THE LOGIC OF CAUSATION

© Avi Sion, 2003. All rights reserved.

Phase Two: Microanalysis

Chapter 12 -  Systematic Microanalysis

1.    Grand Matrices

2.    Moduses in a Two-Item Framework

3.    Catalogue of Moduses, for Three Items  

4.    Enumeration of Moduses, for Three Items.

5.    Comparing Frameworks.

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 2² = 4 rows (with PR sequences 11, 10, 01, 00, conventionally so ordered), and 2⁴ = 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 2³ = 8 rows and 2⁸ = 256 modus columns, conventionally ordered in a similar manner. For four items (P, Q, R, S) we can expect a table with 2⁴ = 16 rows and 2¹⁶ = 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[1] 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:

Table 12.1.            Catalogue of moduses for the four conjunctions of two items (P, R).

** 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.

Table 12.2.     Enumeration of two-item moduses for the strong or absolute weak determinations and their derivatives (form PR).

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_absq_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)[2]. 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_absq_abs), and q_abs = (mq_abs or p_absq_abs).

Again, s = (mn or mq_abs or np_abs), and w_abs = (mq_abs or np_abs or p_absq_abs). Consequently, c_abs = (mn or mq_abs or np_abs or p_absq_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.[3]

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: 

Table 12.3.            Catalogue of moduses for the eight conjunctions of three items (P, Q, R).