THE LOGIC OF CAUSATION
Phase One: Macroanalysis
Chapter 8 – Matricial Analyses.
2. Crucial Matricial Analyses in Figure 1.
3. Crucial Matricial Analyses in Figure 2.
4. Crucial Matricial Analyses in Figure 3.
We will in this chapter show the matricial analyses on the basis of which moods were declared valid or invalid in previous chapters.
Now, the matrix underlying a syllogism may be defined as a table with a listing of all conceivable conjunctions of all the items involved in its premises and conclusion and/or the negations of these items. Thus, for instance, the matrix of three items P, Q, R, will look like this:
P  Q  R 
P  Q  notR 
P  notQ  R 
P  notQ  notR 
notP  Q  R 
notP  Q  notR 
notP  notQ  R 
notP  notQ  notR 
(With four items, the table would be twice as long; with five items, four times as long; and so on.)
Briefly put, matricial analysis is a process which seeks to answer, for each of the conceivable conjunctions in the matrix, the question as to whether it is implied impossible or possible or neither; in the latter case, if the conjunction is neither implied impossible nor implied possible, it is declared open. In other words, matricial analysis is a pursuit of the ‘modus’ of the matrix.
The answers to this question for each row must be derived from the premises, singly or together, by established means; the conclusion is valid only if it may be entirely (with all its implicit clauses) constructed from these answers. As we shall see, this process relies heavily on paradoxical logic, or more simply put, on dilemmatic argument.
The process is, as you will presently discover, long and difficult. I have unfortunately found no better shortcut, maybe other logicians have or eventually do. However, its advantage over reduction is that it provides us with sure results; for with reduction we cannot always be sure to have applied all possible means, whereas with a matricial analysis we know we have exhausted the available information. We are free to choose the appropriate method in each case: certain crucial syllogisms are best subjected to matricial analysis, and then we can use reduction to derive others.
To begin with, here is a step by step description of this method of evaluation. The reader is requested to follow the procedure concretely by referring to one of the examples given in the following sections. It is much less complicated than it sounds.
a. Write down the premises constituting the mood and the putative conclusion(s) to be evaluated. Translate all these causative propositions into conditional or conjunctive propositions, i.e. make their constituent clauses (as elucidated in the chapter on the determinations of causation) explicit. Number the clauses involved for purposes of reference (Roman numerals are used for this, here).
b. Construct a table with a matrix involving all the items and negations of items concerned, in orderly sequences. If there are three items (P, Q, R), the table will have 2^{3} = 8 rows; if there are four items (P, Q, R, S), it will have 2^{4} = 16 rows.
c. Consider first all the positive conditional propositions found in the premises. Every causative proposition contains at least one positive conditional clause; therefore, there will be at least two such clauses per mood. These tell us which conjunctions in the matrix (rows) are implied impossible.
In a threeitem matrix, each such statement (e.g. ‘if P, then Q’, which means ‘P+notQ is impossible’) will imply two rows to be impossible (namely, ‘P+notQ+R’ and ‘P+notQ+notR’). Similarly, in a fouritem matrix, each if/then statement involving two items will imply four rows impossible; while each if/then statement involving three items will imply two rows impossible.
d. Only thereafter, deal with the remaining clauses found in the premises (because their impact will depend on the results of the preceding step), which imply the possibility of certain conjunctions of two or three items in the matrix.
These include negative conditional propositions (e.g. ‘if P, notthen Q’, which means ‘P+notQ is possible’); as well as bare statements of possibility of an item or of a conjunction of items. The latter are to be enlarged with reference to the corresponding positive conditional proposition (e.g. ‘P is possible’ and ‘if P, then Q’, together imply ‘P+Q is possible’).
In a threeitem matrix, a possibility of conjunction of two items will imply that at least one of two rows is possible. In a fouritem matrix, a possibility of conjunction of two items will imply that at least one of four rows is possible; while a possibility of conjunction of three items will imply that at least one of two rows is possible.
Note this well: whereas the impossibility of a conjunction entails the impossibility of all its expressions in the matrix, the possibility of a conjunction is satisfied by only one expression. Thus, the knowledge that two or more rows are collectively possible does not settle the question of the possibility of each of these rows individually.
Only if all but one of these rows are declared impossible by other means (i.e. the preceding step of the procedure), can we declare the remaining one possible. Otherwise, if two or more rows are left unsettled, they must each be considered ‘open’ (i.e. ‘possible or impossible’). That is, even though we know that at least one of them must be possible, we cannot specify which one.
e. When all the information implicit in the premises has been thus systematically included in the table, we can evaluate the putative conclusion(s). Taking one of the clauses at a time, check out whether it can be inferred from the table.
If the clause in question is a positive conditional, every row corresponding to it in the matrix must have been declared impossible to allow us to accept the clause as implied. If the clause in question is a negative conditional or bare statement of possibility, it suffices that one row in the matrix has been declared possible, even if the other(s) was/were declared impossible or left open. (Often, the last clause of the putative conclusion can be inferred directly from a premise, note.)
f. If, and only if, all the clauses of the putative conclusion are thus found to be implied by the data in our table, we may admit that conclusion to be drawable from the premises. If any clause(s) of the putative conclusion is/are left open or worse still denied, by the table, that conclusion must be declared a nonsequitur or antinomy, respectively.
A computer could be programmed to carry out this evaluation process. Once it is understood, it requires no great intelligence to perform. It is tedious detail work, no more.[1]
What is matricial analysis, essentially? A causative proposition is a complex of simpler statements, which affirm the impossibility or possibility of certain conjunctions of items or individual items. But causative propositions differ in their forms and implications, so that comparisons between them are difficult. By recapitulating or recoding all the information in a table, we are better able to judge their mutual impact. The matrix is the common denominator of these disparate forms. The annotations down the comments column of the table comprising it, record the answers to the question we must settle for each row (or conjunction of all the items involved): is this possible, impossible or open (i.e. unsettled)?
The premises collectively structure this table, filling in all or only some of the answers to the question. The mutual impact of the determinations of the premises produces the result. This in turn allows us to judge, with absolute certainty, the logical impact on any of the four putative generic conclusions, and thence evaluate the validity or invalidity of that conclusion.
The main form of reasoning used in matricial analysis is dilemmatic argument – that is, we use paradoxical logic (the branch of logic concerned with paradoxes[2]). This is clear in the above account.
First, with regard to expansion: for example, knowing that P+Q is impossible, we can infer that P+Q+R and P+Q+notR are both impossible, or again knowing that P+Q+R is impossible, we can infer that P+Q+R+S and P+Q+R+notS are both impossible. And conversely, regarding contraction: we can infer a twoitem impossibility from two threeitem ones or a threeitem impossibility from two fouritem ones.
Likewise, by contraposition, when we argue from a twoitem or threeitem possibility to the possibility of at least one of two contrary conjunctions (e.g. from P+Q is possible, to P+Q+R and/or P+Q+notR is/are possible; or from P+Q+R is possible, to P+Q+R+S and/or P+Q+R+notS is/are possible), we rely on dilemma.
Such reasoning is especially productive when, from one clause we know some row(s) of the matrix to be impossible (say, P+Q+R), while from another clause we know that a set of rows including the preceding row(s) is possible (say, P+Q+R and P+Q+notR); for then, if the latter set exceeds the former by only one row, we can infer that remaining row (viz. P+Q+notR, in this example) to be possible.
Matricial analysis can be used to evaluate, and validate or invalidate, any putative conclusion of any mood in any figure of the syllogism (as well as in immediate inferences). It is a universal method.
It could in principle be replaced by use of logical compositions[3]; but we are more likely to be confused in practice by the techniques of symbolic logic; and they are in any event themselves ultimately based on matricial analysis. Similarly, reduction of causative syllogisms to conditional syllogisms[4] is a conceivable method; but in practice we are much more likely to make mistakes with it, so that we will at the end remain uncertain about the reliability of our results; and anyway, matricial analysis is the ultimate basis of conditional syllogism, too.
As we have seen, to avoid having to use matricial analysis everywhere, since it is timeconsuming, it is wise to first identify the minimum number of conclusions, required to be validated or invalidated that way, from which all other conclusions can be derived by direct or indirect reduction. This is the approach adopted here.
Another shortcut resorted to, here, is to identify moods which are ‘mirror images’ of each other, i.e. whose forms are identical in every respect, except that each item in the one is replaced by the contradictory item in the other. In such cases, the matrices of both moods are bound to be identical, except that the polarity of every symbol will be reversed, i.e. notP will replace P wherever it occurs, P will replace notP, and so forth.
Since the great majority of moods have a mirror image (all but four in each figure, viz. mn/mn, mn/pq, pq/mn, and pq/pq), this diminishes the work required by almost half (at most thirty moods per figure, instead of sixty).
We shall in the next three sections evaluate by matricial analysis the positive moods we identified in the preceding chapter as needing evaluation, with respect to some or all (as the case may be) of their conceivable conclusions.
2. Crucial Matricial Analyses in Figure 1.
Evaluation of mood # 117. (Similarly, mutadis mutandis, for mood # 118.)
Major premise: Q is a complete and necessary cause of R:
(i) If Q, then R;
(ii) if notQ, notthen R;
(iii) where: Q is possible.
(iv) If notQ, then notR;
(v) if Q, notthen notR;
(vi) where: notQ is possible.
Minor premise: P (complemented by S) is a partial cause of Q:
(vii) If (P + S), then Q;
(viii) if (notP + S), notthen Q;
(ix) if (P + notS), notthen Q;
(x) where: (P + S) is possible.
Putative conclusion: is P (complemented by S) a partial cause of R?
YES! P is a partial cause of R:
If (P + S), then R  implied by (i) + (vii); 
if (notP + S), notthen R  implied by (iv) + (viii); 
if (P + notS), notthen R  implied by (iv) + (ix); 
where: (P + S) is possible  same as (x). 
Table 8.1. Evaluation of mood # 117.  
P  Q  R  S  implied possible by (i) + (vii) + (x) 
P  Q  R  notS  see (i) + (iii), or (v) 
P  Q  notR  S  implied impossible by (i) 
P  Q  notR  notS  implied impossible by (i) 
P  notQ  R  S  implied impossible by (iv) or (vii) 
P  notQ  R  notS  implied impossible by (iv) 
P  notQ  notR  S  implied impossible by (vii) 
P  notQ  notR  notS  implied possible by (iv) + (ix) 
notP  Q  R  S  see (i) + (iii), or (v) 
notP  Q  R  notS  see (i) + (iii), or (v) 
notP  Q  notR  S  implied impossible by (i) 
notP  Q  notR  notS  implied impossible by (i) 
notP  notQ  R  S  implied impossible by (iv) 
notP  notQ  R  notS  implied impossible by (iv) 
notP  notQ  notR  S  implied possible by (iv) + (viii) 
notP  notQ  notR  notS  see (ii), or (iv) + (vi) 
Evaluation of mood # 124. (Similarly, mutadis mutandis, for mood # 134.)
Major premise: Q is a complete and (complemented by P) a contingent cause of R:
(i) If Q, then R;
(ii) if notQ, notthen R;
(iii) where: Q is possible.
(iv) If (notQ + notP), then notR;
(v) if (Q + notP), notthen notR;
(vi) if (notQ + P), notthen notR;
(vii) where: (notQ + notP) is possible.
Minor premise: P (complemented by S) is a partial and contingent cause of Q:
(viii) If (P + S), then Q;
(ix) if (notP + S), notthen Q;
(x) if (P + notS), notthen Q;
(xi) where: (P + S) is possible.
(xii) If (notP + notS), then notQ;
(xiii) if (P + notS), notthen notQ;
(xiv) if (notP + S), notthen notQ;
(xv) where: (notP + notS) is possible.
Putative conclusion: is P a complete or (complemented by S) a partial cause of R?
NO! P is not implied to be a complete cause of R:
If P, then R?  open; 
if notP, notthen R  impl. by (iv)+(ix), or (iv)+(xii)+(xv); 
where: P is possible  implied by (vi) or (x) or (xi) or (xiii). 
Nor (complemented by S) a partial cause of R:
If (P + S), then R  implied by (i) + (viii); 
if (notP + S), notthen R  implied by (iv) + (ix); 
if (P + notS), notthen R?  open; 
where: (P + S) is possible  same as (xi). 
Table 8.2. Evaluation of mood # 124.  
P  Q  R  S  implied possible by (i) + (viii) + (xi) 
P  Q  R  notS  implied possible by (i) + (xiii) 
P  Q  notR  S  implied impossible by (i) 
P  Q  notR  notS  implied impossible by (i) 
P  notQ  R  S  implied impossible by (viii) 
P  notQ  R  notS  implied possible by (ii) + (iv) + (viii), or (vi) + (viii) 
P  notQ  notR  S  implied impossible by (viii) 
P  notQ  notR  notS  see (ii) or (x) 
notP  Q  R  S  implied possible by (v) + (xii), or (i) + (xiv) 
notP  Q  R  notS  implied impossible by (xii) 
notP  Q  notR  S  implied impossible by (i) 
notP  Q  notR  notS  implied impossible by (i) or (xii) 
notP  notQ  R  S  implied impossible by (iv) 
notP  notQ  R  notS  implied impossible by (iv) 
notP  notQ  notR  S  implied possible by (iv) + (ix) 
notP  notQ  notR  notS  implied possible by (iv) + (xii) + (xv) 
Evaluation of mood # 125. (Similarly, mutadis mutandis, for mood # 136.)
Major premise: Q is a complete and (complemented by S) a contingent cause of R:
(i) If Q, then R;
(ii) if notQ, notthen R;
(iii) where: Q is possible.
(iv) If (notQ + notS), then notR;
(v) if (Q + notS), notthen notR;
(vi) if (notQ + S), notthen notR;
(vii) where: (notQ + notS) is possible.
Minor premise: P is a complete cause of Q:
(viii) If P, then Q;
(ix) if notP, notthen Q;
(x) where: P is possible.
Putative conclusion: is P (complemented by S) a contingent cause of R?
NO! P (complemented by S) is not implied to be a contingent cause of R:
If (notP + notS), then notR?  open; 
if (P + notS), notthen notR?  open; 
if (notP + S), notthen notR  implied by (vi) + (viii); 
where: (notP + notS) is possible  implied by (iv) + (vii) + (viii). 
Table 8.3. Evaluation of mood # 125.  
P  Q  R  S  see (i) + (iii), or (viii) + (x) 
P  Q  R  notS  see (i) + (iii), or (v), or (viii) + (x) 
P  Q  notR  S  implied impossible by (i) 
P  Q  notR  notS  implied impossible by (i) 
P  notQ  R  S  implied impossible by (viii) 
P  notQ  R  notS  implied impossible by (iv) or (viii) 
P  notQ  notR  S  implied impossible by (viii) 
P  notQ  notR  notS  implied impossible by (viii) 
notP  Q  R  S  see (i) + (iii) 
notP  Q  R  notS  see (i) + (iii), or (v) 
notP  Q  notR  S  implied impossible by (i) 
notP  Q  notR  notS  implied impossible by (i) 
notP  notQ  R  S  implied possible by (vi) + (viii) 
notP  notQ  R  notS  implied impossible by (iv) 
notP  notQ  notR  S  see (ii) or (ix) 
notP  notQ  notR  notS  implied possible by (iv) + (vii) + (viii) 
Evaluation of mood # 126. (Similarly, mutadis mutandis, for mood # 135.)
Major premise: Q is a complete and (complemented by S) a contingent cause of R:
(i) If Q, then R;
(ii) if notQ, notthen R;
(iii) where: Q is possible.
(iv) If (notQ + notS), then notR;
(v) if (Q + notS), notthen notR;
(vi) if (notQ + S), notthen notR;
(vii) where: (notQ + notS) is possible.
Minor premise: P is a necessary cause of Q:
(viii) If notP, then notQ;
(ix) if P, notthen notQ;
(x) where: notP is possible.
Putative conclusion: is P a complete or (complemented by S) a contingent cause of R?
NO! P is not implied to be a complete cause of R:
If P, then R?  open; 
if notP, notthen R?  open; 
where: P is possible  implied by (v) + (viii), or (ix). 
Nor (complemented by S) a contingent cause of R:
If (notP + notS), then notR  implied by (iv) + (viii); 
if (P + notS), notthen notR  implied by (v) + (viii); 
if (notP + S), notthen notR?  open; 
where: (notP + notS) is possible?  open. 
Table 8.4. Evaluation of mood # 126.  
P  Q  R  S  see (i) + (iii), or (ix) 
P  Q  R  notS  implied possible by (v) + (viii) 
P  Q  notR  S  implied impossible by (i) 
P  Q  notR  notS  implied impossible by (i) 
P  notQ  R  S  see (vi) 
P  notQ  R  notS  implied impossible by (iv) 
P  notQ  notR  S  see (ii) 
P  notQ  notR  notS  see (ii), or (iv) + (vii) 
notP  Q  R  S  implied impossible by (viii) 
notP  Q  R  notS  implied impossible by (viii) 
notP  Q  notR  S  implied impossible by (i) or (viii) 
notP  Q  notR  notS  implied impossible by (i) or (viii) 
notP  notQ  R  S  see (vi), or (viii) + (x) 
notP  notQ  R  notS  implied impossible by (iv) 
notP  notQ  notR  S  see (ii), or (viii) + (x) 
notP  notQ  notR  notS  see (ii), or (iv) + (vii), or (viii) + (x) 
Evaluation of mood # 127. (Similarly, mutadis mutandis, for mood # 138.)
Major premise: Q is a complete and (complemented by P) a contingent cause of R:
(i) If Q, then R;
(ii) if notQ, notthen R;
(iii) where: Q is possible.
(iv) If (notQ + notP), then notR;
(v) if (Q + notP), notthen notR;
(vi) if (notQ + P), notthen notR;
(vii) where: (notQ + notP) is possible.
Minor premise: P (complemented by S) is a partial cause of Q:
(viii) If (P + S), then Q;
(ix) if (notP + S), notthen Q;
(x) if (P + notS), notthen Q;
(xi) where: (P + S) is possible.
Putative conclusion: is P (complemented by S) a contingent cause of R?
NO! P (complemented by S) is not implied to be a contingent cause of R:
If (notP + notS), then notR?  open; 
if (P + notS), notthen notR  implied by (vi) + (viii); 
if (notP + S), notthen notR?  open; 
where: (notP + notS) is possible?  open. 
Table 8.5. Evaluation of mood # 127.  
P  Q  R  S  implied possible by (i) +(viii) + (xi) 
P  Q  R  notS  see (i) + (iii) 
P  Q  notR  S  implied impossible by (i) 
P  Q  notR  notS  implied impossible by (i) 
P  notQ  R  S  implied impossible by (viii) 
P  notQ  R  notS  implied possible by (vi) + (viii) 
P  notQ  notR  S  implied impossible by (viii) 
P  notQ  notR  notS  see (ii), or (x) 
notP  Q  R  S  see (i) + (iii), or (v) 
notP  Q  R  notS  see (i) + (iii), or (v) 
notP  Q  notR  S  implied impossible by (i) 
notP  Q  notR  notS  implied impossible by (i) 
notP  notQ  R  S  implied impossible by (iv) 
notP  notQ  R  notS  implied impossible by (iv) 
notP  notQ  notR  S  implied possible by (iv) + (ix) 
notP  notQ  notR  notS  see (ii), or (iv) + (vii) 
Evaluation of mood # 128. (Similarly, mutadis mutandis, for mood # 137.)
Major premise: Q is a complete and (complemented by P) a contingent cause of R:
(i) If Q, then R;
(ii) if notQ, notthen R;
(iii) where: Q is possible.
(iv) If (notQ + notP), then notR;
(v) if (Q + notP), notthen notR;
(vi) if (notQ + P), notthen notR;
(vii) where: (notQ + notP) is possible.
Minor premise: P (complemented by S) is a contingent cause of Q:
(viii) If (notP + notS), then notQ;
(ix) if (P + notS), notthen notQ;
(x) if (notP + S), notthen notQ;
(xi) where: (notP + notS) is possible.
Putative conclusion: is P (complemented by S) a contingent cause of R?
YES! P is a contingent cause of R:
If (notP + notS), then notR  implied by (iv) + (viii); 
if (P + notS), notthen notR  implied by (i) + (ix); 
if (notP + S), notthen notR  implied by (v) + (viii), or (i) + (x); 
where: (notP + notS) is possible  same as (xi). 
Table 8.6. Evaluation of mood # 128.  
P  Q  R  S  see (i) + (iii) 
P  Q  R  notS  implied possible by (i) + (ix) 
P  Q  notR  S  implied impossible by (i) 
P  Q  notR  notS  implied impossible by (i) 
P  notQ  R  S  see (vi) 
P  notQ  R  notS  see (vi) 
P  notQ  notR  S  see (ii) 
P  notQ  notR  notS  see (ii) 
notP  Q  R  S  implied possible by (v) + (viii), or (i) + (x) 
notP  Q  R  notS  implied impossible by (viii) 
notP  Q  notR  S  implied impossible by (i) 
notP  Q  notR  notS  implied impossible by (i) or (viii) 
notP  notQ  R  S  implied impossible by (iv) 
notP  notQ  R  notS  implied impossible by (iv) 
notP  notQ  notR  S  see (ii), or (iv) + (vii) 
notP  notQ  notR  notS  implied possible by (iv) + (viii) + (xi) 
Evaluation of mood # 145. (Similarly, mutadis mutandis, for mood # 146.)
Major premise: Q (complemented by S) is a partial and contingent cause of R:
(i) If (Q + S), then R;
(ii) if (notQ + S), notthen R;
(iii) if (Q + notS), notthen R;
(iv) where: (Q + S) is possible.
(v) If (notQ + notS), then notR;
(vi) if (Q + notS), notthen notR;
(vii) if (notQ + S), notthen notR;
(viii) where: (notQ + notS) is possible.
Minor premise: P is a complete cause of Q:
(ix) If P, then Q;
(x) if notP, notthen Q;
(xi) where: P is possible.
Putative conclusion: is P (complemented by S) a partial or contingent cause of R?
NO! P (complemented by S) is not implied to be a partial cause of R:
If (P + S), then R  implied by (i) + (ix); 
if (notP + S), notthen R  implied by (ii) + (ix); 
if (P + notS), notthen R?  open; 
where: (P + S) is possible?  open. 
Nor (complemented by S) a contingent cause of R:
If (notP + notS), then notR?  open; 
if (P + notS), notthen notR?  open; 
if (notP + S), notthen notR  implied by (vii) + (ix); 
where: (notP + notS) is possible  implied by (v) + (viii) + (ix). 
Table 8.7. Evaluation of mood # 145.  
P  Q  R  S  see (i) + (iv), or (ix) + (xi) 
P  Q  R  notS  see (vi), or (ix) + (xi) 
P  Q  notR  S  implied impossible by (i) 
P  Q  notR  notS  see (iii), or (ix) + (xi) 
P  notQ  R  S  implied impossible by (ix) 
P  notQ  R  notS  implied impossible by (v) or (ix) 
P  notQ  notR  S  implied impossible by (ix) 
P  notQ  notR  notS  implied impossible by (ix) 
notP  Q  R  S  see (i) + (iv) 
notP  Q  R  notS  see (vi) 
notP  Q  notR  S  implied impossible by (i) 
notP  Q  notR  notS  see (iii) 
notP  notQ  R  S  implied possible by (vii) + (ix) 
notP  notQ  R  notS  implied impossible by (v) 
notP  notQ  notR  S  implied possible by (ii) + (ix) 
notP  notQ  notR  notS  implied possible by (v) + (viii) + (ix) 
Evaluation of mood # 147. (Similarly, mutadis mutandis, for mood # 148.)
Major premise: Q (complemented by P) is a partial and contingent cause of R:
(i) If (Q + P), then R;
(ii) if (notQ + P), notthen R;
(iii) if (Q + notP), notthen R;
(iv) where: (Q + P) is possible.
(v) If (notQ + notP), then notR;
(vi) if (Q + notP), notthen notR;
(vii) if (notQ + P), notthen notR;
(viii) where: (notQ + notP) is possible.
Minor premise: P (complemented by S) is a partial cause of Q:
(ix) If (P + S), then Q;
(x) if (notP + S), notthen Q;
(xi) if (P + notS), notthen Q;
(xii) where: (P + S) is possible.
Putative conclusion: is P (complemented by S) a partial or contingent cause of R?
YES! P is a partial cause of R:
If (P + S), then R  implied by (i) + (ix); 
if (notP + S), notthen R  implied by (v) + (x); 
if (P + notS), notthen R  implied by (ii) + (ix); 
where: (P + S) is possible  same as (xii). 
NO! P (complemented by S) is not implied to be a contingent cause of R:
If (notP + notS), then notR?  open; 
if (P + notS), notthen notR  implied by (vii) + (ix); 
if (notP + S), notthen notR?  open; 
where: (notP + notS) is possible?  open. 
Table 8.8. Evaluation of mood # 147.  
P  Q  R  S  implied possible by (i) + (ix) + (xii) 
P  Q  R  notS  see (i) + (iv) 
P  Q  notR  S  implied impossible by (i) 
P  Q  notR  notS  implied impossible by (i) 
P  notQ  R  S  implied impossible by (ix) 
P  notQ  R  notS  implied possible by (vii) + (ix) 
P  notQ  notR  S  implied impossible by (ix) 
P  notQ  notR  notS  implied possible by (ii) + (ix) 
notP  Q  R  S  see (vi) 
notP  Q  R  notS  see (vi) 
notP  Q  notR  S  see (iii) 
notP  Q  notR  notS  see (iii) 
notP  notQ  R  S  implied impossible by (v) 
notP  notQ  R  notS  implied impossible by (v) 
notP  notQ  notR  S  implied possible by (v) + (x) 
notP  notQ  notR  notS  see (v) + (viii) 
Evaluation of mood # 152. (Similarly, mutadis mutandis, for mood # 163.)
Major premise: Q is a complete cause of R:
(i) If Q, then R;
(ii) if notQ, notthen R;
(iii) where: Q is possible.
Minor premise: P is a complete and (complemented by S) a contingent cause of Q:
(iv) If P, then Q;
(v) if notP, notthen Q;
(vi) where: P is possible.
(vii) If (notP + notS), then notQ;
(viii) if (P + notS), notthen notQ;
(ix) if (notP + S), notthen notQ;
(x) where: (notP + notS) is possible.
Putative conclusion: is P (complemented by S) a contingent cause of R?
NO! P (complemented by S) is not implied to be a contingent cause of R:
If (notP + notS), then notR?  open; 
if (P + notS), notthen notR  implied by (i) + (viii); 
if (notP + S), notthen notR  implied by (i) + (ix); 
where: (notP + notS) is possible  same as (x). 
Table 8.9. Evaluation of mood # 152.  
P  Q  R  S  see (i) + (iii), or (iv) + (vi) 
P  Q  R  notS  implied possible by (i) + (viii) 
P  Q  notR  S  implied impossible by (i) 
P  Q  notR  notS  implied impossible by (i) 
P  notQ  R  S  implied impossible by (iv) 
P  notQ  R  notS  implied impossible by (iv) 
P  notQ  notR  S  implied impossible by (iv) 
P  notQ  notR  notS  implied impossible by (iv) 
notP  Q  R  S  implied possible by (i) + (ix) 
notP  Q  R  notS  implied impossible by (vii) 
notP  Q  notR  S  implied impossible by (i) 
notP  Q  notR  notS  implied impossible by (i) or (vii) 
notP  notQ  R  S  see (v) 
notP  notQ  R  notS  see (v), or (vii) + (x) 
notP  notQ  notR  S  see (ii) or (v) 
notP  notQ  notR  notS  see (ii), or (v), or (vii) + (x) 
Note that although the above matrix does not show it, the conclusion m of mood 152 (which we uncovered through reduction) is valid. Judging from the last two lines of this matrix, one would think that (notP + notR) is open. However, it is in fact possible. This can be seen as follows.
Clause (ii) above tells us that (notQ + notR) is possible; but two of its possible expressions are implied impossible by (iv); therefore at least one of the remaining two possible expressions has to be possible. This implicit disjunctive result suffices to prove that (notP + notR) is possible, considering that its other two possible expressions are implied impossible by (i).
Compare the matrix of mood 155, whose last line corresponds to the last two lines of the matrix of mood 152. Thus, the above matrix fails to make something significant explicit for us. But this is a mere difficulty of notation, when something about more than one line has to be specified.
The same can be said for the conclusion n of mirror mood 163 and other cases. However, a similar problem does not arise with regard to any of the conclusions tested by matricial analysis in this chapter (as can be verified by reexamining all clauses of tested conclusions which were left open where a possibility was required). So it does not seem worthwhile our trying to remedy this difficulty with more elaborate notational artifices (better than “see (ii)”).
The lesson taught us by this special case is the wisdom of using matricial analysis only for crucial questions and using reduction for all others, as we did. Without awareness of the relation between moods 152 and 155 (or similarly 163 and 166), we might not have spotted the positive conclusion’s validity, unless we had developed a straddling notation.
Evaluation of mood # 153. (Similarly, mutadis mutandis, for mood # 162.)
Major premise: Q is a complete cause of R:
(i) If Q, then R;
(ii) if notQ, notthen R;
(iii) where: Q is possible.
Minor premise: P is a necessary and (complemented by S) a partial cause of Q:
(iv) If (P + S), then Q;
(v) if (notP + S), notthen Q;
(vi) if (P + notS), notthen Q;
(vii) where: (P + S) is possible.
(viii) If notP, then notQ;
(ix) if P, notthen notQ;
(x) where: notP is possible.
Putative conclusion: is P a necessary or (complemented by S) a partial cause of R?
NO! P (complemented by S) is not implied to be a partial cause of R:
If (P + S), then R  implied by (i) + (iv); 
if (notP + S), notthen R?  open; 
if (P + notS), notthen R?  open; 
where: (P + S) is possible  same as (vii). 
Nor a necessary cause of R:
If notP, then notR?  open; 
if P, notthen notR  implied by (i) + (iv) + (vii); 
where: notP is possible  same as (x). 
Table 8.10. Evaluation of mood # 153.  
P  Q  R  S  implied possible by (i) + (iv) + (vii) 
P  Q  R  notS  see (i) + (iii), or (ix) 
P  Q  notR  S  implied impossible by (i) 
P  Q  notR  notS  implied impossible by (i) 
P  notQ  R  S  implied impossible by (iv) 
P  notQ  R  notS  see (vi) 
P  notQ  notR  S  implied impossible by (iv) 
P  notQ  notR  notS  see (ii) or (vi) 
notP  Q  R  S  implied impossible by (viii) 
notP  Q  R  notS  implied impossible by (viii) 
notP  Q  notR  S  implied impossible by (i) or (viii) 
notP  Q  notR  notS  implied impossible by (i) or (viii) 
notP  notQ  R  S  see (v), or (viii) + (x) 
notP  notQ  R  notS  see (viii) + (x) 
notP  notQ  notR  S  see (ii), or (v), or (viii) + (x) 
notP  notQ  notR  notS  see (ii), or (viii) + (x) 
Evaluation of mood # 154. (Similarly, mutadis mutandis, for mood # 164.)
Major premise: Q is a complete cause of R:
(i) If Q, then R;
(ii) if notQ, notthen R;
(iii) where: Q is possible.
Minor premise: P (complemented by S) is a partial and contingent cause of Q:
(iv) If (P + S), then Q;
(v) if (notP + S), notthen Q;
(vi) if (P + notS), notthen Q;
(vii) where: (P + S) is possible.
(viii) If (notP + notS), then notQ;
(ix) if (P + notS), notthen notQ;
(x) if (notP + S), notthen notQ;
(xi) where: (notP + notS) is possible.
Putative conclusion: is P (complemented by S) a contingent cause of R?
NO! P (complemented by S) is not implied to be a contingent cause of R:
If (notP + notS), then notR?  open; 
if (P + notS), notthen notR  implied by (i) + (ix); 
if (notP + S), notthen notR  implied by (i) + (x); 
where: (notP + notS) is possible  same as (xi). 
Table 8.11. Evaluation of mood # 154.  
P  Q  R  S  implied possible by (i) + (iv) + (vii) 
P  Q  R  notS  implied possible by (i) + (ix) 
P  Q  notR  S  implied impossible by (i) 
P  Q  notR  notS  implied impossible by (i) 
P  notQ  R  S  implied impossible by (iv) 
P  notQ  R  notS  see (vi) 
P  notQ  notR  S  implied impossible by (iv) 
P  notQ  notR  notS  see (ii) or (vi) 
notP  Q  R  S  implied possible by (i) + (x) 
notP  Q  R  notS  implied impossible by (viii) 
notP  Q  notR  S  implied impossible by (i) 
notP  Q  notR  notS  implied impossible by (i) or (viii) 
notP  notQ  R  S  see (v) 
notP  notQ  R  notS  see (viii) + (xi) 
notP  notQ  notR  S  see (ii) or (v) 
notP  notQ  notR  notS  see (ii), or (viii) + (xi) 
Evaluation of mood # 155. (Similarly, mutadis mutandis, for mood # 166.)
Major premise: Q is a complete cause of R:
(i) If Q, then R;
(ii) if notQ, notthen R;
(iii) where: Q is possible.
Minor premise: P is a complete cause of Q:
(iv) If P, then Q;
(v) if notP, notthen Q;
(vi) where: P is possible.
Putative conclusion: P is a complete cause of R?
YES! P is a complete cause of R:
If P, then R  implied by (i) + (iv); 
if notP, notthen R  implied by (ii) + (iv); 
where: P is possible  same as (vi). 
Table 8.12. Evaluation of mood # 155.  
P  Q  R  implied possible by (i) + (iv) + (vi) 
P  Q  notR  implied impossible by (i) 
P  notQ  R  implied impossible by (iv) 
P  notQ  notR  implied impossible by (iv) 
notP  Q  R  see (i) + (iii) 
notP  Q  notR  implied impossible by (i) 
notP  notQ  R  see (v) 
notP  notQ  notR  implied possible by (ii) + (iv) 
Evaluation of mood # 171. (Similarly, mutadis mutandis, for mood # 181.)
Major premise: Q (complemented by S) is a partial cause of R:
(i) If (Q + S), then R;
(ii) if (notQ + S), notthen R;
(iii) if (Q + notS), notthen R;
(iv) where: (Q + S) is possible.
Minor premise: P is a complete and necessary cause of Q:
(v) If P, then Q;
(vi) if notP, notthen Q;
(vii) where: P is possible.
(viii) If notP, then notQ;
(ix) if P, notthen notQ;
(x) where: notP is possible.
Putative conclusion: is P (complemented by S) a partial cause of R?
YES! P is a partial cause of R:
If (P + S), then R  implied by (i) + (v); 
if (notP + S), notthen R  implied by (ii) + (v); 
if (P + notS), notthen R  implied by (iii) + (viii); 
where: (P + S) is possible  implied by (i) + (iv) + (viii). 
Table 8.13. Evaluation of mood # 171.  
P  Q  R  S  implied possible by (i) + (iv) + (viii) 
P  Q  R  notS  see (v) + (vii), or (ix) 
P  Q  notR  S  implied impossible by (i) 
P  Q  notR  notS  implied possible by (iii) + (viii) 
P  notQ  R  S  implied impossible by (v) 
P  notQ  R  notS  implied impossible by (v) 
P  notQ  notR  S  implied impossible by (v) 
P  notQ  notR  notS  implied impossible by (v) 
notP  Q  R  S  implied impossible by (viii) 
notP  Q  R  notS  implied impossible by (viii) 
notP  Q  notR  S  implied impossible by (i) or (viii) 
notP  Q  notR  notS  implied impossible by (viii) 
notP  notQ  R  S  see (vi), or (viii) + (x) 
notP  notQ  R  notS  see (vi), or (viii) + (x) 
notP  notQ  notR  S  implied possible by (ii) + (v) 
notP  notQ  notR  notS  see (vi), or (viii) + (x) 
Evaluation of mood # 174. (Similarly, mutadis mutandis, for mood # 184.)
Major premise: Q (complemented by P) is a partial cause of R:
(i) If (Q + P), then R;
(ii) if (notQ + P), notthen R;
(iii) if (Q + notP), notthen R;
(iv) where: (Q + P) is possible.
Minor premise: P (complemented by S) is a partial and contingent cause of Q:
(v) If (P + S), then Q;
(vi) if (notP + S), notthen Q;
(vii) if (P + notS), notthen Q;
(viii) where: (P + S) is possible.
(ix) If (notP + notS), then notQ;
(x) if (P + notS), notthen notQ;
(xi) if (notP + S), notthen notQ;
(xii) where: (notP + notS) is possible.
Putative conclusion: is P (complemented by S) a partial cause of R?
YES! P is a partial cause of R:
If (P + S), then R  implied by (i) + (v); 
if (notP + S), notthen R  implied by (iii) + (ix); 
if (P + notS), notthen R  implied by (ii) + (v); 
where: (P + S) is possible  same as (viii). 
Table 8.14. Evaluation of mood # 174.  
P  Q  R  S  implied possible by (i) + (v) + (viii) 
P  Q  R  notS  implied possible by (i) + (x) 
P  Q  notR  S  implied impossible by (i) 
P  Q  notR  notS  implied impossible by (i) 
P  notQ  R  S  implied impossible by (v) 
P  notQ  R  notS  see (vii) 
P  notQ  notR  S  implied impossible by (v) 
P  notQ  notR  notS  implied possible by (ii) + (v) 
notP  Q  R  S  see (xi) 
notP  Q  R  notS  implied impossible by (ix) 
notP  Q  notR  S  implied possible by (iii) + (ix) 
notP  Q  notR  notS  implied impossible by (ix) 
notP  notQ  R  S  see (vi) 
notP  notQ  R  notS  see (ix) + (xii) 
notP  notQ  notR  S  see (vi) 
notP  notQ  notR  notS  see (ix) + (xii) 
Evaluation of mood # 177. (Similarly, mutadis mutandis, for mood # 188.)
Major premise: Q (complemented by P) is a partial cause of R:
(i) If (Q + P), then R;
(ii) if (notQ + P), notthen R;
(iii) if (Q + notP), notthen R;
(iv) where: (Q + P) is possible.
Minor premise: P (complemented by S) is a partial cause of Q:
(v) If (P + S), then Q;
(vi) if (notP + S), notthen Q;
(vii) if (P + notS), notthen Q;
(viii) where: (P + S) is possible.
Putative conclusion: is P (complemented by S) a partial cause of R?
NO! P (complemented by S) is not implied to be a partial cause of R:
If (P + S), then R  implied by (i)+ (v); 
if (notP + S), notthen R?  open; 
if (P + notS), notthen R  implied by (ii) + (v); 
where: (P + S) is possible  same as (viii). 
Table 8.15. Evaluation of mood # 177.  
P  Q  R  S  implied possible by (i) + (v) + (viii) 
P  Q  R  notS  see (i) + (iv) 
P  Q  notR  S  implied impossible by (i) 
P  Q  notR  notS  implied impossible by (i) 
P  notQ  R  S  implied impossible by (v) 
P  notQ  R  notS  see (vii) 
P  notQ  notR  S  implied impossible by (v) 
P  notQ  notR  notS  implied possible by (ii) + (v) 
notP  Q  R  S 

notP  Q  R  notS 

notP  Q  notR  S  see (iii) 
notP  Q  notR  notS  see (iii) 
notP  notQ  R  S  see (vi) 
notP  notQ  R  notS 

notP  notQ  notR  S  see (vi) 
notP  notQ  notR  notS 

Next section (continuation of same chapter)
[1] Of course, it was not immediately evident; I had to develop the method gradually. The work took me a month or so. I was motivated to do it by the thought that once done by one human being, the formal research would never have to be repeated.
[2] See Future Logic, ch. 31.
[3] See Future Logic, ch. 28.
[4] See Future Logic, ch. 29.