THE LOGIC OF CAUSATION

© Avi Sion, 2003. All rights reserved.

Phase Two: Microanalysis

Chapter 13 -  Some More Microanalyses

1.    Relatives Weaks

2.    Items of Negative Polarity in Two-Item Framework

3.    Items of Negative Polarity in Three-Item Framework

4.    Categoricals and Conditionals

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.

Table 13.1.     Enumeration of three-item moduses for the relative weak determinations and their derivatives (form PQR).

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.[1]

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_relq_rel has only 1 modus, in contrast to the 81 of p_absq_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_relq_rel (11 + 11 + 1 = 23) imply p_absq_abs (81).

Thus, whereas w_abs = mq_abs or np_abs or p_absq_abs, we must equate w_rel to mq_rel or np_rel or p_relq_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_relq_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:

Table 13.2.            Row references in a standard (PR) matrix for different polarities of items.

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:

Table 13.3.     Enumeration of moduses of positive generic forms with different polarities of items, with reference to standard two-item (PR) grand matrix.

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[2]. Thus, “P prevents R” is to mean “P causes notR” (still in the restrictive sense of causation). And just as causation may vary in determination, i.e. be complete, necessary, partial or contingent - so may prevention be subdivided.

Clearly, causation and prevention are both species of ‘causative relations’ in a broad sense. To avoid confusion, we could call the latter genus of both, say, connection[3]. 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.

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[4] 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_absq_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 thr