of another, say X, then any validated conclusion of X, say c1, is also a valid conclusion of Y. (But an invalidated conclusion of X, say c2, cannot be inferred to be an invalid conclusion of Y.)
Proof: Since Y implies X and X implies c1, it follows that Y implies c1.
(But that X does not imply c2, does not mean that Y does not imply c2.)
- If the premises of one of the above moods, say Z, are consistent and imply those of another, say Y, then any invalidated conclusion of Z, say c2, is also an invalid conclusion of Y. (But a validated conclusion of Z, say c1, cannot be inferred to be a valid conclusion of Y.)
Proof: Since Z implies Y and Z does not imply c2, it follows that Y does not imply c2;
for given that Z implies Y, if Y implied c2, Z would imply c2.
(But that Z implies c1, does not mean that Y implies c1.)
One should be careful not to confuse the premises of a mood with a mood as a whole. Referring to the above rules, in case (1), while Y implies X, the validity of X+c1 implies the validity of Y+c1 (this is a direct reduction). In case (2), while Z implies Y, the invalidity of Z+c2 implies the invalidity of Y+c2 (this is an indirect reduction).
Generally, then, to establish a mood Y+c1+notc2 by reduction, we must look for two moods X and Z, such that (1) Y implies X, which concludes c1, and (2) Y is implied by Z, which fails to conclude c2. The following diagram illustrates these principles:
- The above applies to reductions within a given figure, by subalternation. In the special case of direct reduction across figures, by conversion of the major premise (to derive figure 2) or the minor premise (to derive figure 3), the implications between the premises concerned are two-way; it follows in such case that both validity and invalidity are transmitted by the same mood of figure 1.
The following table, based on the preceding one, shows more explicitly the possible sources of validity or invalidity by reduction, for each mood within any figure. It should be noted that we cannot (as far as I can see) predict from it, at the outset for all figures, which moods will require matricial analysis; such knowledge has to be acquired in each figure by judicious trial and error.
Moods implying central mood(if any of them is invalid,the central mood is also invalid) | Mood | Moods implied by central mood(if any of them is valid,the central mood is also valid) | ||||||||||||||
11 | 15 | 16 | 51 | 55 | 56 | 61 | 65 | 66 | ||||||||
12 | 15 | 18 | 52 | 55 | 58 | 62 | 65 | 68 | ||||||||
13 | 16 | 17 | 53 | 56 | 57 | 63 | 66 | 67 | ||||||||
14 | 17 | 18 | 54 | 57 | 58 | 64 | 67 | 68 | ||||||||
11 | 12 | 15 | 55 | 65 | ||||||||||||
11 | 13 | 16 | 56 | 66 | ||||||||||||
13 | 14 | 17 | 57 | 67 | ||||||||||||
12 | 14 | 18 | 58 | 68 | ||||||||||||
21 | 25 | 26 | 51 | 55 | 56 | 81 | 85 | 86 | ||||||||
22 | 25 | 28 | 52 | 55 | 58 | 82 | 85 | 88 | ||||||||
23 | 26 | 27 | 53 | 56 | 57 | 83 | 86 | 87 | ||||||||
24 | 27 | 28 | 54 | 57 | 58 | 84 | 87 | 88 | ||||||||
21 | 22 | 25 | 55 | 85 | ||||||||||||
21 | 23 | 26 | 56 | 86 | ||||||||||||
23 | 24 | 27 | 57 | 87 | ||||||||||||
22 | 24 | 28 | 58 | 88 | ||||||||||||
31 | 35 | 36 | 61 | 65 | 66 | 71 | 75 | 76 | ||||||||
32 | 35 | 38 | 62 | 65 | 68 | 72 | 75 | 78 | ||||||||
33 | 36 | 37 | 63 | 66 | 67 | 73 | 76 | 77 | ||||||||
34 | 37 | 38 | 64 | 67 | 68 | 74 | 77 | 78 | ||||||||
31 | 32 | 35 | 65 | 75 | ||||||||||||
31 | 33 | 36 | 66 | 76 | ||||||||||||
33 | 34 | 37 | 67 | 77 | ||||||||||||
32 | 34 | 38 | 68 | 78 | ||||||||||||
41 | 45 | 46 | 71 | 75 | 76 | 81 | 85 | 86 | ||||||||
42 | 45 | 48 | 72 | 75 | 78 | 82 | 85 | 88 | ||||||||
43 | 46 | 47 | 73 | 76 | 77 | 83 | 86 | 87 | ||||||||
44 | 47 | 48 | 74 | 77 | 78 | 84 | 87 | 88 | ||||||||
41 | 42 | 45 | 75 | 85 | ||||||||||||
41 | 43 | 46 | 76 | 86 | ||||||||||||
43 | 44 | 47 | 77 | 87 | ||||||||||||
42 | 44 | 48 | 78 | 88 | ||||||||||||
11 | 21 | 51 | 55 | 56 | ||||||||||||
12 | 22 | 52 | 55 | 58 | ||||||||||||
13 | 23 | 53 | 56 | 57 | ||||||||||||
14 | 24 | 54 | 57 | 58 | ||||||||||||
11 | 12 | 15 | 21 | 22 | 25 | 51 | 52 | 55 | ||||||||
11 | 13 | 16 | 21 | 23 | 26 | 51 | 53 | 56 | ||||||||
13 | 14 | 17 | 23 | 24 | 27 | 53 | 54 | 57 | ||||||||
12 | 14 | 18 | 22 | 24 | 28 | 52 | 54 | 58 |
Table 7.2 continued.
11 | 31 | 61 | 65 | 66 | ||||||||||||
12 | 32 | 62 | 65 | 68 | ||||||||||||
13 | 33 | 63 | 66 | 67 | ||||||||||||
14 | 34 | 64 | 67 | 68 | ||||||||||||
11 | 12 | 15 | 31 | 32 | 35 | 61 | 62 | 65 | ||||||||
11 | 13 | 16 | 31 | 33 | 36 | 61 | 63 | 66 | ||||||||
13 | 14 | 17 | 33 | 34 | 37 | 63 | 64 | 67 | ||||||||
12 | 14 | 18 | 32 | 34 | 38 | 62 | 64 | 68 | ||||||||
31 | 41 | 71 | 75 | 76 | ||||||||||||
32 | 42 | 72 | 75 | 78 | ||||||||||||
33 | 43 | 73 | 76 | 77 | ||||||||||||
34 | 44 | 74 | 77 | 78 | ||||||||||||
31 | 32 | 35 | 41 | 42 | 45 | 71 | 72 | 75 | ||||||||
31 | 33 | 36 | 41 | 43 | 46 | 71 | 73 | 76 | ||||||||
33 | 34 | 37 | 43 | 44 | 47 | 73 | 74 | 77 | ||||||||
32 | 34 | 38 | 42 | 44 | 48 | 72 | 74 | 78 | ||||||||
21 | 41 | 81 | 85 | 86 | ||||||||||||
22 | 42 | 82 | 85 | 88 | ||||||||||||
23 | 43 | 83 | 86 | 87 | ||||||||||||
24 | 44 | 84 | 87 | 88 | ||||||||||||
21 | 22 | 25 | 41 | 42 | 45 | 81 | 82 | 85 | ||||||||
21 | 23 | 26 | 41 | 43 | 46 | 81 | 83 | 86 | ||||||||
23 | 24 | 27 | 43 | 44 | 47 | 83 | 84 | 87 | ||||||||
22 | 24 | 28 | 42 | 44 | 48 | 82 | 84 | 88 |
Remember that breaks will occur in such implications, if any mood is invalid due to inconsistency between premises.
The following tables summarize the results obtained by such reductions, for each of the figures. Conclusions not validated or invalidated by such means must be evaluated through matricial analysis (which is done in the next chapter). The tables below may be read as follows:
yes = element of conclusion (m, n, p or q) are implied by the given premises.
no = element of conclusion (m, n, p or q) are not implied (which does not mean denied) by the given premises.
by = by any sort of reduction to (number of mood used) or MA (matricial analysis).
Elements of conclusions for which matricial analysis is required are shaded.
since = for given premises, if an element of conclusion is valid (yes), then its contrary element is invalid (no).
** = incompatible premises.
nil = no valid conclusion.
2. Reductions in Figure 1.
First, note that ten moods in subfigure 1d have inconsistent premises. Specifically, if the minor premise (which has form P(S)Q) involves a strong determination, then it conflicts with the weak determination(s) of the major premise (which has form Q(P)R).
For if the minor concerns complete causation, clause (i) of which means that (P + notQ) is impossible – it is incompatible with the major, which implies (P + notQ) is possible, whether it concerns partial causation (see clause (ii) of that) or contingent causation (see clause (iii) of that). Similarly, if the minor concerns necessary causation, clause (i) of which means that (notP + Q) is impossible – it is incompatible with the major, which implies (notP + Q) is possible, whether it concerns partial causation (see clause (iii) of that) or contingent causation (see clause (ii) of that).
Ref. | Mood # | Elements of conclusion implied? | |||
§1 | 111 | m | n | p | q |
major | mn | yes | yes | no | no |
minor | mn | by | by | since | since |
concl. | mn | 155 | 166 | m | n |
§2 | 112 | m | n | p | q |
major | mn | yes | no | no | yes |
minor | mq | by | since | since | by |
concl. | mq | 155 | q | m | 118 |
§2 | 113 | m | n | p | q |
major | mn | no | yes | yes | no |
minor | np | since | by | by | since |
concl. | np | p | 166 | 117 | n |
§6 | 114 | m | n | p | q |
major | mn | no | no | yes | yes |
minor | pq | since | since | by | by |
concl. | pq | p | q | 117 | 118 |
§11 | 115 | m | n | p | q |
major | mn | yes | no | no | no |
minor | m | by | by | since | by |
concl. | m | 155 | 112 | m | 111 |
§11 | 116 | m | n | p | q |
major | mn | no | yes | no | no |
minor | n | by | by | by | since |
concl. | n | 113 | 166 | 111 | n |
§17 | 117 | m | n | p | q |
major | mn | no | no | yes | no |
minor | p | since | by | by | by |
concl. | p | p | 114 | MA | 113 |
§17 | 118 | m | n | p | q |
major | mn | no | no | no | yes |
minor | q | by | since | by | by |
concl. | q | 114 | q | 112 | MA |
Table 7.3 continued.
§3 | 121 | m | n | p | q |
major | mq | yes | no | no | yes |
minor | mn | by | since | since | by |
concl. | mq | 155 | q | m | 181 |
§4 | 122 | m | n | p | q |
major | mq | q of major premise and | |||
minor | mq | m of minor premise | |||
concl. | ** | are incompatible | |||
§5 | 123 | m | n | p | q |
major | mq | q of major premise and | |||
minor | np | n of minor premise | |||
concl. | ** | are incompatible | |||
§8 | 124 | m | n | p | q |
major | mq | no | no | no | yes |
minor | pq | by | since | by | by |
concl. | q | MA | q | MA | 128,184 |
§13 | 125 | m | n | p | q |
major | mq | yes | no | no | no |
minor | m | by | by | since | by |
concl. | m | 155 | 121 | m | MA |
§14 | 126 | m | n | p | q |
major | mq | no | no | no | no |
minor | n | by | by | by | by |
concl. | nil | MA | 121 | 121 | MA |
§19 | 127 | m | n | p | q |
major | mq | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 124 | 124 | 124 | MA |
§20 | 128 | m | n | p | q |
major | mq | no | no | no | yes |
minor | q | by | since | by | by |
concl. | q | 124 | q | 124 | MA |
§3 | 131 | m | n | p | q |
major | np | no | yes | yes | no |
minor | mn | since | by | by | since |
concl. | np | p | 166 | 171 | n |
§5 | 132 | m | n | p | q |
major | np | p of major premise and | |||
minor | mq | m of minor premise | |||
concl. | ** | are incompatible | |||
§4 | 133 | m | n | p | q |
major | np | p of major premise and | |||
minor | np | n of minor premise | |||
concl. | ** | are incompatible | |||
§8 | 134 | m | n | p | q |
major | np | no | no | yes | no |
minor | pq | since | by | by | by |
concl. | p | p | MA | 137,174 | MA |
Table 7.3 continued.
§14 | 135 | m | n | p | q |
major | np | no | no | no | no |
minor | m | by | by | by | by |
concl. | nil | 131 | MA | MA | 131 |
§13 | 136 | m | n | p | q |
major | np | no | yes | no | no |
minor | n | by | by | by | since |
concl. | n | 131 | 166 | MA | n |
§20 | 137 | m | n | p | q |
major | np | no | no | yes | no |
minor | p | since | by | by | by |
concl. | p | p | 134 | MA | 134 |
§19 | 138 | m | n | p | q |
major | np | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 134 | 134 | MA | 134 |
§7 | 141 | m | n | p | q |
major | pq | no | no | yes | yes |
minor | mn | since | since | by | by |
concl. | pq | p | q | 171 | 181 |
§9 | 142 | m | n | p | q |
major | pq | p, q of major premise and | |||
minor | mq | m of minor premise | |||
concl. | ** | are incompatible | |||
§9 | 143 | m | n | p | q |
major | pq | p, q of major premise and | |||
minor | np | n of minor premise | |||
concl. | ** | are incompatible | |||
§10 | 144 | m | n | p | q |
major | pq | no | no | yes | yes |
minor | pq | since | since | by | by |
concl. | pq | p | q | 147,174 | 148,184 |
§23 | 145 | m | n | p | q |
major | pq | no | no | no | no |
minor | m | by | by | by | by |
concl. | nil | 141 | 141 | MA | MA |
§23 | 146 | m | n | p | q |
major | pq | no | no | no | no |
minor | n | by | by | by | by |
concl. | nil | 141 | 141 | MA | MA |
§25 | 147 | m | n | p | q |
major | pq | no | no | yes | no |
minor | p | since | by | by | by |
concl. | p | p | 144 | MA | MA |
§25 | 148 | m | n | p | q |
major | pq | no | no | no | yes |
minor | q | by | since | by | by |
concl. | q | 144 | q | MA | MA |
Table 7.3 continued.
§12 | 151 | m | n | p | q |
major | m | yes | no | no | no |
minor | mn | by | by | since | by |
concl. | m | 155 | 121 | m | 111 |
§15 | 152 | m | n | p | q |
major | m | yes | no | no | no |
minor | mq | by | by | since | by |
concl. | m | 155 | 112 | m | MA |
§16 | 153 | m | n | p | q |
major | m | no | no | no | no |
minor | np | by | by | by | by |
concl. | nil | 113 | MA | MA | 113 |
§24 | 154 | m | n | p | q |
major | m | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | 114 | 114 | 124 | MA |
§27 | 155 | m | n | p | q |
major | m | yes | no | no | no |
minor | m | by | by | since | by |
concl. | m | MA | 112 | m | 111 |
§28 | 156 | m | n | p | q |
major | m | no | no | no | no |
minor | n | by | by | by | by |
concl. | nil | 113 | 121 | 111 | 111 |
§29 | 157 | m | n | p | q |
major | m | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 113 | 114 | 124 | 113 |
§30 | 158 | m | n | p | q |
major | m | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 114 | 112 | 112 | 152 |
§12 | 161 | m | n | p | q |
major | n | no | yes | no | no |
minor | mn | by | by | by | since |
concl. | n | 131 | 166 | 111 | n |
§16 | 162 | m | n | p | q |
major | n | no | no | no | no |
minor | mq | by | by | by | by |
concl. | nil | MA | 112 | 112 | MA |
§15 | 163 | m | n | p | q |
major | n | no | yes | no | no |
minor | np | by | by | by | since |
concl. | n | 113 | 166 | MA | n |
§24 | 164 | m | n | p | q |
major | n | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | 114 | 114 | MA | 134 |
Table 7.3 continued.
§28 | 165 | m | n | p | q |
major | n | no | no | no | no |
minor | m | by | by | by | by |
concl. | nil | 131 | 112 | 111 | 111 |
§27 | 166 | m | n | p | q |
major | n | no | yes | no | no |
minor | n | by | by | by | since |
concl. | n | 113 | MA | 111 | n |
§30 | 167 | m | n | p | q |
major | n | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 113 | 114 | 163 | 113 |
§29 | 168 | m | n | p | q |
major | n | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 114 | 112 | 112 | 134 |
§18 | 171 | m | n | p | q |
major | p | no | no | yes | no |
minor | mn | since | by | by | by |
concl. | p | p | 141 | MA | 131 |
§21 | 172 | m | n | p | q |
major | p | p of major premise and | |||
minor | mq | m of minor premise | |||
concl. | ** | are incompatible | |||
§22 | 173 | m | n | p | q |
major | p | p of major premise and | |||
minor | np | n of minor premise | |||
concl. | ** | are incompatible | |||
§26 | 174 | m | n | p | q |
major | p | no | no | yes | no |
minor | pq | since | by | by | by |
concl. | p | p | 134 | MA | 134 |
§31 | 175 | m | n | p | q |
major | p | no | no | no | no |
minor | m | by | by | by | by |
concl. | nil | 131 | 135 | 135 | 131 |
§32 | 176 | m | n | p | q |
major | p | no | no | no | no |
minor | n | by | by | by | by |
concl. | nil | 131 | 141 | 136 | 131 |
§33 | 177 | m | n | p | q |
major | p | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 134 | 134 | MA | 134 |
§34 | 178 | m | n | p | q |
major | p | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 134 | 134 | 138 | 134 |
Table 7.3 continued.
§18 | 181 | m | n | p | q |
major | q | no | no | no | yes |
minor | mn | by | since | by | by |
concl. | q | 141 | q | 121 | MA |
§22 | 182 | m | n | p | q |
major | q | q of major premise and | |||
minor | mq | m of minor premise | |||
concl. | ** | are incompatible | |||
§21 | 183 | m | n | p | q |
major | q | q of major premise and | |||
minor | np | n of minor premise | |||
concl. | ** | are incompatible | |||
§26 | 184 | m | n | p | q |
major | q | no | no | no | yes |
minor | pq | by | since | by | by |
concl. | q | 124 | q | 124 | MA |
§32 | 185 | m | n | p | q |
major | q | no | no | no | no |
minor | m | by | by | by | by |
concl. | nil | 141 | 121 | 121 | 125 |
§31 | 186 | m | n | p | q |
major | q | no | no | no | no |
minor | n | by | by | by | by |
concl. | nil | 126 | 121 | 121 | 126 |
§34 | 187 | m | n | p | q |
major | q | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 124 | 124 | 124 | 127 |
§33 | 188 | m | n | p | q |
major | q | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 124 | 124 | 124 | MA |
Summary of figure 1.
- 30 valid moods:
111-118, 121, 124-125, 128, 131, 134, 136-137, 141, 144, 147-148, 151-152, 155, 161, 163, 166, 171, 174, 181, 184.
- 24 moods without conclusion (nil):
126-127, 135, 138, 145-146, 153-154, 156-158, 162, 164-165, 167-168, 175-178, 185-188.
- 10 impossible moods (**):
122-123, 132-133, 142-143, 172-173, 182-183.
Total of moods = 30 valid and 34 invalid = 64.
3. Reductions in Figure 2.
First, note that six moods in subfigure 2d have inconsistent premises. Specifically, if the minor premise (which has form P(S)Q) involves a strong determination, then it conflicts with any weak determination of same polarity in the major premise (which has form R(P)Q).
For if the minor concerns complete causation, clause (i) of which means that (P + notQ) is impossible – it is incompatible with the major, which implies (P + notQ) is possible when it concerns partial causation (see clause (ii) of that). Similarly, if the minor concerns necessary causation, clause (i) of which means that (notP + Q) is impossible – it is incompatible with the major, which implies (notP + Q) is possible when it concerns contingent causation (see clause (ii) of that).
Additionally, we may directly reduce a number of moods in figure 2 to figure 1, by converting the major premise. This is feasible when the major premise involves only strong causation; i.e. subfigures 2a and 2b are thus reducible respectively to subfigures 1a and 1b. This is not feasible when the major premise involves weak causation, since its conversion results in negation of the complement; which means that subfigures 2c and 2d have to be evaluated relatively independently (i.e. within the same figure, even if possibly through some moods reduced to figure 1).
Ref. | Mood # | Elements of conclusion implied? | |||
§1 | 211 | m | n | p | q |
major | mn | yes | yes | no | no |
minor | mn | by | by | since | since |
concl. | mn | 111 | 111 | m | n |
§2 | 212 | m | n | p | q |
major | mn | yes | no | no | yes |
minor | mq | by | since | since | by |
concl. | mq | 112 | q | m | 112 |
§2 | 213 | m | n | p | q |
major | mn | no | yes | yes | no |
minor | np | since | by | by | since |
concl. | np | p | 113 | 113 | n |
§6 | 214 | m | n | p | q |
major | mn | no | no | yes | yes |
minor | pq | since | since | by | by |
concl. | pq | p | q | 114 | 114 |
§11 | 215 | m | n | p | q |
major | mn | yes | no | no | no |
minor | m | by | by | since | by |
concl. | m | 115 | 115 | m | 115 |
§11 | 216 | m | n | p | q |
major | mn | no | yes | no | no |
minor | n | by | by | by | since |
concl. | n | 116 | 116 | 116 | n |
Table 7.4 continued.
§17 | 217 | m | n | p | q |
major | mn | no | no | yes | no |
minor | p | since | by | by | by |
concl. | p | p | 117 | 117 | 117 |
§17 | 218 | m | n | p | q |
major | mn | no | no | no | yes |
minor | q | by | since | by | by |
concl. | q | 118 | q | 118 | 118 |
§3 | 221 | m | n | p | q |
major | mq | no | yes | no | no |
minor | mn | by | by | by | since |
concl. | n | MA | 256 | MA | n |
§4 | 222 | m | n | p | q |
major | mq | no | no | no | no |
minor | mq | by | by | by | by |
concl. | nil | MA | MA | MA | MA |
§5 | 223 | m | n | p | q |
major | mq | q of major premise and | |||
minor | np | n of minor premise | |||
concl. | ** | are incompatible | |||
§8 | 224 | m | n | p | q |
major | mq | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | MA | MA | MA | MA |
§13 | 225 | m | n | p | q |
major | mq | no | no | no | no |
minor | m | by | by | by | by |
concl. | nil | 221 | 222 | 221 | 221 |
§14 | 226 | m | n | p | q |
major | mq | no | yes | no | no |
minor | n | by | by | by | since |
concl. | n | 221 | 256 | 221 | n |
§19 | 227 | m | n | p | q |
major | mq | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 224 | 224 | 224 | 224 |
§20 | 228 | m | n | p | q |
major | mq | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 222 | 222 | 222 | 222 |
§3 | 231 | m | n | p | q |
major | np | yes | no | no | no |
minor | mn | by | by | since | by |
concl. | m | 265 | MA | m | MA |
§5 | 232 | m | n | p | q |
major | np | p of major premise and | |||
minor | mq | m of minor premise | |||
concl. | ** | are incompatible |
Table 7.4 continued.
§4 | 233 | m | n | p | q |
major | np | no | no | no | no |
minor | np | by | by | by | by |
concl. | nil | MA | MA | MA | MA |
§8 | 234 | m | n | p | q |
major | np | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | MA | MA | MA | MA |
§14 | 235 | m | n | p | q |
major | np | yes | no | no | no |
minor | m | by | by | since | by |
concl. | m | 265 | 231 | m | 231 |
§13 | 236 | m | n | p | q |
major | np | no | no | no | no |
minor | n | by | by | by | by |
concl. | nil | 233 | 231 | 231 | 231 |
§20 | 237 | m | n | p | q |
major | np | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 233 | 233 | 233 | 233 |
§19 | 238 | m | n | p | q |
major | np | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 234 | 234 | 234 | 234 |
§7 | 241 | m | n | p | q |
major | pq | no | no | no | no |
minor | mn | by | by | by | by |
concl. | nil | MA | MA | MA | MA |
§9 | 242 | m | n | p | q |
major | pq | p of major premise and | |||
minor | mq | m of minor premise | |||
concl. | ** | are incompatible | |||
§9 | 243 | m | n | p | q |
major | pq | q of major premise and | |||
minor | np | n of minor premise | |||
concl. | ** | are incompatible | |||
§10 | 244 | m | n | p | q |
major | pq | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | MA | MA | MA | MA |
§23 | 245 | m | n | p | q |
major | pq | no | no | no | no |
minor | m | by | by | by | by |
concl. | nil | 241 | 241 | 241 | 241 |
Table 7.4 continued.
§23 | 246 | m | n | p | q |
major | pq | no | no | no | no |
minor | n | by | by | by | by |
concl. | nil | 241 | 241 | 241 | 241 |
§25 | 247 | m | n | p | q |
major | pq | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 244 | 244 | 244 | 244 |
§25 | 248 | m | n | p | q |
major | pq | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 244 | 244 | 244 | 244 |
§12 | 251 | m | n | p | q |
major | m | no | yes | no | no |
minor | mn | by | by | by | since |
concl. | n | 161 | 161 | 161 | n |
§15 | 252 | m | n | p | q |
major | m | no | no | no | no |
minor | mq | by | by | by | by |
concl. | nil | 162 | 162 | 162 | 162 |
§16 | 253 | m | n | p | q |
major | m | no | yes | no | no |
minor | np | by | by | by | since |
concl. | n | 163 | 163 | 163 | n |
§24 | 254 | m | n | p | q |
major | m | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | 164 | 164 | 164 | 164 |
§27 | 255 | m | n | p | q |
major | m | no | no | no | no |
minor | m | by | by | by | by |
concl. | nil | 165 | 165 | 165 | 165 |
§28 | 256 | m | n | p | q |
major | m | no | yes | no | no |
minor | n | by | by | by | since |
concl. | n | 166 | 166 | 166 | n |
§29 | 257 | m | n | p | q |
major | m | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 167 | 167 | 167 | 167 |
§30 | 258 | m | n | p | q |
major | m | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 168 | 168 | 168 | 168 |
§12 | 261 | m | n | p | q |
major | n | yes | no | no | no |
minor | mn | by | by | since | by |
concl. | m | 151 | 151 | m | 151 |
Table 7.4 continued.
§16 | 262 | m | n | p | q |
major | n | yes | no | no | no |
minor | mq | by | by | since | by |
concl. | m | 152 | 152 | m | 152 |
§15 | 263 | m | n | p | q |
major | n | no | no | no | no |
minor | np | by | by | by | by |
concl. | nil | 153 | 153 | 153 | 153 |
§24 | 264 | m | n | p | q |
major | n | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | 154 | 154 | 154 | 154 |
§28 | 265 | m | n | p | q |
major | n | yes | no | no | no |
minor | m | by | by | since | by |
concl. | m | 155 | 155 | m | 155 |
§27 | 266 | m | n | p | q |
major | n | no | no | no | no |
minor | n | by | by | by | by |
concl. | nil | 156 | 156 | 156 | 156 |
§30 | 267 | m | n | p | q |
major | n | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 157 | 157 | 157 | 157 |
§29 | 268 | m | n | p | q |
major | n | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 158 | 158 | 158 | 158 |
§18 | 271 | m | n | p | q |
major | p | no | no | no | no |
minor | mn | by | by | by | by |
concl. | nil | 241 | 231 | 231 | 231 |
§21 | 272 | m | n | p | q |
major | p | p of major premise and | |||
minor | mq | m of minor premise | |||
concl. | ** | are incompatible | |||
§22 | 273 | m | n | p | q |
major | p | no | no | no | no |
minor | np | by | by | by | by |
concl. | nil | 233 | 233 | 233 | 233 |
§26 | 274 | m | n | p | q |
major | p | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | 234 | 234 | 234 | 234 |
§31 | 275 | m | n | p | q |
major | p | no | no | no | no |
minor | m | by | by | by | by |
concl. | nil | 241 | 231 | 231 | 231 |
Table 7.4 continued.
§32 | 276 | m | n | p | q |
major | p | no | no | no | no |
minor | n | by | by | by | by |
concl. | nil | 233 | 231 | 231 | 231 |
§33 | 277 | m | n | p | q |
major | p | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 233 | 233 | 233 | 233 |
§34 | 278 | m | n | p | q |
major | p | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 234 | 234 | 234 | 234 |
§18 | 281 | m | n | p | q |
major | q | no | no | no | no |
minor | mn | by | by | by | by |
concl. | nil | 221 | 241 | 221 | 221 |
§22 | 282 | m | n | p | q |
major | q | no | no | no | no |
minor | mq | by | by | by | by |
concl. | nil | 222 | 222 | 222 | 222 |
§21 | 283 | m | n | p | q |
major | q | q of major premise and | |||
minor | np | n of minor premise | |||
concl. | ** | are incompatible | |||
§26 | 284 | m | n | p | q |
major | q | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | 224 | 224 | 224 | 224 |
§32 | 285 | m | n | p | q |
major | q | no | no | no | no |
minor | m | by | by | by | by |
concl. | nil | 221 | 222 | 221 | 221 |
§31 | 286 | m | n | p | q |
major | q | no | no | no | no |
minor | n | by | by | by | by |
concl. | nil | 221 | 241 | 221 | 221 |
§34 | 287 | m | n | p | q |
major | q | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 224 | 224 | 224 | 224 |
§33 | 288 | m | n | p | q |
major | q | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 222 | 222 | 222 | 222 |
Summary of figure 2.
- 18 valid moods:
211-218, 221, 226, 231, 235, 251, 253, 256, 261-262, 265.
- 40 moods without conclusion (nil):
222, 224-225, 227-228, 233-234, 236-238, 241, 244-248, 252, 254-255, 257-258, 263-264, 266-268, 271, 273-278, 281-282, 284-288.
- 6 impossible moods (**):
223, 232, 242-243, 272, 283.
Total of moods = 18 valid and 46 invalid = 64.
4. Reductions in Figure 3.
First, note that ten moods in subfigure 3d have inconsistent premises. Specifically, if the minor premise (which has form Q(S)P) involves a strong determination, then it conflicts with the weak determination(s) of the major premise (which has form Q(P)R).
For if the minor concerns complete causation, clause (i) of which means that (notP + Q) is impossible – it is incompatible with the major, which implies (notP + Q) is possible, whether it concerns partial causation (see clause (iii) of that) or contingent causation (see clause (ii) of that). Similarly, if the minor concerns necessary causation, clause (i) of which means that (P + notQ) is impossible – it is incompatible with the major, which implies (P + notQ) is possible, whether it concerns partial causation (see clause (ii) of that) or contingent causation (see clause (iii) of that).
Additionally, we may directly reduce a number of moods in figure 3 to figure 1, by converting the minor premise. This is feasible when the minor premise involves only strong causation; i.e. subfigures 3a and 3c are thus reducible respectively to subfigures 1a and 1c. This is not feasible when the minor premise involves weak causation, since its conversion results in negation of the complement; which means that subfigures 3b and 3d have to be evaluated relatively independently (i.e. within the same figure, even if possibly through some moods reduced to figure 1).
Ref. | Mood # | Elements of conclusion implied? | |||
§1 | 311 | m | n | p | q |
major | mn | yes | yes | no | no |
minor | mn | by | by | since | since |
concl. | mn | 111 | 111 | m | n |
§2 | 312 | m | n | p | q |
major | mn | no | yes | no | no |
minor | mq | by | by | by | since |
concl. | n | MA | 365 | MA | n |
§2 | 313 | m | n | p | q |
major | mn | yes | no | no | no |
minor | np | by | by | since | by |
concl. | m | 356 | MA | m | MA |
Table 7.5 continued.
§6 | 314 | m | n | p | q |
major | mn | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | MA | MA | MA | MA |
§11 | 315 | m | n | p | q |
major | mn | no | yes | no | no |
minor | m | by | by | by | since |
concl. | n | 116 | 116 | 116 | n |
§11 | 316 | m | n | p | q |
major | mn | yes | no | no | no |
minor | n | by | by | since | by |
concl. | m | 115 | 115 | m | 115 |
§17 | 317 | m | n | p | q |
major | mn | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 314 | 313 | 313 | 313 |
§17 | 318 | m | n | p | q |
major | mn | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 312 | 314 | 312 | 312 |
§3 | 321 | m | n | p | q |
major | mq | yes | no | no | yes |
minor | mn | by | since | since | by |
concl. | mq | 121 | q | m | 121 |
§4 | 322 | m | n | p | q |
major | mq | q of major premise and | |||
minor | mq | m of minor premise | |||
concl. | ** | are incompatible | |||
§5 | 323 | m | n | p | q |
major | mq | q of major premise and | |||
minor | np | n of minor premise | |||
concl. | ** | are incompatible | |||
§8 | 324 | m | n | p | q |
major | mq | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | MA | MA | MA | MA |
§13 | 325 | m | n | p | q |
major | mq | no | no | no | no |
minor | m | by | by | by | by |
concl. | nil | 126 | 126 | 126 | 126 |
§14 | 326 | m | n | p | q |
major | mq | yes | no | no | no |
minor | n | by | by | since | by |
concl. | m | 125 | 125 | m | 125 |
§19 | 327 | m | n | p | q |
major | mq | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 324 | 324 | 324 | 324 |
Table 7.5 continued.
§20 | 328 | m | n | p | q |
major | mq | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 324 | 324 | 324 | 324 |
§3 | 331 | m | n | p | q |
major | np | no | yes | yes | no |
minor | mn | since | by | by | since |
concl. | np | p | 131 | 131 | n |
§5 | 332 | m | n | p | q |
major | np | p of major premise and | |||
minor | mq | m of minor premise | |||
concl. | ** | are incompatible | |||
§4 | 333 | m | n | p | q |
major | np | p of major premise and | |||
minor | np | n of minor premise | |||
concl. | ** | are incompatible | |||
§8 | 334 | m | n | p | q |
major | np | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | MA | MA | MA | MA |
§14 | 335 | m | n | p | q |
major | np | no | yes | no | no |
minor | m | by | by | by | since |
concl. | n | 136 | 136 | 136 | n |
§13 | 336 | m | n | p | q |
major | np | no | no | no | no |
minor | n | by | by | by | by |
concl. | nil | 135 | 135 | 135 | 135 |
§20 | 337 | m | n | p | q |
major | np | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 334 | 334 | 334 | 334 |
§19 | 338 | m | n | p | q |
major | np | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 334 | 334 | 334 | 334 |
§7 | 341 | m | n | p | q |
major | pq | no | no | yes | yes |
minor | mn | since | since | by | by |
concl. | pq | p | q | 141 | 141 |
§9 | 342 | m | n | p | q |
major | pq | p and q of major premise and | |||
minor | mq | m of minor premise | |||
concl. | ** | are incompatible | |||
§9 | 343 | m | n | p | q |
major | pq | p and q of major premise and | |||
minor | np | n of minor premise | |||
concl. | ** | are incompatible |
Table 7.5 continued.
§10 | 344 | m | n | p | q |
major | pq | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | MA | MA | MA | MA |
§23 | 345 | m | n | p | q |
major | pq | no | no | no | no |
minor | m | by | by | by | by |
concl. | nil | 146 | 146 | 146 | 146 |
§23 | 346 | m | n | p | q |
major | pq | no | no | no | no |
minor | n | by | by | by | by |
concl. | nil | 145 | 145 | 145 | 145 |
§25 | 347 | m | n | p | q |
major | pq | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 344 | 344 | 344 | 344 |
§25 | 348 | m | n | p | q |
major | pq | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 344 | 344 | 344 | 344 |
§12 | 351 | m | n | p | q |
major | m | yes | no | no | no |
minor | mn | by | by | since | by |
concl. | m | 151 | 151 | m | 151 |
§15 | 352 | m | n | p | q |
major | m | no | no | no | no |
minor | mq | by | by | by | by |
concl. | nil | 312 | MA | 312 | 312 |
§16 | 353 | m | n | p | q |
major | m | yes | no | no | no |
minor | np | by | by | since | by |
concl. | m | 356 | 313 | m | 313 |
§24 | 354 | m | n | p | q |
major | m | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | 314 | 314 | 314 | 314 |
§27 | 355 | m | n | p | q |
major | m | no | no | no | no |
minor | m | by | by | by | by |
concl. | nil | 156 | 156 | 156 | 156 |
§28 | 356 | m | n | p | q |
major | m | yes | no | no | no |
minor | n | by | by | since | by |
concl. | m | 155 | 155 | m | 155 |
§29 | 357 | m | n | p | q |
major | m | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 314 | 313 | 313 | 313 |
Table 7.5 continued.
§30 | 358 | m | n | p | q |
major | m | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 312 | 314 | 312 | 312 |
§12 | 361 | m | n | p | q |
major | n | no | yes | no | no |
minor | mn | by | by | by | since |
concl. | n | 161 | 161 | 161 | n |
§16 | 362 | m | n | p | q |
major | n | no | yes | no | no |
minor | mq | by | by | by | since |
concl. | n | 312 | 365 | 312 | n |
§15 | 363 | m | n | p | q |
major | n | no | no | no | no |
minor | np | by | by | by | by |
concl. | nil | MA | 313 | 313 | 313 |
§24 | 364 | m | n | p | q |
major | n | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | 314 | 314 | 314 | 314 |
§28 | 365 | m | n | p | q |
major | n | no | yes | no | no |
minor | m | by | by | by | since |
concl. | n | 166 | 166 | 166 | n |
§27 | 366 | m | n | p | q |
major | n | no | no | no | no |
minor | n | by | by | by | by |
concl. | nil | 165 | 165 | 165 | 165 |
§30 | 367 | m | n | p | q |
major | n | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 314 | 313 | 313 | 313 |
§29 | 368 | m | n | p | q |
major | n | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 312 | 314 | 312 | 312 |
§18 | 371 | m | n | p | q |
major | p | no | no | yes | no |
minor | mn | since | by | by | by |
concl. | p | p | 171 | 171 | 171 |
§21 | 372 | m | n | p | q |
major | p | p of major premise and | |||
minor | mq | m of minor premise | |||
concl. | ** | are incompatible | |||
§22 | 373 | m | n | p | q |
major | p | p of major premise and | |||
minor | np | n of minor premise | |||
concl. | ** | are incompatible |
Table 7.5 continued.
§26 | 374 | m | n | p | q |
major | p | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | 334 | 334 | 334 | 334 |
§31 | 375 | m | n | p | q |
major | p | no | no | no | no |
minor | m | by | by | by | by |
concl. | nil | 176 | 176 | 176 | 176 |
§32 | 376 | m | n | p | q |
major | p | no | no | no | no |
minor | n | by | by | by | by |
concl. | nil | 175 | 175 | 175 | 175 |
§33 | 377 | m | n | p | q |
major | p | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 334 | 334 | 334 | 334 |
§34 | 378 | m | n | p | q |
major | p | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 334 | 334 | 334 | 334 |
§18 | 381 | m | n | p | q |
major | q | no | no | no | yes |
minor | mn | by | since | by | by |
concl. | q | 181 | q | 181 | 181 |
§22 | 382 | m | n | p | q |
major | q | q of major premise and | |||
minor | mq | m of minor premise | |||
concl. | ** | are incompatible | |||
§21 | 383 | m | n | p | q |
major | q | q of major premise and | |||
minor | np | n of minor premise | |||
concl. | ** | are incompatible | |||
§26 | 384 | m | n | p | q |
major | q | no | no | no | no |
minor | pq | by | by | by | by |
concl. | nil | 324 | 324 | 324 | 324 |
§32 | 385 | m | n | p | q |
major | q | no | no | no | no |
minor | m | by | by | by | by |
concl. | nil | 186 | 186 | 186 | 186 |
§31 | 386 | m | n | p | q |
major | q | no | no | no | no |
minor | n | by | by | by | by |
concl. | nil | 185 | 185 | 185 | 185 |
Table 7.5 continued.
§34 | 387 | m | n | p | q |
major | q | no | no | no | no |
minor | p | by | by | by | by |
concl. | nil | 324 | 324 | 324 | 324 |
§33 | 388 | m | n | p | q |
major | q | no | no | no | no |
minor | q | by | by | by | by |
concl. | nil | 324 | 324 | 324 | 324 |
Summary of figure 3.
- 18 valid moods:
311-313, 315-316, 321, 326, 331, 335, 341, 351, 353, 356, 361-362, 365, 371, 381.
- 36 moods without conclusion (nil):
314, 317-318, 324-325, 327-328, 334, 336-338, 344-348, 352, 354-355, 357-358, 363-364, 366-368, 374-378, 384-388.
- 10 impossible moods (**):
322-323, 332-333, 342-343, 372-373, 382-383.
Total of moods = 18 valid and 46 invalid = 64.
Chapter 8. Matricial Analyses
1. Matricial Analysis.
We will in this chapter show the matricial analyses on the basis of which moods were declared valid or invalid in previous chapters.
Now, the matrix underlying a syllogism may be defined as a table with a listing of all conceivable conjunctions of all the items involved in its premises and conclusion and/or the negations of these items. Thus, for instance, the matrix of three items P, Q, R, will look like this:
P | Q | R |
P | Q | notR |
P | notQ | R |
P | notQ | notR |
notP | Q | R |
notP | Q | notR |
notP | notQ | R |
notP | notQ | notR |
(With four items, the table would be twice as long; with five items, four times as long; and so on.)
Briefly put, matricial analysis is a process which seeks to answer, for each of the conceivable conjunctions in the matrix, the question as to whether it is implied impossible or possible or neither; in the latter case, if the conjunction is neither implied impossible nor implied possible, it is declared open. In other words, matricial analysis is a pursuit of the ‘modus’ of the matrix.
The answers to this question for each row must be derived from the premises, singly or together, by established means; the conclusion is valid only if it may be entirely (with all its implicit clauses) constructed from these answers. As we shall see, this process relies heavily on paradoxical logic, or more simply put, on dilemmatic argument.
The process is, as you will presently discover, long and difficult. I have unfortunately found no better short-cut, maybe other logicians have or eventually do. However, its advantage over reduction is that it provides us with sure results; for with reduction we cannot always be sure to have applied all possible means, whereas with a matricial analysis we know we have exhausted the available information. We are free to choose the appropriate method in each case: certain crucial syllogisms are best subjected to matricial analysis, and then we can use reduction to derive others.
To begin with, here is a step by step description of this method of evaluation. The reader is requested to follow the procedure concretely by referring to one of the examples given in the following sections. It is much less complicated than it sounds.
- Write down the premises constituting the mood and the putative conclusion(s) to be evaluated. Translate all these causative propositions into conditional or conjunctive propositions, i.e. make their constituent clauses (as elucidated in the chapter on the determinations of causation) explicit. Number the clauses involved for purposes of reference (Roman numerals are used for this, here).
- Construct a table with a matrix involving all the items and negations of items concerned, in orderly sequences. If there are three items (P, Q, R), the table will have 2^{3} = 8 rows; if there are four items (P, Q, R, S), it will have 2^{4} = 16 rows.
- Consider first all the positive conditional propositions found in the premises. Every causative proposition contains at least one positive conditional clause; therefore, there will be at least two such clauses per mood. These tell us which conjunctions in the matrix (rows) are implied impossible.
In a three-item matrix, each such statement (e.g. ‘if P, then Q’, which means ‘P+notQ is impossible’) will imply two rows to be impossible (namely, ‘P+notQ+R’ and ‘P+notQ+notR’). Similarly, in a four-item matrix, each if/then statement involving two items will imply four rows impossible; while each if/then statement involving three items will imply two rows impossible.
- Only thereafter, deal with the remaining clauses found in the premises (because their impact will depend on the results of the preceding step), which imply the possibility of certain conjunctions of two or three items in the matrix.
These include negative conditional propositions (e.g. ‘if P, not-then Q’, which means ‘P+notQ is possible’); as well as bare statements of possibility of an item or of a conjunction of items. The latter are to be enlarged with reference to the corresponding positive conditional proposition (e.g. ‘P is possible’ and ‘if P, then Q’, together imply ‘P+Q is possible’).
In a three-item matrix, a possibility of conjunction of two items will imply that at least one of two rows is possible. In a four-item matrix, a possibility of conjunction of two items will imply that at least one of four rows is possible; while a possibility of conjunction of three items will imply that at least one of two rows is possible.
Note this well: whereas the impossibility of a conjunction entails the impossibility of all its expressions in the matrix, the possibility of a conjunction is satisfied by only one expression. Thus, the knowledge that two or more rows are collectively possible does not settle the question of the possibility of each of these rows individually.
Only if all but one of these rows are declared impossible by other means (i.e. the preceding step of the procedure), can we declare the remaining one possible. Otherwise, if two or more rows are left unsettled, they must each be considered ‘open’ (i.e. ‘possible or impossible’). That is, even though we know that at least one of them must be possible, we cannot specify which one.
- When all the information implicit in the premises has been thus systematically included in the table, we can evaluate the putative conclusion(s). Taking one of the clauses at a time, check out whether it can be inferred from the table.
If the clause in question is a positive conditional, every row corresponding to it in the matrix must have been declared impossible to allow us to accept the clause as implied. If the clause in question is a negative conditional or bare statement of possibility, it suffices that one row in the matrix has been declared possible, even if the other(s) was/were declared impossible or left open. (Often, the last clause of the putative conclusion can be inferred directly from a premise, note.)
- If, and only if, all the clauses of the putative conclusion are thus found to be implied by the data in our table, we may admit that conclusion to be drawable from the premises. If any clause(s) of the putative conclusion is/are left open or worse still denied, by the table, that conclusion must be declared a non-sequitur or antinomy, respectively.
A computer could be programmed to carry out this evaluation process. Once it is understood, it requires no great intelligence to perform. It is tedious detail work, no more.[45]
What is matricial analysis, essentially? A causative proposition is a complex of simpler statements, which affirm the impossibility or possibility of certain conjunctions of items or individual items. But causative propositions differ in their forms and implications, so that comparisons between them are difficult. By recapitulating or recoding all the information in a table, we are better able to judge their mutual impact. The matrix is the common denominator of these disparate forms. The annotations down the comments column of the table comprising it, record the answers to the question we must settle for each row (or conjunction of all the items involved): is this possible, impossible or open (i.e. unsettled)?
The premises collectively structure this table, filling in all or only some of the answers to the question. The mutual impact of the determinations of the premises produces the result. This in turn allows us to judge, with absolute certainty, the logical impact on any of the four putative generic conclusions, and thence evaluate the validity or invalidity of that conclusion.
The main form of reasoning used in matricial analysis is dilemmatic argument – that is, we use paradoxical logic (the branch of logic concerned with paradoxes[46]). This is clear in the above account.
First, with regard to expansion: for example, knowing that P+Q is impossible, we can infer that P+Q+R and P+Q+notR are both impossible, or again knowing that P+Q+R is impossible, we can infer that P+Q+R+S and P+Q+R+notS are both impossible. And conversely, regarding contraction: we can infer a two-item impossibility from two three-item ones or a three-item impossibility from two four-item ones.
Likewise, by contraposition, when we argue from a two-item or three-item possibility to the possibility of at least one of two contrary conjunctions (e.g. from P+Q is possible, to P+Q+R and/or P+Q+notR is/are possible; or from P+Q+R is possible, to P+Q+R+S and/or P+Q+R+notS is/are possible), we rely on dilemma.
Such reasoning is especially productive when, from one clause we know some row(s) of the matrix to be impossible (say, P+Q+R), while from another clause we know that a set of rows including the preceding row(s) is possible (say, P+Q+R and P+Q+notR); for then, if the latter set exceeds the former by only one row, we can infer that remaining row (viz. P+Q+notR, in this example) to be possible.
Matricial analysis can be used to evaluate, and validate or invalidate, any putative conclusion of any mood in any figure of the syllogism (as well as in immediate inferences). It is a universal method.
It could in principle be replaced by use of logical compositions[47]; but we are more likely to be confused in practice by the techniques of symbolic logic; and they are in any event themselves ultimately based on matricial analysis. Similarly, reduction of causative syllogisms to conditional syllogisms[48] is a conceivable method; but in practice we are much more likely to make mistakes with it, so that we will at the end remain uncertain about the reliability of our results; and anyway, matricial analysis is the ultimate basis of conditional syllogism, too.
As we have seen, to avoid having to use matricial analysis everywhere, since it is time-consuming, it is wise to first identify the minimum number of conclusions, required to be validated or invalidated that way, from which all other conclusions can be derived by direct or indirect reduction. This is the approach adopted here.
Another short-cut resorted to, here, is to identify moods which are ‘mirror images’ of each other, i.e. whose forms are identical in every respect, except that each item in the one is replaced by the contradictory item in the other. In such cases, the matrices of both moods are bound to be identical, except that the polarity of every symbol will be reversed, i.e. notP will replace P wherever it occurs, P will replace notP, and so forth.
Since the great majority of moods have a mirror image (all but four in each figure, viz. mn/mn, mn/pq, pq/mn, and pq/pq), this diminishes the work required by almost half (at most thirty moods per figure, instead of sixty).
We shall in the next three sections evaluate by matricial analysis the positive moods we identified in the preceding chapter as needing evaluation, with respect to some or all (as the case may be) of their conceivable conclusions.
2. Crucial Matricial Analyses in Figure 1.
Evaluation of mood # 117. (Similarly, mutatis mutandis, for mood # 118.)
Major premise: Q is a complete and necessary cause of R:
- If Q, then R;
- if notQ, not-then R;
- where: Q is possible.
- If notQ, then notR;
- if Q, not-then notR;
- where: notQ is possible.
Minor premise: P (complemented by S) is a partial cause of Q:
- If (P + S), then Q;
- if (notP + S), not-then Q;
- if (P + notS), not-then Q;
- where: (P + S) is possible.
Putative conclusion: is P (complemented by S) a partial cause of R?
YES! P is a partial cause of R:
If (P + S), then R | implied by (i) + (vii); |
if (notP + S), not-then R | implied by (iv) + (viii); |
if (P + notS), not-then R | implied by (iv) + (ix); |
where: (P + S) is possible | same as (x). |
P | Q | R | S | implied possible by (i) + (vii) + (x) |
P | Q | R | notS | see (i) + (iii), or (v) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | implied impossible by (iv) or (vii) |
P | notQ | R | notS | implied impossible by (iv) |
P | notQ | notR | S | implied impossible by (vii) |
P | notQ | notR | notS | implied possible by (iv) + (ix) |
notP | Q | R | S | see (i) + (iii), or (v) |
notP | Q | R | notS | see (i) + (iii), or (v) |
notP | Q | notR | S | implied impossible by (i) |
notP | Q | notR | notS | implied impossible by (i) |
notP | notQ | R | S | implied impossible by (iv) |
notP | notQ | R | notS | implied impossible by (iv) |
notP | notQ | notR | S | implied possible by (iv) + (viii) |
notP | notQ | notR | notS | see (ii), or (iv) + (vi) |
Evaluation of mood # 124. (Similarly, mutatis mutandis, for mood # 134.)
Major premise: Q is a complete and (complemented by P) a contingent cause of R:
- If Q, then R;
- if notQ, not-then R;
- where: Q is possible.
- If (notQ + notP), then notR;
- if (Q + notP), not-then notR;
- if (notQ + P), not-then notR;
- where: (notQ + notP) is possible.
Minor premise: P (complemented by S) is a partial and contingent cause of Q:
- If (P + S), then Q;
- if (notP + S), not-then Q;
- if (P + notS), not-then Q;
- where: (P + S) is possible.
- If (notP + notS), then notQ;
- if (P + notS), not-then notQ;
- if (notP + S), not-then notQ;
- where: (notP + notS) is possible.
Putative conclusion: is P a complete or (complemented by S) a partial cause of R?
NO! P is not implied to be a complete cause of R:
If P, then R? | open; |
if notP, not-then R | implied by (iv) + (ix), or (iv) + (xii) + (xv); |
where: P is possible | implied by (vi) or (x) or (xi) or (xiii). |
Nor (complemented by S) a partial cause of R:
If (P + S), then R | implied by (i) + (viii); |
if (notP + S), not-then R | implied by (iv) + (ix); |
if (P + notS), not-then R? | open; |
where: (P + S) is possible | same as (xi). |
P | Q | R | S | implied possible by (i) + (viii) + (xi) |
P | Q | R | notS | implied possible by (i) + (xiii) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | implied impossible by (viii) |
P | notQ | R | notS | implied possible by (ii) + (iv) + (viii), or (vi) + (viii) |
P | notQ | notR | S | implied impossible by (viii) |
P | notQ | notR | notS | see (ii) or (x) |
notP | Q | R | S | implied possible by (v) + (xii), or (i) + (xiv) |
notP | Q | R | notS | implied impossible by (xii) |
notP | Q | notR | S | implied impossible by (i) |
notP | Q | notR | notS | implied impossible by (i) or (xii) |
notP | notQ | R | S | implied impossible by (iv) |
notP | notQ | R | notS | implied impossible by (iv) |
notP | notQ | notR | S | implied possible by (iv) + (ix) |
notP | notQ | notR | notS | implied possible by (iv) + (xii) + (xv) |
Evaluation of mood # 125. (Similarly, mutatis mutandis, for mood # 136.)
Major premise: Q is a complete and (complemented by S) a contingent cause of R:
- If Q, then R;
- if notQ, not-then R;
- where: Q is possible.
- If (notQ + notS), then notR;
- if (Q + notS), not-then notR;
- if (notQ + S), not-then notR;
- where: (notQ + notS) is possible.
Minor premise: P is a complete cause of Q:
- If P, then Q;
- if notP, not-then Q;
- where: P is possible.
Putative conclusion: is P (complemented by S) a contingent cause of R?
NO! P (complemented by S) is not implied to be a contingent cause of R:
If (notP + notS), then notR? | open; |
if (P + notS), not-then notR? | open; |
if (notP + S), not-then notR | implied by (vi) + (viii); |
where: (notP + notS) is possible | implied by (iv) + (vii) + (viii). |
P | Q | R | S | see (i) + (iii), or (viii) + (x) |
P | Q | R | notS | see (i) + (iii), or (v), or (viii) + (x) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | implied impossible by (viii) |
P | notQ | R | notS | implied impossible by (iv) or (viii) |
P | notQ | notR | S | implied impossible by (viii) |
P | notQ | notR | notS | implied impossible by (viii) |
notP | Q | R | S | see (i) + (iii) |
notP | Q | R | notS | see (i) + (iii), or (v) |
notP | Q | notR | S | implied impossible by (i) |
notP | Q | notR | notS | implied impossible by (i) |
notP | notQ | R | S | implied possible by (vi) + (viii) |
notP | notQ | R | notS | implied impossible by (iv) |
notP | notQ | notR | S | see (ii) or (ix) |
notP | notQ | notR | notS | implied possible by (iv) + (vii) + (viii) |
Evaluation of mood # 126. (Similarly, mutatis mutandis, for mood # 135.)
Major premise: Q is a complete and (complemented by S) a contingent cause of R:
- If Q, then R;
- if notQ, not-then R;
- where: Q is possible.
- If (notQ + notS), then notR;
- if (Q + notS), not-then notR;
- if (notQ + S), not-then notR;
- where: (notQ + notS) is possible.
Minor premise: P is a necessary cause of Q:
- If notP, then notQ;
- if P, not-then notQ;
- where: notP is possible.
Putative conclusion: is P a complete or (complemented by S) a contingent cause of R?
NO! P is not implied to be a complete cause of R:
If P, then R? | open; |
if notP, not-then R? | open; |
where: P is possible | implied by (v) + (viii), or (ix). |
Nor (complemented by S) a contingent cause of R:
If (notP + notS), then notR | implied by (iv) + (viii); |
if (P + notS), not-then notR | implied by (v) + (viii); |
if (notP + S), not-then notR? | open; |
where: (notP + notS) is possible? | open. |
P | Q | R | S | see (i) + (iii), or (ix) |
P | Q | R | notS | implied possible by (v) + (viii) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | see (vi) |
P | notQ | R | notS | implied impossible by (iv) |
P | notQ | notR | S | see (ii) |
P | notQ | notR | notS | see (ii), or (iv) + (vii) |
notP | Q | R | S | implied impossible by (viii) |
notP | Q | R | notS | implied impossible by (viii) |
notP | Q | notR | S | implied impossible by (i) or (viii) |
notP | Q | notR | notS | implied impossible by (i) or (viii) |
notP | notQ | R | S | see (vi), or (viii) + (x) |
notP | notQ | R | notS | implied impossible by (iv) |
notP | notQ | notR | S | see (ii), or (viii) + (x) |
notP | notQ | notR | notS | see (ii), or (iv) + (vii), or (viii) + (x) |
Evaluation of mood # 127. (Similarly, mutatis mutandis, for mood # 138.)
Major premise: Q is a complete and (complemented by P) a contingent cause of R:
- If Q, then R;
- if notQ, not-then R;
- where: Q is possible.
- If (notQ + notP), then notR;
- if (Q + notP), not-then notR;
- if (notQ + P), not-then notR;
- where: (notQ + notP) is possible.
Minor premise: P (complemented by S) is a partial cause of Q:
- If (P + S), then Q;
- if (notP + S), not-then Q;
- if (P + notS), not-then Q;
- where: (P + S) is possible.
Putative conclusion: is P (complemented by S) a contingent cause of R?
NO! P (complemented by S) is not implied to be a contingent cause of R:
If (notP + notS), then notR? | open; |
if (P + notS), not-then notR | implied by (vi) + (viii); |
if (notP + S), not-then notR? | open; |
where: (notP + notS) is possible? | open. |
P | Q | R | S | implied possible by (i) +(viii) + (xi) |
P | Q | R | notS | see (i) + (iii) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | implied impossible by (viii) |
P | notQ | R | notS | implied possible by (vi) + (viii) |
P | notQ | notR | S | implied impossible by (viii) |
P | notQ | notR | notS | see (ii), or (x) |
notP | Q | R | S | see (i) + (iii), or (v) |
notP | Q | R | notS | see (i) + (iii), or (v) |
notP | Q | notR | S | implied impossible by (i) |
notP | Q | notR | notS | implied impossible by (i) |
notP | notQ | R | S | implied impossible by (iv) |
notP | notQ | R | notS | implied impossible by (iv) |
notP | notQ | notR | S | implied possible by (iv) + (ix) |
notP | notQ | notR | notS | see (ii), or (iv) + (vii) |
Evaluation of mood # 128. (Similarly, mutatis mutandis, for mood # 137.)
Major premise: Q is a complete and (complemented by P) a contingent cause of R:
- If Q, then R;
- if notQ, not-then R;
- where: Q is possible.
- If (notQ + notP), then notR;
- if (Q + notP), not-then notR;
- if (notQ + P), not-then notR;
- where: (notQ + notP) is possible.
Minor premise: P (complemented by S) is a contingent cause of Q:
- If (notP + notS), then notQ;
- if (P + notS), not-then notQ;
- if (notP + S), not-then notQ;
- where: (notP + notS) is possible.
Putative conclusion: is P (complemented by S) a contingent cause of R?
YES! P is a contingent cause of R:
If (notP + notS), then notR | implied by (iv) + (viii); |
if (P + notS), not-then notR | implied by (i) + (ix); |
if (notP + S), not-then notR | implied by (v) + (viii), or (i) + (x); |
where: (notP + notS) is possible | same as (xi). |
P | Q | R | S | see (i) + (iii) |
P | Q | R | notS | implied possible by (i) + (ix) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | see (vi) |
P | notQ | R | notS | see (vi) |
P | notQ | notR | S | see (ii) |
P | notQ | notR | notS | see (ii) |
notP | Q | R | S | implied possible by (v) + (viii), or (i) + (x) |
notP | Q | R | notS | implied impossible by (viii) |
notP | Q | notR | S | implied impossible by (i) |
notP | Q | notR | notS | implied impossible by (i) or (viii) |
notP | notQ | R | S | implied impossible by (iv) |
notP | notQ | R | notS | implied impossible by (iv) |
notP | notQ | notR | S | see (ii), or (iv) + (vii) |
notP | notQ | notR | notS | implied possible by (iv) + (viii) + (xi) |
Evaluation of mood # 145. (Similarly, mutatis mutandis, for mood # 146.)
Major premise: Q (complemented by S) is a partial and contingent cause of R:
- If (Q + S), then R;
- if (notQ + S), not-then R;
- if (Q + notS), not-then R;
- where: (Q + S) is possible.
- If (notQ + notS), then notR;
- if (Q + notS), not-then notR;
- if (notQ + S), not-then notR;
- where: (notQ + notS) is possible.
Minor premise: P is a complete cause of Q:
- If P, then Q;
- if notP, not-then Q;
- where: P is possible.
Putative conclusion: is P (complemented by S) a partial or contingent cause of R?
NO! P (complemented by S) is not implied to be a partial cause of R:
If (P + S), then R | implied by (i) + (ix); |
if (notP + S), not-then R | implied by (ii) + (ix); |
if (P + notS), not-then R? | open; |
where: (P + S) is possible? | open. |
Nor (complemented by S) a contingent cause of R:
If (notP + notS), then notR? | open; |
if (P + notS), not-then notR? | open; |
if (notP + S), not-then notR | implied by (vii) + (ix); |
where: (notP + notS) is possible | implied by (v) + (viii) + (ix). |
P | Q | R | S | see (i) + (iv), or (ix) + (xi) |
P | Q | R | notS | see (vi), or (ix) + (xi) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | see (iii), or (ix) + (xi) |
P | notQ | R | S | implied impossible by (ix) |
P | notQ | R | notS | implied impossible by (v) or (ix) |
P | notQ | notR | S | implied impossible by (ix) |
P | notQ | notR | notS | implied impossible by (ix) |
notP | Q | R | S | see (i) + (iv) |
notP | Q | R | notS | see (vi) |
notP | Q | notR | S | implied impossible by (i) |
notP | Q | notR | notS | see (iii) |
notP | notQ | R | S | implied possible by (vii) + (ix) |
notP | notQ | R | notS | implied impossible by (v) |
notP | notQ | notR | S | implied possible by (ii) + (ix) |
notP | notQ | notR | notS | implied possible by (v) + (viii) + (ix) |
Evaluation of mood # 147. (Similarly, mutatis mutandis, for mood # 148.)
Major premise: Q (complemented by P) is a partial and contingent cause of R:
- If (Q + P), then R;
- if (notQ + P), not-then R;
- if (Q + notP), not-then R;
- where: (Q + P) is possible.
- If (notQ + notP), then notR;
- if (Q + notP), not-then notR;
- if (notQ + P), not-then notR;
- where: (notQ + notP) is possible.
Minor premise: P (complemented by S) is a partial cause of Q:
- If (P + S), then Q;
- if (notP + S), not-then Q;
- if (P + notS), not-then Q;
- where: (P + S) is possible.
Putative conclusion: is P (complemented by S) a partial or contingent cause of R?
YES! P is a partial cause of R:
If (P + S), then R | implied by (i) + (ix); |
if (notP + S), not-then R | implied by (v) + (x); |
if (P + notS), not-then R | implied by (ii) + (ix); |
where: (P + S) is possible | same as (xii). |
NO! P (complemented by S) is not implied to be a contingent cause of R:
If (notP + notS), then notR? | open; |
if (P + notS), not-then notR | implied by (vii) + (ix); |
if (notP + S), not-then notR? | open; |
where: (notP + notS) is possible? | open. |
P | Q | R | S | implied possible by (i) + (ix) + (xii) |
P | Q | R | notS | see (i) + (iv) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | implied impossible by (ix) |
P | notQ | R | notS | implied possible by (vii) + (ix) |
P | notQ | notR | S | implied impossible by (ix) |
P | notQ | notR | notS | implied possible by (ii) + (ix) |
notP | Q | R | S | see (vi) |
notP | Q | R | notS | see (vi) |
notP | Q | notR | S | see (iii) |
notP | Q | notR | notS | see (iii) |
notP | notQ | R | S | implied impossible by (v) |
notP | notQ | R | notS | implied impossible by (v) |
notP | notQ | notR | S | implied possible by (v) + (x) |
notP | notQ | notR | notS | see (v) + (viii) |
Evaluation of mood # 152. (Similarly, mutatis mutandis, for mood # 163.)
Major premise: Q is a complete cause of R:
- If Q, then R;
- if notQ, not-then R;
- where: Q is possible.
Minor premise: P is a complete and (complemented by S) a contingent cause of Q:
- If P, then Q;
- if notP, not-then Q;
- where: P is possible.
- If (notP + notS), then notQ;
- if (P + notS), not-then notQ;
- if (notP + S), not-then notQ;
- where: (notP + notS) is possible.
Putative conclusion: is P (complemented by S) a contingent cause of R?
NO! P (complemented by S) is not implied to be a contingent cause of R:
If (notP + notS), then notR? | open; |
if (P + notS), not-then notR | implied by (i) + (viii); |
if (notP + S), not-then notR | implied by (i) + (ix); |
where: (notP + notS) is possible | same as (x). |
P | Q | R | S | see (i) + (iii), or (iv) + (vi) |
P | Q | R | notS | implied possible by (i) + (viii) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | implied impossible by (iv) |
P | notQ | R | notS | implied impossible by (iv) |
P | notQ | notR | S | implied impossible by (iv) |
P | notQ | notR | notS | implied impossible by (iv) |
notP | Q | R | S | implied possible by (i) + (ix) |
notP | Q | R | notS | implied impossible by (vii) |
notP | Q | notR | S | implied impossible by (i) |
notP | Q | notR | notS | implied impossible by (i) or (vii) |
notP | notQ | R | S | see (v) |
notP | notQ | R | notS | see (v), or (vii) + (x) |
notP | notQ | notR | S | see (ii) or (v) |
notP | notQ | notR | notS | see (ii), or (v), or (vii) + (x) |
Note that although the above matrix does not show it, the conclusion m of mood 152 (which we uncovered through reduction) is valid. Judging from the last two lines of this matrix, one would think that (notP + notR) is open. However, it is in fact possible. This can be seen as follows.
Clause (ii) above tells us that (notQ + notR) is possible; but two of its possible expressions are implied impossible by (iv); therefore at least one of the remaining two possible expressions has to be possible. This implicit disjunctive result suffices to prove that (notP + notR) is possible, considering that its other two possible expressions are implied impossible by (i).
Compare the matrix of mood 155, whose last line corresponds to the last two lines of the matrix of mood 152. Thus, the above matrix fails to make something significant explicit for us. But this is a mere difficulty of notation, when something about more than one line has to be specified.
The same can be said for the conclusion n of mirror mood 163 and other cases. However, a similar problem does not arise with regard to any of the conclusions tested by matricial analysis in this chapter (as can be verified by reexamining all clauses of tested conclusions which were left open where a possibility was required). So it does not seem worthwhile our trying to remedy this difficulty with more elaborate notational artifices (better than “see (ii)”).
The lesson taught us by this special case is the wisdom of using matricial analysis only for crucial questions and using reduction for all others, as we did. Without awareness of the relation between moods 152 and 155 (or similarly 163 and 166), we might not have spotted the positive conclusion’s validity, unless we had developed a straddling notation.
Evaluation of mood # 153. (Similarly, mutatis mutandis, for mood # 162.)
Major premise: Q is a complete cause of R:
- If Q, then R;
- if notQ, not-then R;
- where: Q is possible.
Minor premise: P is a necessary and (complemented by S) a partial cause of Q:
- If (P + S), then Q;
- if (notP + S), not-then Q;
- if (P + notS), not-then Q;
- where: (P + S) is possible.
- If notP, then notQ;
- if P, not-then notQ;
- where: notP is possible.
Putative conclusion: is P a necessary or (complemented by S) a partial cause of R?
NO! P (complemented by S) is not implied to be a partial cause of R:
If (P + S), then R | implied by (i) + (iv); |
if (notP + S), not-then R? | open; |
if (P + notS), not-then R? | open; |
where: (P + S) is possible | same as (vii). |
Nor a necessary cause of R:
If notP, then notR? | open; |
if P, not-then notR | implied by (i) + (iv) + (vii); |
where: notP is possible | same as (x). |
P | Q | R | S | implied possible by (i) + (iv) + (vii) |
P | Q | R | notS | see (i) + (iii), or (ix) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | implied impossible by (iv) |
P | notQ | R | notS | see (vi) |
P | notQ | notR | S | implied impossible by (iv) |
P | notQ | notR | notS | see (ii) or (vi) |
notP | Q | R | S | implied impossible by (viii) |
notP | Q | R | notS | implied impossible by (viii) |
notP | Q | notR | S | implied impossible by (i) or (viii) |
notP | Q | notR | notS | implied impossible by (i) or (viii) |
notP | notQ | R | S | see (v), or (viii) + (x) |
notP | notQ | R | notS | see (viii) + (x) |
notP | notQ | notR | S | see (ii), or (v), or (viii) + (x) |
notP | notQ | notR | notS | see (ii), or (viii) + (x) |
Evaluation of mood # 154. (Similarly, mutatis mutandis, for mood # 164.)
Major premise: Q is a complete cause of R:
- If Q, then R;
- if notQ, not-then R;
- where: Q is possible.
Minor premise: P (complemented by S) is a partial and contingent cause of Q:
- If (P + S), then Q;
- if (notP + S), not-then Q;
- if (P + notS), not-then Q;
- where: (P + S) is possible.
- If (notP + notS), then notQ;
- if (P + notS), not-then notQ;
- if (notP + S), not-then notQ;
- where: (notP + notS) is possible.
Putative conclusion: is P (complemented by S) a contingent cause of R?
NO! P (complemented by S) is not implied to be a contingent cause of R:
If (notP + notS), then notR? | open; |
if (P + notS), not-then notR | implied by (i) + (ix); |
if (notP + S), not-then notR | implied by (i) + (x); |
where: (notP + notS) is possible | same as (xi). |
P | Q | R | S | implied possible by (i) + (iv) + (vii) |
P | Q | R | notS | implied possible by (i) + (ix) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | implied impossible by (iv) |
P | notQ | R | notS | see (vi) |
P | notQ | notR | S | implied impossible by (iv) |
P | notQ | notR | notS | see (ii) or (vi) |
notP | Q | R | S | implied possible by (i) + (x) |
notP | Q | R | notS | implied impossible by (viii) |
notP | Q | notR | S | implied impossible by (i) |
notP | Q | notR | notS | implied impossible by (i) or (viii) |
notP | notQ | R | S | see (v) |
notP | notQ | R | notS | see (viii) + (xi) |
notP | notQ | notR | S | see (ii) or (v) |
notP | notQ | notR | notS | see (ii), or (viii) + (xi) |
Evaluation of mood # 155. (Similarly, mutatis mutandis, for mood # 166.)
Major premise: Q is a complete cause of R:
- If Q, then R;
- if notQ, not-then R;
- where: Q is possible.
Minor premise: P is a complete cause of Q:
- If P, then Q;
- if notP, not-then Q;
- where: P is possible.
Putative conclusion: P is a complete cause of R?
YES! P is a complete cause of R:
If P, then R | implied by (i) + (iv); |
if notP, not-then R | implied by (ii) + (iv); |
where: P is possible | same as (vi). |
P | Q | R | implied possible by (i) + (iv) + (vi) |
P | Q | notR | implied impossible by (i) |
P | notQ | R | implied impossible by (iv) |
P | notQ | notR | implied impossible by (iv) |
notP | Q | R | see (i) + (iii) |
notP | Q | notR | implied impossible by (i) |
notP | notQ | R | see (v) |
notP | notQ | notR | implied possible by (ii) + (iv) |
Evaluation of mood # 171. (Similarly, mutatis mutandis, for mood # 181.)
Major premise: Q (complemented by S) is a partial cause of R:
- If (Q + S), then R;
- if (notQ + S), not-then R;
- if (Q + notS), not-then R;
- where: (Q + S) is possible.
Minor premise: P is a complete and necessary cause of Q:
- If P, then Q;
- if notP, not-then Q;
- where: P is possible.
- If notP, then notQ;
- if P, not-then notQ;
- where: notP is possible.
Putative conclusion: is P (complemented by S) a partial cause of R?
YES! P is a partial cause of R:
If (P + S), then R | implied by (i) + (v); |
if (notP + S), not-then R | implied by (ii) + (v); |
if (P + notS), not-then R | implied by (iii) + (viii); |
where: (P + S) is possible | implied by (i) + (iv) + (viii). |
P | Q | R | S | implied possible by (i) + (iv) + (viii) |
P | Q | R | notS | see (v) + (vii), or (ix) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied possible by (iii) + (viii) |
P | notQ | R | S | implied impossible by (v) |
P | notQ | R | notS | implied impossible by (v) |
P | notQ | notR | S | implied impossible by (v) |
P | notQ | notR | notS | implied impossible by (v) |
notP | Q | R | S | implied impossible by (viii) |
notP | Q | R | notS | implied impossible by (viii) |
notP | Q | notR | S | implied impossible by (i) or (viii) |
notP | Q | notR | notS | implied impossible by (viii) |
notP | notQ | R | S | see (vi), or (viii) + (x) |
notP | notQ | R | notS | see (vi), or (viii) + (x) |
notP | notQ | notR | S | implied possible by (ii) + (v) |
notP | notQ | notR | notS | see (vi), or (viii) + (x) |
Evaluation of mood # 174. (Similarly, mutatis mutandis, for mood # 184.)
Major premise: Q (complemented by P) is a partial cause of R:
- If (Q + P), then R;
- if (notQ + P), not-then R;
- if (Q + notP), not-then R;
- where: (Q + P) is possible.
Minor premise: P (complemented by S) is a partial and contingent cause of Q:
- If (P + S), then Q;
- if (notP + S), not-then Q;
- if (P + notS), not-then Q;
- where: (P + S) is possible.
- If (notP + notS), then notQ;
- if (P + notS), not-then notQ;
- if (notP + S), not-then notQ;
- where: (notP + notS) is possible.
Putative conclusion: is P (complemented by S) a partial cause of R?
YES! P is a partial cause of R:
If (P + S), then R | implied by (i) + (v); |
if (notP + S), not-then R | implied by (iii) + (ix); |
if (P + notS), not-then R | implied by (ii) + (v); |
where: (P + S) is possible | same as (viii). |
P | Q | R | S | implied possible by (i) + (v) + (viii) |
P | Q | R | notS | implied possible by (i) + (x) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | implied impossible by (v) |
P | notQ | R | notS | see (vii) |
P | notQ | notR | S | implied impossible by (v) |
P | notQ | notR | notS | implied possible by (ii) + (v) |
notP | Q | R | S | see (xi) |
notP | Q | R | notS | implied impossible by (ix) |
notP | Q | notR | S | implied possible by (iii) + (ix) |
notP | Q | notR | notS | implied impossible by (ix) |
notP | notQ | R | S | see (vi) |
notP | notQ | R | notS | see (ix) + (xii) |
notP | notQ | notR | S | see (vi) |
notP | notQ | notR | notS | see (ix) + (xii) |
Evaluation of mood # 177. (Similarly, mutatis mutandis, for mood # 188.)
Major premise: Q (complemented by P) is a partial cause of R:
- If (Q + P), then R;
- if (notQ + P), not-then R;
- if (Q + notP), not-then R;
- where: (Q + P) is possible.
Minor premise: P (complemented by S) is a partial cause of Q:
- If (P + S), then Q;
- if (notP + S), not-then Q;
- if (P + notS), not-then Q;
- where: (P + S) is possible.
Putative conclusion: is P (complemented by S) a partial cause of R?
NO! P (complemented by S) is not implied to be a partial cause of R:
If (P + S), then R | implied by (i)+ (v); |
if (notP + S), not-then R? | open; |
if (P + notS), not-then R | implied by (ii) + (v); |
where: (P + S) is possible | same as (viii). |
P | Q | R | S | implied possible by (i) + (v) + (viii) |
P | Q | R | notS | see (i) + (iv) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | implied impossible by (v) |
P | notQ | R | notS | see (vii) |
P | notQ | notR | S | implied impossible by (v) |
P | notQ | notR | notS | implied possible by (ii) + (v) |
notP | Q | R | S | |
notP | Q | R | notS | |
notP | Q | notR | S | see (iii) |
notP | Q | notR | notS | see (iii) |
notP | notQ | R | S | see (vi) |
notP | notQ | R | notS | |
notP | notQ | notR | S | see (vi) |
notP | notQ | notR | notS |
3. Crucial Matricial Analyses in Figure 2.
Evaluation of mood # 221. (Similarly, mutatis mutandis, for mood # 231.)
Major premise: R is a complete and (complemented by S) a contingent cause of Q:
- If R, then Q;
- if notR, not-then Q;
- where: R is possible.
- If (notR + notS), then notQ;
- if (R + notS), not-then notQ;
- if (notR + S), not-then notQ;
- where: (notR + notS) is possible.
Minor premise: P is a complete and necessary cause of Q:
- If P, then Q;
- if notP, not-then Q;
- where: P is possible.
- If notP, then notQ;
- if P, not-then notQ;
- where: notP is possible.
Putative conclusion is P a complete or (complemented by S) a partial cause of R?
NO! P is not implied to be a complete cause of R:
If P, then R? | denied by (vi) + (xi); |
if notP, not-then R | implied by (iv) + (vii) + (viii); |
where: P is possible | same as (x). |
Nor (complemented by S) a partial cause of R:
If (P + S), then R? | denied by (vi) + (xi); |
if (notP + S), not-then R? | open; |
if (P + notS), not-then R? | denied by (iv) + (viii); |
where: (P + S) is possible | implied by (vi) + (xi). |
P | Q | R | S | see (i) + (iii), or (viii) + (x), or (xii) |
P | Q | R | notS | implied possible by (v) + (xi) |
P | Q | notR | S | implied possible by (vi) + (xi) |
P | Q | notR | notS | implied impossible by (iv) |
P | notQ | R | S | implied impossible by (i) or (viii) |
P | notQ | R | notS | implied impossible by (i) or (viii) |
P | notQ | notR | S | implied impossible by (viii) |
P | notQ | notR | notS | implied impossible by (viii) |
notP | Q | R | S | implied impossible by (xi) |
notP | Q | R | notS | implied impossible by (xi) |
notP | Q | notR | S | implied impossible by (xi) |
notP | Q | notR | notS | implied impossible by (iv) or (xi) |
notP | notQ | R | S | implied impossible by (i) |
notP | notQ | R | notS | implied impossible by (i) |
notP | notQ | notR | S | see (ii), or (ix), or (xi) + (xiii) |
notP | notQ | notR | notS | implied possible by (iv) + (vii) + (viii) |
Evaluation of mood # 222. (Similarly, mutatis mutandis, for mood # 233.)
Major premise: R is a complete and (complemented by P) a contingent cause of Q:
- If R, then Q;
- if notR, not-then Q;
- where: R is possible.
- If (notR + notP), then notQ;
- if (R + notP), not-then notQ;
- if (notR + P), not-then notQ;
- where: (notR + notP) is possible.
Minor premise: P is a complete and (complemented by S) a contingent cause of Q:
- If P, then Q;
- if notP, not-then Q;
- where: P is possible.
- If (notP + notS), then notQ;
- if (P + notS), not-then notQ;
- if (notP + S), not-then notQ;
- where: (notP + notS) is possible.
Putative conclusion is P (complemented by S) a cause of R?
NO! P is not implied to be a complete cause of R:
If P, then R? | open; |
if notP, not-then R | implied by (i) + (xi) + (xiv); |
where: P is possible | same as (x). |
Nor (complemented by S) a partial cause of R:
If (P + S), then R? | open; |
if (notP + S), not-then R? | open; |
if (P + notS), not-then R? | open; |
where: (P + S) is possible? | open. |
Nor a necessary cause of R:
If notP, then notR? | denied by (v) + (xi), or (iv) + (xiii); |
if P, not-then notR? | open; |
where: notP is possible | implied by (v) or (vii) or (ix) or (xiii) or (xiv). |
Nor (complemented by S) a contingent cause of R:
If (notP + notS), then notR | implied by (i) + (xi); |
if (P + notS), not-then notR? | open; |
if (notP + S), not-then notR | implied by (v) + (xi), or (iv) + (xiii); |
where: (notP + notS) is possible | same as (xiv). |
P | Q | R | S | see (i) + (iii), or (viii) + (x) |
P | Q | R | notS | see (i) + (iii), or (viii) + (x), or (xii) |
P | Q | notR | S | see (vi), or (viii) + (x) |
P | Q | notR | notS | see (vi), or (viii) + (x), or (xii) |
P | notQ | R | S | implied impossible by (i) or (viii) |
P | notQ | R | notS | implied impossible by (i) or (viii) |
P | notQ | notR | S | implied impossible by (viii) |
P | notQ | notR | notS | implied impossible by (viii) |
notP | Q | R | S | implied possible by (v) + (xi), or (iv) + (xiii) |
notP | Q | R | notS | implied impossible by (xi) |
notP | Q | notR | S | implied impossible by (iv) |
notP | Q | notR | notS | implied impossible by (iv) or (xi) |
notP | notQ | R | S | implied impossible by (i) |
notP | notQ | R | notS | implied impossible by (i) |
notP | notQ | notR | S | see (ii), or (iv) + (vii), or (ix) |
notP | notQ | notR | notS | implied possible by (i) + (xi) + (xiv) |
Evaluation of mood # 224. (Similarly, mutatis mutandis, for mood # 234.)
Major premise: R is a complete and (complemented by P) a contingent cause of Q:
- If R, then Q;
- if notR, not-then Q;
- where: R is possible.
- If (notR + notP), then notQ;
- if (R + notP), not-then notQ;
- if (notR + P), not-then notQ;
- where: (notR + notP) is possible.
Minor premise: P (complemented by S) is a partial and contingent cause of Q:
- If (P + S), then Q;
- if (notP + S), not-then Q;
- if (P + notS), not-then Q;
- where: (P + S) is possible.
- If (notP + notS), then notQ;
- if (P + notS), not-then notQ;
- if (notP + S), not-then notQ;
- where: (notP + notS) is possible.
Putative conclusion is P (complemented by S) a cause of R?
NO! P is not implied to be a complete cause of R:
If P, then R? | denied by (i) + (x); |
if notP, not-then R | implied by (i) + (ix), or (i) + (xii) + (xv); |
where: P is possible | implied by (vi) or (x) or (xi) or (xiii). |
Nor (complemented by S) a partial cause of R:
If (P + S), then R? | open; |
if (notP + S), not-then R | implied by (i) + (ix); |
if (P + notS), not-then R | implied by (i) + (x); |
where: (P + S) is possible | same as (xi). |
Nor a necessary cause of R:
If notP, then notR? | denied by (iv) + (xiv), or (v) + (xii); |
if P, not-then notR? | open; |
where: notP is possible | implied by (v) or (vii) or (ix) or (xiv) or (xv). |
Nor (complemented by S) a contingent cause of R:
If (notP + notS), then notR | implied by (i) + (xii); |
if (P + notS), not-then notR? | open; |
if (notP + S), not-then notR | implied by (iv) + (xiv), or (v) + (xii); |
where: (notP + notS) is possible | same as (xv). |
P | Q | R | S | see (i) + (iii), or (viii) + (xi) |
P | Q | R | notS | see (i) + (iii), or (xiii) |
P | Q | notR | S | see (vi), or (viii) + (xi) |
P | Q | notR | notS | see (vi), or (xiii) |
P | notQ | R | S | implied impossible by (i) or (viii) |
P | notQ | R | notS | implied impossible by (i) |
P | notQ | notR | S | implied impossible by (viii) |
P | notQ | notR | notS | implied possible by (i) + (x) |
notP | Q | R | S | implied possible by (iv) + (xiv), or (v) + (xii) |
notP | Q | R | notS | implied impossible by (xii) |
notP | Q | notR | S | implied impossible by (iv) |
notP | Q | notR | notS | implied impossible by (iv) or (xii) |
notP | notQ | R | S | implied impossible by (i) |
notP | notQ | R | notS | implied impossible by (i) |
notP | notQ | notR | S | implied possible by (i) + (ix) |
notP | notQ | notR | notS | implied possible by (i) + (xii) + (xv) |
Evaluation of mood # 241.
Major premise: R (complemented by S) is a partial and contingent cause of Q:
- If (R + S), then Q;
- if (notR + S), not-then Q;
- if (R + notS), not-then Q;
- where: (R + S) is possible.
- If (notR + notS), then notQ;
- if (R + notS), not-then notQ;
- if (notR + S), not-then notQ;
- where: (notR + notS) is possible.
Minor premise: P is a complete and necessary cause of Q:
- If P, then Q;
- if notP, not-then Q;
- where: P is possible.
- If notP, then notQ;
- if P, not-then notQ;
- where: notP is possible.
Putative conclusion is P (complemented by S) a cause of R?
NO! P is not implied to be a complete cause of R:
If P, then R? | denied by (vii) + (xii); |
if notP, not-then R | implied by (ii) + (ix), or (v) + (viii) + (ix); |
where: P is possible | same as (xi). |
Nor (complemented by S) a partial cause of R:
If (P + S), then R? | denied by (vii) + (xii); |
if (notP + S), not-then R | implied by (ii) + (ix); |
if (P + notS), not-then R? | denied by (v) + (ix); |
where: (P + S) is possible | implied by (i) + (iv) + (xii), or (vii) + (xii). |
Nor a necessary cause of R:
If notP, then notR? | denied by (iii) + (ix); |
if P, not-then notR | implied by (i) + (iv) + (xii), or (vi) + (xii); |
where: notP is possible | same as (xiv). |
Nor (complemented by S) a contingent cause of R:
If (notP + notS), then notR? | denied by (iii) + (ix); |
if (P + notS), not-then notR | implied by (vi) + (xii); |
if (notP + S), not-then notR? | denied by (i) + (xii); |
where: (notP + notS) is possible | implied by (iii) + (ix), or (v) + (viii) + (ix). |
P | Q | R | S | implied possible by (i) + (iv) + (xii) |
P | Q | R | notS | implied possible by (vi) + (xii) |
P | Q | notR | S | implied possible by (vii) + (xii) |
P | Q | notR | notS | implied impossible by (v) |
P | notQ | R | S | implied impossible by (i) or (ix) |
P | notQ | R | notS | implied impossible by (ix) |
P | notQ | notR | S | implied impossible by (ix) |
P | notQ | notR | notS | implied impossible by (ix) |
notP | Q | R | S | implied impossible by (xii) |
notP | Q | R | notS | implied impossible by (xii) |
notP | Q | notR | S | implied impossible by (xii) |
notP | Q | notR | notS | implied impossible by (v) or (xii) |
notP | notQ | R | S | implied impossible by (i) |
notP | notQ | R | notS | implied possible by (iii) + (ix) |
notP | notQ | notR | S | implied possible by (ii) + (ix) |
notP | notQ | notR | notS | implied possible by (v) + (viii) + (ix) |
Evaluation of mood # 244.
Major premise: R (complemented by P) is a partial and contingent cause of Q:
- If (R + P), then Q;
- if (notR + P), not-then Q;
- if (R + notP), not-then Q;
- where: (R + P) is possible.
- If (notR + notP), then notQ;
- if (R + notP), not-then notQ;
- if (notR + P), not-then notQ;
- where: (notR + notP) is possible.
Minor premise: P (complemented by S) is a partial and contingent cause of Q:
- If (P + S), then Q;
- if (notP + S), not-then Q;
- if (P + notS), not-then Q;
- where: (P + S) is possible.
- If (notP + notS), then notQ;
- if (P + notS), not-then notQ;
- if (notP + S), not-then notQ;
- where: (notP + notS) is possible.
Putative conclusion is P (complemented by S) a cause of R?
NO! P is not implied to be a complete cause of R:
If P, then R? | denied by (i) + (xi), or (ii) + (ix); |
if notP, not-then R? | open; |
where: P is possible | implied by (ii) or (iv) or (vii) or (xi) or (xii) or (xiv). |
Nor (complemented by S) a partial cause of R:
If (P + S), then R? | open; |
if (notP + S), not-then R? | open; |
if (P + notS), not-then R | implied by (i) + (xi), or (ii) + (ix); |
where: (P + S) is possible | same as (xii). |
Nor a necessary cause of R:
If notP, then notR? | denied by (v) + (xv), or (vi) + (xiii); |
if P, not-then notR? | open; |
where: notP is possible | implied by (iii) or (vi) or (viii) or (x) or (xv) or (xvi). |
Nor (complemented by S) a contingent cause of R:
If (notP + notS), then notR? | open; |
if (P + notS), not-then notR? | open; |
if (notP + S), not-then notR | implied by (v) + (xv), or (vi) + (xiii); |
where: (notP + notS) is possible | same as (xvi). |
P | Q | R | S | see (i) + (iv), or (ix) + (xii) |
P | Q | R | notS | see (i) + (iv), or (xiv) |
P | Q | notR | S | see (vii), or (ix) + (xii) |
P | Q | notR | notS | see (vii), or (xiv) |
P | notQ | R | S | implied impossible by (i) or (ix) |
P | notQ | R | notS | implied impossible by (i) |
P | notQ | notR | S | implied impossible by (ix) |
P | notQ | notR | notS | implied possible by (i) + (xi), or (ii) + (ix) |
notP | Q | R | S | implied possible by (v) + (xv), or (vi) + (xiii) |
notP | Q | R | notS | implied impossible by (xiii) |
notP | Q | notR | S | implied impossible by (v) |
notP | Q | notR | notS | implied impossible by (v) or (xiii) |
notP | notQ | R | S | see (iii) or (x) |
notP | notQ | R | notS | see (iii), or (xiii) + (xvi) |
notP | notQ | notR | S | see (v) + (viii), or (x) |
notP | notQ | notR | notS | see (v) + (viii), or (xiii) + (xvi) |
4. Crucial Matricial Analyses in Figure 3.
Evaluation of mood # 312. (Similarly, mutatis mutandis, for mood # 313.)
Major premise: Q is a complete and necessary cause of R:
- If Q, then R;
- if notQ, not-then R;
- where: Q is possible.
- If notQ, then notR;
- if Q, not-then notR;
- where: notQ is possible.
Minor premise: Q is a complete and (complemented by S) a contingent cause of P:
- If Q, then P;
- if notQ, not-then P;
- where: Q is possible.
- If (notQ + notS), then notP;
- if (Q + notS), not-then notP;
- if (notQ + S), not-then notP;
- where: (notQ + notS) is possible.
Putative conclusion: is P a complete or (complemented by S) a partial cause of R?
NO! P is not implied to be a complete cause of R:
If P, then R? | denied by (iv) + (xii); |
if notP, not-then R | implied by (iv) + (x) + (xiii); |
where: P is possible | implied by (vii) + (ix), or (xi), or (xii). |
Nor (complemented by S) a partial cause of R:
If (P + S), then R? | denied by (iv) + (xii); |
if (notP + S), not-then R? | open; |
if (P + notS), not-then R? | denied by (i) + (x); |
where: (P + S) is possible | implied by (xii). |
P | Q | R | S | see (i) + (iii), or (v), or (vii) + (ix) |
P | Q | R | notS | implied possible by (i) + (xi) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | implied impossible by (iv) |
P | notQ | R | notS | implied impossible by (iv) or (x) |
P | notQ | notR | S | implied possible by (iv) + (xii) |
P | notQ | notR | notS | implied impossible by (x) |
notP | Q | R | S | implied impossible by (vii) |
notP | Q | R | notS | implied impossible by (vii) |
notP | Q | notR | S | implied impossible by (i) or (vii) |
notP | Q | notR | notS | implied impossible by (i) or (vii) |
notP | notQ | R | S | implied impossible by (iv) |
notP | notQ | R | notS | implied impossible by (iv) |
notP | notQ | notR | S | see (ii), or (iv) + (vi), or (viii) |
notP | notQ | notR | notS | implied possible by (iv) + (x) + (xiii) |
Evaluation of mood # 314.
Major premise: Q is a complete and necessary cause of R:
- If Q, then R;
- if notQ, not-then R;
- where: Q is possible.
- If notQ, then notR;
- if Q, not-then notR;
- where: notQ is possible.
Minor premise: Q (complemented by S) is a partial and contingent cause of P:
- If (Q + S), then P;
- if (notQ + S), not-then P;
- if (Q + notS), not-then P;
- where: (Q + S) is possible.
- If (notQ + notS), then notP;
- if (Q + notS), not-then notP;
- if (notQ + S), not-then notP;
- where: (notQ + notS) is possible.
Putative conclusion: is P (complemented by S) a cause of R?
NO! P is not implied to be a complete cause of R:
If P, then R? | denied by (iv) + (xiii); |
if notP, not-then R | implied by (iv) + (viii), or (iv) + (xi) + (xiv); |
where: P is possible | implied by (vii) + (x), or (xii), or (xiii). |
Nor (complemented by S) a partial cause of R:
If (P + S), then R? | denied by (iv) + (xiii); |
if (notP + S), not-then R | implied by (iv) + (viii); |
if (P + notS), not-then R? | denied by (i) + (xi); |
where: (P + S) is possible | implied by (vii) + (x), or (xiii). |
Nor a necessary cause of R:
If notP, then notR? | denied by (i) + (ix); |
if P, not-then notR | implied by (i) + (vii) + (x), or (i) + (xii); |
where: notP is possible | implied by (viii), or (ix), or (xi) + (xiv). |
Nor (complemented by S) a contingent cause of R:
If (notP + notS), then notR? | denied by (i) + (ix); |
if (P + notS), not-then notR | implied by (i) + (xii); |
if (notP + S), not-then notR? | denied by (iv) + (vii); |
where: (notP + notS) is possible | implied by (ix), or (xi) + (xiv). |
P | Q | R | S | implied possible by (i) + (vii) + (x) |
P | Q | R | notS | implied possible by (i) + (xii) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | implied impossible by (iv) |
P | notQ | R | notS | implied impossible by (iv) or (xi) |
P | notQ | notR | S | implied possible by (iv) + (xiii) |
P | notQ | notR | notS | implied impossible by (xi) |
notP | Q | R | S | implied impossible by (vii) |
notP | Q | R | notS | implied possible by (i) + (ix) |
notP | Q | notR | S | implied impossible by (i) or (vii) |
notP | Q | notR | notS | implied impossible by (i) |
notP | notQ | R | S | implied impossible by (iv) |
notP | notQ | R | notS | implied impossible by (iv) |
notP | notQ | notR | S | implied possible by (iv) + (viii) |
notP | notQ | notR | notS | implied possible by (iv) + (xi) + (xiv) |
Evaluation of mood # 324. (Similarly, mutatis mutandis, for mood # 334.)
Major premise: Q is a complete and (complemented by P) a contingent cause of R:
- If Q, then R;
- if notQ, not-then R;
- where: Q is possible.
- If (notQ + notP), then notR;
- if (Q + notP), not-then notR;
- if (notQ + P), not-then notR;
- where: (notQ + notP) is possible.
Minor premise: Q (complemented by S) is a partial and contingent cause of P:
- If (Q + S), then P;
- if (notQ + S), not-then P;
- if (Q + notS), not-then P;
- where: (Q + S) is possible.
- If (notQ + notS), then notP;
- if (Q + notS), not-then notP;
- if (notQ + S), not-then notP;
- where: (notQ + notS) is possible.
Putative conclusion: is P (complemented by S) a cause of R?
NO! P is not implied to be a complete cause of R:
If P, then R? | open; |
if notP, not-then R | implied by (iv) + (ix), or (iv) + (xii) + (xv); |
where: P is possible | implied by (vi), or (viii) + (xi), or (xiii), or (xiv). |
Nor (complemented by S) a partial cause of R:
If (P + S), then R? | open; |
if (notP + S), not-then R | implied by (iv) + (ix); |
if (P + notS), not-then R? | denied by (i) + (xii); |
where: (P + S) is possible | implied by (vi) + (xii), or (viii) + (xi), or (xiv). |
Nor a necessary cause of R:
If notP, then notR? | denied by (v) + (viii), or (i) + (x); |
if P, not-then notR | implied by (i) + (viii) + (xi), or (i) + (xiii), or (vi) + (xii); |
where: notP is possible | implied by (v), or (ix), or (x), or (xii) + (xv). |
Nor (complemented by S) a contingent cause of R:
If (notP + notS), then notR? | denied by (v) + (viii), or (i) + (x); |
if (P + notS), not-then notR | implied by (i) + (xiii); |
if (notP + S), not-then notR? | denied by (iv) + (viii); |
where: (notP + notS) is possible | implied by (v) + (viii), or (x), or (xii) + (xv). |
P | Q | R | S | implied possible by (i) + (viii) + (xi) |
P | Q | R | notS | implied possible by (i) + (xiii) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | implied possible by (vi) + (xii) |
P | notQ | R | notS | implied impossible by (xii) |
P | notQ | notR | S | see (ii) or (xiv) |
P | notQ | notR | notS | implied impossible by (xii) |
notP | Q | R | S | implied impossible by (viii) |
notP | Q | R | notS | implied possible by (v) + (viii), or (i) + (x) |
notP | Q | notR | S | implied impossible by (i) or (viii) |
notP | Q | notR | notS | implied impossible by (i) |
notP | notQ | R | S | implied impossible by (iv) |
notP | notQ | R | notS | implied impossible by (iv) |
notP | notQ | notR | S | implied possible by (iv) + (ix) |
notP | notQ | notR | notS | implied possible by (iv) + (xii) + (xv) |
Evaluation of mood # 344.
Major premise: Q (complemented by P) is a partial and contingent cause of R:
- If (Q + P), then R;
- if (notQ + P), not-then R;
- if (Q + notP), not-then R;
- where: (Q + P) is possible.
- If (notQ + notP), then notR;
- if (Q + notP), not-then notR;
- if (notQ + P), not-then notR;
- where: (notQ + notP) is possible.
Minor premise: Q (complemented by S) is a partial and contingent cause of P:
- If (Q + S), then P;
- if (notQ + S), not-then P;
- if (Q + notS), not-then P;
- where: (Q + S) is possible.
- If (notQ + notS), then notP;
- if (Q + notS), not-then notP;
- if (notQ + S), not-then notP;
- where: (notQ + notS) is possible.
Putative conclusion: is P (complemented by S) a cause of R?
NO! P is not implied to be a complete cause of R:
If P, then R? | denied by (ii) + (xiii); |
if notP, not-then R | implied by (iii) + (ix), or (v) + (x), or (v) + (xiii) + (xvi); |
where: P is possible | implied by (ii), or (iv), or (vii) + (xiii), or (ix) + (xii), or (xiv) or (xv). |
Nor (complemented by S) a partial cause of R:
If (P + S), then R? | denied by (ii) + (xiii); |
if (notP + S), not-then R | implied by (v) + (x); |
if (P + notS), not-then R? | denied by (i) + (xiii); |
where: (P + S) is possible | implied by (ii) + (xiii), or (vii) + (xiii), or (ix) + (xii), or (xv). |
Nor a necessary cause of R:
If notP, then notR? | denied by (vi) + (ix); |
if P, not-then notR | implied by (i) + (ix) + (xii), or (i) + (xiv), or (vii) + (xiii); |
where: notP is possible | implied by (iii), or (vi), or (viii), or (x), or (xi), or (xiii) + (xvi). |
Nor (complemented by S) a contingent cause of R:
If (notP + notS), then notR? | denied by (vi) + (ix); |
if (P + notS), not-then notR | implied by (i) + (xiv); |
if (notP + S), not-then notR? | denied by (v) + (ix); |
where: (notP + notS) is possible | implied by (iii) + (ix), or (vi) + (ix), or (xi), or (xiii) + (xvi). |
P | Q | R | S | implied possible by (i) + (ix) + (xii) |
P | Q | R | notS | implied possible by (i) + (xiv) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | implied possible by (vii) + (xiii) |
P | notQ | R | notS | implied impossible by (xiii) |
P | notQ | notR | S | implied possible by (ii) + (xiii) |
P | notQ | notR | notS | implied impossible by (xiii) |
notP | Q | R | S | implied impossible by (ix) |
notP | Q | R | notS | implied possible by (vi) + (ix) |
notP | Q | notR | S | implied impossible by (ix) |
notP | Q | notR | notS | implied possible by (iii) + (ix) |
notP | notQ | R | S | implied impossible by (v) |
notP | notQ | R | notS | implied impossible by (v) |
notP | notQ | notR | S | implied possible by (v) + (x) |
notP | notQ | notR | notS | implied possible by (v) + (xiii) + (xvi) |
Evaluation of mood # 352. (Similarly, mutatis mutandis, for mood # 363.)
Major premise: Q is a complete cause of R:
- If Q, then R;
- if notQ, not-then R;
- where: Q is possible.
Minor premise: Q is a complete and (complemented by S) a contingent cause of P:
- If Q, then P;
- if notQ, not-then P;
- where: Q is possible.
- If (notQ + notS), then notP;
- if (Q + notS), not-then notP;
- if (notQ + S), not-then notP;
- where: (notQ + notS) is possible.
Putative conclusion: is P a necessary cause of R?
NO! P is not implied to be a necessary cause of R:
If notP, then notR? | open; |
if P, not-then notR | implied by (i) + (viii); |
where: notP is possible? | implied by (v), or (vii) + (x). |
P | Q | R | S | see (i) + (iii), or (iv) + (vi) |
P | Q | R | notS | implied possible by (i) + (viii) |
P | Q | notR | S | implied impossible by (i) |
P | Q | notR | notS | implied impossible by (i) |
P | notQ | R | S | see (ix) |
P | notQ | R | notS | implied impossible by (vii) |
P | notQ | notR | S | see (ii) or (ix) |
P | notQ | notR | notS | implied impossible by (vii) |
notP | Q | R | S | implied impossible by (iv) |
notP | Q | R | notS | implied impossible by (iv) |
notP | Q | notR | S | implied impossible by (i) or (iv) |
notP | Q | notR | notS | implied impossible by (i) or (iv) |
notP | notQ | R | S | see (v) |
notP | notQ | R | notS | see (v), or (vii) + (x) |
notP | notQ | notR | S | see (ii) or (v) |
notP | notQ | notR | notS | see (ii) or (v), or (vii) + (x) |
Chapter 9. Squeezing Out More Information
1. The Interactions of Determinations.
Before considering the possibility of other inferences from causative propositions, let us summarize and extend the results obtained thus far, and especially try and understand them in a global perspective. We have in the preceding chapters identified, in the three figures, 66 valid positive conclusions obtainable from positive premises, out of 192 (3*8*8) possible combinations of generic and joint premises. We thus found a validity rate of 34.4% – meaning that reasoning with causative propositions cannot be left to chance, since we would likely be wrong two times out of three! The table shows the distribution of valid and invalid moods in the three figures:
Figure | Valid Moods(positive) | Invalid Moods(impossible or nil) |
1 | 30 | 34 |
2 | 18 | 46 |
3 | 18 | 46 |
Total | 66 | 126 |
Moreover, not all of the valid moods have equal significance. As the table below shows, some moods (20, shaded) are conceptually basic, while others (46) are mere derivatives of these, in the sense of compounds (16) or subalterns (30) of them. We shall call the former ‘primary’ moods, and the latter ‘secondary’ moods. Note that these terms are not intended as references to validation processes, but to comparisons of results. By which I mean that some of the moods here classed as ‘primary’ (such as #217, to cite one case) were validated by reduction to others; whereas some of the moods here classed as ‘secondary’ (such as #117, for example) were among those that had to be validated by matricial analysis.
A primary mood teaches us a lesson in reasoning. For instance, mood 1/m/m/m (#155) teaches us that in Figure 1, the premises m and m yield the conclusion m. A secondary (subaltern or compound) mood has premises that teach us nothing new (compared to the corresponding primary), except to tell us that no additional information is implied. For instances, 1/m/mq/m (#152) is equivalent (subaltern) to 1/m/m/m; and 1/mn/mn/mn (#111) is equivalent to (a compound of) 1/m/m/m plus 1/n/n/n.
Such equivalencies are due to the fact that the premises of the secondary mood imply those of the primary mood(s), while the conclusion(s) of the latter imply that of the former. We can thus ‘reconstruct’ the derivative mood from its conceptual source(s). Effectively, primary moods represent general truths, of which secondary moods are specific expressions. This ordering of the valid moods signifies that we do not have to memorize them all, but only 20 out of 66.
In the following table, the valid positive moods of causative syllogism are listed for each figure in order of the strength of their conclusions (joint determinations before generics). Within each group of moods yielding a given conclusion, moods are ordered in the reverse order with reference to their premises (the weakest premises capable of yielding a certain conclusion being listed first, so far as possible – some are of course incomparable). Explanations will be given further on.
Primary moods (shaded) are distinguished from compounds and subalterns, and the primary sources of the secondaries are specified. Notice that all moods with a joint determination as conclusion are compounds.
No. | Major | Minor | Conclusion | Relation | to mood |
Figure 1 (12 primaries, 8 compounds and 10 subalterns) | |||||
111 | mn | mn | mn | compound | 155 + 166 |
121 | mq | mn | mq | compound | 155 + 181 |
112 | mn | mq | mq | compound | 155 + 118 |
131 | np | mn | np | compound | 166 + 171 |
113 | mn | np | np | compound | 166 + 117 |
141 | pq | mn | pq | compound | 171 + 181 |
114 | mn | pq | pq | compound | 117 + 118 |
144 | pq | pq | pq | compoundcompound | 147 + 148,174 + 184 |
155 | m | m | m | primary | |
152 | m | mq | m | subaltern | 155 |
125 | mq | m | m | subaltern | 155 |
151 | m | mn | m | subaltern | 155 |
115 | mn | m | m | subaltern | 155 |
166 | n | n | n | primary | |
163 | n | np | n | subaltern | 166 |
136 | np | n | n | subaltern | 166 |
161 | n | mn | n | subaltern | 166 |
116 | mn | n | n | subaltern | 166 |
147 | pq | p | p | primary | |
174 | p | pq | p | primary | |
137 | np | p | p | primary | |
134 | np | pq | p | subaltern | 137 or 174 |
171 | p | mn | p | primary | |
117 | mn | p | p | primary | |
148 | pq | q | q | primary | |
184 | q | pq | q | primary | |
128 | mq | q | q | primary | |
124 | mq | pq | q | subaltern | 128 or 184 |
181 | q | mn | q | primary | |
118 | mn | q | q | primary |
Table 9.2 continued.
Figure 2 (4 primaries, 4 compounds and 10 subalterns) | |||||
211 | mn | mn | mn | compound | 256 + 265 |
212 | mn | mq | mq | compound | 265 + 218 |
213 | mn | np | np | compound | 256 + 217 |
214 | mn | pq | pq | compound | 217 + 218 |
265 | n | m | m | primary | |
262 | n | mq | m | subaltern | 265 |
235 | np | m | m | subaltern | 265 |
261 | n | mn | m | subaltern | 265 |
215 | mn | m | m | subaltern | 265 |
231 | np | mn | m | subaltern | 265 |
256 | m | n | n | primary | |
253 | m | np | n | subaltern | 256 |
226 | mq | n | n | subaltern | 256 |
251 | m | mn | n | subaltern | 256 |
216 | mn | n | n | subaltern | 256 |
221 | mq | mn | n | subaltern | 256 |
217 | mn | p | p | primary | |
218 | mn | q | q | primary | |
Figure 3 (4 primaries, 4 compounds and 10 subalterns) | |||||
311 | mn | mn | mn | compound | 356 + 365 |
321 | mq | mn | mq | compound | 356 + 381 |
331 | np | mn | np | compound | 365 + 371 |
341 | pq | mn | pq | compound | 371 + 381 |
356 | m | n | m | primary | |
353 | m | np | m | subaltern | 356 |
326 | mq | n | m | subaltern | 356 |
351 | m | mn | m | subaltern | 356 |
316 | mn | n | m | subaltern | 356 |
313 | mn | np | m | subaltern | 356 |
365 | n | m | n | primary | |
362 | n | mq | n | subaltern | 365 |
335 | np | m | n | subaltern | 365 |
361 | n | mn | n | subaltern | 365 |
315 | mn | m | n | subaltern | 365 |
312 | mn | mq | n | subaltern | 365 |
371 | p | mn | p | primary | |
381 | q | mn | q | primary |
As already stated, we need only keep in mind the 20 primaries, the remaining 46 secondaries being obvious corollaries. It is implicitly understood that, had any of the latter been primary (e.g. if 1/m/mq had concluded mq, say, instead of just m), it would have been classified as such among the former.
We can further cut down the burden on memory by taking stock of ‘mirror’ moods. As we can see on the table above, among the primaries (shaded): in Figure 1, mood 166 is a mirror of mood 155, 148 of 147, 184 of 174, 128 of 137, 118 of 117, and 181 of 171. In Figure 2, mood 256 is a mirror of mood 265, and 218 of 217. In Figure 3, mood 365 is a mirror of mood 356, and 381 of 371. In this way, we need only remember 10 primary moods (6 in the first figure, 2 in the second and 2 in the third), and the 10 others follow by mirroring.
To better understand the results obtained, we ought to notice the phenomenon of transposition of determinations in the premises. Moods can be paired-off if they have the same premises in reverse order. Note that, for each pair, the figure number (hundreds) is the same, while the numbers of the major and minor premises (tens and units, respectively) are transposed.
- Thus, among primaries, we should mentally pair off the following: 147 and 174, 117 and 171, 148 and 184, 118 and 181. In these paired cases, the combination of the determinations involved has the same conclusion, however ordered in the premises. Take, for instance, moods 147 and 174, i.e. 1/pq/p/p and 1/p/pq/p; the conclusion has the same determination p, whether the determinations of the premises are pq/p or p/pq. This allows us to regard, in such cases, the determination of the conclusion as a product of the determinations of the premises, irrespective of their ordering. (We can similarly pair off many secondary moods: for instances, 125 and 152, 115 and 151, etc.)
- In the case of the following transposed pairs, 265 and 256, 356 and 365, the conclusions are of similar strength, but not identical determination. Thus, for instance, 1/n/m/m (#265) and 1/m/n/n (#256) are comparable although only by way of mirroring. (We can similarly pair off some secondaries, like 226 and 262, 235 and 253, etc.)
- Some individual moods have the same determination in both premises, and thus cannot be paired with others. These, we might say, pair off with themselves. Thus with Nos. 155, 166 among primaries; and some likewise among secondaries.
- But, note well, some moods are not similarly paired; specifically, the primary moods 128, 137, 217, 218, 371, 381 are not; similarly some of the secondaries. For instance, mood 1/np/p/p (#137) is valid, but mood 1/p/np/p (#173) is invalid. This teaches us not to indiscriminately look upon the order of the determinations in the premises as irrelevant.
Moreover, transposition of the determinations of the premises should not be confused with transposition of the premises themselves. For if the premises are transposed, the conclusion obtained from them is converted. Additionally, in the case of the first figure, transposition of the premises would take us out of the first figure (into the so-called fourth figure[49]), since the middle item changes position in them. As for the second and third figures, though transposition of premises does not entail a change of figure (the middle item remains in the same position either way), it entails a change of determination in the conclusion (since the items in the premises change place); see for instances moods 256 and 265, or 356 and 365.
Nevertheless, awareness of the phenomenon of transposition of determinations is valuable, because it allows us to make an analogy with composition of forces in mechanics. Syllogism in general may be viewed as a doctrine concerning the interactions of different propositional forms. With regard to the determinations of causation, we learn from the cases mentioned above something about the interactions of determinations, i.e. how their ‘forces’ combine.
We can push this insight further, with reference to the hierarchies between the significant moods (primaries) and their respective derivatives (secondaries). Consider, for instance, the primary mood 1/m/m, which has conclusion m; if we gradually increase the strength of the major premise (to mq or mn), while keeping the minor premise the same (m), or vice versa, the determination of the conclusion remains unaffected (m). In contrast, if we increase the strength of both premises at once to mn, the conclusion increases in strength to mn. Similarly in many other cases. Thus, some increases in strength in the premises produce no additional strength in the conclusion; but at some threshold, the intensification may get sufficient to produce an upward shift in determination.
We can in like manner view changes in conclusion from p to m or from q to n (and likewise to the joint determinations mq or np). For instance, compare moods 1/mn/p/p and 1/mn/m/m; here, keeping the major premise constant (mn), as we upgrade the minor premise from p to m, we find the conclusion upgraded from p to m. Similarly with moods 1/p/mn/p and 1/m/mn/m, keeping the minor constant while varying the major. Let us not forget that the determinations of causation were conceived essentially as modalities: p and m, though defined as mutually exclusive, are meant as different degrees of positive causation; similarly for the negative aspects of causation, q and n. Thus, some such transitions were to be expected.
We can in this way interpret our list of valid moods as a map of the changing topography in the field of determination. This gives us an interesting overview of the whole domain of causation. This is the intent of Table 9.2, above.
2. Negative Moods.
Thus far, we have only validated causative syllogisms with positive premises and positive conclusions. We will now look into the possibility of obtaining, at least by derivation from the foregoing, additional valid moods involving a negative premise and, consequently, a negative conclusion.
This is made possible by using Aristotle’s method of indirect reduction, or reduction ad absurdum. To begin with, let us describe the various reduction processes involved. Note the changed positions of items P, Q, R, in each situation. The mood to be validated (left) involves a positive premise (indicated by a + sign) and a negative premise (-) yielding a negative putative conclusion. The reduction process keeps one of the original premises (the positive one), and shows that contradicting the putative conclusion would result, through an already validated positive mood (right), in contradiction of the other premise (the negative one). Notice the figure used for validation purposes depends on which original premise is the positive one, staying constant in the process.
Figure 1 | Reduction process: | Figure 2 | |
major premise | +QR | keeping the same major, | +QR |
minor premise | -PQ | if we deny the conclusion, | +PR |
Conclusion | -PR | then we deny the minor. | +PQ |
Figure 1 | Reduction process: | Figure 3 | |
major premise | -QR | if we deny the conclusion, | +PR |
minor premise | +PQ | keeping the same minor, | +PQ |
Conclusion | -PR | then we deny the major. | +QR |
Figure 2 | Reduction process: | Figure 1 | |
major premise | +RQ | keeping the same major, | +RQ |
minor premise | -PQ | if we deny the conclusion, | +PR |
Conclusion | -PR | then we deny the minor. | +PQ |
Figure 2 | Reduction process: | Figure 3 | |
major premise | -RQ | keeping the minor as a major, | +PQ |
minor premise | +PQ | if we deny the conclusion, | +PR |
Conclusion | -PR | then we deny the major. | +RQ |
Figure 3 | Reduction process: | Figure 1 | |
major premise | -QR | if we deny the conclusion, | +PR |
minor premise | +QP | keeping the same minor, | +QP |
Conclusion | -PR | then we deny the major. | +QR |
Figure 3 | Reduction process: | Figure 2 | |
major premise | +QR | keeping the major as a minor, | +PR |
minor premise | -QP | if we deny the conclusion, | +QR |
Conclusion | -PR | then we deny the minor. | +QP |
Consider, for instance, a first figure syllogism QR/PQ/PR, which we wish to reduce ad absurdum to a second figure syllogism of established validity. Knowing that the given major premise (QR), and the negation of the putative conclusion (PR), together imply (in Figure 2) the negation of the given minor premise (PQ) – we are logically forced to admit the putative conclusion from the given premises (in Figure 1). Similar arguments apply to the other three cases, as indicated above.
Using these reduction arguments, we can validate the following moods, in the three figures. In the following table, all I have done is apply indirect reduction to the primary moods listed in Table 9.2. I ignored all subaltern and compound moods in it, since they would only give rise to other derivatives.
Major | Minor | Conclusion | Source |
Figure 1 from Figure 2 – keep same major | |||
n | not-m | not-m | 265 |
m | not-n | not-n | 256 |
mn | not-p | not-p | 217 |
mn | not-q | not-q | 218 |
Figure 1 from Figure 3 – keep same minor | |||
not-m | n | not-m | 356 |
not-n | m | not-n | 365 |
not-p | mn | not-p | 371 |
not-q | mn | not-q | 381 |
Table 9.3 continued.
Figure 2 from Figure 1 – keep same major | |||
m | not-m | not-m | 155 |
n | not-n | not-n | 166 |
mn | not-p | not-p | 117 |
np | not-p | not-p | 137 |
pq | not-p | not-p | 147 |
mn | not-q | not-q | 118 |
mq | not-q | not-q | 128 |
pq | not-q | not-q | 148 |
p | not-p | not(mn) | 171 |
q | not-q | not(mn) | 181 |
p | not-p | not(pq) | 174 |
q | not-q | not(pq) | 184 |
Figure 2 from Figure 3 – keep the minor as a major | |||
not-n | n | not-m | 365 |
not-m | m | not-n | 356 |
not-p | p | not(mn) | 371 |
not-q | q | not(mn) | 381 |
Figure 3 from Figure 2 – keep the major as a minor | |||
n | not-n | not-m | 256 |
m | not-m | not-n | 265 |
p | not-p | not(mn) | 217 |
q | not-q | not(mn) | 218 |
Figure 3 from Figure 1 – keep same minor | |||
not-m | m | not-m | 155 |
not-n | n | not-n | 166 |
not-p | mn | not-p | 171 |
not-p | pq | not-p | 174 |
not-q | mn | not-q | 181 |
not-q | pq | not-q | 184 |
not-p | p | not(mn) | 117 |
not-q | q | not(mn) | 118 |
not-q | q | not(mq) | 128 |
not-p | p | not(np) | 137 |
not-p | p | not(pq) | 147 |
not-q | q | not(pq) | 148 |
Obviously, the significance of not-p or not-q in a premise or conclusion must be carefully assessed in each case. This is best done by writing it out in full.
Take for example 1/mn/not-p/not-p, which we derived ad absurdum from mood 217, i.e. 2/mn/p/p. The major premise in both cases has form QR. The Figure 1 mood has minor premise of form P(S)R and conclusion of form P(S)Q. The Figure 2 minor premise and conclusion have form P(S)Q and P(S)R, respectively. We thus indirectly reduce subfigure 1b to subfigure 2b. The complement is S everywhere and the negative propositions not-p can be read as not-p_{S}. We may also generalize this argument to all complements, since whatever the complement happen to be it will return in the conclusion. It follows that if the minor premise is absolute, so is the conclusion.
In some other cases, however, the transition is not so simple. For example, when 2/p/not-p/not(mn) is reduced ad absurdum to 1/p/mn/p, we apparently have subfigure 2d (say) derived from subfigure 1c. But the number of complements does not match, so this case is rather artificial in construction. But I will not delve further into such issues here, not wanting to complicate matters unnecessarily. The conscientious reader will find personal investigation of these details a rewarding exercise.
Nevertheless, many of the above results are not without practical interest and value. For a start, they allow us to squeeze a bit more information out of causative propositions, and thus tell us a little more about the topography of the field of determination mentioned earlier. Most importantly, all the moods listed in this table involve a negative generic premise. Until now, we have only managed to validate moods with positive premises, i.e. positive moods. These are the first negative moods we manage to validate, by indirect reduction to (primary) positive moods.
This supplementary class of valid moods yields negative conclusions, whether the negation of a generic determination or that of a joint determination. Remember that the conclusions not(mn), not(mq), not(np), or not(pq) can be interpreted as disjunctive propositions involving all remaining (i.e. not negated) formal possibilities. Thus, for instance, not(mn) means “either mq or np or pq or non-causation”.
Summarizing, we have a total of 20 valid moods with a negative major premise, and 20 with a negative minor premise, making a total of 40 new moods. In Figure 1, the statistics are 4 + 4 = 8; in Figure2, they are 4 + 12 = 16; and in Figure 3, they are 12+ 4 = 16. We could similarly derive additional negative moods, by indirect reduction to compound and subaltern moods: this exercise is left to the reader.
3. Negative Conclusions from Positive Moods.
We have in the preceding chapters evaluated all conceivable positive conclusions from positive moods, i.e. from moods both of whose premises are positive (generic and/or joint) causative propositions. But we have virtually ignored negative conclusions from these (positive) moods, effectively lumping them with ‘non-conclusions’ (labeled nil), which they are not. We shall consider the significance of negative conclusions now[50].
In this context, it is important to keep in mind the distinction between a mood not implying a certain conclusion (which is therefore a non-sequitur, an ‘it does not follow’, which is invalid, but whose contradictory may yet be a valid or invalid conclusion), and a mood implying the negation of (i.e. denying) a certain conclusion (which is therefore more specifically an antinomy, so that not only is it invalid, but moreover it is so because its contradictory is a valid conclusion).
- For a start, we have to note that wherever a positive mood yields a valid positive conclusion, it also incidentally yields a valid negative conclusion, namely one denying the contrary determination(s). Thus, for example, mood 111 (mn/mn) yields the positive conclusions “P is a complete and necessary cause of Q” (mn); it therefore also yields as negative conclusions “P is not a partial and not a contingent cause of Q” (not-p and not-q). We thus have at least as many valid negative conclusions as we have valid positive ones. Such syllogisms with negative conclusions are, of course, mere subalterns of those with positive conclusions they are derived from.
- Moreover, we may notice that some of the crucial matricial analyses developed in the previous chapter invalidated certain conclusions, not merely by leaving one or more of their constituent clauses open, but more radically by denying, i.e. implying the negation of, some clause(s). Specifically, this occurred in the 14 cases listed in the following table (where ‘+’ means implied, ‘-’ means denied, and ‘?’ means neither implied nor denied).
Notice that this table concerns negations of p or q relative to the complement S (whence my use here of the notation p_{s} or q_{s}), which is not the same as absolute negation. It is very important to specify the complement, otherwise contradictions might wrongly be thought to appear at later stages. In the case of negations of m or n, they are absolute anyway since there are no complements for them. Also note that:
- Where m or n is affirmed (as in moods 221, 231, 312, 313), then p or q (respectively) may be denied absolutely, i.e. whatever complement (S, notS or any other) be considered for p or q. That is, m implies not-p and n implies not-q. This can also be stated as m = mn or mq and n = mn or np, wherein the complement is unspecified (possibly but not necessarily S, or notS, or any other).
- Although not-m by itself does not imply p, not-m + n = np (moods 221, 312). Likewise, although not-n by itself does not imply q, not-n + m = mq (moods 231, 313). This is evident from the fact that absolute lone determinations are impossible. Here again, note well, the complement concerned is not specified (i.e. it may be, but need not be, S, or notS, or any other, say T).
- Furthermore, where m and/or n is/are denied (as occurs in all 14 cases to some extent), the additional denial if any of p and/or q (as in 221, 231, 241, and the six moods of Figure 3) has to initially be understood as a restricted negation, i.e. as not-p_{S} or not-q_{S}. Additional work is required to prove radical negation of the weak determinations.
- Since causation is by joint determination or not at all, not-m + not-n = pq or no-causation. But, not-m + not-n + not-p_{S} + not-q_{S} may not offhand be interpreted as no-causation, since pq remains conceivable as p_{notS}q_{notS} or relative to some other complement T. Note well that p+ not-p_{S} does not imply p_{notS} and likewise q+ not-q_{S} does not imply q_{notS}.
No. | Premises | m | n | p_{s} | q_{s} | Full conclusion | Comments | |
Figure 1 | ||||||||
None | ||||||||
Figure 2 | ||||||||
231 | np | mn | + | – | – | – | mq and not-q_{s} | Since m + not-n = mq |
221 | mq | mn | – | + | – | – | np and not-p_{s} | Since n + not-m = np |
233 | np | np | – | ? | ? | ? | not-m | Many outcomes possible |
222 | mq | mq | ? | – | ? | ? | not-n | Many outcomes possible |
224 | mq | pq | – | – | ? | ? | pq or no-causation | Since not-m + not-n |
234 | np | pq | – | – | ? | ? | pq or no-causation | Since not-m + not-n |
244 | pq | pq | – | – | ? | ? | pq or no-causation | Since not-m + not-n |
241 | pq | mn | – | – | – | – | pq but not-p_{s} + not-q_{s}or no-causation | Since if causation, thennot-m + not-n = pq |
Figure 3 | ||||||||
313 | mn | np | + | – | – | – | mq and not-q_{s} | Since m + not-n = mq |
312 | mn | mq | – | + | – | – | np and not-p_{s} | Since n + not-m = np |
324 | mq | pq | ? | – | – | – | not-n and not-p_{s} + not-q_{s} | Various outcomes possible |
334 | np | pq | – | ? | – | – | not-m and not-p_{s} + not-q_{s} | Various outcomes possible |
314 | mn | pq | – | – | – | – | pq but not-p_{s} + not-q_{s}or no-causation | Since if causation, thennot-m + not-n = pq |
344 | pq | pq | – | – | – | – | pq but not-p_{s} + not-q_{s}or no-causation | Since if causation, thennot-m + not-n = pq |
There are thus 8 moods in the second figure and 6 in the third figure with additional negative conclusions (as revealed by matricial analysis in the preceding chapter). The differences between these two figures are simply due to moods 322 and 342 being self-contradictory, as already seen.
Note in passing that the conclusions of moods 231, 313 and 221, 312 may be read as the relative to S “lone determinations” m-alone_{rel} and n-alone_{rel}, respectively; but it of course does not follow from this that absolute lone determinations exist – indeed we see here that in absolute terms the respective conclusion is mq or np. The latter imply that relative to some item other than S, be it notS or some other item T, q or p (as applicable) is true. That is of course not much information, but better than nothing.
It should be noted that none of these moods is implied by others, so that the negative conclusions implied by them are not repeated in such putative other moods. (See Diagram 7.1 and Table 7.2, in chapter 7, on reduction.) An issue nevertheless arises, as to whether the moods mentioned, above under (a) and (b), exhaust negative conclusions drawable from positive moods. The answer seems to be yes, we have covered all negative conclusions. This may be demonstrated as follows.
Suppose a mood (i.e. premises) labeled ‘A’ is found by matricial analysis to not-imply some positive conclusion ‘C’. Consider another mood ‘B’, such that A implies B. It follows that B does not imply C, since if B implied C, then A would imply C – in contradiction to what was given. But our question is: may B still formally imply notC? Well, suppose B indeed implied notC, then A would imply notC, in conflict with the subalternative result of our matricial analysis that A does not imply C. Granting that matricial analysis yields the maximum result, such conflict is unacceptable. Therefore, it is not logically conceivable that B imply notC as a rule.
We can thus remain confident that the negative conclusions of positive moods mentioned above make up an exhaustive list, provided of course that we remain conscious of the complement under discussion at all times.
In any case, we have in this way succeeded in squeezing some more information out of causative propositions occurring in syllogistic conjunctions. No moods of this sort were found in Figure 1. In Figure 2, two moods (221, 231) were already valid in the sense of yielding positive conclusions; their validity has now been reinforced with additional information; six other moods in this figure (222, 224, 233, 234, 241, 244) were previously classed as ‘invalid,’ in the sense of yielding no positive conclusions; but here they have been declared ‘valid’ with regard to certain negative conclusions. Similarly, in Figure 3, two moods (312, 313) have increased in validity, while another four (314, 324, 334, 344) have acquired some validity. So, in sum, we have four moods with reinforced validity and ten with newly acquired validity.[51]
We can derive additional valid moods from these, as we did before, by use of indirect reduction, or reduction ad absurdum. If we focus, for the purpose of illustration, on the negative conclusions not-m and/or not-n in Table 9.4, we obtain the following:
Major | Minor | Conclusion | Source |
Figure 1 from Figure 2 – keep same major | |||
pq | n and/or m | not(mn) | 241 |
mq | m | not(mn) | 221 |
np | n | not(mn) | 231 |
mq | n | not(mq) | 222 |
np | m | not(np) | 233 |
mq or np or pq | n and/or m | not(pq) | 224, 234, 244 |
Figure 1 from Figure 3 – keep same minor | |||
n and/or m | pq | not(mn) and not(pq) | 314, 344 |
n | mq | not(mn) | 312 |
m | np | not(mn) | 313 |
n | pq | not(mq) | 324 |
m | pq | not(np) | 334 |
Table 9.5 continued.
Figure 2 from Figure 3 – keep the minor as a major | |||
m | mn | not(mq) | 312 |
n | mn | not(np) | 313 |
n and/or m | mn or pq | not(pq) | 314, 344 |
n | mq | not(pq) | 324 |
m | np | not(pq) | 334 |
Figure 3 from Figure 2 – keep the major as a minor | |||
pq | n and/or m | not(mq) and not(np) | 224, 234 |
mn | m | not(mq) | 221 |
mq | n | not(mq) | 222 |
mn | n | not(np) | 231 |
np | m | not(np) | 233 |
mn or pq | n and/or m | not(pq) | 241, 244 |
We can analyze these results as follows, for examples.
With regard to the Figure 1 moods in the above table derived ad absurdum from Nos. 222 and 233, namely mq/n/not(mq) and np/m/not(np), they correspond respectively to moods 126 and 135. Until here, these moods were invalid, because we had no positive conclusions from them. But here we have found some very vague conclusions, which negate joint determinations (a relatively indefinite result, since it signifies a disjunction of possible conclusions: i.e. either the remaining joint determinations or no-causation).
The same moods in Figure 3, correspond to the moods 326 and 335. In their case, however, we had positive conclusions from them, namely m from mq/n and n from np/m. The additional negative conclusions obtained from them here, namely not(mq) and not(np), respectively, constitute further information extraction, since they are not formally implied by the previous conclusions.
Note well that p and q in these four cases mean p_{s} and q_{s}, respectively, since we are in subfigure (c). Therefore, in Figure 3, we should not go on to infer that m + not(mq) = mn, or that n + not(np) = mn, i.e. that both moods 326 and 335 yield the full conclusion mn! They only in fact yield m and n in absolute terms, the rest of the conclusions being only relative to S. It would not be reasonable to expect more determination than that, because it would mean we are getting more out of our syllogism than we put in to it, contrary to the rules of inference.
4. Imperfect Moods.
Imperfect moods[52] of causative syllogism are those involving negative items as terms. That is, instead of directly concerning P, Q, R, S, they might relate to notP, notQ, notR and/or notS. We would not expect the investigation of such negative terms to enrich us with any new formal information, but rather to unnecessarily burden us with useless repetition. All the logic of such propositions can be derived quite easily from that of propositions with positive terms. We certainly will not engage in that exercise here (although some logician may be tempted to develop this field once and for all for the record). But we need to point out a couple of interesting facets of this issue.
- a) As pointed out in a footnote in the chapter on immediate inferences, we commonly use positive forms with a negative intent, i.e. whose terms are positive on the surface but negative under it. Thus, the expression “P prevents Q” may be explicated as “P causes notQ”. Rather than work out all the logical properties of this new copula called “prevention,” we can simply reduce it to that of causation, by changing all occurrences of Q in causative logic to notQ. We could thus speak of complete or partial prevention, necessary or contingent prevention; and we could correlate such various forms with each other, in oppositions, eductions and syllogisms.
However, we could additionally correlate the forms of prevention in every which way with the forms of causation. It is in the event that we wish to do this, that the need to develop a logic of imperfect moods would arise. Such an enlarged logic would concern not only forms like “P causes Q” (causation) and “P causes notQ” (prevention), but also forms like “notP causes Q” and “notP causes notQ.” The latter two may be called inverse forms of causation and prevention, respectively.
- b) A particularly interesting negative term is when a partial or contingent causative proposition involves a negative complement. For example, the proposition “P (with complement notR) is a partial cause of Q,” involving the negative complement notR, needs to be investigated to fully comprehend the proposition “P (with complement R) is a partial cause of Q,” involving the positive complement R. Some of this work has been done in the chapter on immediate inferences.
We saw there that the ‘absolute’ proposition p_{abs} “P is a partial cause of Q” (irrespective of complement) is implied by either of those ‘relative’ propositions p_{R} or p_{notR} (that specify the complement). It follows of course that the negation of the absolute implies the negation of both the relatives. Also, p_{abs} may be true while only one of p_{R} or p_{notR} is true and the other is false. That is, the conjunctions ‘p_{abs} + not-p_{R}’ or ‘p_{abs} + not-p_{notR}’ are logically possible. Similarly with regard to contingent causation, q.
Now, what shall arouse our interest in syllogistic theory are occurrences of a negative minor or subsidiary item. As the reader may recall, in Table 5.2 we identified four ‘subfigures’ (labeled a, b, c, d) for each of the three figures of causative syllogism, according to the presence and position of a positive complement in either premise or in the conclusion. We can here identify five more subfigures (to be labeled e, f, g, h, i) for each of the three figures. These ‘imperfect’ subfigures are clarified in the table below:
Subfigures | e | f | g | h | i |
Figure 1 | QR | Q(S)R | Q(P)R | Q(notP)R | Q(notP)R |
P(S)Q | PQ | P(S)Q | P(S)Q | P(S)Q | |
P(notS)R | P(notS)R | P(notS)R | P(S)R | P(notS)R | |
Figure 2 | RQ | R(S)Q | R(P)Q | R(notP)Q | R(notP)Q |
P(S)Q | PQ | P(S)Q | P(S)Q | P(S)Q | |
P(notS)R | P(notS)R | P(notS)R | P(S)R | P(notS)R | |
Figure 3 | QR | Q(S)R | Q(P)R | Q(notP)R | Q(notP)R |
Q(S)P | QP | Q(S)P | Q(S)P | Q(S)P | |
P(notS)R | P(notS)R | P(notS)R | P(S)R | P(notS)R |
Subfigures ‘e’ and ‘f’ are the most interesting. In both, the complement in the conclusion is negative compared to its origin in one of the premises; the subsidiary term has thus changed polarity. In subfigure ‘e’, the original complement is in the minor premise; in ‘f’, it is in the major premise. Subfigures ‘g,’ ‘h,’ ‘i’ are more complicated, since they involve the minor item or its negation as complement in the major premise. This is a conceivable situation, though one we are not likely to encounter often.
The layouts described by ‘e’ and ‘f’ are relatively common in our causative reasoning, inasmuch as we often have to distinguish between absolute and relative partial or contingent causation, or their negations. To make such distinctions, and decide just how much can be inferred from given premises, we have to refer to these subfigures. Logicians are therefore called upon to develop this particular field further, although the information is already tacit in the results of the subfigures we have already dealt with.
This work will not be pursued further here, except for the following general contribution. The table below predicts how subfigures may be derived from others by direct reduction (i.e. conversion of major or minor premise), i.e. it shows the logical interrelationships between the various subfigures in the different figures. Included in this table are indications for the reduction of perfect as well as imperfect subfigures of Figures 2 and 3 to subfigures of Figure 1. In one case, we reduce a subfigure of Figure 1 to subfigures of Figures 2, 3. This table, obtained by reflection on Tables 5.2 and 9.4, can be viewed as a guide to action for a future logician who may volunteer to finish this job.
Stages of developmentof study | If mood is evaluated in subfigure | Then mood is derivable in subfigure | |
Firstly, | 2a | 1a | |
perfect | 2b | 1b | |
moods | 2c | 1f | |
2d | 1h | ||
3a | 1a | ||
3b | 1e | ||
3c | 1c | ||
3d | 1g | ||
Secondly, | 2e | 1e | |
main | 2f | 1c | |
imperfect | 2h | 1d | |
moods | 2i | 1g | |
3e | 1b | ||
3f | 1f | ||
3g | 1d | ||
3i | 1h | ||
Thirdly,remainingimperfect moods | 1i | 2g, 3h |
Chapter 10. Wrapping Up Phase One
1. Highlights of Findings.
I will stop the first phase of my research on the logic of causation at this point. Not just because I do not think it is worth going further into minutiae. I in fact do not consider that all the important formal issues have been covered. However, I do regard the logical techniques applied so far to have come close to the limits of their utility. That is why I have been developing more precise techniques, which I will publish eventually as Phase Two. Let us meanwhile review some of our main findings thus far in Phase One, and what information we are still missing.
We have succeeded in defining the various determinations of causation, by means of propositional forms already known to logic. These forms involve conjunctions (‘and’), conditionings (‘if-then’), modalities (‘possibly’, ‘actually’), and of course negations of all those (‘not’).
The mechanics of these various source forms are thoroughly treated in my work Future Logic, and need not be reviewed here. Since we already know the deductive properties of these underlying forms (how they logically interact) and how they can ultimately be induced from experience (abstraction, adduction, generalization and particularization, factorial analysis, factor selection and formula revision), these formal problems are in principle already solved for causative propositions. It is only a question of finding ways and means to extract the implicit information systematically and reliably.
I have tried to perform just this job in the preceding pages. The difficulties encountered are never such as to put the whole enterprise in doubt, note well. They are only due to the complexity of forms involved, since each positive causative is a conjunctive compound of several simpler forms, and all the more so in the case of negative propositions, which are disjunctive compounds of such simpler forms. The main problem is thus one of volume of information to be treated; there is so much data to sort out, order and organize, that we can easily get lost, forget things, make minor errors with numerous hidden repercussions.
I am only human, and may well have made some mistakes in this process. A major annoyance for me is that I am often forced to interrupt my research work due to the need to earn my living by other means. In such circumstances, my attention is diverted for long periods; my mind loses its thorough concentration on the subject matter, and I have to later re-learn it all. Hopefully, I have nevertheless succeeded in spotting and removing all eventual inconsistencies. Certainly, I have tried: always making consistency checks, painstakingly reviewing large bodies of data and long chains of reasoning, doing what I call “quality control”.
The best way to do this is to arrive at the same results using different means. That is one reason why, although the above Phase One work apparently stands up well on its own, I will not be entirely satisfied until Phase Two is complete and I arrive there at consistent results. But to return for now to our findings thus far…
It must be understood that this research has not been idle reshuffling of information and symbols. It had both practical and theoretical purposes in mind.
The practical questions relate to everyday reasoning about causes and effects. One of the principal questions we posed, you will recall, was whether the cause of the cause of something is itself a cause of that thing or not, and if it is, to whether it is so to the same degree or a lesser degree. This issue of causal (or effectual) chains is what the investigation of causal syllogism is all about. What our dispassionate research has shown is that it is absurd to expect ordinary reasoning, unaided by such patient formal reflections, to arrive at accurate results. The answer to the question about chains is resounding and crucial: the cause of a cause is not necessarily itself a cause, and if it is a cause it need not be one to the same degree. Once the scientific impact of this is understood, the importance of such research becomes evident.
But this syllogistic issue has not been the only one dealt with. We have in the process engaged in many other investigations of practical value. The definitions of the determinations causation by means of matrixes can help both laypeople and scientists to classify particular causative relations, simply by observing conjunctions of presences and absences of various items. Generalizations may occur thereafter, but they should always be checked by further empirical observation (at least, a readiness to notice; eventually, active experiment) and adjusted as new data appears (or is uncovered).
Another interesting finding has been the clarification of the relationships between positive and negative, absolute and relative causative propositions: for instance, that we may affirm partial or contingent causation, while denying it of a particular complement. One very important principle – that we have assumed in this volume, but not proved, because the proof is only possible in the later phase of research – is that (absolute) “lone determinations” are logically impossible. This means that we may in practice consider that if there is causation at all, it must be in one or the other of the four “joint” determinations.
Another finding worth highlighting is that non-causation is denial of the four genera (or four species) of causation, and before these can be definitely denied we have to go through a long process of empirical verification, observing presences and absences of items or their negations in all logically possible conjunctions. It is thus in practice as difficult to prove non-causation as to prove causation! Indeed, to be concluded the former requires a lot more careful analysis of data than the latter. Of course, in practice (as with all induction) we assume causation absent, except where it is proved present. But if we want to check the matter out closely, a more sustained effort is required.
With regard to the theoretical significance of our findings, now. By theoretical, here, I mean: relevant to philosophical discussions and debates about causality. Obviously, so far we have only treated causation, and said nothing about volition and allied cause-effect relations, so we cannot talk about causality in its broadest sense.
What our perspective makes clear is that the existence of “causation” is indubitable, once we apprehend it as a set of experiential yes or no answers to simple questions, leaving aside references to some underlying “force” or “connection” (which might be discussed as a later explanatory hypothesis). If we look upon causation in a positivistic manner, and avoid metaphysical discussions that tend to mystify, it is a simple matter. Causation is an abstraction, in response to phenomenologically evident data. It is a summary of data.
It is not purely empirical, in the sense of a concept only summarizing presences of phenomena. It involves a rational element, in that it also summarizes absences of phenomena. Affirmation may only be acknowledgment of the empirically apparent. But negation, as I have stressed in my work Phenomenology[53], is a partly rational act (a question is asked: is the thing I remember or imagine now present to my senses?), as well as a partly empirical act (the answer is no: I see or hear or otherwise sense nothing equivalent to that image!). Absence does not exist independently like presence, but signifies an empirically disappointed mental expectation.
Reading debates between philosophers (for example, David Hume’s discussions), one might get the impression that non-causation is an obvious concept, while causation needs to be defined and justified. But, as we have seen here, non-causation can only be understood and proven with reference to causation. Before we can project a world without causation, we have to first understand what we mean by causation, its different determinations, their interactions, and so forth. But the moment we do that, the existence of causation is already obvious. However, this does not mean that non-causation does not exist. Quite the contrary. Since, as we have seen, some formal processes like syllogism with premises of causation are inconclusive, we may say that the existence of causation implies that of non-causation! This finding has two aspects:
- The more immediate aspect is inferred from the fact that the cause of a cause of something is not necessarily itself a cause of it: taking any two things at random, they may or not be causatively related. This implication is valuable to contradict the Buddhist notion that “everything is caused by everything”. But the possibility of independence from some things does not exclude dependence on other Each of the two things taken at random may well have other causes and effects than each other.
- A more radical aspect is the issue of spontaneity, or no causation by anything at all. We can only touch upon this issue here, since we have only dealt with causation so far. But what our formal study of causation has made clear is that we cannot say offhand whether or not spontaneity in this sense is possible. There is no “law of causation” that spontaneity is impossible, i.e. that “everything has a cause”, as far as I can see. Nothing we have come across so far implies such a universal law; it can only be affirmed by generalization. Spontaneity (chance, the haphazard) remains conceivable.
I think the point is made: that formal research such as the present one has both practical and theoretical value. Let us now explain why the research undertaken so far is insufficient.
2. The Modes of Causation.
The observant reader will have noticed that throughout the present study we have concentrated on logical causation, i.e. on causative propositions based on logical conditioning. But of course, this is but one aspect of human aetiological reasoning. To be thorough, we need to consider not only such “de dicta” forms, but also the “de re” modes of causation, i.e. natural, temporal, extensional and spatial causation. In many ways, the latter are more interesting than the former. We have focused our attention on logical causation because it is the most widely known theoretically, although not necessarily the most widely used in practice.
Each of these modes of causation is derived on one of the modes of conditioning. A thorough study of the underlying forms of conditioning may be found in my work Future Logic (Part IV, Chapters 33-42)[54]. What is evident from that study is that natural, temporal, extensional and spatial conditioning, are in most respects similar to logical conditioning, but in significant respects different. The difference is essentially due to the fact that logical conditional propositions (like “if P then Q”) distinctively cannot be made to universally imply the “bases” (i.e. “P is possible, Q is possible”) – because if they were made to, we would not be able to express paradoxes[55]. From this structural difference, various differences in behavior (during inference) emerge.
However, this distinction dissolves in the context of causation, because here logical causation like all other types implies the bases. We have specified this fact as the last clause of each of the definitions of the determinations. Complete or partial causation implied the cause, or the conjunction of causes, and therefore the effect, to be possible; necessary and contingent causation implied them to be unnecessary. It follows that all the logical properties of the different modes of causation will be comparable. The subdivision of each mode of causation into different determinations will be the same, as will the underlying interplay of presences and absences, possibilities and impossibilities, in every conceivable combination and permutation. All the matrixes of their forms will be identical and all arguments will have the same conclusions.
The only difference between these different logics is simply that the “possibility” and “impossibility” referred to in the definitions and matrices have a different sense in each case. In logical causation, they refer to logical modalities; in natural causation, to natural modality; in extensional causation, to extensional modality; and so forth. The only task left to logicians, therefore, is to more closely examine the interrelationships between these different modes of causation. That is, for instance, how any two natural and extensional causative propositions are opposed to each other, and how they behave in combination (i.e. within arguments). This complex work will not be attempted here.
Nevertheless, I have already in Future Logic clarified the following essential relationships. Logical necessity implies but is not implied by the de re necessities. Logical possibility is implied by but does not imply the de re possibilities. Similarly on the negative side, for impossibility and unnecessity. Thus, the logical mode lies on the outer edges of rectangles of oppositions including the de re modes.
For now, let us only clarify in what context each mode is used. Logical (or de dicta) causation is concerned with causes in the literal sense of “reasons”; that is to say, it helps us to order our discourse and eventual knowledge with reference to logical implications, presuppositions, disconnections, contradictions, or consistencies, between hypotheses and/or apparent evidences. In contrast, the de re modes of causation are more directly object-oriented.
- The paradigm of natural causation is:
When the individual X actually is, has or does C (the cause),
then it (or some other individual Y) must (i.e. in all circumstances) be, have or do E (the effect);
and when C is not actual, neither is E.
In this context, C and E are qualities, properties or activities of any sort, relative to some individual entity X (or pair of individuals X, Y, respectively). Presence, here, is called “actuality” to refer us to the underlying natural modality. Necessity, here, means in all circumstances relative to this X in the antecedent. The implied basis of such propositions is that “this X can both C and E” (or “X+C and Y+E is potential for the individual(s) concerned”, as appropriate) – no need of additional clauses in that respect. The antecedent and consequent may be static or dynamic, and may or may not be temporally separated.
- The paradigm of temporal causation is very similar, save that “must” becomes “always” (all units of time) in the body of time concerned. The form is “When… at some time, then… at all times”.
- The paradigm of extensional causation is a bit different:
In such cases as class X in some instance is, has or does C (the cause),
then it (or another instance of class X or an instance of some other class Y) must (i.e. in all instances) be, have or do E (the effect);
and in such cases as C does not have an instance, neither does E.
In this context, C and E are qualities, properties or activities of any sort, relative to some class of entities X (or pair of classes X, Y, respectively). Presence, here, is called “instancing” to refer us to the underlying extensional modality. Necessity, here, means in all instances of X in the antecedent. The implied basis of such propositions is that “some X are both C and E” (or “X+C and Y+E is extensionally possible for the class(es) concerned”, as appropriate) – no need of additional clauses in that respect. The antecedent and consequent may be static or dynamic, and may or may not be temporally separated. They distinctively need not be actualities, but may be potentialities or necessities, note well, since extensional conditioning refers only to quantity.
The paradigm of spatial causation is very similar, except that “must” becomes “everywhere” (all units of space) in the body of space concerned. The form is “Where… at some place, there… at all places”.
What I want to make sure here is that the reader understands that there are different modes of causation, and that the differences between them are significant to ordinary and scientific thought or discourse.
For example, the theory of Evolution is based partly on observation or experiment on individual biological specimens (spatial, temporal and natural causation) and partly on putting together the jigsaw puzzle of scattered findings relating to a class of individuals in different times and places (extensional causation), as well as partly on theoretical insights about consistency and implications between postulates and experiences (logical causation). All these involve induction and deduction, hypothetical reasoning and generalizations, but their focal center changes.
When, for instance, we take note of the structural or even genetic similarities of all vertebrates, and presume them to have a common ancestor, we are engaged in extensional causative reasoning. We would be engaged in natural causative reasoning, only if we could trace the ascendancy from individual child to individual parent all the way back to the first vertebrate specimen. In the extensional mode, the different individuals (e.g. paleontological findings) are regarded as expressions of a single class (genus, species, variation, whatever). In the natural mode, our focus is on the life of individuals as such (irrespective of their class appurtenance).
People, and even scientists, often confuse these different ways of thinking, and remain unaware that they may lead to different conclusions, or at least nuance our conclusions considerably. For this reason, the study of the modes of causation needs to be carried out in appropriate detail.
3. Gaps and Loose Ends.
The main characteristics and limits of Phase One of our research into the logic of (logical) causation are two:
- The methodology of matricial analysis used for validation of inferences is cumbersome, bulky, manual, and therefore susceptible to human error.
- We are only able to deal systematically and exhaustively with positive causative propositions; negative causative propositions can only be treated incidentally, not directly.
For all the achievements of our research so far, these two defects leave us with an aftertaste of dissatisfaction. We have not till here succeeded in completely automating validation: human attention and intelligence are required at every step to ensure consistency and exhaustiveness. This does not prevent us from a thorough and reliable treatment of positive propositions, provided we have the requisite patience and carefulness. But the task becomes too daunting when dealing with negatives, in view of their disjunctive nature and of the sheer volume of data involved.
To overcome these handicaps, we have to greatly simplify matricial analysis, make it so digital that a computer program could operate it. This is what we shall endeavor to do in Phase Two. There we shall refer to the method of matricial analysis used in the present Phase One as macroanalysis, in contrast to the more pointed methodology of microanalysis used in Phase Two. In the latter case, we shall be able to develop a versatile logical mechanics, wherein any conjunctive, conditional or causative proposition, positive or negative, individually or in combination with any other(s), can be fully interpreted or evaluated in a matricial analysis and in ordinary language. This is no promise or vain boast: it is already largely done, needing only to be completed.
One important practical consequence of this new approach is our ability to freely handle negative causative propositions, and draw inferences from them (if they imply anything) in any arguments wherein they appear. Another is the crucial finding that absolute “lone” determinations are logically impossible; this refers, the reader will recall, to propositions involving only one positive generic determination, all three others being denied. But most importantly, it allows us to demonstrate everything demonstrable in causative logic without a drop of lingering doubt, since human error is eliminated.
Appendix 1: J. S. Mill’s Methods: A Critical Analysis.
Revised version. The present essay was originally written, or at least published, in 1999; but I decided to rewrite almost all of it in March 2005, when I found the time to engage in more detailed hermeneutics. Following an analysis that could be characterized as almost Talmudic (though not as mere ‘pilpul’), my conclusions about Mill’s methods are considerably more severe.
Preamble
Below, I list John Stuart Mill’s five “Methods of Experimental Inquiry”[56]; then I try to expose and evaluate them. It should be noted that though my approach is at times critical, my main intent is to clarify; I am more interested in Mill’s achievements, than in his apparent mistakes. (All symbols used below are mine – introduced to facilitate and clarify discussion.)
Mill’s terminology is a bit obscure, but can be interpreted with some effort.
In the paradigm (the first method), he seems to be looking out at the world, or a specific domain of it, and observing something (say, X) occurring in some things or events, in scattered places and times, and not occurring in others; and also observing some second thing (say, Y) occurring in some things or events, in scattered places and times, and not occurring in others; and he wonders at how two such events can be causally related.
In the first three methods, Mill verbally differentiates the two things under study by naming one “the phenomenon” (X, for us) and the other “the circumstance[57]” (Y, for us), suggesting that in his mind’s eye the former is the effect and the latter its cause, although note well in his conclusions he rightly (usually) considers the two items interchangeable, so that either might be the cause or effect of the other. In the fourth method, Y is viewed as a “part” of the “phenomenon” X. In the last method, Mill refers to both items with the same word, viz. “phenomenon”. Whatever the words used for X and Y, it is clear that Mill has no intent to prejudice the conclusion. These terms are intended very broadly to mean any thing or event, i.e. (since he is considering experimental inquiry) any object of perception. (I prefer the very neutral – purely logical – term “item” for this.)
Now, these items (X, Y, or their negations) are found scattered in the world, or some segment thereof, in various things or events, in scattered places and times – this is what Mill means by “instances”. Wherever X, Y, or their negations occur, that is one of the “instances” or cases under consideration. Thus, the instances might be instances of a kind of thing (e.g. humans or water)[58], and X, Y, or their negations, might be predicates (in a broad sense, including any attribute or movement or situation or quantitative property or relation or whatever) of that subject.
The goal of Mill’s present study, as its name implies, is methodological: he seeks to correlate phenomena, i.e. to identify how we can establish one thing to be a cause or effect of another, to whatever extent. Given certain facts about X and Y, what conclusions can be drawn as to their causal relation? The causal relation investigated is evidently causation (of whatever mode), rather than volition (although once volitions have taken place, their results become causatives), note.
He has in mind experimental inquiry – but in fact, his arguments could equally be applied to passive observations. His are (ideally, at least) universal inductive principles, which effectively define various causative relations, as well as offer practical guidance for their discovery.
As we shall see, Mill apparently makes numerous mistakes; and overall, his treatment of causation is not as systematic and exhaustive as it should have been. For all that, his doctrine is instructive, as is the discussions it stimulates.
1. The Joint Method of Agreement and Difference
Mill stated:
If two or more instances (A, B…) in which the phenomenon (X) occurs have only one circumstance in common (Y), while two or more instances (C, D…) in which it (X) does not occur have nothing in common save the absence of that circumstance (Y), the circumstance (Y) in which alone the two sets of instances differ, is the effect, or the cause, or an indispensable part of the cause [of the given phenomenon] (X).
Let X be the phenomenon, and A, B… be instances in which it occurs, and C, D… be instances in which it does not occur; and let Y be a circumstance the former instances (A, B…) have in common exclusively, and the latter instances (C, D…) lack in common exclusively. Then, according to Mill:
Instances A, B… have X and have Y (exclusively); and
Instances C, D… lack X and lack Y (exclusively);
Therefore: Y is the effect, or the cause (or an indispensable part of the cause), of X.
This may be considered as an inductive argument, with two compound premises and a disjunctive conclusion (i.e. a set of three possible conclusions). I have here put in brackets and in italics those parts of the premises and the conclusion that I consider mistaken, for reasons I shall presently discuss.
A simplified and corrected version of Mill’s statement would look as follows:
If a phenomenon (X) is invariably accompanied by another (Y), and its absence (not-X) is invariably accompanied by the other’s absence (not-Y) –– we may infer that X is the cause of Y, or Y is the cause of X, in the sense of complete and necessary causation.
This simple statement is an apt description of the strongest causative relation possible between two items X and Y or between their negations. It corresponds to what David Hume earlier called “constant conjunction”, between two phenomena and between their negations. That this is for Mill the essence of the method under consideration is evident in the name he gave it: “agreement and difference”. This name also shows his awareness that causation has both a positive and a negative aspect.
Had Mill contented himself with such a simple statement, I would have congratulated him for providing scientists with an excellent research tool. I do not therefore quite know why Mill chose to complicate the matter by adding an extraneous condition in each premise and proposing an inaccurate alternative conclusion. Before we consider these problems, however, let me further analyze the intent of Mill’s main statement.
The terms “agreement” and “difference” in the title of this method refer respectively to having in common or lacking in common some feature (namely the “circumstance” Y). The expression “joint” method here is due to these terms recurring separately in the next two methods.
When Mill refers to “two or more instances” in each premise, he must in fact be referring to the two or more instances, i.e. all (the known) instances. Clearly, this must be the case, otherwise it would be conceivable that we encountered instances of X without Y, or instances of not-X without not-Y; and if that were the case, the proposed strong conclusion would not be valid. Mill ought to have added the definite article “the” to avoid all misunderstanding.
Mill’s use of the expression “two or more” is due to his trying to say several (too many) things at once. First, that one instance is hardly sufficient to establish causation; there must be repetition of the conjunctions. Second, the number of repetitions is indefinite, because we are here (except when dealing with finite sets) concerned with open-ended induction. We can never know all the instances directly, but can only arrive at general premises through generalization from all known cases to all cases period. The conclusion is only as valid as those generalizations.
The form “If X, then Y, and if not X, then not Y” (= “Y is the effect of X”) and its contraposite “if Y, then X, and if not Y, then not X” (= “Y is the cause X”) are both generalizations from the forms “X and Y are universally conjoined, and not-X and not-Y are universally conjoined”. If, upon further inquiry, the latter generalities turn out to be inaccurate, the inferences drawn from them must also be attenuated.
Mill should have specified all that explicitly (I do not know if he did so somewhere else). But there is little doubt in my mind that he tacitly intended it. He might also have pointed out that the “two sets of instances” involved (here symbolized as A, B… and C, D…) once generalized, together exhaustively cover the whole world.
Another implicit detail worth highlighting is that “X is contingent and Y is contingent”. This is inferable from the observation and mention of occurrences of X and of not-X, and likewise regarding Y and not-Y. Note also that although Mill speaks of “the phenomenon” or “the circumstance” – the predicates X and Y are general terms, and not one-time happenstances, since each occurs in “two or more instances”.
Finally, looking at Mill’s conclusion, we may add that his uncertainty as to which of the two items X and Y causes the other (at least in our main conclusion) is justified. Since the relationships described in the premises are symmetrical with regard to X and Y (apart from purely verbal differences), the conclusion cannot differentiate between them. At this level, then, the words “cause” and “effect” have no formal difference; some other condition (such as time’s arrow or the degree of abstraction) must be specified before we can identify a direction of causation.
Now, let us turn to criticism of Mill’s formula.
Mill’s first inexplicable complication is his requirement that the “circumstance in common” (viz. Y or not-Y) in the premises be exclusive. In the first premise, he says Y is the “only one”; and in the second, there is “nothing save” not-Y. Moreover, these circumstances must “alone” differentiate the two sets of instances, for the conclusion to follow.
Mill apparently fears that some third item, say Z, might come into play and affect the projected strong relation between X and Y. However, this fear is formally unjustified. Let us consider the extreme case where three items X, Y, Z are constantly conjoined, and their negations not-X, not-Y, not-Z likewise always occur in tandem. In such a situation, all the following propositions (and their respective contraposites) are true:
If X then Y, and if not X then not Y.
If X then Z, and if not X then not Z.
If Y then Z, and if not Y then not Z.
The truth of the latter two propositions does not impinge upon the truth of the first one. The causative relation between X and Y remains the same, even if some third factor like Z comes into play. The same can be argued if only one of these extra propositions is true. In such situations, we would simply conclude that there are parallel causations, or again causative chains.
Mill apparently failed to develop these concepts, and inserted an extraneous requirement of exclusivity in a vague attempt to insure against possible third-factor interference. In truth, the relation between any two variables X and Y can be determined without reference to any other variables.
If – as indeed does occur – the two variables under consideration are affected by others, to the extent that their relation is weaker than here concluded, we will soon notice the fact by observing that X is not always with Y, and/or that not-X is not always with not-Y. But in such case, the stated premise(s) about constant conjunction will simply not be true! In other words, in such case, Mill’s conception of the premise(s) would be self-contradictory.
Perhaps, someone might interject, Mill was here trying to account for the scientific methodology of “keeping all other things equal”? No – because: this refers to a situation where there are two or more partial causes to an effect, and to establish each of the partial causes as such, we have to consider each one in turn without the other – and in such case, complete causation could not be a putative conclusion for any of the partial causes.
The second inexplicable complication in Mill’s formula is his reference in the conclusion to a third alternative, viz. that Y might be “an indispensable part of the cause” of X. This clause is interesting, first of all, because it indicates that when Mill initially states that Y might be “the effect, or the cause” of X, he has in mind complete causation (as distinct from the partial causation in the third alternative).
With regard to this third alternative, let us first notice that Mill does not mention that X might equally be “an indispensable part of the cause of” Y, even though he has granted that X and Y are interchangeable in the first two alternatives. Why this asymmetry? I suspect it is not intended to convey some radical insight, but merely reflects Mill’s terminology and the gradual development of his formula.
He started by referring to Y as a “circumstance”, suggesting that he viewed it as the precondition or cause of X, “the phenomenon” under investigation. Then, it probably occurred to him that he could not formally distinguish between X and Y, as to which is the cause and which is the effect – so he added the possibility that Y might be the effect of X. Then, he got to thinking Y could be a partial (necessary) cause of X, so he added that in; but he simply forgot to recover symmetry and suggest the reverse to be possible.
Now, the big issue: the phrase “an indispensable part of the cause” clearly refers to partial necessary causation. Given that X and Y are indeed constantly conjoined and that their negations are constantly conjoined, no conclusion is formally permissible other than complete necessary causation. It follows that it was an error for Mill to insert this additional disjunct in his conclusion.
Note parenthetically, Mill does not anywhere give us a clue as to how partial necessary causation might be distinguished from complete necessary causation. Supposing such alternative conclusion had been correct, he would have been obliged to a detail practical methodology for resolving the issue.
I suspect that Mill resorted to the said third alternative conclusion due to his lingering doubt concerning some possible third factor (which we above labeled Z) weakening the relation between X and Y. Apparently, Mill considered that Z might diminish the degree of causation of X by Y from complete to partial; i.e. he viewed Z as a complementary partial cause imbedded with Y in some larger cause.
This explanation is appealing, because it suggests a correlation between the said complications in premises and conclusion. However, as already shown, Z might equally well be a parallel or concatenated complete cause – so we must still fault Mill for imprecision and confusion. In any case, logically, Mill could not have his cake and eat it too. If in the premises he has firmly excluded circumstances besides Y, there is no reason for him to make allowance in his conclusion for an eventual complement Z!
Another objection we could raise here is: if Mill considered the possibility here of partial necessary causation, why not equally that of complete contingent causation, or for that matter, the possibility of partial contingent causation? If he felt (perhaps because of their inductive basis) his premises were shaky, then why did he not foresee all possible modifications of the main conclusion (complete necessary causation)?
The answer to the latter question(s) is simply that although Mill conceived of partial causation, he apparently never grasped the inverse concept of contingent causation. This will become evident as we continue our analysis of his methods, and find no mention anywhere of that weak alternative to necessary causation. Mill’s omission suggests that, in his mind, only “indispensable” things could be causatives (although if asked the question he might well have denied it).
Another deficiency in Mill’s viewpoint is his failure to consider that in some cases, though X and Y and their negations exhibit perfect regularities of conjunction as described in the premises, we (i.e. people in general) do not conclude that Y causes X or X causes Y, but conclude that “X and Y are both effects of some third thing”. This alternative conclusion is admittedly inexplicable formally, just as the distinction between cause and effect is difficult to pinpoint. But there may in practice be indices that encourage the former, just as there are indices for the latter. Granting this, it would have been more appropriate for Mill to use that clause as his third alternative.
To sum up: what is manifest from all our above analysis is that Mill had an unclear idea of causation, mixing its paradigm up with its possible variations. He failed to first clearly distinguish and separately consider all the determinations of causation (both generically and specifically). Consequently, when he faced the inductive issue – the issue of how in practice to identify causation – his confusion was compounded by the need to consider the fact of generalization and the possibility of particularization.
2. The Method of Agreement
Mill stated:
If two or more instances (A, B…) of the phenomenon (X)… have only one circumstance (Y) in common, the circumstance (Y) in which alone all the instances agree is the cause (or effect) of the given phenomenon (X).
Let X be the phenomenon, and A, B… be instances in which it occurs; and let Y be the only circumstance they have in common. Then, according to Mill:
Instances A, B… have X and have Y (exclusively); and
Therefore: Y is the cause, or the effect, of X.
This may be considered as an inductive argument, with a compound premise and a disjunctive conclusion (i.e. a set of two possible conclusions). In view of the name given to this method, the conclusion may be taken to refer to the positive aspect of causation, i.e. complete causation. I have here put in brackets and in italics the ‘exclusive’ demand of the premise, which I consider mistaken for reasons to be presently given.
The essence of this argument is generalization, from the constant conjunction of two items, X and Y, wherever and whenever they are observed to occur (the instances A, B…), to all existing or possible instances. The conclusion from such universal repetition is either that “if Y, then X” (whence, Y completely causes X) or that “if X, then Y” (whence, X completely causes Y).
Such generalization is logically possible, note well, provided that the “two or more instances” (A, B…) are all the encountered instances of X and of Y. Mill obviously intended that, but he should have made it clear – e.g. by saying the two or more – to preempt his formula being construed as allowing for unspecified instances in which X occurs without Y or Y occurs without X.
Mill should have mentioned this to show his awareness of the formalities involved, notably that the form “if X, then Y” means “X is impossible without Y” (and similarly, “if Y, then X” means “Y is impossible without X”). The most significant aspect (for a causative conclusion) of the constant conjunction of the two items is the implied denial of possible conjunction between one item and the negation of the other.
We could offer a generous reading Mill’s statement to cover this issue. We could suppose that Mill confused circumstances other than Y with circumstances contrary to Y, and suggest that the clause “only one circumstance (Y) in common” is intended to mean that there are no instances with X accompanied by some negation of Y. Likewise, the exclusive word “alone” could be taken to refer to X rather than Y, meaning that the two or more instances involving X, are the only ones among “all the instances” to have Y, implying that there are no instances without X that have Y. However, I do not seriously think Mill intended this interpretation.
Another tacit proviso for drawing our conclusion is that each of the items X and Y be contingent. Strictly speaking, a conditional proposition like “if X, then Y” or “if Y, then X” can be taken to imply causation only if we know that “X is possible, but unnecessary” and “Y is possible, but unnecessary”[59]. In Mill’s statement, here (unlike in the joint method), the occurrence of X and Y is implied in the premise, but their non-occurrence is not mentioned. This omission is noteworthy, suggesting that Mill was not fully aware of these requirements for validity.
It should be said, too, that once the contingency of the theses is granted, a hypothetical proposition could be contraposited. That is, “if X, then Y” would imply “if not Y, then not X”; similarly, “if Y, then X” would imply “if not X, then not Y”. Thus, although the intent of Mill’s formula (judging by its title) was an inference of complete causation, strictly speaking his formula allows for one of necessary causation. That is, the valid conclusion from his premise is a disjunction of four possible conclusions.
Thus, Mill’s formula leaves us uncertain, not only as to which item is the cause and which is the effect (as he admits), but also as to whether we are dealing with complete or necessary causation (which he fails to notice). One thing is sure, however, is that the conclusion is a strong determination. This is tacitly suggested by Mill in his use of the definite article “the” in “the cause” or in “the effect”. If he had had in mind weak determination (i.e. partial or contingent causation), he would have probably written “a cause” and “an effect”.
This brings us to Mill’s requirement that the instances where the phenomenon (X) have “only one” circumstance (Y) in common, which he repeats when we says that the latter is “alone” that in which the instances agree. Why such exclusiveness? We have seen a similar, mystifying concern in Mill’s joint method. In the present case, again, Mill seems worried that there may be circumstances other than Y that will weaken the causative relation between Y and X; i.e. he is trying to preempt any possibility of partial (or contingent) causation.
In his mind’s eye, apparently, if some other circumstance (say, Z) was also (like Y) constantly conjoined with the phenomenon (X), a doubt would arise as to which of the two circumstances, Y or Z, caused X. But this is formally unjustified: the possible truth of “if Y, then X” would not be affected by the eventual truth of any other proposition like “if Z, then X”; if X, Y and Z are compatible, as our premise confirms, the two hypotheticals are quite compatible. Mill here again has apparently not considered the possibility of parallel causations or causative chains.
We might add that Mill’s attempt to limit the number of accompanying circumstances to just ‘one’ is ontologically open to doubt. Are there anywhere in the world two or more things (instances in which X occurs) having literally only ‘one’ circumstance (Y) in common? I very much doubt it! If there is such a set of things, it must be very exceptional. Most things have many (innumerable) common factors. There are always large predicates like existence, location in space and time, size, shape, etc. to consider, for a start.
Usually, when we say something so exclusive, we do not really mean it. For example, saying “the only similarity between these two individuals is their wealth” – we do not really mean to imply that the individuals do not both have a spinal cord, a heart, a brain, etc. Such misleading language is not accurate in scientific statements; at least, we should think twice before ever using it or taking it literally.
3. The Method of Difference
Mill stated:
If an instance (A) in which the phenomenon (X)… occurs, and an instance (B) in which it (X) does not occur, have every circumstance in common save one (Y)… [, that circumstance] (Y) is the effect, or the cause, or an indispensable part of the cause… [of the given phenomenon] (X).
Let X be the phenomenon, and A be an instance in which it occurs and B be an instance in which it does not occur; and let Y be the only circumstance they do not have in common. Then, according to Mill:
Instance A has X and has Y; and
Instance B lacks X and lacks Y and
(Instances A and B, have every other circumstance in common;)
Therefore: Y is the effect, or the cause (or an indispensable part of the cause), of X.
This was intended as an inductive argument, with two compound premises and a disjunctive conclusion (i.e. a set of three possible conclusions). As we shall demonstrate below, this argument is a rather gauche depiction of necessary causation. I have here put in brackets and in italics those parts of the premises (here treated as a third premise) and the conclusion that I consider mistaken, for reasons I shall presently discuss.
It should first be noted that Mill’s formulation does not make clear whether the presence of X is accompanied by the presence or absence of Y, and inversely what the absence of X is accompanied by. I have assumed symmetry, i.e. presence with presence, and absence with absence, in order that the conclusion be expressed wholly in positive terms. It is not a very important issue, but still a puzzling imprecision on Mill’s part.
Next, let us notice that Mill’s formula mentions only one instance (A) of X’s occurrence (presumably with Y) and only one instance (B) of X’s (and Y’s) non-occurrence – without this time in any way suggesting plurality, let alone universality. Mill’s wording as it stands does not exclude the possibility of some third instance where X occurs with not-Y, and of some fourth instance where not-X occurs with Y. In such cases, how would Mill dare claim a causative relation?
This is very intriguing[60]: I find it hard to suppose that Mill considers that causation can be induced from single instances. One may from single occurrences deny that some causation is applicable, but one could in nowise affirm it. In order for the premises to allow the conclusion he proposes, we would have to replace “an instance” with “all (known) instances” in at least one of the premises. Causation is about patterns of conjunction, not about coincidences. Mere occasional agreement or difference does not establish a pattern.
One wonders what Mill possibly had in mind! (I suspect he had eaten or drunk too much the day he wrote this.)
Perhaps Mill considered the constancy of surrounding circumstances as the requisite pattern, somehow. Why does he at all refer to the two instances (A and B) having “every circumstance in common”[61] save one? This is a redundancy: the very uniformity of surrounding circumstances makes them irrelevant. In any case, uniformity in only two instances is hardly significant.
I presume, here again, he imagined that if the surrounding circumstances had not been uniform, they would have somehow impinged on the causative relation between X and Y. For this reason, he insists on their distinctive uniformity whether X or not-X is the case. He is apparently not aware of the possibility of parallel causations or of causative chains.
In any case, there are always innumerable surrounding circumstances, behaving in quite random fashion, that are totally unconnected with the phenomena at hand; non-uniformity is not proof of causation. And moreover, the circumstances that are here uniform (in the instances A, B) might behave more erratically in other instances.
Is the exceptive (“save one”) clause in Mill’s formula, i.e. the contrasting behavior of Y, his main focus, perhaps? The given fact that one circumstance (Y) differs from all other circumstances in that it is uncommon, i.e. present in one instance (say, A) but absent in the other (say, B), just makes Y stand out from the rest; it does not signify a causative relation to X. This is all the more true when, as here, only a couple of instances are under consideration.
But finally, it occurs to me that there is one way we can at least in part redeem Mill’s statement. That is by supposing that, when he here referred to “an instance” he subconsciously had in mind “a kind of instance”! In that case, A and B are each a set of instances, corresponding respectively to the occurrences of X (with Y) and those of not-X (with not-Y). From these (experimentally) encountered instances, we may by generalization assume the same regularities hold universally.
Granting this supposition, and ignoring the extraneous mention of uniform surrounding conditions and insistence that Y be the only non-uniform circumstance, a causative relation between X and Y can indeed be inferred. However, in such case the premises and conclusion of this method would seem identical to those of the joint method! This is obviously not Mill’s intention.
Considering the title of the ‘method of difference’, we can safely suppose that it refers to something found in part in the ‘joint method of agreement and difference’ and not found in the ‘method of agreement’. Mill was apparently struggling to split necessary complete causation (the ‘joint method’) into its two components, complete causation (agreement) and necessary causation (difference). He managed to formulate the former, positive aspect readily enough, but had considerable trouble putting his finger on the latter, negative aspect.
A further confirmation of our supposition is to be found by comparison of the conclusions of the three methods. Note first that whereas the method of agreement concludes that Y is “the cause (or effect) of” X, the other two methods conclude in reverse order that Y is “the effect, or the cause… of” X. Moreover, the joint method and the method of difference, distinctively from the method of agreement, propose as an alternative conclusion that Y might be “an indispensable part of the cause” of X.
This latter possibility obviously refers to partial necessary causation, as earlier pointed out. “Indispensable” means that one cannot do without it, it is a sine qua non, a necessity; and “part of the cause” means a fraction of the sufficient cause. All this suggests that, in Mill’s mind, the causation found by the method of agreement is essentially positive and whole, whereas that found in the other two ways may be negative and fractional.
But since, as already said, the joint method and the method of difference cannot be identical, the latter must be assumed to focus on necessary causation only. We should, by combination of the methods of agreement and difference, arrive at the same result as with the joint method. So, our task is to isolate the ‘difference’ component (necessary causation) from the ‘agreement’ component (complete causation).
Mill might have achieved this by proposing some sort of negative ‘mirror image’ of his formula for the method of agreement, one about “two or more instances (A, B) in which the phenomenon (X) does not occur” having “the absence of one circumstance (Y) in common”. Some such more analogous statement could be constructed for the method of difference, but I will not even try, because of all the difficulties in the earlier statements already discussed.
Moreover, if we attempt such a reconstruction, we soon realize the title “method of difference” to be a misnomer, in view of the use of the term “agree” within Mill’s formula for the method of agreement. His method of difference is really just another application of the method of agreement, except that we focus in it on the absences, instead of presences, of the items (X, Y) concerned. “Difference” (i.e. disagreement) can only really be claimed in the joint method, where we switch from presence to absence or vice-versa. In this perspective, the titles ‘method of agreement of positives’ and ‘method of agreement of negatives’ might be more appropriate.
Whatever the name used for it, and the language used to formulate it, it is evident for reasons of symmetry that the method of difference aims at the negative aspect of causation, i.e. necessary causation. It follows that the premise(s) must be such that by generalization we can ideally conclude that “if not-X, then not-Y” or “if not-Y, then not-X”. This would in practice be based on observed constant conjunction between not-X and not-Y. The matter is that simple!
Mill realizes this at some level, but goes quite astray in his attempt to put it in words. His statement of the method of difference is incredibly garbled. He not only repeats some of the mistakes he made in formulating the preceding two methods, but also makes many more.
Before leaving this topic, it should be added that the said constant conjunction of negations only formally implies causation after generalization if the terms concerned are known contingent, i.e. if X is possible and Y is possible. Moreover, given such contingency, the inferred conditional propositions can be contraposited to “if Y, then X” and “if X, then Y”; so that strictly speaking, the conclusion formally allows for complete causation as well necessary causation (whether of X by Y, or of Y by X).
Observed constant conjunction of negations does not, however, formally allow as alternative conclusion partial necessary causation – or for that matter, complete contingent causation or partial contingent causation. Mill’s proposition that Y may be “an indispensable part of the cause” of X is artificial and erroneous. Needless to say, reversing its direction would also be erroneous, as would inverting the polarities of the terms. Anyway, as already pointed out, Mill apparently completely misses out on the possibility of contingent causation.
I have already discussed the issue of partial causation with regard to the joint method, and will not repeat my comments here. These are commendable attempts by Mill to insert it in his analyses, but his approach so far is unequal to the task. He makes arbitrary claims in his conclusions, which are incompatible with his premises; and even supposing consistency, he provides no means to decide between his alternative conclusions. He does, however, offer some more precise means for identifying partial causes in his next method, that of ‘residues’.
4. The Method of Residues
Mill stated:
Subduct from any phenomenon (F) such part (D) as is known by previous inductions to be the effect of certain antecedents (A), and the residue (E) of the phenomenon (F) is the effect of the remaining antecedents (B).
Here, Mill is attempting to deal with partial causation. He is saying:
Suppose: D is a part of F; and E is the rest of F (i.e. D + E = F).
And suppose: A causes D (i.e. presumably, If A, then D, etc.)
It follows that: B causes E (i.e. presumably, If B, then E, etc.)
Note that a tacit assumption, here (suggested by the reference in the conclusion to “remaining” antecedents), which we can readily grant, is that A and B together (as C, say) cause F (the compound of D and E), i.e. that:
(A + B) = C; and C causes F (i.e. presumably, If C, then F, etc.)
Note also that I presume that the kind of causation by A of D, and by B of E, intended by Mill, is complete causation[62], i.e. a relation including positive implication by the cause of the effect (i.e. if the cause, then the effect), plus strictly speaking a negation of the inverse implication (i.e. if not the cause, not-then not the effect).
The causations mentioned and tacit in Mill’s statement are considered as already established, as he admits by saying “as is known by previous inductions”. The means of induction used is not specified; he presumably intends one of the other four ‘methods’ (probably the second). His formula is only intended to infer a causation from within other, given causations. This is a purely deductive argument.
Moreover, Mill appeals to the relation between whole and parts without really defining it. We could briefly express that relation by saying that D and E together imply and are implied by F. But to fully clarify this relation, we ought to mention that D without E or E without D, as well as not-D + not-E, amount to not-F. Similarly, with regard to A + B versus C.
Mill’s process of “subduction” is thus essentially based on the following reasoning:
If A+B (= C), then D+E (= F) – call this the major premise.
But: If A, then D – call this the minor premise.
Therefore, If B, then E – the putative conclusion.
This argument is, I hasten to add, formally invalid, although a common error of inference! This can be seen by splitting the major premise into the two hypotheticals:
If A + B, then D
If A + B, then E
Clearly, the minor premise “if A, then D” overrides the first proposition, “if A + B, then D”, which has the same consequent, showing the component “B” of the antecedent to be extraneous. However, the second proposition, “if A + B, then E”, whose consequent is different, is unaffected by the minor premise; i.e. its antecedent remains compound. We can, if we wish, “nest” this eduction, putting our result in the form “if A, then if B, then D”. But this inference still leaves “A” conditional.
Whence, Mill’s putative conclusion “if B, then E” is pretentious. The only way we could draw it would be to confirm “A” to be categorically true. It does not suffice to mention the element “A” conditionally, as in “if A, then D”. Thus, Mill’s present account of partial causation is not strictly correct.
Partial causation can readily be defined and in practice identified, but the appropriate formula for it is a bit more complicated than Mill suggests. It requires a more radical understanding and more systematic treatment of causation. There is no need to go into it here, since I treat it in detail in my main text on the subject.
The notion of a “residue” (or remainder or leftover) is a mathematical one, rooted in the relation of whole and part: if you have a basket with three fruits and you remove one, you still have two left. A similar idea can be used in causation – but only to say:
If one of the partial causes is found to be present, then we can anticipate that as soon as the remaining partial causes are also found to be present, all their collective effects will follow on their heels.
Mill’s ‘method of residues’ subconsciously appeals to this obvious truth. But he confuses the issue, when he considers that things (like D) that the present phenomenon (A) causes by itself (i.e. things it alone suffices to bring about) can be counted as among the collective effects (like E) of all the causal phenomena under consideration (A and B). This is his essential error. Here again (as with his previous attempts to infer partial causations), his premises and conclusion are not consistent with each other.
Note finally that Mill’s language is positive, suggesting that he had in mind specifically partial causation. Here again, as in the preceding methods, he does not apparently consider the other form of weak causation, that involving negative theses, viz. contingent causation.
Moreover, even supposing that Mill had successfully identified partial causatives, he does not here specify that such causes might be necessary or contingent. Perhaps, having spoken (although out of place) about necessary partial causation in the joint method and the method of difference (mentioning “an indispensable part of the cause”), he might be supposed here to be focusing on contingent partial causation. But this would be reading into Mill’s treatment something he has given no sign he has awareness of.
One more point worth adding, concerning the appeal to “residues” in reasoning about causes. There is indeed a method that can be so named, one commonly used by scientists and ordinary thinkers. This method was known to Francis Bacon already, long before Mill. It consists simply of disjunctive apodosis – i.e. the gradual elimination of alternative hypotheses. Such reasoning has the form:
Either P or Q or R or… is the cause of S;
these (P, Q, R,…) are all the conceivable causes of S.
The cause of S is not …; and it is not R; and it is not Q.
Therefore, the cause of S must be P (i.e. the only remaining alternative).
For example, Sherlock Holmes might say: “the culprit is either Jack or Jill; it can’t be Jack, since he has an alibi; therefore, it has to be Jill.”
5. The Method of Concomitant Variations
Mill stated:
Whatever phenomenon (X) varies in any manner whenever another phenomenon (Y) varies in some particular manner, (X) is either a cause or an effect of that phenomenon (Y), or is connected with it through some fact of causation.
Let X be “whatever phenomenon”, and Y be “another phenomenon”; let X1, X2, X3… be variants of X, and Y1, Y2, Y3… be corresponding variants of Y. Then:
- Whenever Y varies from Y1 to Y2, X varies from X1 to X2;
- Whenever Y varies from Y2 to Y3, X varies from X2 to X3;
- ;
- therefore, X is “either a cause or an effect of” Y, “or is connected with it through some fact of causation”.
Notice Mill’s use of “whenever”: he is correctly referring to unvarying relations, not mere random coincidences. That is, we may suppose he was implying that if the variations of the two phenomena (the kinds of events we labeled X and Y) are not concomitant, they may be assumed independent of each other.
Mill does not explicitly tell us what degree of causation may be inferred – whether complete and necessary, or only the one or the other, or neither. He is seemingly open to all possibilities, since he vaguely mentions that “some fact of causation” may in some cases be the best conclusion we can draw. Granting this phrase refers to the weaker determinations, we may suppose that when he refers to “a cause or an effect” he means a stronger determination. However, since he here uses the indefinite article “a”, instead of his usual definite article “the”, this supposition is debatable. In sum, Mill concludes some sort of causation to be inferable, but is vague as to which sort and when.
Whereas in the first three methods, changes from presence to absence or vice versa are concerned – in concomitant variations, every incremental change in measure or degree of the cause is accompanied by a corresponding incremental change in the measure or degree of the effect; and/or vice-versa. In some cases, the correspondences between two phenomena are in this way very regular; but in other cases, additional phenomena have to be taken into consideration to clarify the more complex relationship involved.
In any case, the fact of concomitant variation may be considered an ontological derivative of that of causation, dealing with quantitative instead of merely qualitative relationships between two or more phenomena. That is, in Mill’s terms, this fifth method is a corollary, or frequent further development, of the preceding four.
In the case of ‘agreement’ (interpreted as complete causation), we would expect changes in the cause to be invariably followed by concomitant variations in the effect. In the case of ‘difference’ (interpreted as necessary causation), we would expect changes in the effect indicative of predictable concomitant variations in the cause. In the ‘joint’ case (i.e. the strongest possible causative relation), both these directions of inference would be applicable.
Note that these alternatives are not made clear in Mill’s formula, where X’s variations follow Y’s variations, yet X is concluded to be “either a cause or an effect” of Y. Given regular variation of X with Y, the more probable conclusion would be that Y is a complete cause of X; although a second possible conclusion would be that X is a necessary cause of Y. Mill presumably does not mention the reverse case, where Y varies with X, simply because he considers that in such case we would just place each term in the other’s position in his formula. Fair enough, but then he should at least have mentioned in his formula that in some cases variations are concomitant in one direction only, and in others in both directions!
In the last case (‘residues’ – interpreted as partial and/or contingent causation), we would have to use more a elaborate technique: to identify and monitor all the factors involved, and observe how and how much each varies with whatever changes occur, or are experimentally produced, in the other factors. This is generally achieved using the cunning method of “keeping all other things equal” while investigating just two factors at a time, until all the factors have successively been played off against each other and we obtain a full picture of their multilateral quantitative relationship.
In my view, Mill should have mentioned all that explicitly in his formula. It is reasonable to assume he knew it, since the method was oft used in scientific experiments in his day. Why didn’t he, then? Let us go on, anyway, and analyze these matters a bit more.
We can theoretically express concomitant variations by means of series of causal propositions, either through statements mentioning changes (as above initially done) or more radically through statements mentioning states, like:
If A=A1, then B=B1;
if A=A2, then B=B2;
etc.
However, often in practice, these innumerable, point-by-point correlations between various quantities are plotted on a graph and then summarized in a mathematical equation. For example, if B is directly proportional to A, we would write (where k is some constant):
B = kA.
Actually, we have to be careful in this matter, because such a mathematical equation implies/presupposes fully convertible relations. Thus, the following would also have to be true:
If A=A1, then B=B1; if B=B1, then A=A1; where B1=kA1.
If A=A2, then B=B2; if B=B2, then A=A2; where B2=kA2.
etc.
This does not have to imply the causation involved to be reversible, only that A be a complete and necessary cause of B. Thus, while in common language we can readily express concomitant variation between a merely complete cause and its effect (or conceivably between a merely necessary cause and its effect) – in the language of mathematical equations, necessary as well as complete causation is implied (although, I believe, modern mathematics can readily overcome this difficulty).
Note well that if we just say “If A=A1, then B=B1”, it does not exclude that for another value of A (say, A8), B may have the same value (B1) again. Such reiterations of value will translate mathematically into more complex formulas than mere proportionality.
More complex relationships may, but do not in all cases, signify partial and/or contingent causation, involving more than two items (at least two causes and one effect). Thus, note well, Mill’s statement of this method need not be limited to two variables; he presumably had this in mind when he wrote the alternative conclusion “or is connected with it through some fact of causation”.
Note, finally: the idea of comparing variations between two or more variables was proposed long before Mill, by Francis Bacon.
Concluding Remarks
John Stuart Mill (1806-73) was an English philosopher, a highly educated man whose interests ranged very widely, including all aspects of logic. He published the work in which he presents the above ‘methods of experimental inquiry’, A System of Logic, when he was 37. He sought for a pragmatic, empiricist, inductive approach to knowledge; an updated logic, but one that would “supplement and not supersede” Aristotle’s.
Mill’s five methods have generally been well received, and I acknowledge them as having been an inspiration to me. However, as the above analysis shows, though his intentions were laudable, his performance was often woefully inadequate. I take no pleasure in saying this; but I am somewhat consoled by the knowledge that others have before me also sharply criticized him.
If these methods had been developed before the dawn of modern science – say before the publication of Isaac Newton’s Principia (1687) – I would have congratulated their author for having provided researchers with potentially valuable cognitive tools. But Mill’s work is dated 1843 – almost the mid-19^{th} Century!
At that late date in modern science and philosophy, one could no longer discover these research tools, but one could at least give an ex post facto exposé and validation of them. Mill’s effort in that direction was, in the last analysis, surprisingly confused, considering his broad knowledge of science and philosophy till his day.
As we have seen, Mill’s methods could just as well be characterized as techniques ‘for identifying causation’, because that is the form of their conclusions; and also, because experimental data is not essential to them, i.e. they can be applied as well to passive observations. His method of residues, unlike the others, is deductive rather than inductive. Whether this list of methods, without regard to its internal imperfections, constitutes an exhaustive summary of actual scientific techniques is open to debate.
What is clear, anyhow, is that Mill did not fully understand the relations of causation. Flaws are evident in his treatment of each of his five methods. Briefly put:
- In the ‘joint method’, he seemingly tries and succeeds defining or identifying the paradigm of causation, complete necessary causation. However, his understanding is put in doubt by his mention of irrelevant conditions (exclusiveness of the circumstances) in the premises, and his drawing of an alternative conclusion (“an indispensable part of the cause”) logically contrary to the given premises.
- In the ‘agreement method’, he seemingly tries and succeeds defining or identifying complete causation. However, his understanding is put in doubt by his mention of irrelevant conditions (exclusiveness) in the premises, and his failure to specify the unnecessity of the theses (as needed to infer causation from constant conjunction).
- In the ‘difference method’, he seemingly tries but quite fails defining or identifying necessary causation. His understanding is put in doubt by his appeal to single instances (instead of kinds), his mention of extraneous conditions (the uniformity of surrounding circumstances), and his drawing of an alternative conclusion (“an indispensable part of the cause”) logically contrary to the seemingly intended premises.
- In the ‘residues method’, he seemingly tries to deduce a partial causation from two complete causations. His understanding is here again put in doubt, by his proposing conflicting premises (the same thing cannot be both a complete and a partial cause of a given phenomenon), and his suggesting an excessive conclusion (i.e. more than the givens allow).
- In the ‘concomitant variations method’, he seemingly tries and vaguely succeeds defining or identifying the quantitative aspect of causation. This is logically his soundest method, but he fails to mention and distinguish the various degrees of causation that may be involved.
Mill obviously had difficulty with the concept of plurality of causes; i.e. distinguishing between parallelism and composition of causes. The inclusion of redundancies concerning surrounding circumstances in some of his statements indicates that he did not have an entirely accurate picture of causation. His resort to seemingly last minute inserts at the tail end of certain conclusions leads to the same suspicion. Moreover, in none of the five methods does he so much as hint he has heard of contingent causation.
Mill’s first four methods may be taken to essentially refer to the causative forms mn, m, n, and p, respectively. The first and third methods mention the specific determination np, but give us no clue as to how such causation might be established, i.e. concluded rather than mn or n, respectively. Since he apparently ignores the generic determination q, he misses the specific determinations mq and pq. His treatment is thus neither symmetrical nor exhaustive.
I should also point out that Mill does not clearly distinguish between generic and specific determinations. I assume he does not intend the generic determinations that he separately as well as jointly affirms (namely: m, n) as absolute lone determinations; but the issue is not to my knowledge explicitly raised by him so we cannot be sure what he imagined.
As we have seen, Mill’s formulations are open to further criticism. His language is often ambiguous and its intent difficult to fathom. He did not always manage to capture in words what he was trying to say. His logic is in places dubious, if not downright self-contradictory. He may propose mutually incoherent premises and/ conclusions that contradict explicit premises.
The main reason for the weaknesses in Mill’s treatment is perhaps his attempt to deal with definition and induction simultaneously. He would have been more successful if he had, more systematically, first defined the various forms of causation (ratio essendi) and then investigated how their contents may be induced (ratio cognoscendi). Perhaps due to his association with the Utilitarian school of philosophy, he was ideologically inclined towards a rather heuristic approach, eschewing a more theoretical treatment of causation.
The logician’s main task is to describe and validate forms of reasoning. While Mill took some pains to describe causal arguments, he made little effort to validate them. At times, his treatment seems like a sham – not out of malice, but due to negligence. He does not seem to intentionally lie (as some do); but one gets the impression he has not really done his best to do a good job, and he does not expect anyone to notice or care.
The logician’s role is also to provide methodological aids for scientists, students, and indeed thinking people in general. Whether Mill’s contributions to causal logic ever actually affected anyone’s investigation of nature in a positive or negative way is hard to say. Nevertheless, some of his thoughts on the subject were misleading, and the fact should be made public.
This is all very disappointing, considering J. S. Mill’s status in British intellectual history. How could a man of his social standing and educational caliber have made such mistakes, and moreover gotten away with them, one wonders.
After all, causation and its varieties were pretty well known to the ancients; this is even evident in commonly used Latin legal terms, like causa sufficiens or sine qua non. And Mill was very well read in ancient thought; he was brought up with it by his father, James.
Major British philosophers had already discussed causality at considerable length. John Locke (1632-1704), in An Essay Concerning Human Understanding (1690), put forward a theory of induction based on regularities of sequence between phenomena. David Hume (1711-1776), for all his avowed skepticism in An Enquiry Concerning Human Understanding (1748), had clearly expounded constant conjunction. Mill’s views about causation were frankly influenced by Hume’s.
Most shocking, is the realization that Mill’s logical treatise (1843) was published 238 years after the founding father of British Empiricism, Francis Bacon (1561-1626), published his Novum Organon (1605). Mill was aware of Bacon’s work, too, since he (rightly) criticized Bacon’s view of causation as simplistic in various respects. But he manifestly failed to notice and learn the important lessons taught by this unsung (or insufficiently sung) hero of the modern scientific method; namely, Bacon’s programme of adduction and matricial analysis (to use my terminology).
Suffices to quote the Encyclopaedia Britannica (2004) description of Bacon’s “new method” for this failure of Mill’s to be clear:
The crucial point, Bacon realized, is that induction must work by elimination not, as it does in common life and the defective scientific tradition, by simple enumeration. Thus he stressed “the greater force of the negative instance”—the fact that while “all A are B” is only very weakly confirmed by “this A is B,” it is shown conclusively to be false by “this A is not B.” He devised tables, or formal devices for the presentation of singular pieces of evidence, in order to facilitate the rapid discovery of false generalizations. What survives this eliminative screening, Bacon assumes, may be taken to be true.
Bacon presents tables of presence, of absence, and of degree. Tables of presence contain a collection of cases in which one specified property is found. They are then compared to each other to see what other properties are always present. Any property not present in just one case in such a collection cannot be a necessary condition of the property being investigated. Second, there are tables of absence, which list cases that are as alike as possible to the cases in the tables of presence except for the property under investigation. Any property that is found in the second case cannot be a sufficient condition of the original property. Finally, in tables of degree proportionate variations of two properties are compared to see if the proportion is maintained.
Avi Sion
The Logic of Causation:
Phase Two:
Microanalysis
Phase II: Microanalysis. Seeing various difficulties encountered in the first phase, and the fact that some issues were left unresolved in it, a more precise method is developed in the second phase, capable of systematically answering most outstanding questions. This improved matricial analysis (microanalysis) is based on tabular prediction of all logically conceivable combinations and permutations of conjunctions between two or more items and their negations (grand matrices). Each such possible combination is called a ‘modus’ and is assigned a permanent number within the framework concerned (for 2, 3, or more items). This allows us to identify each distinct (causative or other, positive or negative) propositional form with a number of alternative moduses.
This technique greatly facilitates all work with causative and related forms, allowing us to systematically consider their eductions, oppositions, and syllogistic combinations. In fact, it constitutes a most radical approach not only to causative propositions and their derivatives, but perhaps more importantly to their constituent conditional propositions. Moreover, it is not limited to logical conditioning and causation, but is equally applicable to other modes of modality, including extensional, natural, temporal and spatial conditioning and causation. From the results obtained, we are able to settle with formal certainty most of the historically controversial issues relating to causation.
Chapter 11. Piecemeal Microanalysis.
1. Binary Coding and Unraveling.
We have developed a theory of causative propositions and arguments (eductions and syllogisms) by means of an analysis of the possibilities and impossibilities implied for the various combinations of the items concerned. This was characterized as ‘matricial analysis’, because of our recourse to tables for assessing and recording results.
But thus far we have only really engaged in elementary matricial analysis, which may be called macroanalysis. We shall now introduce a more advanced approach, which may be called microanalysis. They are not different methods. Microanalysis is based on macroanalysis; it is merely a more detailed examination, digging deeper into the issues concerned, in an attempt to solve outstanding problems.
As we have seen, the determinations of causation are best expressed through a matrix, a table composed of ‘items’ and ‘moduses’. The items are the terms or theses related by the causative proposition concerned. Each conceivable conjunction of these items, in positive or negative form, defines a row of the matrix. The modus for each such conjunction is a statement regarding its logical possibility or impossibility, or ‘openness’ (the latter in cases where the conjunction is in some unspecified contexts possible and in others impossible, so that an uncertainty remains). The moduses for the various conjunctions of items together constitute an additional column of the matrix.[63]
If we array the items of a matrix in a conventional arrangement (presenting the same row always in the same place), then the modus columns of all matrices will be comparable. By such standardization, we can express a determination of causation by merely writing down a string of moduses (i.e. its modus column), which we may call the modus of the determination concerned as a whole, or (for reasons we shall see presently) its summary modus.
To simplify things, we may revert to binary codes. We may express the presence or absence of each item in the matrix by a 1 or 0 notation. Similarly, we may code the modus for each row by a 1, 0 or · (dot – meaning blank). The zeros or ones have different meanings in the items and modus cells of the matrix, note well:
Binary codes:
In the items columns: | 1 = present | 0 = absent | |
In the modus column(s): | 1 = possible | 0 = impossible | · = open |
Such notation is merely convenient abbreviation, allowing us to express the summary modus of any determination as a relatively short string of digits and see the whole matrix in one sweep of the eyes. It is also, obviously, useful for computer programming purposes. Of course, if we are dealing with two items (say, P, R), the modus string will have 2^{2} = 4 digits; if with three items (say, P, Q, R), it will have 2^{3} = 8 digits; and so forth. Whether the string of digits is distinctive for each determination, we shall look into further on[64].
As we said, the rows of a matrix are defined and (conventionally) located by combinations of items. Thus, for two items, P and R, the four possible PR sequences are 11, 10, 01, 00, which may be labeled a, b, c, d if need be. We may choose this order of combinations as our standard arrangement (any other permutation is equally conceivable, but we conventionally settle on this one[65]). Similarly, for three items, P, Q and R, there are eight possible PQR sequences, which may be labeled a-h if need be. And so forth, for more items.
We may thus, to begin with, present the matrices of the generic determinations of causation as in the following tables. These include (in the first two or three columns) the items in positive (1) or negative (0) forms, arrayed in standard combinations; followed by the summary modus for each propositional form (shaded column, symbol S), which you will recall we developed at the beginning of our research (in Phase 1, chapter 2) by analyzing the meaning of each of its constituent clauses and assessing the result of their interactions.
New columns are then introduced, which present all the conceivable realizations of the summary modus. These realizations, called alternative moduses, are obtained simply by substituting, successively, a 0 (for ‘impossible’) or 1 (for ‘possible’) for each dot (‘open’ position) encountered in the summary modus, so that no dots are leftover. This process can be called unraveling. The alternative moduses thus make explicit all cases inherent in the summary modus; and conversely, the latter is a summary of all the information contained in the former.
Note that the alternative moduses are themselves, ultimately, summaries, too. For while a zero (for impossibility) signifies that the combination of items concerned is in every context or always absent, a one (for possibility) signifies that it is in some contexts or sometimes present[66]. Thus, to remove all implicit modality, and consider only actualities, we would have to dissect each such modus into an unspecifiable number of actualizations, where ‘0’ means absent and ‘1’ means present, simply. However, such further analysis is not needed for our purposes; the moduses as above defined are sufficiently informative.[67]
Consideration of a summary modus constitutes macroanalysis; that of alternative moduses, microanalysis. That is all the difference between these two methods of matricial analysis: one of degree of detail. In the former, we have a rough idea of the relations involved; in the latter, it is as if we scrutinize them under a microscope.
The similar strings of zeros and ones used by computer programmers to code letters of the alphabet and symbols (I am thinking of ASCII codes) were arbitrary, pure conventions. But here, note well, once the meanings of zeros and ones, and the order of their presentation, are decided, there is nothing conventional about the string for each determination; it is a logical property of it, objectively given information.
2. The Generic Determinations.
In the four tables below, the precise significance of the numbers heading the columns of alternative moduses will be made clear in the next chapter; for now, just consider them as arbitrary labels. It should be stressed at the outset that these modus numbers are not to be confused with the determination numbers or mood numbers used in earlier chapters. Note also that Tables 11.1 and 11.2 concern two items (P, R), whereas Tables 11.3 and 11.4 concern three items (P, Q, R)[68]; the summary moduses of these two sets are therefore not directly comparable, the former being within a ‘two-item framework’, the latter within a ‘three-item framework’.
The two-item modus of complete causation of form PR (symbolized by m, or more precisely m_{PR}) was previously established to be “10.1”. This is, through the following table, worked out to have two conceivable realizations, namely “1001” or “1011” (labeled respectively Nos. 10, 12).
Items | S | 2 alternative moduses | ||
P | R | m | 10 | 12 |
1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 0 |
0 | 1 | · | 0 | 1 |
0 | 0 | 1 | 1 | 1 |
In contrast, the two-item modus, summarily put, of necessary causation of form PR (symbolized by n, or most precisely n_{PR}) was previously established to be “1.01”. This is, through the following table, worked out to have two conceivable realizations, namely “1001” or “1101” (labeled respectively Nos. 10, 14).
Items | S | 2 alternative moduses | ||
P | R | n | 10 | 14 |
1 | 1 | 1 | 1 | 1 |
1 | 0 | · | 0 | 1 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 1 | 1 | 1 |
The three-item summary modus of relative partial causation of form P(Q)R (symbolized, according to context, by p or p_{rel}, or p_{Q} or most precisely p_{PQR}) was previously established to be “10.1.1..” . This is, through the following table, worked out to have sixteen conceivable realizations, as shown below (labeled respectively Nos. 149-152, 157-160, 181-184, 189-192). Note well that this is true relative to complement Q; we shall consider absolute partial causation further on.
Items | S | 16 alternative moduses | |||||||||||||||||
P | (Q) | R | p | 149 | 150 | 151 | 152 | 157 | 158 | 159 | 160 | 181 | 182 | 183 | 184 | 189 | 190 | 191 | 192 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | · | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | · | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | · | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | · | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
The three-item summary modus (column S) of contingent causation of form P(Q)R (symbolized, according to context, by q or q_{rel}, or q_{Q} or most precisely q_{PQR}) was previously established to be “..1.1.01”. This is, through the following table, worked out to have sixteen conceivable realizations, as shown below (labeled respectively Nos. 42, 46, 58, 62, 106, 110, 122, 126, 170, 174, 186, 190, 234, 238, 250, 254). Note well that this is true relative to complement Q; we shall consider absolute contingent causation further on.
Items | S | 16 alternative moduses | |||||||||||||||||
P | (Q) | R | q | 42 | 46 | 58 | 62 | 106 | 110 | 122 | 126 | 170 | 174 | 186 | 190 | 234 | 238 | 250 | 254 |
1 | 1 | 1 | · | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | · | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | · | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | · | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
As above stated, we shall presently look into the summary moduses of absolute partial and contingent causation. As we shall see, they are much wider than those relative to a given complement, dealt with above. This is natural, since absolute weak causations are vaguer forms than relative weak causations.
Comparing the summary moduses of complete and necessary causation with two identical items, 10.1 and 1.01, we see more clearly in what sense they are ‘mirror images’ of each other: the strings are identical, viewing one from left to right and the other from right to left. Similarly for the summary moduses of partial and contingent causation with three identical items, 10.1.1.. and ..1.1.01.
It should also be noted that a weak cause and its complement have the same summary modus. That is, partial causation of forms P(Q)R and Q(P)R have the same 10.1.1.. summary; and contingent causation of forms P(Q)R and Q(P)R have the same ..1.1.01 summary. This was obvious from the original definitions of these determinations, in which P and Q had the same relations to each other and to R; the distinction of one or the other of P, Q as complement was purely one of convenience or focus.[69]
Another observation we can make at this stage is that a generic determination and its (appropriate) converse would have one and the same summary modus.
That is true for the strong and absolute weak[70] determinations, which all concern two items. For instance, “P is a complete cause of R” and “R is a necessary cause of P” (note the change of determination, as well as that of item positions) are both here described by the string “10.1”. This can be ascertained by reading Table 11.2 in a different order, starting with the first row, then the third, then the second, then the fourth.
That is also true for the weak determinations, which involve three items. For instance, “P (complemented by Q) is a partial cause of R” is convertible to “R (complemented by notQ) is a contingent cause of P” (note well the change of complement polarity, as well as of determination and item positions) are both here described by the string “10.1.1..”. Again, we can prove this by rereading Table 11.4 in a different order, starting with the third row, then the seventh, then the first, then the fifth, then the fourth, then the eighth, then the second, then the sixth.
Indeed, we can say that convertibility is to be explained by such identity of moduses. Clearly, it follows that we cannot express direction of causation by reference to summary moduses. The orientations “from P to R” and “from R to P” must have some meaning – they are not empty verbal distinctions – but that meaning is not apparent in the way of a difference between moduses. It has to be sought in other properties, as already argued.
3. Contraction and Expansion.
Now, the above account does not allow us to compare the moduses of the strong determinations with those of the weak ones, nor tell us how to distinguish absolute from relative weak determinations. To enable such comparisons, we need to develop two processes: (a) contraction of a three-item modus into a two-item modus, and (b) expansion of a two-item modus into a three-item modus….
Let us first consider contraction of the three-item moduses of p or q (in their relative forms). Take first the case of partial causation by P of R, with reference to Table 11.3, above.
- The conjunction (P + Q + R) is possible, since the first row is always coded 1, whereas (P + notQ + R) is open, since the third row is sometimes coded 0 and sometimes 1. Nevertheless, it follows that the conjunction (P + R) is possible (i.e. to be coded 1), since “(P + Q + R) is possible” implies that “(P + R) is possible”. Note that if we regarded (P + R) as merely open, we would fail to record that there is no column with 0s in both the first and third cells.
Note this well: it is a finding we altogether missed in macroanalysis, and which may therefore affect some of our results.
- The same reasoning applies for the conjunctions (P + notR), comprising the second and fourth rows, and (notP + notR), comprising the sixth and eighth rows. They are both possible conjunctions, and not merely open.
- On the other hand, the conjunction (notP + R), comprising the fifth and seventh rows, must be declared open (i.e. be coded ), since it is conceivable for both (notP + Q + R) and (notP + notQ + R) to be found impossible (as in the columns numbered 149, 150, 181, 182).
In this way the three-item modus for relative partial causation “10.1.1..” becomes the two-item modus “11.1”. Similarly with contingent causation: its three-item modus “..1.1.01” becomes the two-item modus “1.11”.
Thus, in case of need, we can contract a three-item modus into a two-item one, by changing a combination of 1 and 0, or 1 and ·, in corresponding locations, into a 1. Also, a combination of two dots yields one dot. Note well the rule of contraction:
- Where there is a 1 in the 3-item modus, there must be a 1 in the 2-item modus.
Additionally note, though we have not yet encountered cases:
- The only way we could obtain a 0 in a two-item modus, from a three-item modus, would be to find only 0s along both rows of the latter.
- If we find cases of ‘11’,’10’ and/or ‘01’ mixed with cases of ‘00’ in the three-item modus, we must conclude a dot () in the two-item modus.
Now, what have we found here? We started with weak causations by P of R, relative to some complement Q specifically, and ended with weak causations by P or R, without specification of Q, i.e. absolutely. The three-item modus for p or q relative to Q could not be equated to the same relative to some other complement, say Q_{1}; their matrices are superficially similar, but the items involved (namely PQR and PQ_{1}R) are quite different. But if we contract both kinds to two-item moduses, they would be indistinguishable, since the items involved (namely PR) are exactly identical.
Thus, the two-item modus of absolute (which includes relative) partial causation of form PR (symbolized, according to context, by p or p_{abs}, or most precisely p_{PR}) is by contraction found to be “11.1”. This is, through the following table, worked out to have two conceivable realizations, namely “1101” or “1111” (labeled respectively Nos. 14, 16).
Items | S | 2 alternative moduses | ||
P | R | p | 14 | 16 |
1 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 1 | 1 |
0 | 1 | · | 0 | 1 |
0 | 0 | 1 | 1 | 1 |
In contrast, the two-item modus, summarily put, of absolute (which includes relative) contingent causation of form PR (symbolized, according to context, by q or q_{abs}, or most precisely q_{PR}) is by contraction found to be “1.11”. This is, through the following table, worked out to have two conceivable realizations, namely “1011” or “1111” (labeled respectively Nos. 12, 16).
Items | S | 2 alternative moduses | ||
P | R | q | 12 | 16 |
1 | 1 | 1 | 1 | 1 |
1 | 0 | · | 0 | 1 |
0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 1 | 1 |
Notice the similarity between the summary moduses of m (10.1) and p_{abs} (11.1), or those of n (1.01) and q_{abs} (1.11). Where for the strong determination we have a ‘0’ code, in the corresponding absolute weak determination we have a ‘1’; the remaining codes being identical. This, as we shall see in a later chapter[71], allows us to define the absolute weak determinations in formal terms.
Also note that the absolute weak determinations are convertible, just like the strong ones (as we pointed out in the previous section). For instance, “P is a partial cause of R” converts to “R is a contingent cause of P” (note the change in determination, as well as that of item positions), since these two forms have the same summary modus “11.1”.
The next question to ask is: what are the three-item moduses of m or n, or of p or q in their absolute forms? We can answer this question by means of expansion, as follows.
Consider, to begin with, the strong determinations. In the case of complete causation by P of R, the following can be said:
- Knowing the conjunction (P + notR) is impossible, it follows that both (P + Q + notR) and (P + notQ + notR) are impossible conjunctions (whence the initial modus 0 becomes two moduses 0).
- Whereas, since the conjunction (P + R) is possible, it follows only that at least one of (P + Q + R) and (P + notQ + R) is a possible conjunction (i.e. they cannot both be impossible) – but we cannot predict which one is possible, so both conjunctions must be declared open (whence, the initial modus 1 becomes two moduses ). Similarly, mutatis mutandis, for (notP + notR).
- Lastly, since the conjunction (notP + R) is open, so will a fortiori its two derivatives be (i.e. the initial modus becomes two moduses ·).
In this way the two-item modus “10.1” for complete causation becomes the three-item modus “.0.0….”. Similarly with necessary causation: its two-item modus “1.01” becomes the three-item modus “….0.0.”. Notice the loss of information occasioned by the change in each case, due to the fact that ones become dots; the results of such expansions are vaguer than their sources.
Thus, in case of need, we can expand a two-item modus into a three-item one, by changing a zero into two zeros in the appropriate locations, and a one or dot into two dots as appropriate. Here, note well the ‘appropriate locations’ are not adjacent rows: they are the first and third, the second and fourth, the fifth and sixth, etc., reflecting a correspondence in combination of items – such as (P + notR) becoming (P + Q + notR) or (P + notQ + notR), which means moving from a PR sequence 10 to the PQR sequences 110 and 100, which signify the second and fourth rows of the matrix.
However, note well the restrictions implied in the following rules of expansion:
- Where there is a 0 in the 2-item modus, there must be two 0s in the 3-item modus.
- Where there is a 1 in the 2-item modus, there cannot be two 0s in the 3-item modus.
- Where there is a dot in the 2-item modus, there might be any combinations of 0s and/or 1s in the 3-item modus.
With regard to ‘zero’ moduses (impossibility), they are universalized as it were from the initial row to the corresponding expanded rows. With regard to ‘ones’ (possibility), what is universalized from the single initial row to the two subsumed rows is the interdiction of zeros: just as in the two-item modus 0 is excluded by 1, so the three-item expansion cannot include columns (moduses) having 0s in both the corresponding rows. A fortiori, in the case of ‘dots’ (which might include zeros or ones), we cannot predict combinations in the two cells concerned, since all pairs are allowed, i.e. 0 and 0, 0 and 1, 1 and 0, 1 and 1.
Consider now the weak determinations, in accord with the rules of expansion just ascertained. If we similarly expand the two-item modus of absolute (or relative) partial causation, namely 11.1, into a three-item modus, we obtain “……..”, since all ones or zeros become dots. Likewise, by expansion of the two-item modus of absolute (or relative) contingent causation, namely 1.11, into a three-item modus, we obtain “……..”.
Note well that the result in both these cases is a string of dots, signifying complete uncertainty, the least possible amount of information. Each initial one or dot was expanded into two dots, so that all remaining specificity in the initial string was dissolved in its derivatives.
Note also the marked difference between the three-item strings of the absolute weak determinations “……..”, and those of the corresponding relative forms, namely “10.1.1..” and “..1.1.01” respectively.
Clearly, for all four generic determinations, expansion of a two-item modus “1” (possible) into two three-item moduses “·“ (open) results in a loss of data; i.e. the information that ‘at least one of the two conjunctions concerned must be possible: i.e. they cannot both be impossible’ is no longer coded in our table. A calculus of causation should be so designed as to avoid all loss of information due to mere linguistic inadequacies[72]. Thus, we have to find a way to express, through a special code in the modus, say µ (Gk. letter mu), that at least one of the two (or more) conjunctions so coded is implicitly a “1”.[73]
Complete causation | = µ0µ0.µ.µ |
Necessary causation | = µ.µ.0µ0µ |
Partial causation (absolute) | = µµµµ.µ.µ |
Contingent causation (absolute) | = µ.µ.µµµµ |
This measure by itself is not enough; to save all available information, we would have to specify the rows concerned, say by labeling them a-h. For instance, if at least one of rows ‘a’ and ‘c’ has to have modus 1, each would have to be coded more specifically as µ_{ac}. Such coding means that the PQR sequences signified by the labels a and c (namely, 111 and 101) may have moduses with 0 and 1, 1 and 0, 1 and 1 – but they may not have the pair 0 and 0. It follows that:
- if a = µ_{ac} and c = 0, then a = 1 (since 00 is inconceivable), and
- if a = µ_{ac} and c = 1, then a = (since 01 and 11 are both conceivable);
and likewise, of course, if c = µ_{ac} and a = 0 then c = 1, and if c = µ_{ac} and a = 1 then c = ·.
All this information can be considered implicit in a table like the following (the relative weak determinations of form PQR are included for comparison):
Row | Items | m | n | p_{abs} | q_{abs} | p_{rel} | q_{rel} | ||
label | P | Q | R | PR | PR | PR | PR | PQR | PQR |
a | 1 | 1 | 1 | µ_{ac} | µ_{ac} | µ_{ac} | µ_{ac} | 1 | · |
b | 1 | 1 | 0 | 0 | · | µ_{bd} | · | 0 | · |
c | 1 | 0 | 1 | µ_{ac} | µ_{ac} | µ_{ac} | µ_{ac} | · | 1 |
d | 1 | 0 | 0 | 0 | · | µ_{bd} | · | 1 | · |
e | 0 | 1 | 1 | · | 0 | · | µ_{eg} | · | 1 |
f | 0 | 1 | 0 | µ_{fh} | µ_{fh} | µ_{fh} | µ_{fh} | 1 | · |
g | 0 | 0 | 1 | · | 0 | · | µ_{eg} | · | 0 |
h | 0 | 0 | 0 | µ_{fh} | µ_{fh} | µ_{fh} | µ_{fh} | · | 1 |
All this may seem pretty complicated, but as we shall see it simplifies a lot of things. Through the summary moduses in the above table, we can identify precisely the alternative moduses in a three-item framework implied by each of the four determinations (for the items PR or PQR).
As we shall see in the next chapter, the strong determinations m and n turn out to have 36 such alternative moduses each, while the weak determinations p and q in absolute form (as here), have 108 alternative moduses each (to compare to the 16 moduses of relative weaks). We shall list these moduses in the next chapter, so no need to do so here; they are easy to unravel by substituting zeros and ones for dots as previously explained.
4. Intersection, Nullification and Merger.
We shall now consider certain inferences from the above data.
The joining of generic determinations can be considered as the intersection of their respective summary moduses. By such conjunction of two propositions (or more), two classes (generic determinations) are used to express a more restrictive class (a joint determination), with whatever they have in common.
By this process, in a two-item framework (where the weak determinations are absolute), the joint determinations are found to have the following summary moduses:
- complete-necessary causation, mn = 10.1 + 1.01 = 1001 (modus No. 10);
- complete-contingent causation, mq = 10.1 + 1.11 = 1011 (modus No. 12);
- necessary-partial causation, np = 1.01 + 11.1 = 1101 (modus No. 14);
- partial-contingent causation, pq = 11.1 + 1.11 = 1111 (modus No. 16).
As can be seen, the result in each case is a single alternative modus (mentioned in brackets), which represents what the joined generics have in common. Thus, for instance, m has moduses 10 and 12, and n has moduses 10 and 14; therefore mn (meaning m + n) will have modus 10. The resulting summary modus is more defined than its sources, i.e. there are less dots, there are less uncertainties in the relation between the items.
This operation is merely an application of the well-known rule of class logic, that the logical product of two classes (such as m and n, each of which subsumes two subclasses, namely 10 and 12 for m and 10 and 14 for n) is the elements they have in common (namely, modus 10, in the case of mn). This can be seen for example in an Euler diagram, comprising two circles which overlap: their common area is the outcome of their product, and usually smaller than the circles (in our example, modus 10).
Note that by ‘logical product’ logicians mean that the two (or more) classes are conjoined together (i.e. mn means m + n)[74]. It must be stressed that modus lists are disjunctive not conjunctive, so that underlying this formula is another one (namely mn = ‘modus 10 or modus 12’ and ‘modus 10 or modus 14’, which means ‘in any event, modus 10’, i.e. it refers to the leftover after removing from consideration the elements ‘modus 12’ and ‘modus 14’, which are exclusive in either disjunct.
Similarly, in a three-item framework (where the weak determinations may be absolute or relative), intersection of the generic determinations yields the joint determinations, with the following summary moduses:
- complete-necessary causation:
mn | = µ0µ0.µ.µ + µ.µ.0µ0µ | = µ0µ00µ0µ (9 alternative moduses) |
- complete-contingent causation,
mq_{abs} | = µ0µ0.µ.µ + µ.µ.µµµµ | = µ0µ0µµµµ (27 alternative moduses) |
mq_{rel} | = µ0µ0.µ.µ + ..1.1.01 | = .0101.01 (4 alternative moduses) |
- necessary-partial causation:
np_{abs} | = µ.µ.0µ0µ + µµµµ.µ.µ | = µµµµ0µ0µ (27 alternative moduses) |
np_{rel} | = µ.µ.0µ0µ + 10.1.1.. | = 10.1010. (4 alternative moduses) |
- partial-contingent causation:
p_{abs}q_{abs} | = µµµµ.µ.µ + µ.µ.µµµµ | = µµµµµµµµ (81 alternative moduses) |
p_{rel}q_{rel} | = 10.1.1.. + ..1.1.01 | = 10111101 (1 alternative modus) |
Here again, the result signifies the alternative moduses that the joined generics have in common; we shall not list them at this stage: the list will be given in the next chapter. In the case of p_{rel}q_{rel}, exceptionally, the result is a fully specifying summary modus, i.e. a single alternative modus (that labeled #190, as we shall see later). The resulting summary modus fuses together the most definite elements of the initial summary moduses; some dots become µ’s, and some dots or µ’s become more specifically a 0 or a 1. The µ’s concerned are in pairs like µ_{ac} remember; the subscripts are not mentioned here for brevity.
Some of these results correspond to those obtained by macroanalysis, note. To grasp the rules of intersection, let us review the examples shown above:
- The summary moduses are never conflicting in a given position (1 in one case and 0 in the other); this simply means that the determinations joined are compatible.
- For each position identical in both generic summary moduses, or more definite (µ or 1 or 0) in one and indefinite (· or µ) in the other, the resulting corresponding position in the joint summary modus has that equal or more definite value.
We cannot join two determinations whose summary moduses have conflicting elements in the same position (a 0 in one and a 1 in the other): they are incompatible propositions, it is an impossible conjunction. In alternative modus terms, it means that these determinations do not have even one modus in common; in class logic terms, it means that the given classes (generic determinations) do not overlap: they have no intersection. Such logically empty concepts are known as null classes; we might therefore refer to the act of judging a class to be null as nullification.
For instances, the conjunctions mp, nq are null classes. Since m has moduses 10, 12 and p has moduses 14, 16, they have no common ground, no modus in which to coexist. Similarly for n and q, mutatis mutandis. This we know already from macroanalysis. More interesting, is the capacity nullification gives us to judge the feasibility of lone determinations, as we shall see in the next chapter.
Let us now consider another logical composition, that of merger, which disjoins two (or more) propositions, to obtain a single, vaguer proposition. In alternative modus terms, this process puts together all the alternative moduses listed for the given propositions in a larger list for the merged proposition. In class logic terms, this means that the two (or more) classes together become a single class covering all the areas they have exclusively as well as those they have in common.
This corresponds to the ‘logical sum’ of classes, where the two initial classes merge into a larger class by inclusive disjunction (expressed by operator or, which here means ‘and/or’, i.e. ‘not both not’; this is often symbolized by a ‘v’, or in some computer languages by a ‘½‘)[75]. Inclusive disjunction means that all the elements subsumed by the given classes are to be included in the larger class; if a subclass subsumes x elements and another involves y elements, then the larger class covers (x + y) elements. In contrast, in conjunction, only the elements subsumed by all the given classes are selected, forming a narrower class.[76]
We can, for instance, merge joint determinations into generics; thus, “mn v mq” is equivalent to just “m”, “mn v np” becomes “n”, “np v pq” becomes “p”, and “mq v pq” results in “q”. We can likewise merge generic determinations into broader concepts, such as strong or weak causation or causation, as shown below. Merger is easy if we work directly with alternative moduses; but it becomes very complicated if we refer to summary moduses, due to the inadequacies of the ad hoc notation system we have used so far.
In a two-item framework, it is feasible if we introduce an additional symbol, say l (Gk. letter lambda), signifying that the two positions in the formula where it occurs cannot both be coded ‘1’ (in contrast to µ, which signifies that they cannot both be ‘0’, remember). In such case, we can predict the summary moduses of the following vague propositions (s, w, c) on the basis of the generics merged in them:
- strong causation, symbol s = m or n = 10.1 v 1.01 = 1ll1 (moduses Nos. 10, 12, 14)[77];
- absolute weak causation, symbol w_{abs} = p_{abs} or q_{abs} = 11.1 v 1.11 = 1µµ1 (moduses Nos. 12, 14, 16);
- relative weak causation, symbol w_{rel} = p_{rel} or q_{rel} = same two-item summary modus as for absolute weak causation;
- causation, symbol c = m or n or p_{abs} or q_{abs} = 10.1 v 1.01 v 11.1 v 1.11 = 1..1 (moduses Nos. 10, 12, 14, 16).
Note that ‘causation’ here means some causation, causation of any determination whatever, whether m, n, p_{abs} or q_{abs}. As we will show in the next chapter, ‘contributory causation’ (m or p) and ‘possible causation’ (n or q) are different from it only with reference to relatives; in absolute terms, they are identical to each other and to causation (because m implies not-p_{abs}, and n implies not-q_{abs}).
The same four operations in a three-item system all apparently yield one and the same conclusion, namely “µ.µ..µ.µ” (try and see) – which is the summary modus of causation, covering 144 alternative moduses, as we shall see. This is of course an absurd result, because, as we shall see in the next chapter, strong causation in fact covers 63 moduses; absolute weak causation, 135 moduses; and relative weak causation, 31 moduses! It follows that our notation system is inadequate for merger operations other than:
c = µ0µ0.µ.µ v µ.µ.0µ0µ v µµµµ.µ.µ v µ.µ.µµµµ = µ.µ..µ.µ (144 moduses).
What this means is that the symbolic language developed so far is too simple to express more complex relations than those intended by a 0, 1, · or µ (or even l, just introduced to enable merger of the two-item summary moduses of strong determinations[78]). It does not generate a distinctive summary modus for each and every form. However, I will not bother to attempt improving on it, not wishing to get bogged down in inessential matters. For our primary goal here is not to develop a calculus of summary moduses, but to ascertain how generic propositions can be merged into vaguer forms. And this we can readily do with reference to the underlying alternative moduses, which is good enough.
5. Negation.
Before moving on, let us review the ground covered thus far. We started with binary coding of the summary moduses of the generic determinations known to us thanks to macroanalyses performed at the very start of our research into causative propositions. We saw that these summary moduses involved uncertainties (coded ·). To eliminate these information gaps, we had to unravel the summary moduses, that is, identify the underlying alternative moduses (involving 0 or 1 codes exclusively). We thus introduced microanalysis.
However, the strong determinations m, n were expressed in a two-item (PR) framework, while the relative weaks p_{rel}, q_{rel} were expressed in a three-item (PQR) framework – so these two sets of forms were not comparable. We therefore had to work out the means for contraction and expansion of their summary moduses (the latter process required that we introduce a fourth code, µ). This also allowed us to ascertain the two- and three- item summary moduses of absolute weak determinations p_{abs}, q_{abs} – first by contracting those of the relative weaks p_{rel}, q_{rel}, then by expanding these results.
Having thus obtained both the two- and three- item summary moduses of all six generic determinations, we had all the information we need to work out the matrices of all derivative propositions. Indeed, by means of intersection we can readily identify the alternative moduses of any conjunction of determinations: they are the alternative moduses the latter have in common. A special case of this is nullification: if the propositions we wish to conjoin have no alternative moduses in common, they are incompatible. And by means of merger we can readily identify the alternative moduses of any disjunction of determinations: they are the alternative moduses the latter have all taken together.
We thus dispose of the basic data and logical processes we need for microanalysis of all positive forms, be they generic, joint (i.e. narrower than the generics) or vague (i.e. broader than the generics). But we still lack the alternative moduses of negative forms of whatever breadth. We cannot obtain their summary moduses by macroanalysis, as we did for the generic positive forms, because of the underlying complexity of negative causative propositions. So we must look for more profound means.
Thus far, we have engaged in microanalysis that may be characterized as piecemeal. In the next chapter, we shall approach this topic with a more holistic perspective, which we may refer to as systematic microanalysis. That consists in considering all conceivable alternative moduses in a given framework (fixed by the number of items under consideration), and then locating the determination(s) under consideration within this full range of possibilities.
The alternative moduses of negative forms become easy to identify thereby. Having the list of all conceivable alternative moduses in a given framework, and the alternative moduses of a positive form, we can readily infer those of the corresponding negative form: they are all the remaining alternative moduses! This process, which we shall simply call negation[79], is akin to subtraction. If a class subsumes x elements and a subclass of it involves y elements, then the remaining area covers (x – y) elements.
Microanalysis thus ultimately enables us to distinctively define any and every causative proposition (and other, related forms, as we shall see), with little effort. Furthermore, such detailed matricial analysis turns out to be a panacea, providing us with resolutions to all deductive issues in causation.
In particular, note that once we identify the moduses of negative generics, we can ascertain those of lone determinations, which conjoin one positive generic with the negations of all others. As we shall see in the next chapter, absolute lones are nullified. However, as we shall see in a subsequent chapter, relative lones are not nullified. Let us here mention for the record their summary moduses, which may be constructed knowing their alternative moduses, there identified (check and see for yourself that these summaries give rise to the correct alternatives):
- m-alone_{rel} = µ0µ0µµµµ
- n-alone_{rel} = µµµµ0µ0µ
- p-alone_{rel} = 10.1µ1µ.
- q-alone_{rel} = .µ1µ1.01
If we compare these to the summary moduses of m, n, p_{rel} and q_{rel}, respectively (which are given in Table 11.7 above), as well as to those of joint determinations mn, mq_{rel}, np_{rel}, p_{rel}q_{rel} (given in the previous section), we may observe the following mutations.
A code 0 or 1 for a generic is retained in a joint or lone including it. A µ found in m (or n, as the case may be) is retained in mn, and in m-alone_{rel} (or n-alone_{rel}), but not in mq_{rel} (or np_{rel}), because in the latter one µ is superseded by the 1 found in the corresponding position in p_{rel} (or q_{rel}), so that the remaining µ becomes a dot. There are no dots left in p_{rel}q_{rel} because all the dots in p_{rel} or q_{rel} have all been superseded by a 1 or 0. A dot in m or n becomes a µ in m-alone_{rel} or n-alone_{rel}, respectively. As for p-alone_{rel} or q-alone_{rel}, the dots in p_{rel} or q_{rel} not paired-off with a 1 become µ, whereas those paired-off with a 1 remain dots.
As already explained, a µ signifies that the pair of cells containing it (the first and third, the second and fourth, the fifth and seventh, or the sixth and eighth) may separately be 0 or 1, but cannot together be 0. No such restriction occurs where there are mere dots. Thus, what the above teaches us, especially, is that a relative lone determination has a slightly more restrictive modus than the corresponding generic determination, but is in all other respects identical.
Chapter 12. Systematic Microanalysis.
1. Grand Matrices.
Our study of causative propositions, in a first phase, consisted in conception of positive forms, their dissection into defining clauses, and their matricial analysis, or more precisely their macroanalysis. That provided us with the means to solve various problems, including many syllogistic issues; but it left us without practical means to answer questions concerning negative forms. We consequently, in a second phase, opted for a more detailed and deep method of study, microanalysis. We thus somewhat improved our predictive abilities; but serious difficulties remained, due to our approach being piecemeal.
To resolve outstanding issues, we must approach microanalysis in a more systematic manner. Instead of constructing matrices for each propositional form, we shall proceed in the opposite direction and conceive a grand matrix for the items concerned in which each and every propositional form can be located. A grand matrix tabulates all conceivable moduses for a given number of items, and assigns a numerical label (an address, as it were) to each such logical possibility. Once this is developed, we can identify the places of the various determinations within such a broad framework, and easily predict all their interactions.
Through grand matrices, we have an overview of all possible relations between the items concerned. We can then focus on particular segments of the matrix as signifying this or that specific relation.
Two items (P, R) give rise to a table with 2^{2} = 4 rows (with PR sequences 11, 10, 01, 00, conventionally so ordered), and 2^{4} = 16 modus columns (conventionally ordered with the maximum number of ‘zeros’ on the left and the maximum number of ‘ones’ on the right, then numbered 1-16). Such a table defines the general relation of any pair of items, and is the same whatever they happen to be.
A specific relation proposed for two particular items is then expressed by highlighting the modus column(s) corresponding to that specific relation (or by stating their numerical labels). The degree of determination involved is visually represented by the pattern of zeros and ones which stand out against the background of the grand matrix in which they are imbedded.
The grand matrix prefigures all ‘potential’ configurations for the number of items involved; while the highlighted alternative(s) depict the apparent or supposed ‘actual’ configuration for the particular items under scrutiny, which constitutes the distinctive determination relating them with each other.
In the case of three items (P, Q, R), the table has 2^{3} = 8 rows and 2^{8} = 256 modus columns, conventionally ordered in a similar manner. For four items (P, Q, R, S) we can expect a table with 2^{4} = 16 rows and 2^{16} = 65,536 modus columns. And so forth. Note well that the concrete content of the items is irrelevant to the structure of the grand matrix; it looks the same for any given number of items.
From an epistemological and ontological point of view, a grand matrix depicts the universe of imaginable relations between any two (or more) items in the world or in knowledge taken at random. In reality, i.e. in the experienced world or at a given stage of knowledge development, only some of these relations (alternative moduses, i.e. conjunctions of presences and absences) will be found applicable to the items under scrutiny.
Thus, we can visualize the ‘distance’ (their separation in space-time, or their conceptual difference) between any two or more items in the world or in knowledge as inhabited by a belt[80] with strips of zeros and ones (a grand matrix with alternative moduses), of which some are highlighted or potent in the case concerned, and the rest are neutralized or inactive. We thus propose a very binary structure for the world and for knowledge, appealing by its universality and simplicity.
Indeed, in this perspective, we can even conceive of a ‘universal matrix’, comprising the umpteen items in the world or in knowledge, and an enormous tapestry of logically possible relations with zillions of zeros and ones in their every combination and permutation. For x items, this matrix would have y = 2^{x} rows and z = 2^{y} columns.
With this image in mind, the pursuit of knowledge can be considered as an attempt to pinpoint – on the basis of sensory and other experience, as well as of mental speculation and logical insight – the applicable moduses within such broad ranges, for the items concerned. A specific relation like ‘causation’ or ‘complete causation’ is thus a selection of moduses proposed as applicable to the concrete items concerned. The applicable alternative moduses constitute the ‘bond’ (of some degree) between the items in a given case.
Identification of applicable moduses proceeds gradually, inductively (with deduction as but a tool of induction). They are not known immediately, without residual doubts. Intellectual work is required.
We start with a mass of phenomena in flux. Appearances are presented to consciousness, perceptually (concretes) or conceptually (abstracts). We stratify some as ‘given’ (pure) and others as ‘speculative’ (mental projections about the pure), and try through logical insight to judge the hypotheses most fitting for the overall context of currently available data.
Much of our ‘thinking’ in relation to causation consists simply in trying to encapsulate the data available in the different forms of causation. This is a trial and error process, which may be characterized as successive formulation and (if need arise) elimination of hypotheses. Our approach may be passive, unconscious; or proactive, purposeful.
Normally, we first try out the strongest form of causation (mn), then lesser forms (mq or np), and finally the weakest (pq); if none of these work, we conclude with non-causation. Alternatively, we may proceed on a deeper level, with reference to if-then statements or, more cautiously, to moduses, before we build up comprehensive causative propositions.
As the empirical context changes, growing and becoming more focused, our opinion may vary. We may also discover, through deductive reasoning, inconsistencies between different conclusions. What seemed previously a successful summary of information then has to be reviewed. But eventually things seem to settle down and solidify, and we may presume that our opinion at last corresponds to (or more closely than ever approaches) the ‘real’ state of affairs, and may be regarded as knowledge.
Logic, after working out matricial configurations, immediately imposes one universal restriction: the alternative modus in any grand matrix consisting only of zeros, with no ones, cannot be true. Whatever the grand matrix, i.e. for any number and content of items, only alternative moduses involving at least one ‘1’ code are at all credible; in every such matrix, the first modus, composed entirely of ‘0’ codes, has no credibility.
This is just a restatement with regard to matricial analysis of the Laws of Non-Contradiction and of the Excluded Middle. Since the rows of our matrix already predict every conceivable combination of the items in their positive and negative forms, at least one of these rows has to possibly exist; if a column means that none of these combinations may occur, it contradicts that setup and lays claim to yet another combination of items. Such a claim would be absurd, and may be rejected at the outset.
All other moduses are logically sound per se, though they might well be excluded within a given context. Indeed, the knowledge enterprise may be viewed as a search for good reasons for the elimination of as many moduses as we can, so as to be left with a limited number of moduses which signify an interesting specific relation like causation. We thus move from the vaguely conceivable, to a more focused and pondered evaluation.
We cannot say at the outset which relation (expressed by one or more moduses) applies in a given case. There is bound to be some relation, but as we shall soon see logic does not insist on a specifically causative relation, it allows for a non-causative relation. Ab initio, all logic stipulates is that the modus consisting only of zeros can never apply.
This is the nearest thing to a ‘law of causation’ we can foresee at this stage; which by itself implies that there is no law of causation in the traditional senses, or that if there is one it must be sought for in other ways. We shall, of course, return to this topic in more detail, in a later chapter.
2. Moduses in a Two-Item Framework.
We shall first consider a two-item framework, and catalogue all its conceivable moduses, then enumerate those applicable to each category of proposition. In the following table, P is looked upon as a putative cause, while R is looked upon as a putative effect. Their conceivable combinations define rows, and columns refer to all initially conceivable alternative moduses for them.
In a two-item grand matrix, there are 4 rows and 16 columns, as we have seen, and therefore 64 cells. Each cell may equally be coded 0 (impossible) or 1 (possible), so that each code will occur a total of 32 times. The matrix is constructed by coding: in the first row, 0 in the first 8 cells then 1 in the last 8 cells; for the second row, 0 in the first and third set of 4 cells then 1 in the second and fourth set of 4 cells; in the third row, we have a succession of pairs, 00, 11, 00, 11, and so forth; finally, in the fourth row, we coded 0, 1, 0, 1, in succession. We are thus sure to have foreseen every possible interplay of 0 and 1 codes.
Take the time to notice that we have ordered the alternative moduses in a progressive manner, starting with a maximum number of 0s in a column (no cell coded 1) and ending with a maximum number of 1s in a column (no cell coded 0). We then conventionally number (or label) the columns so ordered, 1-16. The rows, note well, are also in a conventional arrangement, with four PR sequences 11, 10, 01, 00, respectively (labeled a-d, if need be).
Now, the column labeled No. 1 is an impossible modus, since at least one row has to have a ‘1’, by the Laws of Non-Contradiction and of the Excluded Middle. Significantly, this is the only combination excluded universally by those logical laws, as already explained. Concerning the remaining 15 possible moduses, they are exhaustive (one of them must be true) and mutually exclusive (no more than one may be true at once).
Here, then, is the grand matrix for two items, a catalogue of all conceivable alternative moduses for any two items, like P, R:
Row | Items | ** | Possible moduses, labeled 2-15 | |||||||||||||||
label | P | R | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
a | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
b | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
c | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
d | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
** Column labeled No. 1 is an impossible modus.
The following table interprets the preceding, by enumeration of the alternative moduses of the main causative forms. It is based on the known characteristics of positive strong and absolute weak generics, i.e. the moduses given in Tables 1, 2, 5 and 6 of the previous chapter. From this initial information, we can, using the processes of negation, intersection and merger, infer the alternative moduses of derivative forms, i.e. negatives, as well as joints and vaguer forms (s, w, c), and their negations.
Note that relative weak determinations are not dealt with here, because, in a two-item framework, they have the same moduses as absolutes. They can only be distinguished as of a three-item framework, so we cannot analyze them and their derivatives till we get there.
Determination | Column number(s) | Comment |
Strongs and their negations: | ||
m | 10, 12 | 2 alternatives, by macroanalysis. |
n | 10, 14 | 2 alternatives, by macroanalysis. |
not-m | 2-9, 11, 13-16 | All alternatives but those of m;i.e. 13 cases. |
not-n | 2-9, 11-13, 15-16 | All alternatives but those of n;i.e. 13 cases. |
Absolute weaks and their negations: | ||
p_{abs} | 14, 16 | 2 alternatives, by macroanalysis of p_{rel} and contraction. |
q_{abs} | 12, 16 | 2 alternatives, by macroanalysis of q_{rel} and contraction. |
not-p_{abs} | 2-13, 15 | All alternatives but those of p_{abs};i.e. 13 cases. |
not-q_{abs} | 2-11, 13-15 | All alternatives but those of q_{abs};i.e. 13 cases. |
Table 12.2 continued.
Joints (absolute) and their negations: | ||
mn | 10 | Their one common alternative, by intersection. |
mq_{abs} | 12 | Their one common alternative, by intersection. |
np_{abs} | 14 | Their one common alternative, by intersection. |
p_{abs}q_{abs} | 16 | Their one common alternative, by intersection. |
not(mn) | 2-9, 11-16 | All alternatives but that of mn;i.e. 14 cases. |
not(mq_{abs}) | 2-11, 13-16 | All alternatives but that of mq_{abs};i.e. 14 cases. |
not(np_{abs}) | 2-14, 15-16 | All alternatives but that of np_{abs};i.e. 14 cases. |
not(p_{abs}q_{abs}) | 2-15 | All alternatives but that of p_{abs}q_{abs}; i.e. 14 cases. |
Strong causation and its negation: | ||
s = m or n | 10, 12, 14 | All their 3 alternatives, by merger. |
not-s = not-m + not-n | 2-9, 11, 13, 15-16 | All alternatives but the preceding; i.e. 12 cases. |
Absolute weak causation and its negation: | ||
w_{abs} = p_{abs} or q_{abs} | 12, 14, 16 | All their 3 alternatives, by merger. |
not- w_{abs} = not-p_{abs} + not-q_{abs} | 2-11, 13, 15 | All alternatives but the preceding; i.e. 12 cases. |
Causation (absolute) and its negation: | ||
c_{abs} = m or n or p_{abs} or q_{abs} | 10, 12, 14, 16 | All their four alternatives, by merger. |
not-c_{abs} = not-m + not-n + not-p_{abs} + not-q_{abs} | 2-9, 11, 13, 15 | All alternatives but the preceding; i.e. 11 cases. |
Let us highlight some of the information in the above table. First, take note of the ease with which we are now able to define any negative form, given the moduses of the corresponding positive form, by simply listing the leftover moduses. We can also readily define vaguer positive forms, like s, w, c, by merging the modus lists of their components. These forms were until here very difficult to define, remember.
Second, we can see at a glance that compatible forms are those which have a common modus (or more); for instance, m and n, m and q_{abs}, n and p_{abs}, p_{abs} and q_{abs} can be joined, because they share a modus (respectively, 10, 12, 14 and 16). Incompatibilities are also made evident by such a table; thus, m and p_{abs} have no common modus, nor do n and q_{abs}; so these are incompatible pairs and give rise to no form.
Third, certain compounds of positives and negatives have not been listed in the above table, because they are equivalent to already listed forms, i.e. all their moduses are the same. Implication signifies that every modus of the implying form is a modus of the implied form; this is not mere overlap, note, but full inclusion of one form in the other.
Two one-way implications (and their contraposites) must be noted:
- that p_{abs} implies not-m (or m implies not-p_{abs}), and
- that q_{abs} implies not-n (or n implies not-q_{abs}).
This is because the moduses Nos. 14, 16 of p_{abs} are both also moduses of not-m, and the moduses Nos. 12, 16 of q_{abs} are both also moduses of not-n. Given that m implies not-p_{abs}, it follows that (m + not-p_{abs}) is identical to m. Similarly, (n + not-q_{abs}) = n; (not-m + p_{abs}) = p_{abs}; and (not-n + q_{abs}) = q_{abs}. There is therefore no need to list these four conjunctions separately.
Mutual implication or equivalence occurs when the forms compared have the very same alternative modus list. Thus,
- (m + not-q_{abs}) = (n + not-p_{abs}) = mn (modus 10);
- (m + not-n) = (not-p_{abs} + q_{abs}) = mq_{abs} (modus 12);
- (not-m + n) = (p_{abs} + not-q_{abs}) = np_{abs} (modus 14); and
- (not-m + q_{abs}) = (not-n + p_{abs}) = p_{abs}q_{abs} (modus 16).
There is therefore no need to list these various conjunctions separately. In contrast, for instance, m and n do not imply each other, though they have one modus in common (No. 10), because each has a modus the other lacks. Likewise for p_{abs} and q_{abs}, they overlap only in one of their moduses (No. 16) and both have a distinct additional modus.
Fourth, some compositions have not been listed in the above table, because they do not constitute an interesting concept. Falling in this category are m or q_{abs} (moduses 10, 12, 16) and its negation (not-m + not-q_{abs}), or again n or p_{abs} (moduses 10, 14, 16) and its negation (not-n + not-p_{abs}).
Fifth, certain conjunctions of positives and negatives have not been listed in the above table, because they give rise to no forms. Note especially that (absolute) lone determinations are excluded from consideration (or nullified) by this technique. That is, we cannot form the following conjunctions of positive and negatives, because they do not share a single common alternative modus:
- m-alone_{abs} = m + not-n + not-p_{abs} + not-q_{abs} = null-class;
- n-alone_{abs} = n + not-m + not-p_{abs} + not-q_{abs} = null-class;
- p-alone_{abs} = p_{abs} + not-m + not-n + not-q_{abs} = null-class;
- q-alone_{abs} = q_{abs} + not-m + not-n + not-p_{abs} = null-class.
Thus, for instance, m shares modus 12 with not-n and (needless to say, since it implies it) with not-p_{abs}, but this modus is absent in not-q_{abs}. And so forth, for the other absolute lones. These symbolically contrived conjunctions are therefore impossible in fact: by reference to the moduses we can definitively establish this fact and understand it.
This is an important formal principle, which may be looked upon as a ‘law of causation’ (among others)[81]. Had (absolute) lone determinations been possible, our view of the causative relation would have been much less deterministic. Before microanalysis, we could not ascertain whether or not the generic determinations m, n, p_{abs} or q_{abs} may logically exist without intersection; now we know for sure that they can only exist within joint determinations.
The following equations follow from the nullification of lones:
- m = (mn or mq_{abs}), and n = (mn or np_{abs});
- p_{abs} = (np_{abs} or p_{abs}q_{abs}), and q_{abs} = (mq_{abs} or p_{abs}q_{abs}).
Again, s = (mn or mq_{abs} or np_{abs}), and w_{abs} = (mq_{abs} or np_{abs} or p_{abs}q_{abs}). Consequently, c_{abs} = (mn or mq_{abs} or np_{abs} or p_{abs}q_{abs}); and it is equivalent to (m or p_{abs}) and to (n or q_{abs}). Also, by negation, not-c_{abs} is equivalent to (not-m + not-p_{abs}) and to (not-n + not-q_{abs}).
These various compounds are therefore implicit in the above table, and need not be listed.
Lastly, we should notice the genus-species relations between forms. Thus, mn is a species of m and a species of n, because it shares a modus (No. 10) with each of them, and has none they lack; the latter forms are more generic or less definite, since they involve additional alternatives. Similarly, s is vaguer or broader in possibilities than m or n, and therefore a genus of theirs; likewise, p_{abs} and q_{abs} are species of w_{abs}. Causation (c) is clearly the summum genus for all the positive forms. Negatives can be examined in the same perspective.
It is also worth noticing what underlies the relative strengths of determinations. Note that the alternative moduses of the strong determinations (10, 12, 14) involve more zeros than those of the weaks (12, 14, 16). In particular, ignoring the common moduses (12, 14), compare modus 10 (two 0s) with modus 16 (no 0s). Clearly, m and n are stronger than p and q, because they involve more impossibility (two extra zeros); zeros more firmly delimit a relation. Similarly, comparing joints with each other; the more zeros in the modus, the stronger the determination.
3. Catalogue of Moduses, for Three Items.
Let us now consider a three-item framework. We shall here catalogue all its conceivable moduses; and in the next section, we shall enumerate those applicable to each category of proposition. In the following table, P and Q are looked upon as putative causes, while R is looked upon as a putative effect. Their conceivable combinations define rows, and columns refer to all initially conceivable alternative moduses for them.
In a three-item grand matrix, there are 8 rows and 256 columns, as we have seen, and therefore 2048 cells. Each cell may equally be coded 0 (impossible) or 1 (possible), so that each code will occur a total of 1024 times. This matrix is constructed in the same manner as the preceding one, by coding 0s and 1s progressively throughout it, so symmetrically that we can be sure it is exhaustive.
The columns (representing the alternative moduses), so ordered, are then numbered (or labeled) 1-256. Since the order of the rows is also fixed conventionally, with eight PQR sequences 111, 110, 101, 100, 011, 010, 001, 000 (which can, if need be, be labeled a-h, respectively), the modus number suffices to symbolize the modus concerned.[82]
Now, the column labeled No. 1 is an impossible modus, since at least one row has to have a ‘1’, by the Laws of Non-Contradiction and of the Excluded Middle. Significantly, this is the only combination excluded universally by those logical laws, as already explained. Concerning the remaining 255 possible moduses, they are exhaustive (one of them must be true) and mutually exclusive (no more than one may be true at once).
Here, then, is the grand matrix for three items, a catalogue of all conceivable alternative moduses for any three items, such as P, Q, R:
Items | ** | Possible moduses, labeled 2-16 | ||||||||||||||||
P | (Q) | R | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
** Column labeled No. 1 is an impossible modus.
Same table continued.
Items | Moduses, labeled 17-32 | |||||||||||||||||
P | (Q) | R | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 |
1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Same table continued.
Items | Moduses, labeled 33-48 | |||||||||||||||||
P | (Q) | R | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 |
1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Same table continued.
Items | Moduses, labeled 49-64 | |||||||||||||||||
P | (Q) | R | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 |
1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Table 12.3 continued.
Items | Moduses, labeled 65-80 | |||||||||||||||||
P | (Q) | R | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Same table continued.
Items | Moduses, labeled 81-96 | |||||||||||||||||
P | (Q) | R | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 |
1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Same table continued.
Items | Moduses, labeled 97-112 | |||||||||||||||||
P | (Q) | R | 97 | 98 | 99 | 100 | 101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 | 110 | 111 | 112 |
1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Same table continued.
Items | Moduses, labeled 113-128 | |||||||||||||||||
P | (Q) | R | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 | 121 | 122 | 123 | 124 | 125 | 126 | 127 | 128 |
1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Table 12.3 continued.
Items | Moduses, labeled 129-144 | |||||||||||||||||
P | (Q) | R | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | 137 | 138 | 139 | 140 | 141 | 142 | 143 | 144 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Same table continued.
Items | Moduses, labeled 145-160 | |||||||||||||||||
P | (Q) | R | 145 | 146 | 147 | 148 | 149 | 150 | 151 | 152 | 153 | 154 | 155 | 156 | 157 | 158 | 159 | 160 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Same table continued.
Items | Moduses, labeled 161-176 | |||||||||||||||||
P | (Q) | R | 161 | 162 | 163 | 164 | 165 | 166 | 167 | 168 | 169 | 170 | 171 | 172 | 173 | 174 | 175 | 176 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Same table continued.
Items | Moduses, labeled 177-192 | |||||||||||||||||
P | (Q) | R | 177 | 178 | 179 | 180 | 181 | 182 | 183 | 184 | 185 | 186 | 187 | 188 | 189 | 190 | 191 | 192 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Table 12.3 continued.
Items | Moduses, labeled 193-208 | |||||||||||||||||
P | (Q) | R | 193 | 194 | 195 | 196 | 197 | 198 | 199 | 200 | 201 | 202 | 203 | 204 | 205 | 206 | 207 | 208 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Same table continued.
Items | Moduses, labeled 209-224 | |||||||||||||||||
P | (Q) | R | 209 | 210 | 211 | 212 | 213 | 214 | 215 | 216 | 217 | 218 | 219 | 220 | 221 | 222 | 223 | 224 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Same table continued.
Items | Moduses, labeled 225-240 | |||||||||||||||||
P | (Q) | R | 225 | 226 | 227 | 228 | 229 | 230 | 231 | 232 | 233 | 234 | 235 | 236 | 237 | 238 | 239 | 240 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Same table continued.
Items | Moduses, labeled 241-256 | |||||||||||||||||
P | (Q) | R | 241 | 242 | 243 | 244 | 245 | 246 | 247 | 248 | 249 | 250 | 251 | 252 | 253 | 254 | 255 | 256 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
4. Enumeration of Moduses, for Three Items.
The following table interprets the preceding, by enumeration of the alternative moduses of the main causative forms. It is based on the known characteristics of positive strong and weak generics, i.e. the moduses given in Tables 1-6 of the previous chapter. From this initial information, we can, using the processes of negation, intersection and merger, infer the alternative moduses of derivative forms, i.e. negatives, as well as joints and vaguer forms (s, w, c), and their negations.
We shall deal here only with the absolute weak determinations and their derivatives; relative weaks and their derivatives will be considered in the next chapter.
Determination | Modus numbers | Comment |
Strongs and their negations: | ||
m | 34, 36-40, 42, 44-48, 130, 132-136, 138, 140-144, 162, 164-168, 170, 172-176 | 36 alternatives,by macroanalysis. |
n | 34, 37-38, 50, 53-54, 98, 101-102, 114, 117-118, 130, 133-134, 146, 149-150, 162, 165-166, 178, 181-182, 194, 197-198, 210, 213-214, 226, 229-230, 242, 245-246 | 36 alternatives,by macroanalysis. |
not-m | 2-33, 35, 41, 43, 49-129, 131, 137, 139, 145-161, 163, 169, 171, 177-256 | All alternatives but those of m, i.e. 219 cases. |
not-n | 2-33, 35-36, 39-49, 51-52, 55-97, 99-100, 103-113, 115-116, 119-129, 131-132, 135-145, 147-148, 151-161, 163-164, 167-177, 179-180, 183-193, 195-196, 199-209, 211-212, 215-225, 227-228, 231-241, 243-244, 247-256 | All alternatives but those of n, i.e. 219 cases. |
Absolute weaks and their negations: | ||
p_{abs} | 50, 52-56, 58, 60-64, 98, 100-104, 106, 108-112, 114, 116-120, 122, 124-128, 146, 148-152, 154, 156-160, 178, 180-184, 186, 188-192, 194, 196-200, 202, 204-208, 210, 212-216, 218, 220-224, 226, 228-232, 234, 236-240, 242, 244-248, 250, 252-256 | 108 alternatives,by macroanalysis of p_{rel} then contraction and expansion. |
q_{abs} | 36, 39-40, 42, 44-48, 52, 55-56, 58, 60-64, 100, 103-104, 106, 108-112, 116, 119-120, 122, 124-128, 132, 135-136, 138, 140-144, 148, 151-152, 154, 156-160, 164, 167-168, 170, 172-176, 180, 183-184, 186, 188-192, 196, 199-200, 202, 204-208, 212, 215-216, 218, 220-224, 228, 231-232, 234, 236-240, 244, 247-248, 250, 252-256 | 108 alternatives,by macroanalysis of q_{rel} then contraction and expansion. |
not-p_{abs} | 2-49, 51, 57, 59, 65-97, 99, 105, 107, 113, 115, 121, 123, 129-145, 147, 153, 155, 161-177, 179, 185, 187, 193, 195, 201, 203, 209, 211, 217, 219, 225, 227, 233, 235, 241, 243, 249, 251 | All alternatives but those of p_{abs}, i.e. 147 cases. |
not-q_{abs} | 2-35, 37-38, 41, 43, 49-51, 53-54, 57, 59, 65-99, 101-102, 105, 107, 113-115, 117-118, 121, 123, 129-131, 133-134, 137, 139, 145-147, 149-150, 153, 155, 161-163, 165-166, 169, 171, 177-179, 181-182, 185, 187, 193-195, 197-198, 201, 203, 209-211, 213-214, 217, 219, 225-227, 229-230, 233, 235, 241-243, 245-246, 249, 251 | All alternatives but those of q_{abs}, i.e. 147 cases. |
Table 12.4 continued.
Joints (absolute) and their negations: | ||
mn | 34, 37-38, 130, 133-134, 162, 165-166 | Their 9 common alternatives, by intersection. |
mq_{abs} | 36, 39-40, 42, 44-48, 132, 135-136, 138, 140-144, 164, 167-168, 170, 172-176 | Their 27 common alternatives, by intersection. |
np_{abs} | 50, 53-54, 98, 101-102, 114, 117-118, 146, 149-150, 178, 181-182, 194, 197-198, 210, 213-214, 226, 229-230, 242, 245-246 | Their 27 common alternatives, by intersection. |
p_{abs}q_{abs} | 52, 55-56, 58, 60-64, 100, 103-104, 106, 108-112, 116, 119-120, 122, 124-128, 148, 151-152, 154, 156-160, 180, 183-184, 186, 188-192, 196, 199-200, 202, 204-208, 212, 215-216, 218, 220-224, 228, 231-232, 234, 236-240, 244, 247-248, 250, 252-256 | Their 81 common alternatives, by intersection. |
not(mn) | 2-33, 35-36, 39-129, 131-132, 135-161, 163-164, 167-256 | All alternatives but those of mn; i.e. 246 cases. |
not(mq_{abs}) | 2-35, 37-38, 41, 43, 49-131, 133-134, 137, 139, 145-163, 165-166, 169, 171, 177-256 | all alternatives but those of mq_{abs}; i.e. 228 cases. |
not(np_{abs}) | 2-49, 51-52, 55-97, 99-100, 103-113, 115-116, 119-145, 147-148, 151-177, 179-180, 183-193, 195-196, 199-209, 211-212, 215-225, 227-228, 231-241, 243-244, 247-256 | All alternatives but those of np_{abs}; i.e. 228 cases. |
not(p_{abs}q_{abs}) | 2-51, 53-54, 57, 59, 65-99, 101-102, 105, 107, 113-115, 117-118, 121, 123, 129-147, 149-150, 153, 155, 161-179, 181-182, 185, 187, 193-195, 197-198, 201, 203, 209-211, 213-214, 217, 219, 225-227, 229-230, 233, 235, 241-243, 245-246, 249, 251 | All alternatives but those of p_{abs}q_{abs}; i.e. 174 cases. |
Strong causation and its negation: | ||
s = m or n | 34, 36-40, 42, 44-48, 50, 53-54, 98, 101-102, 114, 117-118, 130, 132-136, 138, 140-144, 146, 149-150, 162, 164-168, 170, 172-176, 178, 181-182, 194, 197-198, 210, 213-214, 226, 229-230, 242, 245-246 | Their 63 separate and common alternatives (including overlap, i.e. mn), by merger. |
not-s = not-m + not-n | 2-33, 35, 41, 43, 49, 51-52, 55-97, 99-100, 103-113, 115-116, 119-129, 131, 137, 139, 145, 147-148, 151-161, 163, 169, 171, 177, 179-180, 183-193, 195-196, 199-209, 211-212, 215-225, 227-228, 231-241, 243-244, 247-256 | All alternatives but the preceding; i.e. 192 cases. |
Absolute weak causation and its negation: | ||
w_{abs} = p_{abs} or q_{abs} | 36, 39-40, 42, 44-48, 50, 52-56, 58, 60-64, 98, 100-104, 106, 108-112, 114, 116-120, 122, 124-128, 132, 135-136, 138, 140-144, 146, 148-152, 154, 156-160, 164, 167-168, 170, 172-176, 178, 180-184, 186, 188-192, 194, 196-200, 202, 204-208, 210, 212-216, 218, 220-224, 226, 228-232, 234, 236-240, 242, 244-248, 250, 252-256 | Their 135 separate and common alternatives (including overlap, i.e. p_{abs}q_{abs}), by merger. |
not- w_{abs} = not-p_{abs} + not-q_{abs} | 2-35, 37-38, 41, 43, 49, 51, 57, 59, 65-97, 99, 105, 107, 113, 115, 121, 123, 129-131, 133-134, 137, 139, 145, 147, 153, 155, 161-163, 165-166, 169, 171, 177, 179, 185, 187, 193, 195, 201, 203, 209, 211, 217, 219, 225, 227, 233, 235, 241, 243, 249, 251 | All alternatives but the preceding; i.e. 120 cases. |
Table 12.4 continued.
Causation (absolute) and its negation: | ||
c_{abs} = m or n or p_{abs} or q_{abs} | 34, 36-40, 42, 44-48, 50, 52-56, 58, 60-64, 98, 100-104, 106, 108-112, 114, 116-120, 122, 124-128, 130, 132-136, 138, 140-144, 146, 148-152, 154, 156-160, 162, 164-168, 170, 172-176, 178, 180-184, 186, 188-192, 194, 196-200, 202, 204-208, 210, 212-216, 218, 220-224, 226, 228-232, 234, 236-240, 242, 244-248, 250, 252-256 | Their 144 separate and common alternatives (including overlap). |
not-c_{abs} = not-m + not-n + not-p_{abs} + not-q_{abs} | 2-33, 35, 41, 43, 49, 51, 57, 59, 65-97, 99, 105, 107, 113, 115, 121, 123, 129, 131, 137, 139, 145, 147, 153, 155, 161, 163, 169, 171, 177, 179, 185, 187, 193, 195, 201, 203, 209, 211, 217, 219, 225, 227, 233, 235, 241, 243, 249, 251 | All alternatives but the preceding; i.e. 111 cases. |
The results obtained in Table 12.4 can be made to conveniently stand out by color coding each form’s moduses in Table 12.3. This is left to the reader to do.
We need not repeat here what was said before, with reference to the similar table for a two-item framework (Table 12.2); the same comments apply, because the relationships there established are true irrespective of framework. We will, however, highlight something which was less visible before, namely the consistency between various results.
There are never overlaps between contradictory propositions, and their alternatives sum up to 255; also, each generic sums up to two joints (since absolute lones do not exist). For instance, m comprises 36 alternative moduses, the 9 of mn plus the 27 of mq_{abs}; while not-m has the 219 remaining alternatives. Similarly, with regard to n. Likewise, p_{abs} comprises 108 alternatives, the 27 of np_{abs} plus the 81 of p_{abs}q_{abs}; while not-p_{abs} has the 147 remaining alternatives. Similarly, with regard to q_{abs}.
Moreover, the number of moduses corresponding to the vaguer forms are predictable. Thus, s (= m or n) comprises the 36 moduses of m plus the 36 of n, less the 9 of mn[83], a total of 63 alternatives; and its negation has 255 – 63 = 192 alternatives. We can similarly predict the moduses of w_{abs} (= p_{abs} or q_{abs}) to be 108 + 108 – 81 = 135; and a residue of 120 alternatives for its negation. For c (= s or w_{abs}) we have 63 + 135 – 2*27 = 144 (the 54 subtracted being those of mq_{abs} and np_{abs} – i.e. of sw_{abs}); for its negation, 111.
Thus, incidentally, causation in all its forms covers more than half the matrix, but still leaves a large space to non-causation.
5. Comparing Frameworks.
Now let us compare the results in Tables 2 and 4. They are essentially the same tables, except that each modus of the first is, as it were, further subdivided into a number of moduses in the second. However, the subdivision is evidently not proportional, say in the ratio 16:256; you cannot just say that to each two-item modus there corresponds 16 three-item ones. The following table makes this disproportionality clear:
Framework | m,n | p_{abs},q_{abs} | mn | mq_{abs},np_{abs} | p_{abs}q_{abs} | s | w_{abs} | c |
Two-Item | 2 | 2 | 1 | 1 | 1 | 3 | 3 | 4 |
Three-Item | 36 | 108 | 9 | 27 | 81 | 63 | 135 | 144 |
The explanation is easy. Expansion of a two-item alternative modus into a number of three-item moduses depends on how many zero or one codes it involves. For, as we saw in the previous chapter (with the proviso of appropriate locations), each ‘0’ in a two-item framework has a single expression (‘0 0’) in the three-item framework; whereas each ‘1’ in the former has three expressions in the latter (‘0 1’, ‘1 0’ or ‘1 1’ – i.e. any but ‘0 0’).
Thus, if a two-item modus involves four ‘zeros’ and no ‘one’, its three-item equivalent will consist of 1*1*1*1 = 1 (equally impossible) modus; if the former involves three zeros and a one, the latter will consist of 1*1*1*3 = 3 moduses; if the former involves two zeros and two ones, the latter will consist of 1*1*3*3 = 9 moduses; if the former involves one zero and three ones, the latter will consist of 1*3*3*3 = 27 moduses; and if the former involves no zero and four ones, the latter will consist of 3*3*3*3 = 81 moduses.
Whence, the strongs m, n, which each involves two two-item moduses, one with two zeros (No. 10) and one with a single zero (no. 12 or 14), will have 9 + 27 = 36 three-item moduses; whereas, the weaks p_{abs}, q_{abs}, which each involves two two-item moduses, one with a single zero (no. 12 or 14) and one with no zero (No. 16) and will have 27 + 81 = 108 three-item moduses.
The numbers of three-item moduses for the conjunctions and disjunctions of these forms follow. The joint mn (two-item modus No. 10) will have 9 of them; mq_{abs} (modus No. 12) and np_{abs} (modus No. 14) will each have 27; and p_{abs}q_{abs} (modus 16) will have 81. The vague form s (moduses 10, 12, 14) will have 9 + 2*27 = 63; w_{abs} (moduses 12, 14, 16) will have 2*27 + 81 = 135; and c (moduses 10, 12, 14, 16) will have 9 + 2*27 + 81 = 144.
We can proceed in a like manner to predict expansions of negative forms. Furthermore, given the two-item modus(es) of a form, we can predict not only how many moduses it will have in a three-item framework, but precisely which moduses it will have. Thus, a table of equivalencies between the two frameworks can be constructed without difficulty. In short, we have here a functioning calculus.
The precise three-item modus(es) corresponding to each two-item modus are given in the following table:
Two-item modus | No. of zerosin it | Correspondingthree-itemmodus numbers | No. of moduses |
1 | 4 | 1 | 1 |
2 | 3 | 2, 5, 6 | 3 |
3 | 3 | 3, 9, 11 | 3 |
4 | 2 | 4, 7, 8, 10, 12-16 | 9 |
5 | 3 | 17, 65, 81 | 3 |
6 | 2 | 18, 21-22, 66, 69-70, 82, 85-86 | 9 |
7 | 2 | 19, 25, 27, 67, 73, 75, 83, 89, 91 | 9 |
8 | 1 | 20, 23-24, 26, 28-32, 68, 71-72, 74, 76-80, 84, 87-88, 90, 92-96 | 27 |
9 | 3 | 33, 129, 161 | 3 |
10 | 2 | 34, 37-38, 130, 133, 134, 162, 165-166 | 9 |
11 | 2 | 35, 41, 43, 131, 137, 139, 163, 169, 171 | 9 |
12 | 1 | 36, 39-40, 42, 44-48, 132, 135-136, 138, 140-144, 164, 167-168, 170, 172-176 | 27 |
13 | 2 | 49, 97, 113, 145, 177, 193, 209, 225, 241 | 9 |
Table 12.6 continued.
14 | 1 | 50, 53-54, 98, 101-102, 114, 117-118, 146, 149-150, 178, 181-182, 194, 197-198, 210, 213-214, 226, 229-230, 242, 245-246 | 27 |
15 | 1 | 51, 57, 59, 99, 105, 107, 115, 121, 123, 147, 153, 155, 179, 185, 187, 195, 201, 203, 211, 217, 219, 227, 233, 235, 243, 249, 251 | 27 |
16 | 0 | 52, 55-56, 58, 60-64, 100, 103-104, 106, 108-112, 116, 119-120, 122, 124-128, 148, 151-152, 154, 156-160, 180, 183-184, 186, 188-192, 196, 199-200, 202, 204-208, 212, 215-216, 218, 220-224, 228, 231-232, 234, 236-240, 244, 247-248, 250, 252-256 | 81 |
16 | Total number of moduses | 256 |
Needless to say, each modus will occur only once in the above table, making a total of 16 or 256 moduses, according to the framework. Clearly, if we had developed this table earlier, we could have derived Table 12.4 from Table 12.2.[84]
Obviously, we can follow the same procedures to expand three-item alternative moduses (of which there are 256) into four-item alternative moduses (of which there are 65,536 – as seen earlier).
The number and configuration of the latter will emerge from the each of the former, in accordance with the number of zero and one codes it contains and the way they are arrayed within it (i.e. the incidence, prevalence and locations of zero and one codes in it). A table of correspondences can thus be constructed, which details the results obtained in each case.
We have above identified the main lines of what might be called the two-three (2/3) table of correspondences, emerging from the operation of expansion of ‘0’ into ‘0 0’ and ‘1’ into ‘0 1’, ‘1 0’, ‘1 1’ (all pairs but ‘0 0’). We could thereafter, step by step, build similar tables of correspondence of size 3/4 or 4/5… and so forth on to infinity, if need arise to resolve eventual issues.
For instance, from a three-item matrix (which has 8 rows) to a four-item matrix, each combination of zeros and ones will result in a product of eight factors of 1 (for ‘0’ codes) or 3 (for ‘1’ codes) – e.g., a modus with 1 zero and 7 ones will become 1*3*3*3*3*3*3*3 = 2187 moduses, in various possible permutations. These are long-winded techniques, which may or may not be needed.
Chapter 13. Some More Microanalyses.
1. Relative Weaks.
We have in the previous chapter identified the alternative moduses of the absolute weak determinations and their derivatives. We will here ascertain those of relative weaks and their derivatives. In a two-item framework, relatives are of course indistinguishable from absolutes; they arise only as of a three-item framework.
The following table may be viewed as a continuation of Table 12.4 of the previous chapter; and the modus numbers listed in it refer to the grand matrix in Table 12.3 of the previous chapter. Note well that p_{rel} and q_{rel} (and their derivatives with the same suffix), below, refer to partial or contingent causation between P and R relative to Q; that is, P with complement Q are putative causes of R.
Determination | Modus numbers | Comment |
Relative weaks and their negations: | ||
p_{rel} | 149-152, 157-160, 181-184, 189-192 | 16 alternatives, by macroanalysis. |
q_{rel} | 42, 46, 58, 62, 106, 110, 122, 126, 170, 174, 186, 190, 234, 238, 250, 254 | 16 alternatives, by macroanalysis. |
not-p_{rel} | 2-148, 153-156, 161-180, 185-188, 193-256 | All alternatives but those of p_{rel}, i.e. 239 cases. |
not-q_{rel} | 2-41, 43-45, 47-57, 59-61, 63-105, 107-109, 111-121, 123-125, 127-169, 171-173, 175-185, 187-189, 191-233, 235-237, 239-249, 251-253, 255-256 | All alternatives but those of q_{rel}, i.e. 239 cases. |
Joints (relative) and their negations: | ||
mn | 34, 37-38, 130, 133-134, 162, 165-166 | Their 9 common alternatives. |
mq_{rel} | 42, 46, 170, 174 | Their 4 common alternatives. |
np_{rel} | 149-150, 181-182 | Their 4 common alternatives. |
p_{rel}q_{rel} | 190 | Their 1 common alternatives. |
not(mn) | 2-33, 35-36, 39-129, 131-132, 135-161, 163-164, 167-256 | All alternatives but those of mn; i.e. 246 cases. |
not(mq_{rel}) | 2-41, 43-45, 47-169, 171-173, 175-256 | All alternatives but those of mq_{rel}; i.e. 251 cases. |
not(np_{rel}) | 2-148, 151-180, 183-256 | All alternatives but those of np_{rel}; i.e. 251 cases. |
not(p_{rel}q_{rel}) | 2-189, 191-256 | All alternatives but those of p_{rel}q_{rel}; i.e. 254 cases. |
Relative lones and their negations: | ||
m-alone_{rel} | 36, 39-40, 44-45, 47-48, 132, 135-136, 138, 140-144, 164, 167-168, 172-173, 175-176 | The 23 common alternatives of m, not-n, and not-q_{rel}. |
n-alone_{rel} | 50, 53-54, 98, 101-102, 114, 117-118, 146, 178, 194, 197-198, 210, 213-214, 226, 229-230, 242, 245-246 | The 23 common alternatives of n, not-m, and not-p_{rel}. |
Table 13.1 continued.
p-alone_{rel} | 151-152, 157-160, 183-184, 189, 191-192 | The 11 common alternatives of p_{rel}, not-n, and not-q_{rel}. |
q-alone_{rel} | 58, 62, 106, 110, 122, 126, 186, 234, 238, 250, 254 | The 11 common alternatives of q_{rel}, not-m, and not-p_{rel}. |
not(m-alone_{rel}) | 2-35, 37-38, 41-43, 46, 49-131, 133-134, 137, 139, 145-163, 165-166, 169-171, 174, 177-256 | All alternatives but those ofm-alone_{rel}; i.e. 232 cases. |
not(n-alone_{rel}) | 2-49, 51-52, 55-97, 99-100, 103-113, 115-116, 119-145, 147-177, 179-193, 195-196, 199-209, 211-212, 215-225, 227-228, 231-241, 243-244, 247-256 | All alternatives but those ofn-alone_{rel}; i.e. 232 cases. |
not(p-alone_{rel}) | 2-150, 153-156, 161-182, 185-188, 190, 193-256 | All alternatives but those ofp-alone_{rel}; i.e. 244 cases. |
not(q-alone_{rel}) | 2-57, 59-61, 63-105, 107-109, 111-121, 123-125, 127-185, 187-233, 235-237, 239-249, 251-253, 255-256 | All alternatives but those ofq-alone_{rel}; i.e. 244 cases. |
Relative weak causation and its negation: | ||
w_{rel} = p_{rel} or q_{rel} | 42, 46, 58, 62, 106, 110, 122, 126, 149-152, 157-160, 170, 174, 181-184, 186, 189-192, 234, 238, 250, 254 | Their 31 separate and common alternatives (including overlap, i.e. p_{rel}q_{rel} = 1). |
p_{rel} + not-q_{rel} | 149-152, 157-160, 181-184, 189, 191-192 | Their 15 common alternatives. |
not-p_{rel} + q_{rel} | 42, 46, 58, 62, 106, 110, 122, 126, 170, 174, 186, 234, 238, 250, 254 | Their 15 common alternatives. |
not-w_{rel} = not-p_{rel} + not-q_{rel} | 2-41, 43-45, 47-57, 59-61, 63-105, 107-109, 111-121, 123-125, 127-148, 153-156, 161-169, 171-173, 175-180, 185, 187-188, 193-233, 235-237, 239-249, 251-253, 255-256 | All alternatives but those of w_{rel}; i.e. 224 cases. |
Contributory causation (relative) and its negation: | ||
m or p_{rel} | 34, 36-40, 42, 44-48, 130, 132-136, 138, 140-144, 149-152, 157-160, 162, 164-168, 170, 172-176, 181-184, 189-192 | Their 52 separate alternatives (no overlap). |
not-m + not-p_{rel} | 2-33, 35, 41, 43, 49-129, 131, 137, 139, 145-148, 153-156, 161, 163, 169, 171, 177-180, , 185-188, 193-256 | All alternatives but the preceding; i.e. 203 cases. |
Possible causation (relative) and its negation: | ||
n or q_{rel} | 34, 37-38, 42, 46, 50, 53-54, 58, 62, 98, 101-102, 106, 110, 114, 117-118, 122, 126, 130, 133-134, 146, 149-150, 162, 165-166, 170, 174, 178, 181-182, 186, 190, 194, 197-198, 210, 213-214, 226, 229-230, 234, 238, 242, 245-246, 250, 254 | Their 52 separate alternatives (no overlap). |
not-n + not-q_{rel} | 2-33, 35-36, 39-41, 43-45, 47-49, 51-52, 55-57, 59-61, 63-97, 99-100, 103-105, 107-109, 111-113, 115-116, 119-121, 123-125, 127-129, 131-132, 135-145, 147-148, 151-161, 163-164, 167-169, 171-173, 175-177, 179-180, 183-185, 187-189, 191-193, 195-196, 199-209, 211-212, 215-225, 227-228, 231-233, 235-237, 239-241, 243-244, 247–249, 251-253, 255-256 | All alternatives but the preceding; i.e. 203 cases. |
Table 13.1 continued.
Causation (relative) and its negation: | ||
c_{rel} = m or n or p_{rel} or q_{rel} | 34, 36-40, 42, 44-48, 50, 53-54, 58, 62, 98, 101-102, 106, 110, 114, 117-118, 122, 126, 130, 132-136, 138, 140-144, 146, 149-152, 157-160, 162, 164-168, 170, 172-176, 178, 181-184, 186, 189-192, 194, 197-198, 210, 213-214, 226, 229-230, 234, 238, 242, 245-246, 250, 254 | Their 86 separate and common alternatives (including overlap). |
not-c_{rel }= not-m + not-n+ not-p_{rel } + not-q_{rel} | 2-33, 35, 41, 43, 49, 51-52, 55-57, 59-61, 63-97, 99-100, 103-105, 107-109, 111-113, 115-116, 119-121, 123-125, 127-129, 131, 137, 139, 145, 147-148, 153-156, 161, 163, 169, 171, 177, 179-180, 185, 187-188, 193, 195-196, 199-209, 211-212, 215-225, 227-228, 231-233, 235-237, 239-241, 243-244, 247-249, 251-253, 255-256 | All alternatives but the preceding; i.e. 169 cases. |
Now, let us compare the above results for relative weaks to those for absolute weaks in Table 12.4 of the previous chapter. The logical properties of these forms are quite distinct. When we unravel the summary modus µµµµ.µ.µ of p_{abs}, we obtain 108 alternative moduses; similarly, the summary modus µ.µ.µµµµ of q_{abs} yields 108 alternative moduses. In contrast, the summaries of p_{rel} and q_{rel} – namely, 10.1.1.. and ..1.1.01 – give rise to 16 alternatives each.
The first thing to note is that the 16 moduses of p_{rel} are all included in the 108 of p_{abs}; and likewise, the 16 of q_{rel} are among the 108 of q_{abs}. Look at the tables, and see this for yourself. What this means is that the positive relative weaks imply and are species of the positive absolute weaks.
Moreover, note that the latter are more than twice as broad in possibilities than the former. This reveals to us that p_{PR} is not merely the sum of p_{Q} and p_{notQ}, i.e. that “P (with whatever complement) is a partial cause of R” means more than “P (whether with complement Q or notQ) is a partial cause of R”; similarly, regarding q. We shall list the precise moduses of p_{notQ} and q_{notQ} further on; but we can predict at the outset that they will be 16 in number in each case, by the demands of symmetry. Therefore, absolute weak causation between P and R can occur with complements other than Q or notQ; and we cannot engage in dilemmatic arguments, saying that if Q is not the complement, notQ must be it. It is wise to keep that in mind.
Consequently, the negations of the relative weaks are broader than those of the corresponding absolute weaks; the former involve 239 (255 – 16) alternative moduses each, the latter only 147 (255 – 108) among these.[85]
Consider now the relative joint determinations: mq_{rel} and np_{rel} have only 4 moduses each, while the corresponding absolute joints mq_{abs} and np_{abs} have 27 each; and p_{rel}q_{rel} has only 1 modus, in contrast to the 81 of p_{abs}q_{abs}. Thus, as we move from absolute to relative determination, we narrow down the possibilities, we get more specific. On the negative side, the possibilities are broadened, from 228 to 251 or 174 to 254.
We saw in the previous chapter that absolute lone determinations do not exist, for the simple reason that their constituents have no common modus. On the other hand, as can be seen above, relative lone determinations do indeed exist, since their constituents have common moduses, 23 for the strongs and 11 for the weaks.
But the latter concepts are of course not as significant as the former. For as we can see with reference to the moduses involved, the relative lones – together with the relative joints – are merely species of (i.e. are all included in) the absolute joints; that is:
- m-alone_{rel} + mq_{rel} (23 + 4) = mq_{abs} (27, i.e. the 36 of m less the 9 of mn);
- n-alone_{rel} + np_{rel} (23 + 4) = np_{abs} (27, i.e. the 36 of n less the 9 of mn);
- p-alone_{rel} + q-alone_{rel} + p_{rel}q_{rel} (11 + 11 + 1 = 23) imply p_{abs}q_{abs} (81).
Thus, whereas w_{abs} = mq_{abs} or np_{abs} or p_{abs}q_{abs}, we must equate w_{rel} to mq_{rel} or np_{rel} or p_{rel}q_{rel} or p-alone_{rel} or q-alone_{rel}; check it out with reference to the moduses involved. Note that w_{rel} involves only 31 moduses, the 15 of p_{rel} + not-q_{rel}, the 15 of not-p_{rel} + q_{rel}, and the 1 of p_{rel}q_{rel}. This is in contrast to w_{abs} which has 135 (the same 31, and 103 more besides). Consequently, not-w_{rel} has 224 moduses, including all 120 of not-w_{abs}.
We saw in the previous chapter that contributory causation, possible causation and causation tout court are one and the same concept with regard to absolute weaks, all with the same 144 moduses. But with regard to relative weaks, they are different concepts, as the above table clearly shows.
The relative form of contributory causation “m or p_{rel}” has 52 moduses, and that of possible causation “n or q_{rel}” has 52, while relative causation “m or n or p_{rel} or q_{rel}” involves 86. The latter 86 moduses comprise the preceding 52 + 52, minus the 18 moduses of the four relative joint determinations (their overlaps); and all these moduses are of course included in the list of 144 for absolute causation.
The moduses of the negations of these three relative forms follow, as shown in our table. Note especially that negation of relative causation, not-c_{rel }(169 moduses), does not imply negation of absolute causation, not-c_{abs} (111 moduses); but instead, the latter implies and is a species of the former, including all its moduses and more.
We need not mention in the above table the combinations (m + not-p_{rel}), (n + not-q_{rel}), (not-m + p_{rel}), (not-n + q_{rel}), because, as can be seen with reference to the common moduses of the positive and negative forms constituting them, they are respectively equivalent to m, n, p_{rel}, q_{rel}.
The remaining combinations are not mentioned because they are not particularly interesting. This refers to (m or q_{rel}), comprising the 4 moduses of mq_{rel} plus the 32 of “m + not-q_{rel}” plus the 12 of “not-m + q_{rel}”, a total 48 alternatives; and to “n or p_{rel}”, comprising the 4 moduses of np_{rel} plus the 32 of “n + not-p_{rel}” plus the 12 of “not-n + p_{rel}”, a total 48 alternatives; as well as to their respective negations, “not-m + not-q_{rel}” and “not-n + not-p_{rel}”, which involve 207 moduses each.
2. Items of Negative Polarity in Two-Item Framework.
The grand matrices, in which the various forms of causative propositions are embedded, are equally the habitats of similar propositions involving like items but of negative polarity. Such propositions need also to be microanalyzed, for reasons which will be become apparent after we do so. The job is rather easy, involving a mere reshuffling of the summary moduses of propositions with items of positive polarity.
Let us consider, to begin with, the positive generic forms in a two-item framework (strongs or absolute weaks only – relative weaks being indistinguishable here), with reference to Table 12.1 of the previous chapter (turn to it, and note well that it has P and R as column headings for items).
We have previously ascertained the summary moduses of generics with items ‘P.R’; our task here is to find out those for the same forms with items ‘notP.notR’, ‘P.notR’ and ‘notP.R’. Symbolically, such forms can be distinguished by changes in suffix. Thus, for complete causation, symbol m, we would write m_{PR}, m_{notPnotR}, m_{PnotR}, and m_{notPR}, according to the sequence of items intended; similarly for n, p, q – each form gives rise to four.
Now, if we changed the column headings of the said table from P.R to some other combination (notP.notR, P.notR or notP.R), the modus numbers (labels) applicable to each form would remain the same but change meanings (i.e. refer to different arrays of an equal number of 0 and 1 codes), and we would not be able to compare same forms with different suffixes.
What we need to do, rather, is retain the same grand matrix (the one for positive items P.R), and locate within it the moduses of the forms we want to compare. This grand matrix has four rows, which we may label a-d, in which the PR sequences are 11 (both present), 10 (P present, R absent), 01 (P absent, R present), and 00 (both absent).
If we wish to refer to this same matrix as our standard framework, for forms with an item of different polarity, we must refer to a different rows. Clearly, notP = 1 is the same as P = 0, and notP = 0 is the same as P = 1; similarly with respect to notR. Thus, the reshuffling of rows is therefore predictable, as follows:
Row in | Row | Sequences for different polarities of items | |||||||
PR matrix | label | PR | notPnotR | PnotR | notPR | ||||
PR | a | 11 | a | 00 | d | 10 | b | 01 | c |
PnotR | b | 10 | b | 01 | c | 11 | a | 00 | d |
notPR | c | 01 | c | 10 | b | 00 | d | 11 | a |
notPnotR | d | 00 | d | 11 | a | 01 | c | 10 | b |
Consider m, for instance. Whereas the summary modus for m_{PR} is abcd = 10.1 (as previously ascertained by macroanalysis, yielding alternative moduses Nos. 10, 12 after unraveling) – for m_{notPnotR} it will be the mirror image dcba = 1.01 (moduses 10, 14); for m_{PnotR} it will be badc = 011. (moduses 7, 8); and for m_{notPR} it will be the mirror image cdab = .110 (moduses 7, 15). That is, knowing the summary modus for m_{PR} to be 10.1 (1 in row a, 0 in row b, · in row c, and 1 in row d), we can predict it for all the other forms of m by merely reshuffling the rows as indicated in the above table. Similarly, with regard to n, p, q.
We can in this manner, without much effort, identify the summary and alternative moduses in a standard two-item grand matrix of the positive generic forms (and thence, if need be, of all other forms, using the processes of negation, intersection and merger). The following table presents the desired information without further ado:
Causation | Prevention | ||||
Determination | Moduses | PR | notPnotR | PnotR | notPR |
m | summary | 10.1 | 1.01 | 011. | .110 |
alternative | 10, 12 | 10, 14 | 7, 8 | 7, 15 | |
n | summary | 1.01 | 10.1 | .110 | 011. |
alternative | 10, 14 | 10, 12 | 7, 15 | 7, 8 | |
p_{abs} | summary | 11.1 | 1.11 | 111. | .111 |
alternative | 14, 16 | 12, 16 | 15, 16 | 8, 16 | |
q_{abs} | summary | 1.11 | 11.1 | .111 | 111. |
alternative | 12, 16 | 14, 16 | 8, 16 | 15, 16 |
All the above table is inferable from the preceding table, given the summary moduses of m and p_{abs}. Notice the identities between the moduses of pairs of forms with different suffixes. Thus, m_{PR} and n_{notPnotR} are identical; as are m_{notPnotR} and n_{PR}; likewise, m_{PnotR} = n_{notPR}, and m_{notPR} = n_{PnotR}. Similarly with regard to the weaks, p_{PR} and q_{notPnotR}, etc. These identities simply signify that, as we already know, these pairs of forms are inverses of each other. Notice also the mirror images (same string in opposite directions), like for example m_{PR} and n_{PR}, which have the same significance.
These equations allow us to see that forms in PR and notPnotR are closely associated, by mirroring; and similarly for forms in PnotR and notPR. Furthermore, that the former and latter pairs are in turn associated, in another sense, insofar as the first and last digits of the summary modus for the one are identical to the middle digits of it for the other, and vice-versa. Clearly, whatever the respective polarities of the items, their relations remain essentially causative.
All these forms therefore embody similar concepts in different guises, signifying various types and degrees of bondage or cohesion between the items concerned; they have common aspects and are all logically or structurally interrelated. They form a family of propositions. We have so far in our study concentrated on items PR or notPnotR, but given little attention to items PnotR or notPR in view of their similarities and the derivability of their logical properties. But now let us look upon them as distinct paradigms.
All these forms may be classified as ‘causative relations’, in the broad sense we ultimately understand for this term. Yet we have in the present study gotten used to a more restrictive sense of the term ‘causation’, as meaning specifically PR or notPnotR relations. Granting this, we need another term to refer specifically to PnotR and notPR relations; and yet another term to refer to the broad, all-inclusive sense.
Therefore, I propose the following convention, in the appropriate contexts. PR or notPnotR causative relations will be called causation (restrictive sense), while PnotR and notPR causative relations will be called prevention[86]. Thus, “P prevents R” is to mean “P causes notR” (still in the restrictive sense of causation). Both causation and prevention are species of causative relations in a broad sense; but when we want to avoid confusion let us call the latter genus of both, say, connection[87].
We would thus say that two items P and R are connected, if either item or its negation causes (in the restrictive sense) or prevents the other item or its negation. And just as causation may vary in determination, i.e. be complete, necessary, partial or contingent – so may prevention be subdivided.
My purpose here is to make the reader aware that when we speak of causation in a wide sense, we must mentally include both causation in a narrow sense and its family relative prevention. Similarly, note well, if we speak of noncausation, we must know whether we mean negation of causation in a restrictive sense (which does not imply negation of prevention) or negation of all causative relation, i.e. of connection (which implies negation of both causation and prevention).
However, before we adopt such loaded terminology, let us examine the relationships involved more closely. As will be seen, we will have to qualify our statement somewhat.
As we stressed from the word go, causation (and similarly, of course, prevention) formally implies the contingency of the items it involves: i.e. each of the items considered separately must be possible but not necessary[88] for a causative relation between them to be conceivable. If one or more of the items involved is/are not contingent, the other item(s) cannot be causing or caused by it. But it does not follow that any two contingent items are causatively related.
Now, according to our analysis so far, the two-item moduses of causation are four, viz. Nos. 10, 12, 14, 16 (and of noncausation are eleven: Nos. 2-9, 11, 13, 15), those of prevention are four, viz. Nos. 7-8, 15-16 (and of nonprevention are eleven: Nos. 2-6, 9-14. Note that these positives have one common modus, No. 16 (1111), which means that causation and prevention are, in this instance (namely, p_{abs}q_{abs}, i.e. absolute pq, note well), overlapping and compatible. It follows that the two-item moduses of connection are seven, viz. Nos. 7-8, 10, 12, 14-16 (and of nonconnection are eight: Nos. 2-6, 9, 11, 13).
Next, look again at Table 12.1 of the previous chapter. The question may well be asked: what is so special about the above-mentioned moduses of connection (as tentatively defined)? That is, what distinguishes them from the moduses of nonconnection? Let us look for an answer in the number of cells coded 1 or 0 in their alternative moduses.
Connection refers to moduses with four 1s (No. 16), three 1s and one 0 (Nos. 8, 12, 14-15), or two 1s and two 0s (Nos. 7, 10). Nonconnection has moduses with two 1s and two 0s (Nos. 4, 6, 11, 13), or one 1 and three 0s (Nos. 2, 3, 5, 9). Thus, though connection is distinguishable by its comprising moduses with three or four 1s, and nonconnection through moduses with only one 1, they both have moduses with two 1s!
However, we need not be surprised or alarmed. For moduses #s 2, 3, 4 mean that P is impossible (they have code 0 for it, with or without R), and moduses #s 5, 9, 13 mean that P is necessary (i.e. that notP is impossible). Similarly, moduses #s 2, 5, 6 mean that R is impossible (coded 0, whether P is present or absent), and moduses #s 3, 9, 11 mean that R is necessary (i.e. that notR is impossible).
Thus, all the moduses of nonconnection refer to situations where one or two items is/are incontingent, which means present or absent (as the case may be) independently of any other item. In its moduses with three zeros (Nos. 2, 3 5, 9), two items are incontingent; in those with two zeros (Nos. 4, 6, 11, 13), one item is incontingent. In contrast, connection never involves an incontingent item.
Therefore, by this reasoning, connection could be conceptually distinguished from nonconnection with reference to the contingency of both items or to the incontingency of one or the other of them, respectively. But this is nonsensical: it would mean that any two contingent items are necessarily causatively related! Clearly, we must have misinterpreted some relevant fact.
It is this: the last modus of any grand matrix, i.e. the modus involving only 1s, i.e. modus #16 in a two-item framework (similarly, modus #256 for three items, or #65,536 for four items), does not necessarily signify causation (or prevention or connection). For no matter whether the items concerned or their negations are together or apart, the combination is always ‘possible’ (i.e. coded 1) in this modus. So we cannot in fact tell with reference to this uniform modus alone whether the items concerned have any impact on each other.
It follows that in this special case, we must interpret the modus as indicative of possible causation (or prevention or connection); but there may also in some cases turn out to be neither causation nor prevention (i.e. nonconnection). That is to say, the last modus (with all 1s) is indefinite with regard to connection (or causation or prevention) or nonconnection (or noncausation or non prevention). The last modus is in all frameworks included in the form p_{abs}q_{abs}, and indeed in c_{abs}, but when we consider more than two items, it is not part of p_{rel}q_{rel}, or of c_{rel}.
This new finding is in agreement with common sense. Taking any two items at random, we cannot reasonably say that they are either (a) both contingent and causatively connected or (b) one or both incontingent and therefore not causatively connected. There is still another possibility: that (c) they are both contingent and yet not causatively connected. This possibility is inherent, as already stated, in the ‘last modus’ of any matrix, which being composed only of 1s, cannot be definitely interpreted one way or the other.
This realization leaves us a window of opportunity for eventual development of a concept of spontaneity (i.e. chance, and perhaps also freewill). For if we are unable to find for some contingent item any other contingent item with which we may causatively relate it in some way, we may be in the long run allowed to inductively generalize from this “failure to find despite due diligence in searching” to a presumed “spontaneity”. Obviously, if we opt for the postulate of a “law of universal causation”, such a movement of thought becomes illicit. But granting that such a law is itself a product of generalization, we have some freedom of choice in the matter. These important insights will naturally affect our later investigations.
(See discussion in Chapter 16.2, including Table 16.1)
3. Items of Negative Polarity in Three-Item Framework.
All the above can be repeated in a three-item framework. In following table, which concerns strongs and absolute weaks (relative weaks will be dealt with further on), the summary moduses are obtained from those given in Table 13.3 above, by expansion[89]; and the alternative moduses are derived from those given in that table, by applying the correspondences between two- and three- item frameworks developed in Table 12.6 of the previous chapter.
Causation | Prevention | |||
Determination | PR | notPnotR | PnotR | notPR |
m | .0.0…. | ….0.0. | 0.0….. | …..0.0 |
34, 36-40, 42, 44-48, 130, 132-136, 138, 140-144, 162, 164-168, 170, 172-176 | 34, 37-38, 50, 53-54, 98, 101-102, 114, 117-118, 130, 133-134, 146, 149-150, 162, 165-166, 178, 181-182, 194, 197-198, 210, 213-214, 226, 229-230, 242, 245-246 | 19-20, 23-32, 67-68, 71-80, 83-84, 87-96 | 19, 25, 27, 51, 57, 59, 67, 73, 75, 83, 89, 91, 99, 105, 107, 115, 121, 123, 147, 153, 155, 179, 185, 187, 195, 201, 203, 211, 217, 219, 227, 233, 235, 243, 249, 251 | |
n | ….0.0. | .0.0…. | …..0.0 | 0.0….. |
34, 37-38, 50, 53-54, 98, 101-102, 114, 117-118, 130, 133-134, 146, 149-150, 162, 165-166, 178, 181-182, 194, 197-198, 210, 213-214, 226, 229-230, 242, 245-246 | 34, 36-40, 42, 44-48, 130, 132-136, 138, 140-144, 162, 164-168, 170, 172-176 | 19, 25, 27, 51, 57, 59, 67, 73, 75, 83, 89, 91, 99, 105, 107, 115, 121, 123, 147, 153, 155, 179, 185, 187, 195, 201, 203, 211, 217, 219, 227, 233, 235, 243, 249, 251 | 19-20, 23-32, 67-68, 71-80, 83-84, 87-96 | |
p_{abs} | …….. | …….. | …….. | …….. |
50, 52-56, 58, 60-64, 98, 100-104, 106, 108-112, 114, 116-120, 122, 124-128, 146, 148-152, 154, 156-160, 178, 180-184, 186, 188-192, 194, 196-200, 202, 204-208, 210, 212-216, 218, 220-224, 226, 228-232, 234, 236-240, 242, 244-248, 250, 252-256 | 36, 39-40, 42, 44-48, 52, 55-56, 58, 60-64, 100, 103-104, 106, 108-112, 116, 119-120, 122, 124-128, 132, 135-136, 138, 140-144, 148, 151-152, 154, 156-160, 164, 167-168, 170, 172-176, 180, 183-184, 186, 188-192, 196, 199-200, 202, 204-208, 212, 215-216, 218, 220-224, 228, 231-232, 234, 236-240, 244, 247-248, 250, 252-256 | 51-52, 55-64, 99-100, 103-112, 115-116, 119-128, 147-148, 151-160, 179-180, 183-192, 195-196, 199-208, 211-212, 215-224, 227-228, 231-240, 243-244, 247-256 | 20, 23-24, 26, 28-32, 52, 55-56, 58, 60-64, 68, 71-72, 74, 76-80, 84, 87-88, 90, 92-96, 100, 103-104, 106, 108-112, 116, 119-120, 122, 124-128, 148, 151-152, 154, 156-160, 180, 183-184, 186, 188-192, 196, 199-200, 202, 204-208, 212, 215-216, 218, 220-224, 228, 231-232, 234, 236-240, 244, 247-248, 250, 252-256 | |
q_{abs} | …….. | …….. | …….. | …….. |
36, 39-40, 42, 44-48, 52, 55-56, 58, 60-64, 100, 103-104, 106, 108-112, 116, 119-120, 122, 124-128, 132, 135-136, 138, 140-144, 148, 151-152, 154, 156-160, 164, 167-168, 170, 172-176, 180, 183-184, 186, 188-192, 196, 199-200, 202, 204-208, 212, 215-216, 218, 220-224, 228, 231-232, 234, 236-240, 244, 247-248, 250, 252-256 | 50, 52-56, 58, 60-64, 98, 100-104, 106, 108-112, 114, 116-120, 122, 124-128, 146, 148-152, 154, 156-160, 178, 180-184, 186, 188-192, 194, 196-200, 202, 204-208, 210, 212-216, 218, 220-224, 226, 228-232, 234, 236-240, 242, 244-248, 250, 252-256 | 20, 23-24, 26, 28-32, 52, 55-56, 58, 60-64, 68, 71-72, 74, 76-80, 84, 87-88, 90, 92-96, 100, 103-104, 106, 108-112, 116, 119-120, 122, 124-128, 148, 151-152, 154, 156-160, 180, 183-184, 186, 188-192, 196, 199-200, 202, 204-208, 212, 215-216, 218, 220-224, 228, 231-232, 234, 236-240, 244, 247-248, 250, 252-256 | 51-52, 55-64, 99-100, 103-112, 115-116, 119-128, 147-148, 151-160, 179-180, 183-192, 195-196, 199-208, 211-212, 215-224, 227-228, 231-240, 243-244, 247-256 |
The negations, intersections and mergers of these forms can easily be worked out, if need arise.
Notice repetitions (there are only eight sets of moduses for sixteen forms); they signify inversions (with change in polarity of both items and change in determination). But more broadly, note well all the compatibilities and incompatibilities between these various forms, which tell us which of them can occur in tandem and which cannot. The following tables, derived from the above, highlight these oppositions for m and p_{abs}; needless to say, similar tables can be constructed for n and q_{abs}, mutatis mutandis.
Forms compared | Compatibility | Common moduses | |
PR | PR | ||
m | m | yes | all 36 |
m | n | yes | the 9 of mn |
m | p_{abs} | no | None |
m | q_{abs} | yes | the 27 of mq_{abs} |
PR | notPnotR | ||
m | m | yes | the 9 of mn |
m | n | yes | all 36 |
m | p_{abs} | yes | the 27 of mq_{abs} |
m | q_{abs} | no | None |
PR | PnotR | ||
m | m | no | None |
m | n | no | None |
m | p_{abs} | no | None |
m | q_{abs} | no | None |
PR | notPR | ||
m | m | no | None |
m | n | no | None |
m | p_{abs} | no | None |
m | q_{abs} | no | None |
Similarly for n, mutatis mutandis. Notice that the forms of strong causation and of prevention have no moduses in common, and are therefore incompatible. But within either causation or prevention, there are certain compatibilities.
Forms compared | Compatibility | Common moduses | |
PR | PR | ||
p_{abs} | m | no | None |
p_{abs} | n | yes | the 27 of np_{abs} |
p_{abs} | p_{abs} | yes | all 108 |
p_{abs} | q_{abs} | yes | the 81 of p_{abs}q_{abs} |
PR | notPnotR | ||
p_{abs} | m | yes | the 27 of np_{abs} |
p_{abs} | n | no | None |
p_{abs} | p_{abs} | yes | the 81 of p_{abs}q_{abs} |
p_{abs} | q_{abs} | yes | all 108 |
PR | PnotR | ||
p_{abs} | m | no | None |
p_{abs} | n | no | None |
p_{abs} | p_{abs} | yes | the 81 of p_{abs}q_{abs} |
p_{abs} | q_{abs} | yes | the 81 of p_{abs}q_{abs} |
PR | notPR | ||
p_{abs} | m | no | None |
p_{abs} | n | no | None |
p_{abs} | p_{abs} | yes | the 81 of p_{abs}q_{abs} |
p_{abs} | q_{abs} | yes | the 81 of p_{abs}q_{abs} |
Similarly for q_{abs}, mutatis mutandis. Notice that the weak forms of causation and prevention have moduses in common, always the same 81, which are none other than the three-item moduses corresponding to the two-item modus No. 16 (see Table 12.6 of the previous chapter). This is consistent with our earlier finding, that p_{abs}q_{abs} has the same modus whatever the polarities of its two items (except where the two forms involved are equivalent).
Now let us consider relative weak determinations, which only arise as of a three-item framework. For each PR sequence, and each determination, there are two complements to consider: both Q and notQ. To identify the alternative moduses of each form, we may proceed by consideration of their summary moduses.
We know, from Tables 11.3 and 11.4 of the chapter on piecemeal microanalysis, the summary modus of p_{PQR} to be “10.1.1..” and that of q_{PQR} to be “..1.1.01”. These are mirror images of each other, note.
Now, the summary moduses of p_{PnotQR} and q_{PnotQR} are bound to have the same numbers of zeros, ones and dots; only they will be in a different order, such that Q = 1 (i.e. Q) and Q = 0 (i.e. notQ) are in each other’s place. If the eight rows of our matrix are labeled a-h, then keeping the values (1 or 0) of P and R constant, row a will be replaced by c, row b will swap places with d, and likewise e with g and f with h. Thus, we can infer the summary moduses of p_{PnotQR} and q_{PnotQR} to be respectively “.110…1” and “1…011.”; once again these are mirrors, notice.
Next consider forms with items PQnotR. Using similar reasoning with regard to the change from R to notR, we can predict the pairs of rows which replace each other to be: a b, c d, e f, and g h. Thus, the summary modus of p_{PQnotR} has to be “011.1…” and that of q_{PQnotR} “…1.110”. Concerning forms with items PnotQnotR, it follows that the summary modus of p_{PnotQnotR} has to be “1.01..1.” and that of q_{PnotQnotR} “.1..10.1”.
Similarly arguing with regard to a change from PQR to notPQR, the pairs are seen to be a e, b f, c g, and d h, so that the summary modus for p_{notPQR} is “.1..10.1” and that of q_{notPQR} is “1.01..1.”. Concerning forms with items notPnotQR, it follows that the summary modus for p_{notPnotQR} is “…1.110” and that of q_{notPnotQR} is “011.1…”.
Finally, the forms p_{notPQnotR} and q_{notPQnotR} may be derived from, say, those with suffix notPQR (by transposition of adjacent rows); which yields summary moduses “1…011.” and “.110…1”. We may thence infer the summary moduses of the forms with items notPnotQnotR, to be “..1.1.01” in the case of p_{notPnotQnotR} and “10.1.1..” for q_{notPnotQnotR}.
We have thus obtained the summary moduses of all forms of p and q for the items concerned, and can now readily unravel and list their respective alternative moduses. The following table, which may be viewed as a continuation of the preceding, is thereby obtained with reference to the three-item grand matrix (see Table 12.3 of the previous chapter).
Causation | Prevention | |||
Determination | PR | notPnotR | PnotR | notPR |
p_{Q} | 10.1.1.. | 1…011. | 011.1… | .1..10.1 |
149-152, 157-160, 181-184, 189-192 | 135-136, 151-152, 167-168, 183-184, 199-200, 215-216, 231-232, 247-248 | 105-112, 121-128 | 74, 76, 90, 92, 106, 108, 122, 124, 202, 204, 218, 220, 234, 236, 250, 252 | |
q_{notQ} | 1…011. | 10.1.1.. | .1..10.1 | 011.1… |
135-136, 151-152, 167-168, 183-184, 199-200, 215-216, 231-232, 247-248 | 149-152, 157-160, 181-184, 189-192 | 74, 76, 90, 92, 106, 108, 122, 124, 202, 204, 218, 220, 234, 236, 250, 252 | 105-112, 121-128 | |
q_{Q} | ..1.1.01 | .110…1 | …1.110 | 1.01..1. |
42, 46, 58, 62, 106, 110, 122, 126, 170, 174, 186, 190, 234, 238, 250, 254 | 98, 100, 102, 104, 106, 108, 110, 112, 226, 228, 230, 232, 234, 236, 238, 240 | 23, 31, 55, 63, 87, 95, 119, 127, 151, 159, 183, 191, 215, 223, 247, 255 | 147-148, 151-152, 155-156, 159-160, 211-212, 215-216, 219-220, 223-224 | |
p_{notQ} | .110…1 | ..1.1.01 | 1.01..1. | …1.110 |
98, 100, 102, 104, 106, 108, 110, 112, 226, 228, 230, 232, 234, 236, 238, 240 | 42, 46, 58, 62, 106, 110, 122, 126, 170, 174, 186, 190, 234, 238, 250, 254 | 147-148, 151-152, 155-156, 159-160, 211-212, 215-216, 219-220, 223-224 | 23, 31, 55, 63, 87, 95, 119, 127, 151, 159, 183, 191, 215, 223, 247, 255 |
The negations, intersections and mergers of these forms, with each other and with strongs, can easily if need arise be worked out.
Notice repetitions (there are eight sets for sixteen forms); they signify inversions (with change in polarity of all three items and change in determination). But more broadly, note well all the compatibilities or incompatibilities between the various forms of relative weak connection, which tell us which of them can occur in tandem and which cannot. The following table shows, for example, which forms can be conjoined or not with p_{PQR}.
Forms compared | Compatibility | Common moduses | |
PR | PR | ||
p_{Q} | p_{Q} | yes | all |
p_{Q} | q_{Q} | yes | 190 |
p_{Q} | p_{notQ} | no | none |
p_{Q} | q_{notQ} | yes | 151-152, 183-184 |
PR | notPnotR | ||
p_{Q} | p_{Q} | yes | 151-152, 183-184 |
p_{Q} | q_{Q} | no | none |
p_{Q} | p_{notQ} | yes | 190 |
p_{Q} | q_{notQ} | yes | all |
PR | PnotR | ||
p_{Q} | p_{Q} | no | none |
p_{Q} | q_{Q} | yes | 151, 159, 183, 191 |
p_{Q} | p_{notQ} | yes | 151-152, 159-160 |
p_{Q} | q_{notQ} | no | none |
PR | notPR | ||
p_{Q} | p_{Q} | no | none |
p_{Q} | q_{Q} | yes | 151-152, 159-160 |
p_{Q} | p_{notQ} | yes | 151, 159, 183, 191 |
p_{Q} | q_{notQ} | no | none |
Similar tables can be constructed in relation to each partial or contingent form, till all conceivable combinations are exhausted, of course[90]. Some of these results are very significant. Look at each case and reflect on its practical meaning for causative reasoning.
For instance, that p_{PQR} and p_{PnotQR} are incompatible, since they have no moduses in common, means that something cannot be a partial cause of something else with both a certain complement (Q) and its negation (notQ) – if it is so with the one, it is certainly not so with the other; on the other hand, p_{PQR} is conjoinable with p_{notPQnotR} or p_{notPnotQnotR}. Or again, causation of form p_{PQR} excludes prevention of form p_{PQnotR} or p_{notPQR}, whereas it may well occur with prevention of form p_{PnotQnotR} or p_{notPnotQR}. And so forth.
4. Categoricals and Conditionals.
Matricial analysis is applicable not only to causative propositions, but to their constituent conditional and categorical propositions. It is a universal method, as already stated. We initially, you will recall, defined causative propositions through specific combinations (conjunctions or disjunctions) of clauses, consisting of positive and negative conditionals and possible categoricals or conjunctions of categoricals.
Thus, for instances, complete causation was defined as the conjunction of “if P, then R”, “if notP, not-then R” and “P is possible”; partial causation as that of “if (P + Q), then R”, “if (notP + Q), not-then R”, “if (P + notQ), not-then R” and “(P + Q) is possible”; and so forth. The negations of these conjunctions of clauses were then definable as inclusive disjunctions the negations of the clauses.
Eventually, we arrived at definitions of such causative propositions through lists of moduses. But each of their constituent clauses can themselves also be defined through moduses, i.e. microanalyzed; their conjunctions are then inferable by intersection and their disjunctions by merger. We could thus have begun our study by microanalyzing the constituent clauses, and then constructed the determinations with reference to their alternative moduses. By doing so, we shall close the circle, and demonstrate the completeness and consistency of the whole system.
Let us begin with categorical propositions.
An item P, whatever its form, can be considered as a categorical proposition in this context. If we construct a one-item grand matrix for it, we obtain the following table:
P | 1 | 2 | 3 | 4 |
1 | 0 | 0 | 1 | 1 |
0 | 0 | 1 | 0 | 1 |
Column No. 1, which states that both P (first row) and notP (second row) are impossible, is an impossible modus, by the laws of logic. Columns 3-4 (in which the first row is coded ‘1’, i.e. possible) represent the proposition “P is possible”, while columns 2, 4 (in which the second row is coded ‘1’, i.e. possible) represent the proposition “notP is possible”. The common modus of these, No. 4, signifies that both P and notP are possible, i.e. that P is contingent[91]; while modus 3 means that only P is possible (i.e. P is necessary) and modus 2 means that only notP is possible (i.e. P is impossible).
We thus see that all modalities are expressed in the grand matrix.
Note that “P is necessary” is equivalent to the proposition “P but not notP”, i.e. it refers to P to the exclusion of notP, or more simply put to “P”. Similarly, “P is impossible” can be written “notP”. We may thus refer to the non-modal forms “P” or “notP” as exclusive categoricals, to distinguish them from the modal forms “P is possible” or “notP is possible”; note well the differences in moduses for them. “P” (modus 3) is included in “P is possible” (moduses 3-4), but more specific in scope.
Let us now consider the moduses of single items within a two-item framework, with reference to Table 12.1 of the previous chapter. They are:
Proposition | Column number(s) | Comment |
(necessarily) P | 5, 9, 13 | Three alternatives. |
possibly P | 5-16 | All alternatives but those ofnotP; i.e. 12 cases. |
(necessarily) notP | 2-4 | Three alternatives. |
possibly notP | 2-4, 6-8, 10-12, 14-16 | All alternatives but those ofP; i.e. 12 cases. |
(necessarily) R | 3, 9, 11 | Three alternatives. |
possibly R | 3-4, 7-16 | All alternatives but those ofnotR; i.e. 12 cases. |
(necessarily) notR | 2, 5-6 | Three alternatives. |
possibly notR | 2, 4-8, 10, 12-16 | All alternatives but those ofR; i.e. 12 cases. |
These results are obtained by reasoning in a similar manner. For instance, for the moduses of “P”, select the columns where the two rows with P = 0 are both coded ‘0’ (namely, Nos. 5, 9, 13); for the moduses of “P is possible”, select the columns where one or both rows with P = 1 is/are coded ‘1’ (namely, Nos. 5-16) or simply negate the three moduses corresponding to “notP”. Similarly with regard to forms concerning item R.[92]
With regard to non-modal (i.e. necessary) conjunctions of (the positive or negative forms of) the items P and R, they may be obtained by appropriate intersections. Thus, for instance, “P and R” (or “PR”), being the conjunction of “P” (moduses 5, 9, 13) and “R” (moduses 3, 9, 11), yields a single common modus, viz. No. 9; and the negation of that conjunction, viz. “not(PR)”, yields the leftover fourteen possible moduses. Similarly in the other cases; the following table lists results for all such cases, for the record[93]:
Proposition | Column number(s) | Comment |
P + R | 9 | One common alternative. |
P + notR | 5 | One common alternative. |
notP + R | 3 | One common alternative. |
notP + notR | 2 | One common alternative. |
not(P + R) | 2-8, 10-16 | All alternatives but that ofPR; i.e. 14 cases. |
not(P + notR) | 2-4, 6-16 | All alternatives but that ofPnotR; i.e. 14 cases. |
not(notP + R) | 2, 4-16 | All alternatives but that ofnotPR; i.e. 14 cases. |
not(notP + notR) | 3-16 | All alternatives but that ofnotPnotR; i.e. 14 cases. |
Note that, since by “PR” we really mean “P is necessary and R is necessary” or “(P + R) is necessary”, as already explained, the negation of such a conjunction, i.e. “not(PR)”, is a modal proposition of the form “(P + R) is unnecessary”.
Regarding modal conjunctions of the form “(P + R) is possible”, they are equivalent to negative conditional propositions, which have the form “if P, not-then notQ”. They will therefore make their appearance, implicitly, in the next table.
Let us now deal with conditional propositions (here logical conditionals, i.e. hypotheticals), whether positive (in the form if/then) or negative (in the form if/not-then). Their alternative moduses are listed in the following table, again with reference to a standard two-item grand matrix (i.e. Table 12.1 of the previous chapter):
Proposition | Column number(s) | Comment |
If P, then R | 2-4, 9-12 | Seven alternatives. |
If P, then notR | 2-8 | Seven alternatives. |
If notP, then R | 3, 5, 7, 9, 11, 13, 15 | Seven alternatives. |
If notP, then notR | 2, 5-6, 9-10, 13-14 | Seven alternatives. |
If P, not-then R | 5-8, 13-16 | All alternatives but those of“if P, then R”, i.e. 8 cases. |
If P, not-then notR | 9-16 | All alternatives but those of“if P, then notR”, i.e. 8 cases. |
If notP, not-then R | 2, 4, 6, 8, 10, 12, 14, 16 | All alternatives but those of“if notP, then R”, i.e. 8 cases. |
If notP, not-then notR | 3-4, 7-8, 11-12, 15-16 | All alternatives but those of“if notP, then notR”, i.e. 8 cases. |
The above information is obtained as follows. Take for instance “if P, then R”; it is understood to mean that the conjunction (P + notR) is impossible. Thus, referring to the said grand matrix, we must select the columns (alternative moduses) in which, for the PR sequence ‘10’ (second row), this single condition is satisfied, i.e. the corresponding cells are coded ‘0’ (impossible). This is true of the columns labeled 2-4, 9-12 (also of column 1, but that one is universally impossible, as we saw); so these are the applicable moduses, which we have listed in the table. The moduses of “if P, not-then R”, meaning that (P + notR) is possible, follow by negation. Similarly in the other cases, mutatis mutandis.[94]
Let us in this context look at the special cases of hypothetical form known as paradoxical propositions.
First consider dilemmatic argument, to which paradoxical propositions may be assimilated. We can use the information in Table 13.12 to analyze it. For instance, if both “if P, then R” and “if notP, then R” are true, the common moduses are 3, 9, 11. The conclusion of such conjunction being “R”, it is clear that “R” must include these three alternative moduses (at least). That is exactly what we found earlier (Table 13.10).
Now look at Table 12.1, in the previous chapter. Rename R as P in this two-item grand matrix. Here, modus 1 is eliminated from the start because the PP sequences 11 and 00 cannot both be impossible (i.e. coded 0), by the law of contradiction. Moduses 3-8, 11-16 are all also eliminated because the PP sequences 10 or 01 cannot be possible (i.e. coded 1), by the law of contradiction. This leaves us only with the alternative moduses 2, 9 11. Given “if notP, then P” (i.e. ‘notP and notP’ is impossible), we can eliminate moduses 2 and 10, leaving modus 9 (= P). Similarly, given “if P, then notP” (i.e. ‘P and P’ is impossible), we can eliminate moduses 9 and 10, leaving modus 2 (= notP).
In this way, paradoxical forms are made perfectly comprehensible under systematic microanalysis.
We can now interrelate the above forms with those of causative propositions, as follows.
Consider first the strong determinations, m and n. We may define m as the intersection of the moduses of “if P, then R” (namely, 2-4, 9-12), those of “if notP, not-then R” (2, 4, 6, 8, 10, 12, 14, 16) and those of “P is possible” (5-16) – which results in the common moduses 10, 12, as previously ascertained. Similarly, mutatis mutandis, for n (moduses 10, 14).
We see from the above table that m implies or is a species of “if P, then R” (which includes both its moduses 10 and 12)[95], is merely compatible with “if notP, then notR” (specifically, in modus 10), and is excluded from “if P, then notR” and from “if notP, then R” (which both lack moduses 10, 12). With regard to the negatives, m implies “if notP, not-then R” and “if P, not-then notR” (the latter implying that P is possible, note), is merely compatible with “if notP, not-then notR” (specifically, in modus 12), and is excluded from “if P, not-then R”. We can similarly compare n.
Concerning now the absolute weak determinations, p_{abs} and q_{abs}. Their moduses are respectively 14, 16 and 12, 16, so evidently neither of them implies a positive conditional proposition. Regarding p_{abs}, it is excluded from three of them (which lack its two moduses) and is merely compatible with the fourth “if notP, then notR” (in modus 14, but not in modus 16). Accordingly, it implies three negative conditionals (which include both its moduses), while being merely compatible with the fourth “if notP, not-then notR” (in modus 16, but not in modus 14). We can similarly compare q_{abs}.
We may therefore at last formally define absolute partial causation p_{abs} as the conjunction of the three negative conditionals (i) “if P, not-then R”, (ii) “if notP, not-then R” and (iii) “if P, not-then notR”, since their intersection results solely in its moduses 14, 16. Similarly, we may define absolute contingent causation q_{abs} as (i) “if notP, not-then notR”, (ii) “if P, not-then notR” and (iii) “if notP, not-then R”, whose common moduses are 12, 16. Note well these are two interesting equations: we had not previously established or even guessed them.[96]
If, by the way, we recall the summary moduses of p_{abs} and q_{abs}, respectively “11.1” and “1.11”, we realize that this is precisely what they mean, since every code “1” signifies that the PR sequence concerned cannot be “0”. Thus, the first “1” means that the sequence PR = 11 is possible, and so that “if P, then notR” is false; the last “1” means that the sequence PR = 00 is possible, and so that “if notP, then R” is false; and similarly for the middle two positions (which differ in the two forms).
We can similarly treat, mutatis mutandis, the negative forms not-m, not-n, not-p_{abs} and not-q_{abs}. This is left to t