www.TheLogician.net © Avi Sion  all rights reserved 
The Logician © Avi Sion All rights reserved

THE LOGIC OF CAUSATION © Avi Sion, 2003. All rights reserved.
Phase Two: Microanalysis Chapter 13  Some More Microanalyses 2.
Items of Negative Polarity in
TwoItem Framework 3.
Items of Negative Polarity in
ThreeItem Framework 4. Categoricals and Conditionals 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 twoitem framework, relatives are of course indistinguishable from absolutes; they arise only as of a threeitem 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 threeitem 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_{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: · malone_{rel} + mq_{rel} (23 + 4) = mq_{abs} (27, i.e. the 36 of m less the 9 of mn); · nalone_{rel} + np_{rel} (23 + 4) = np_{abs} (27, i.e. the 36 of n less the 9 of mn); · palone_{rel} + qalone_{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 palone_{rel} or qalone_{rel}; check it out with reference to the moduses involved. Note that w_{rel} involves only 31 moduses, the 15 of p_{rel} + notq_{rel}, the 15 of notp_{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, notw_{rel} has 224 moduses, including all 120 of notw_{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, notc_{rel }(169 moduses), does not imply negation of absolute causation, notc_{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 + notp_{rel}), (n + notq_{rel}), (notm + p_{rel}), (notn + 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 + notq_{rel}” plus the 12 of “notm + q_{rel}”, a total 48 alternatives; and to “n or p_{rel}”, comprising the 4 moduses of np_{rel} plus the 32 of “n + notp_{rel}” plus the 12 of “notn + p_{rel}”, a total 48 alternatives; as well as to their respective negations, “notm + notq_{rel}” and “notn + notp_{rel}”, which involve 207 moduses each. 2.
Items of Negative Polarity in TwoItem 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 twoitem 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 ad, 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 twoitem 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 twoitem (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 viceversa. 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, allinclusive 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 twoitem moduses of causation are four, viz. Nos. 10, 12, 14, 16 (and of noncausation are eleven: Nos. 29, 11, 13, 15), those of prevention are four, viz. Nos. 78, 1516 (and of nonprevention are eleven: Nos. 26, 914. 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 twoitem moduses of connection are seven, viz. Nos. 78, 10, 12, 1416 (and of nonconnection are eight: Nos. 26, 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 abovementioned 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, 1415), 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 twoitem 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. 3.
Items of Negative Polarity in ThreeItem Framework. All the above can be repeated in a threeitem 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[5]; 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.
Table 13.4. Enumeration of moduses of strong and absolute weak determinations with different polarities of items, with reference to standard threeitem (PQR) grand matrix.
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}, mutadis mutandis. Table 13.5. Oppositions between m_{PR} and the other generic forms.
Similarly for n, mutadis 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. Table 13.6. Oppositions between p_{PR} and the other generic forms.
Similarly for q_{abs}, mutadis mutandis. Notice that the weak forms of causation and prevention have moduses in common, always the same 81, which are none other than the threeitem moduses corresponding to the twoitem 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 threeitem 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 ah, 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 threeitem grand matrix (see Table 12.3 of the previous chapter). Table 13.7. Enumeration of moduses of relative weak determinations with different polarities of items, with reference to standard threeitem (PQR) grand matrix.
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}. Table 13.8. Oppositions between p_{PQR} and the other relative weaks.
Similar tables can be constructed in relation to each partial or contingent form, till all conceivable combinations are exhausted, of course[6]. 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, notthen R” and “P is possible”; partial causation as that of “if (P + Q), then R”, “if (notP + Q), notthen R”, “if (P + notQ), notthen 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 oneitem grand matrix for it, we obtain the following table: Table 13.9. Catalogue of moduses for a single item (P).
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 34 (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[7]; 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 nonmodal 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 34), but more specific in scope. Let us now consider the moduses of single items within a twoitem framework, with reference to Table 12.1 of the previous chapter. They are: Table 13.10. Enumeration of moduses of positive and negative categoricals in a twoitem (PR) framework.
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. 516) or simply negate the three moduses corresponding to “notP”. Similarly with regard to forms concerning item R.[8] With regard to nonmodal (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[9]: Table 13.11. Enumeration of moduses of positive and negative conjunctions in a twoitem (PR) framework.
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, notthen 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/notthen). Their alternative moduses are listed in the following table, again with reference to a standard twoitem grand matrix (i.e. Table 12.1 of the previous chapter): Table 13.12. Enumeration of moduses of positive and negative conditionals in a twoitem (PR) framework.
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 24, 912 (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, notthen R”, meaning that (P + notR) is possible, follow by negation. Similarly in the other cases, mutadis mutandis.[10] 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 twoitem 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 38, 1116 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, 24, 912), those of “if notP, notthen R” (2, 4, 6, 8, 10, 12, 14, 16) and those of “P is possible” (516)  which results in the common moduses 10, 12, as previously ascertained. Similarly, mutadis 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)[11], 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, notthen R” and “if P, notthen notR” (the latter implying that P is possible, note), is merely compatible with “if notP, notthen notR” (specifically, in modus 12), and is excluded from “if P, notthen 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, notthen 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, notthen R”, (ii) “if notP, notthen R” and (iii) “if P, notthen notR”, since their intersection results solely in its moduses 14, 16. Similarly, we may define absolute contingent causation q_{abs} as (i) “if notP, notthen notR”, (ii) “if P, notthen notR” and (iii) “if notP, notthen R”, whose common moduses are 12, 16. Note well these are two interesting equations: we had not previously established or even guessed them.[12] 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, mutadis mutandis, the negative forms notm, notn, notp_{abs} and notq_{abs}. This is left to the reader as an exercise. Note additionally that an exclusive categorical such as “P” (moduses 5, 9, 13) is incompatible with all forms of causation by P (c), since it has no common moduses with them (moduses 10, 12, 14, 16). Causation requires an underlying contingency for the items concerned (in the mode concerned), and is excluded at the outset where there is categorical necessity. Yet, “P” is compatible with “if P, then Q” (moduses 24, 912), for instance; taken together, they yield common modus 9, which means that R is also necessary. All the above modus lists can easily be restated in terms of threeitem moduses, by using Table 12.6 of the previous chapter. For examples, the latter moduses of “P + R” will be 33, 129, 161 (3 alternatives); those of “if P, then R” will be 216, 3348, 129144, 161176 (63 alternatives); and so forth. We may skip indicating all correspondences; the reader is invited to work them out as an exercise. We must, however, examine conjunctives or conditionals with three items in more detail, with reference to a threeitem grand matrix. For this purpose, we need to know the alternative moduses of “P”, “Q”, “R”, and their respective negations. With regard to “P” and “R”, we need only expand the moduses given in Table 13.10 above, using Table 12.6 of the previous chapter. For “Q”, we must in the usual manner refer directly to Table 12.3 of the previous chapter. The results are given in the following table: Table 13.13. Enumeration of moduses of positive and negative categoricals in a threeitem (PQR) framework.
Propositions of the form “possibly P”, etc., can be microanalyzed by negation[13]; they will have 255  15 = 240 alternative moduses, note. By combining the forms in the above table in every which way, we obtain the following results for conjunctions; and by negating the latter, for denials of conjunctions. Table 13.14. Enumeration of moduses of three item positive and negative conjunctives in a threeitem (PQR) framework.
The following table, concerning conditionals and their negations, is constructed with reference to Table 12.3 of the previous chapter, in the usual manner. For instance, “if (P + Q), then R” means that (P + Q + notR) is impossible; therefore, we select the moduses which register a zero along the row for the PQR sequence 110. Similarly in other positive cases; then negatives are derived by listing the leftover moduses in each case. Table 13.15. Enumeration of moduses of three item positive and negative conditionals in a threeitem (PQR) framework.
We can make similar comments here as before, elucidating the oppositions between causative and the less specific forms. This is left as an exercise for the reader. In particular, the reader should compare the moduses of the relative weak determinations, given in Table 13.1 of the present chapter, with those derived from the two above tables and the original definitions of weak causation. For instance, note that “if (P + Q), then R” includes all 16 moduses of p_{PQR} (and so is a genus of it and serves in its definition); similarly for q_{PQR} in relation to “if (notP + notQ), then notR”. Additionally, observe that all the threeitem moduses of “if P, then R” are included by “if (P + Q), then R” (but not viceversa, of course), so that the former is a species of the latter. Note that the PR form is more restrictive with only 63 moduses, while the PQR form is broader in possibilities with 127 moduses. Similarly in other cases. We have thus finished demonstrating that our grand matrices
have universal utility, enabling us to express any form, whatever its breadth or
polarity. We shall now move on to syllogistic applications, and show that all
issues are resolvable by such matricial analysis. [1] We can see here why relative weaks should not be listed in a twoitem framework. In their positive generic forms, they would have the same alternative moduses as the absolute weaks (though in fact, as we know with reference to the threeitem framework, covering only part of these moduses). However, when such twoitem moduses are negated, the similarity between relatives and absolutes would cease, and we would be led astray, unaware that negative relatives are broader than negative absolutes. [2] Any synonym, like hindrance, obstruction, forestalling, inhibition, counteraction, etc., would do as well; though some of these have slightly different connotations  more active or passive, or psychological or ethical, rather than natural, and so forth. Prevention is to be understood in a very general sense, here. [3] I use this term in another (though not unrelated) sense in Future Logic (see p. 124), with reference to conditional propositions. [4] In the mode concerned. [5] The dots in all the summary moduses of this table are of course meant as m  as explained in the chapter on piecemeal analysis, in the section on expansion and contraction. [6] The results are all either explicit or implicit in the above table. [7] Note well that codes 1 and 0 in the moduses here signify possibility and impossibility, respectively. At a deeper level, that of 'radical' moduses, where they acquire the values of presence or absence (see Piecemeal Microanalysis, Section 1), the situation is of course different. In the latter case, modus 11 is impossible by the Law of NonContradiction (P and notP cannot be both present) and modus 00 is impossible by the Law of the Excluded Middle (P and notP cannot be both absent). [8] It follows, incidentally, that the summary modus of P is '..00' (or, more precisely, 'µµ00') and that of R is '.0.0' (or, more precisely, 'µ0µ0'). Similarly for other cases. [9] As regards summary moduses of the positive conjunctions, they are the same as the alternative moduses, since there is only one in each case. Thus, for instance, the summary of PR would be '1000'. [10] The summary moduses can be worked out from the alternative moduses, here too. [11] Clearly, though m is included in "if P, then R", it is not coextensive with it. The mere discovery of an implication does not signify causation; the other conditions have also to be fulfilled. [12] Compare these definitions to those of m, n. Remember, too, that the negation of a conditional may be expressed as a possibility of conjunction. Thus (after reshuffling the three clauses), p_{abs} also means "(P + R) is possible, (notP + notR) is possible, and (P + notR) is possible"; and q_{abs} also means "(P + R) is possible, (notP + notR) is possible, and (notP + R) is possible". In each case, one conjunction remains open. The conjunction of these two forms, p_{abs}q_{abs}, therefore means that all four conjunctions of the items are possible. [13] For instance, "possibly P" is the negation of "necessarily notP", and therefore has moduses 17256.
