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The Logician © Avi Sion All rights reserved

THE LOGIC OF CAUSATION © Avi Sion, 2010. All rights reserved.
Phase Three: Software Assisted Analysis Chapter 20  Concerning Complements.
1. Reducing Numerous Complements to Just Two.To fully understand partial and contingent causation, we need to return to the issue of complementary causes. In chapter 2.3, where these concepts were first introduced, we showed how any number of complements can be reduced to just two. I wish to here review this important doctrine and further develop it with reference to matricial analysis. Why this doctrine is important is worth reiterating here. Remember, we originally defined partial and/or contingent causation with reference to only two complementary causes (say, P and Q) for a certain effect (say, R). Focusing on one of the complements (say, P) as the first item and ‘main’ cause (or the cause mainly of interest to us in a given context), the other complement (here, Q) could be regarded as embodying the ‘surrounding conditions’ for the partial and/or contingent causation of the third item (R). That is, even though the second item Q is in our definition presented as a single item, it is intended to signify any number of ‘surrounding conditions’ Q1, Q2, Q3, etc. The question arises: what is the exact relation between the underlying numerous surrounding conditions, and their single representative or standin term Q? Obviously we mean Q to be a putative ‘collective effect’ of Q1, Q2, Q3... – we conceive of a causative relation between the numerous underlying causes and their single representative. Indeed, very often we invent a new term Q to stand in for a number of terms Q1, Q2, Q3…, viewing Q as an ‘abstraction’ implied by its constituent items Q1, Q2, Q3, etc. We define Q by saying: “let that be the collective effect of Q1, Q2, Q3, etc.” This is indeed one of the usual avenues of concept formation. In other words, Q need not be something concretely observed in isolation, but may be an abstraction produced ad hoc to facilitate a causative statement. It may, however, thenceforth acquire a life of its own in our discourse. For example, the definition of force as mass times acceleration tells us that the abstraction called ‘force’ can be calculated by measuring the mass of the physical body concerned and the acceleration the force is assumed to effect in it, and multiplying the two quantities together; we do not observe and measure ‘force’ as such, but derive it from more concrete items. But henceforth, force becomes an oftused term in other equations, as if it was directly experienced. Thus, a complementary item such as Q may be viewed as itself the effect of a partial and/or contingent causation whose causes are, say, Q1 and Q2. If there are more than these two underlying items, then one of them (say Q2) may in turn be viewed as the product of two deeper items, say Q3 and Q4. And so on, successively, till all the relevant surrounding conditions are exhausted. Conversely, any number of ‘underlying items’ or ‘surrounding conditions’ can, by successive mergers of two into one, be represented by just one overall representative item Q. This is at least true theoretically, because this is not really how we proceed in practice. In practice, we rather think more globally, as already described in chapter 2.3 (see there for more details). This is the thought process and inductive method of ‘changing one thing at a time, while keeping all other things equal, and observing the effect of that single change’. This can be described in more formal terms as follows for partial causation:
For contingent causation, similar clauses can be used, with the polarities of all the terms involved must be reversed. Thus, the first clause would be: ‘If (notQ1 + notQ2 + notQ3 + …), then notQ’, and so on. That is to say: where Q is a partial cause, with P of R, then Q1, Q2, Q3, etc. are in turn partial causes of Q; and where Q is a contingent cause, with P of R, then Q1, Q2, Q3, etc. are in turn contingent causes of Q. In cases where both partial and contingent causation are involved, Q is the collective effect of the underlying items Q1, Q2, Q3, etc., on both the positive and negative sides. We can also describe such multiple weak causations by means of nesting; this is of course the meaning of the successive reductions of many conditions to just one mentioned above. Nesting of ‘if (Q1 + Q2 + Q3 + …), then Q’ would have the form ‘if (Q1 + (Q2 + (Q3 + …))), then Q’; and similarly for the other clauses. But to repeat, the nesting approach complicates matters perhaps unduly, and is here mentioned just to promote theoretical understanding rather than as a practical means. Of course, nothing forces us to limit ourselves to just two complementary causes, other than the limitations in our computer (hardware and software) resources. A large organization, such as WHO (the World Health Organization), for which I worked for some years long ago, does I think have the computer, personnel and financial resources to investigate any number of complementary causes of any health factor or disease, social problem or solution, or whatever. Note that nothing is said or intended here with regard to the relative part played by the various complements in causing the effect concerned. The quantitative role of these factors is not being examined here, only the fact that they are factors in the causation. They may be very widely different in their degrees of involvement; one complement may be the major determinant, while the other(s) is/are minor factors, or they may all be more or less on an equal footing[1]. I leave aside the issue of proportion, here – without, however, intending to deny its great importance. It is, rightly, regarded as crucial in modern science, but our study here is only concerned with the ‘whether’, not the ‘how much’.
2. Dependence Between Complements.While on the topic of complementary causes, a question worth asking is: what of interconnected complements – i.e. can the two complements be causatively related at all, or are they always independent of each other? That is, given that P, Q are partial and/or contingent causes of R, does it follow that P, Q are unconnected (i.e. compatible every which way, as explained in the previous chapter) – or may they in some cases be connected? This question can be answered by looking at the 3item moduses found in relative partial and contingent causation and prevention (90 moduses, all told), and seeing whether any of them signify some implication between the complements P, Q and/or their negations. It is found that in 32 cases, one or the other of the four possible implications are involved (i.e. 8 cases for each, symmetrically), the remaining 58 cases signifying only that the various conjunctions of the complements and their negations are just contingent. The list of cases concerned may be seen in Table 20.1, posted at the website:
Let us take one of these cases for further examination, say modus 23 (ditto for 31, 55, 63). This concerns relative contingent prevention, i.e. “P with complement Q is a contingent cause of notR”, whose clauses are “if notP and notQ then R; if P and notQ, notthen R; if notP and Q, notthen R; and notP and notQ is possible”. Now, according to our table, modus 23 also corresponds to “P and Q is impossible (i.e. if P, then notQ)”. As can be seen, these two propositions are not in conflict; meaning that the relation of dependence between P and Q does not impinge on their stated causative relation to notR (which nowhere mentions “P and Q”). Obviously, this is not the sort of dependence we are looking for; we seek implications between the complements that affect the causative relation to the third item somewhat. Similarly, modus 42 (ditto for 46, 58, 62), which refers to relative contingent causation, i.e. “P with complement Q is a contingent cause of R” and to “P implies notQ” has no notable impact. And likewise, mutadis mutandis, in six other sets of four moduses (I won’t bother listing them; their first ones are highlighted on the said table). So in fact, by this simple method, we find no significant dependence between the complements, i.e. one having a differential impact on the relative causation or prevention concerned. Obviously, when such generic relative causations or preventions are conjoined to strong determinations, nothing is changed, since the latter involve only two items (e.g. P and R). However, what happens when they are conjoined to each other (when compatible, of course)? We know that this option (i.e. pq rel to Q or notQ, for R and notR) concerns only four 3item moduses altogether, viz. 127, 190, 220, 232; and these, as our table shows, are not among the 32 relevant moduses; therefore, no significant impact arises here, either. Thus, to conclude, although conjunctions of the complements (P, Q, notP, notQ) are not always possible, the cases where they are impossible do not affect the causative relation concerned. Note that this only concerns implication and does not exclude the possibility that the complements might have a weaker causative relation mediated by some additional item. This issue might be further investigated using the 4item matrix, but I will not attempt it here, having shown how the job can be done. There is no real need, for this investigation is moved merely by curiosity – since all the valid moduses of forms generated by matricial analysis are thereby known to involve no internal inconsistency.
3. Exclusive Weak Causation.Another, more valuable, investigation I wish to launch here is into the features of exclusive causation. With regard to the strong determinations, this would take the forms: “If and only if P, then R” or “If and only if notP, then notR”. If we think about them, we realize these both mean “If P, then R; and if notP, then notR” – i.e. mn. Nothing new here, since we have already studied the properties of mn in considerable detail. I would just take this opportunity to remind readers of the danger of ambiguity when we say: “Only if P, R” or “Only if notP, notR” (notice my removal of the words “if and” and “then”). Though statements of this sort often signify exclusive strong causation in the sense just defined (i.e. “if and only if –, then –), often what is intended is much weaker, namely: “If P, possibly R”; and if notP, necessarily notR” and “If notP, possibly notR; and if P, necessarily R”, respectively. In such cases, we are only informing that the consequent R (or notR, as the case may be) is only possible with the antecedent P (or notP) – but we are not claiming that P brings about R (or notP brings about notR). For this reason, it is wise to use the more precise wording (which modern logicians abbreviate to “iff –, then –”[2]). Let us now turn our attention to the weak determinations, and ask what is meant by “If and only if P and Q, then R” or “If and only if notP and notQ, then notR”, which forms we will respectively label as p ex and q ex (or p_{ex} and q_{ex}) – the suffix ‘ex’ standing for exclusive, of course. Note my use here of the harder “iff” sort of exclusion just explained; also, to avoid all ambiguity, note that our intent here is to apply this operator to the conjunction (P and Q) or (notP and notQ), and not merely to the first mentioned complement (i.e. to P or notP). Thus, what I have in mind is, roughly put, the following propositions:
I have (for our purposes here) numbered these forms 7 and 8, to indicate continuation of the list given in Chapter 19.1. They are necessarily ‘relative’ (i.e. have at least 3 items); they do not have ‘absolute’ (2item) versions. Needless to say, the number of complements involved in them need not only be two; any number might be considered, but we shall here focus our investigation on just two complements as usual, so that we can refer to 3item matricial analysis to answer questions that arise. Clearly, the second clauses of forms 7 and 8 can each be expanded into three clauses, as can be proved by means of syllogisms using the clause not(P + Q) or not(notP + notQ) as our middle thesis as the case may be. Furthermore, though we do not mention this above, each implication in causation has a base (i.e. the possibility that the three terms it mentions be conjoined). Thus, each of the above two forms could have been defined more precisely and usefully with reference to eight clauses, as follows:
Now, these definitions show us that of the eight possible combinations of P, Q, R and their negations, four combinations are impossible and four others are possible, in each form. We see that, although some clauses are identical in both the forms p ex and q ex, there are serious conflicts between them; namely, clauses b and c of each are incompatible with clauses i, h respectively of the other. Thus, these two forms are contrary and can never be conjoined as pq ex for the same items PQR. This is a reasonable result, the essence of the forms p ex and q ex being that they mimic completenecessary causation (mn), with reference to more than two items; they are thus intermediate degrees of causation, behaving somewhat like strongs and somewhat like weaks. Let us now compare these forms to their predecessors, listed in Chapter 19.1. We see that, as we would expect, p ex is incompatible with m (see 1a and 7i) and q rel (see 4b and 7c, 4c and 7b), but compatible with n and p rel (they have no conflicting clauses). Indeed, p ex implies n since (7b + 7d) = 2a[3], 7g => 2b, and 7h or 7j => 2c; and p ex also implies p rel since 7a = 3a, 7h = 3b, 7i = 3c, and 7g = 3d. Whence it follows that p ex implies the joint form np rel. Similarly, q ex is incompatible with n and p rel, but compatible with m and q rel. Indeed, q ex implies m and q rel, i.e. q ex implies mq rel. Can we now prove the converse, i.e. that np rel implies p ex and that mq rel implies q ex? The answer is no! The clauses 7c and 7j cannot be drawn from np rel; and similarly, the clauses 8c and 8j cannot be drawn from mq rel. Therefore, the forms p ex and q ex are in fact stronger determinations than the specific forms np rel and mq rel (and not just stronger than the generic forms p rel and q rel). The next obvious question is: what are the oppositions between these various forms when the items concerned are given different polarities? This is best investigated more mechanically by means of matricial analysis. The results are given in the following table, which can be viewed at the website:
The results of this table are interesting, since they show that each of these exclusive forms yields only one modus. Thus, the above mentioned two initial forms p ex and q ex have respectively modus #s 150 and 170. This is comparable in degree of specificity to the single modus of pq rel. The full list of forms and their corresponding moduses can be summarized as follows:
Summary of Table 20.2 – Moduses of exclusive weak forms.
Notice that the inverses have the same items with opposite polarities; and that the modus of their form p ex becomes that of q ex, and vice versa. Now, the fact that each of these exclusive forms is expressive of only one modus should be useful for working out their oppositions and interpretations. For a start, we note that all 8 main forms are contrary to each other, since they have no modus in common; the inverses are of course their respective equivalents, with the already stated changes. For the rest, our macroanalysis above seems after all to suffice; microanalysis adds nothing much more. For instance, regarding modus 150 we see, in Table 18.6, pages 34, that it is one of four moduses (the others being 149, 181, 182) which mean “np rel to Q (and nalone rel to notQ only) in causation”. This does not mean that modus 150 is identical to the other three, but only that it has this common implication, i.e. np rel (etc.) which we knew already (save for the implied lone). What this does tell us, however, is that our interpretations of the moduses thus far were somewhat lacking, since they reveal no difference between the more restrictive exclusive forms (like #150)) and their more ordinary cousins (viz. #s 149, 181, 182, in this example). This shows that our introduction of these additional specifications was useful and important. Upon reflection, we should have expected the exclusive forms to be represented by only one modus, since they are defined by eight clauses! Indeed, any modus could be represented in words by eight clauses concerning the possibility or impossibility of each combination of the three items and their negations. The peculiarity of the exclusive forms is that they do this succinctly and are popularly used.
4. The Need for an Additional Item (or Two).The important thing to note in the first section of the present chapter is that our 3item format of partial and/or contingent causative propositions was from the start intended to cover all eventual numbers of complements. We have not used it as merely the simplest, most accessible, format – but as an allinclusive format, to which all other weak causations can in principle be reduced when necessary. Thus, our investigation into the logic of causation with reference to only one complement Q (to P in the causation of R) was not intended to be supplemented later by consideration of more and more complements. Three items were supposed to do the trick. Why then do we need to consider a fourth (and even possibly a fifth) item, now, in phase III ? For the simple reason that, when we consider causative syllogism we must look into cases with the major and/or the minor premise involving a complement. Since the minor, middle and major terms of our syllogism already take up three items (P, Q, R), we need an additional item S (and maybe even two of them, S and T) to investigate syllogisms with one (or both) premises about relative weak causation.[4] Note well that eventual 4item (or even 5item) syllogisms are all composed of 3item propositions (at least, as regards their premises, though some conclusions may conceivably involve four items). A syllogism requires at least three terms (the major, the middle and then minor) deployed in two premises (the major and minor premises, which share the middle term) and a conclusion (which relates the major and minor terms). This allows for only two terms per proposition. If one (or both) of the premises has a third term (i.e. a complement of weak causation), then the syllogism will have four (or respectively, five) terms. The conclusion will then be expected to have a third (and even fourth) term. Based on past experience with syllogistic reasoning, we certainly need at least one additional item, the fourth (or subsidiary) term S; for we can well expect a weak premise combined with a strong one to yield a weak conclusion. Regarding a possible fifth item T, it is probable that we do not need one, because it is unlikely that two weak premises can yield any conclusion at all; but this must of course be formally established in some way (and I doubt any way other than microanalysis can do the trick). The introduction of a fourth item (S) means dealing with a grand matrix of 65,536 moduses each of which is defined by 16 digits; this is in the realm of the possible given my current computer resources (hardware and software capabilities) – just about. But these material resources are quite insufficient to deal with a fifth item (T), which would require a grand matrix of 2^{32} = 4,294,967,296 moduses of 2^{5} = 32 digits each; therefore I can only speculate about the probable results of a study of the latter. More will be said concerning the fourth item S in the next chapter, when we consider 4item syllogisms.
[1] Eventual variations in proportions in time and/or space should, I think, be considered as due to more phenomenal underlying factors. For example, the ‘age’ of an organism may be a causal factor; but the significance of aging at the cellular level or deeper would have to be investigated to understand why ‘time’ seems to play a role. [2] This valuable word, “iff”, has unfortunately not passed over into general usage. The reason for that is, I think, obvious: it is a word that is distinguishable in written language, but not in spoken language. [3] This is easily established by dilemmatic argument : given “if (notP + Q), then notR” and “if (notP + notQ), then notR”, the conclusion is “if notP, then notR”. [4] The subsidiary term (S) is mentioned in phases I and II in the following places: chapters 5.3 and 9.4, where the various possible subfigures of the syllogism are tabulated; chapters 14.3 and 15.1, where it is stressed that the syllogisms here developed are not 4item ones – i.e. that their full elucidation requires 4item research; chapter 16.1, where the problem and the way to the solution of 4item syllogism are presented.
