THE LOGIC OF CAUSATION
Phase One: Macroanalysis
Chapter 9 – Squeezing Out More Information
1. The Interactions of Determinations.
3. Negative Conclusions from Positive Moods.
1. The Interactions of Determinations.
Before considering the possibility of other inferences from causative propositions, let us summarize and extend the results obtained thus far, and especially try and understand them in a global perspective. We have in the preceding chapters identified, in the three figures, 66 valid positive conclusions obtainable from positive premises, out of 192 (3*8*8) possible combinations of generic and joint premises. We thus found a validity rate of 34.4% – meaning that reasoning with causative propositions cannot be left to chance, since we would likely be wrong two times out of three! The table shows the distribution of valid and invalid moods in the three figures:
Table 9.1. Valid and Invalid Moods  
Figure  Valid Moods (positive)  Invalid Moods (impossible or nil) 
1  30  34 
2  18  46 
3  18  46 
Total  66  126 
Moreover, not all of the valid moods have equal significance. As the table below shows, some moods (20, shaded) are conceptually basic, while others (46) are mere derivatives of these, in the sense of compounds (16) or subalterns (30) of them. We shall call the former ‘primary’ moods, and the latter ‘secondary’ moods. Note that these terms are not intended as references to validation processes, but to comparisons of results. By which I mean that some of the moods here classed as ‘primary’ (such as #217, to cite one case) were validated by reduction to others; whereas some of the moods here classed as ‘secondary’ (such as #117, for example) were among those that had to be validated by matricial analysis.
A primary mood teaches us a lesson in reasoning. For instance, mood 1/m/m/m (#155) teaches us that in Figure 1, the premises m and m yield the conclusion m. A secondary (subaltern or compound) mood has premises that teach us nothing new (compared to the corresponding primary), except to tell us that no additional information is implied. For instances, 1/m/mq/m (#152) is equivalent (subaltern) to 1/m/m/m; and 1/mn/mn/mn (#111) is equivalent to (a compound of) 1/m/m/m plus 1/n/n/n.
Such equivalencies are due to the fact that the premises of the secondary mood imply those of the primary mood(s), while the conclusion(s) of the latter imply that of the former. We can thus ‘reconstruct’ the derivative mood from its conceptual source(s). Effectively, primary moods represent general truths, of which secondary moods are specific expressions. This ordering of the valid moods signifies that we do not have to memorize them all, but only 20 out of 66.
In the following table, the valid positive moods of causative syllogism are listed for each figure in order of the strength of their conclusions (joint determinations before generics). Within each group of moods yielding a given conclusion, moods are ordered in the reverse order with reference to their premises (the weakest premises capable of yielding a certain conclusion being listed first, so far as possible – some are of course incomparable). Explanations will be given further on.
Primary moods (shaded) are distinguished from compounds and subalterns, and the primary sources of the secondaries are specified. Notice that all moods with a joint determination as conclusion are compounds.
Table 9.2. Valid Positive Moods, Primaries and Secondaries.  
No.  Major  Minor  Conclusion  Relation  to mood 
Figure 1 (12 primaries, 8 compounds and 10 subalterns)  
111  mn  mn  mn  compound  155 + 166 
121  mq  mn  mq  compound  155 + 181 
112  mn  mq  mq  compound  155 + 118 
131  np  mn  np  compound  166 + 171 
113  mn  np  np  compound  166 + 117 
141  pq  mn  pq  compound  171 + 181 
114  mn  pq  pq  compound  117 + 118 
144  pq  pq  pq  compound compound  147 + 148, 174 + 184 
155  m  m  m  primary 

152  m  mq  m  subaltern  155 
125  mq  m  m  subaltern  155 
151  m  mn  m  subaltern  155 
115  mn  m  m  subaltern  155 
166  n  n  n  primary 

163  n  np  n  subaltern  166 
136  np  n  n  subaltern  166 
161  n  mn  n  subaltern  166 
116  mn  n  n  subaltern  166 
147  pq  p  p  primary 

174  p  pq  p  primary 

137  np  p  p  primary 

134  np  pq  p  subaltern  137 or 174 
171  p  mn  p  primary 

117  mn  p  p  primary 

148  pq  q  q  primary 

184  q  pq  q  primary 

128  mq  q  q  primary 

124  mq  pq  q  subaltern  128 or 184 
181  q  mn  q  primary 

118  mn  q  q  primary 

Figure 2 (4 primaries, 4 compounds and 10 subalterns)  
211  mn  mn  mn  compound  256 + 265 
212  mn  mq  mq  compound  265 + 218 
213  mn  np  np  compound  256 + 217 
214  mn  pq  pq  compound  217 + 218 
265  n  m  m  primary 

262  n  mq  m  subaltern  265 
235  np  m  m  subaltern  265 
261  n  mn  m  subaltern  265 
215  mn  m  m  subaltern  265 
231  np  mn  m  subaltern  265 
256  m  n  n  primary 

253  m  np  n  subaltern  256 
226  mq  n  n  subaltern  256 
251  m  mn  n  subaltern  256 
216  mn  n  n  subaltern  256 
221  mq  mn  n  subaltern  256 
217  mn  p  p  primary 

218  mn  q  q  primary 

Figure 3 (4 primaries, 4 compounds and 10 subalterns)  
311  mn  mn  mn  compound  356 + 365 
321  mq  mn  mq  compound  356 + 381 
331  np  mn  np  compound  365 + 371 
341  pq  mn  pq  compound  371 + 381 
356  m  n  m  primary 

353  m  np  m  subaltern  356 
326  mq  n  m  subaltern  356 
351  m  mn  m  subaltern  356 
316  mn  n  m  subaltern  356 
313  mn  np  m  subaltern  356 
365  n  m  n  primary 

362  n  mq  n  subaltern  365 
335  np  m  n  subaltern  365 
361  n  mn  n  subaltern  365 
315  mn  m  n  subaltern  365 
312  mn  mq  n  subaltern  365 
371  p  mn  p  primary 

381  q  mn  q  primary 

As already stated, we need only keep in mind the 20 primaries, the remaining 46 secondaries being obvious corollaries. It is implicitly understood that, had any of the latter been primary (e.g. if 1/m/mq had concluded mq, say, instead of just m), it would have been classified as such among the former.
We can further cut down the burden on memory by taking stock of ‘mirror’ moods. As we can see on the table above, among the primaries (shaded): in Figure 1, mood 166 is a mirror of mood 155, 148 of 147, 184 of 174, 128 of 137, 118 of 117, and 181 of 171. In Figure 2, mood 256 is a mirror of mood 265, and 218 of 217. In Figure 3, mood 365 is a mirror of mood 356, and 381 of 371. In this way, we need only remember 10 primary moods (6 in the first figure, 2 in the second and 2 in the third), and the 10 others follow by mirroring.
To better understand the results obtained, we ought to notice the phenomenon of transposition of determinations in the premises. Moods can be pairedoff if they have the same premises in reverse order. Note that, for each pair, the figure number (hundreds) is the same, while the numbers of the major and minor premises (tens and units, respectively) are transposed.
· Thus, among primaries, we should mentally pair off the following: 147 and 174, 117 and 171, 148 and 184, 118 and 181. In these paired cases, the combination of the determinations involved has the same conclusion, however ordered in the premises. Take, for instance, moods 147 and 174, i.e. 1/pq/p/p and 1/p/pq/p; the conclusion has the same determination p, whether the determinations of the premises are pq/p or p/pq. This allows us to regard, in such cases, the determination of the conclusion as a product of the determinations of the premises, irrespective of their ordering. (We can similarly pair off many secondary moods: for instances, 125 and 152, 115 and 151, etc.)
· In the case of the following transposed pairs, 265 and 256, 356 and 365, the conclusions are of similar strength, but not identical determination. Thus, for instance, 1/n/m/m (#265) and 1/m/n/n (#256) are comparable although only by way of mirroring. (We can similarly pair off some secondaries, like 226 and 262, 235 and 253, etc.)
· Some individual moods have the same determination in both premises, and thus cannot be paired with others. These, we might say, pair off with themselves. Thus with Nos. 155, 166 among primaries; and some likewise among secondaries.
· But, note well, some moods are not similarly paired; specifically, the primary moods 128, 137, 217, 218, 371, 381 are not; similarly some of the secondaries. For instance, mood 1/np/p/p (#137) is valid, but mood 1/p/np/p (#173) is invalid. This teaches us not to indiscriminately look upon the order of the determinations in the premises as irrelevant.
Moreover, transposition of the determinations of the premises should not be confused with transposition of the premises themselves. For if the premises are transposed, the conclusion obtained from them is converted. Additionally, in the case of the first figure, transposition of the premises would take us out of the first figure (into the socalled fourth figure[1]), since the middle item changes position in them. As for the second and third figures, though transposition of premises does not entail a change of figure (the middle item remains in the same position either way), it entails a change of determination in the conclusion (since the items in the premises change place); see for instances moods 256 and 265, or 356 and 365.
Nevertheless, awareness of the phenomenon of transposition of determinations is valuable, because it allows us to make an analogy with composition of forces in mechanics. Syllogism in general may be viewed as a doctrine concerning the interactions of different propositional forms. With regard to the determinations of causation, we learn from the cases mentioned above something about the interactions of determinations, i.e. how their ‘forces’ combine.
We can push this insight further, with reference to the hierarchies between the significant moods (primaries) and their respective derivatives (secondaries). Consider, for instance, the primary mood 1/m/m, which has conclusion m; if we gradually increase the strength of the major premise (to mq or mn), while keeping the minor premise the same (m), or vice versa, the determination of the conclusion remains unaffected (m). In contrast, if we increase the strength of both premises at once to mn, the conclusion increases in strength to mn. Similarly in many other cases. Thus, some increases in strength in the premises produce no additional strength in the conclusion; but at some threshold, the intensification may get sufficient to produce an upward shift in determination.
We can in like manner view changes in conclusion from p to m or from q to n (and likewise to the joint determinations mq or np). For instance, compare moods 1/mn/p/p and 1/mn/m/m; here, keeping the major premise constant (mn), as we upgrade the minor premise from p to m, we find the conclusion upgraded from p to m. Similarly with moods 1/p/mn/p and 1/m/mn/m, keeping the minor constant while varying the major. Let us not forget that the determinations of causation were conceived essentially as modalities: p and m, though defined as mutually exclusive, are meant as different degrees of positive causation; similarly for the negative aspects of causation, q and n. Thus, some such transitions were to be expected.
We can in this way interpret our list of valid moods as a map of the changing topography in the field of determination. This gives us an interesting overview of the whole domain of causation. This is the intent of Table 9.2, above.
Thus far, we have only validated causative syllogisms with positive premises and positive conclusions. We will now look into the possibility of obtaining, at least by derivation from the foregoing, additional valid moods involving a negative premise and, consequently, a negative conclusion.
This is made possible by using Aristotle’s method of indirect reduction, or reduction ad absurdum. To begin with, let us describe the various reduction processes involved. Note the changed positions of items P, Q, R, in each situation. The mood to be validated (left) involves a positive premise (indicated by a + sign) and a negative premise () yielding a negative putative conclusion. The reduction process keeps one of the original premises (the positive one), and shows that contradicting the putative conclusion would result, through an already validated positive mood (right), in contradiction of the other premise (the negative one). Notice the figure used for validation purposes depends on which original premise is the positive one, staying constant in the process.
 Figure 1  Reduction process:  Figure 2 
major premise  +QR  keeping the same major,  +QR 
minor premise  PQ  if we deny the conclusion,  +PR 
conclusion  PR  then we deny the minor.  +PQ 




 Figure 1  Reduction process:  Figure 3 
major premise  QR  if we deny the conclusion,  +PR 
minor premise  +PQ  keeping the same minor,  +PQ 
conclusion  PR  then we deny the major.  +QR 




 Figure 2  Reduction process:  Figure 1 
major premise  +RQ  keeping the same major,  +RQ 
minor premise  PQ  if we deny the conclusion,  +PR 
conclusion  PR  then we deny the minor.  +PQ 




 Figure 2  Reduction process:  Figure 3 
major premise  RQ  keeping the minor as a major,  +PQ 
minor premise  +PQ  if we deny the conclusion,  +PR 
conclusion  PR  then we deny the major.  +RQ 




 Figure 3  Reduction process:  Figure 1 
major premise  QR  if we deny the conclusion,  +PR 
minor premise  +QP  keeping the same minor,  +QP 
conclusion  PR  then we deny the major.  +QR 




 Figure 3  Reduction process:  Figure 2 
major premise  +QR  keeping the major as a minor,  +PR 
minor premise  QP  if we deny the conclusion,  +QR 
conclusion  PR  then we deny the minor.  +QP 
Consider, for instance, a first figure syllogism QR/PQ/PR, which we wish to reduce ad absurdum to a second figure syllogism of established validity. Knowing that the given major premise (QR), and the negation of the putative conclusion (PR), together imply (in Figure 2) the negation of the given minor premise (PQ) – we are logically forced to admit the putative conclusion from the given premises (in Figure 1). Similar arguments apply to the other three cases, as indicated above.
Using these reduction arguments, we can validate the following moods, in the three figures. In the following table, all I have done is apply indirect reduction to the primary moods listed in Table 9.2. I ignored all subaltern and compound moods in it, since they would only give rise to other derivatives.
Table 9.3. Valid Negative Moods, Primaries only.  
Major  Minor  Conclusion  Source 
Figure 1 from Figure 2 – keep same major  
n  notm  notm  265 
m  notn  notn  256 
mn  notp  notp  217 
mn  notq  notq  218 
Figure 1 from Figure 3 – keep same minor  
notm  n  notm  356 
notn  m  notn  365 
notp  mn  notp  371 
notq  mn  notq  381 
Figure 2 from Figure 1 – keep same major  
m  notm  notm  155 
n  notn  notn  166 
mn  notp  notp  117 
np  notp  notp  137 
pq  notp  notp  147 
mn  notq  notq  118 
mq  notq  notq  128 
pq  notq  notq  148 
p  notp  not(mn)  171 
q  notq  not(mn)  181 
p  notp  not(pq)  174 
q  notq  not(pq)  184 
Figure 2 from Figure 3 – keep the minor as a major  
notn  n  notm  365 
notm  m  notn  356 
notp  p  not(mn)  371 
notq  q  not(mn)  381 
Figure 3 from Figure 2 – keep the major as a minor  
n  notn  notm  256 
m  notm  notn  265 
p  notp  not(mn)  217 
q  notq  not(mn)  218 
Figure 3 from Figure 1 – keep same minor  
notm  m  notm  155 
notn  n  notn  166 
notp  mn  notp  171 
notp  pq  notp  174 
notq  mn  notq  181 
notq  pq  notq  184 
notp  p  not(mn)  117 
notq  q  not(mn)  118 
notq  q  not(mq)  128 
notp  p  not(np)  137 
notp  p  not(pq)  147 
notq  q  not(pq)  148 
Obviously, the significance of notp or notq in a premise or conclusion must be carefully assessed in each case. This is best done by writing it out in full.
Take for example 1/mn/notp/notp, which we derived ad absurdum from mood 217, i.e. 2/mn/p/p. The major premise in both cases has form QR. The Figure 1 mood has minor premise of form P(S)R and conclusion of form P(S)Q. The Figure 2 minor premise and conclusion have form P(S)Q and P(S)R, respectively. We thus indirectly reduce subfigure 1b to subfigure 2b. The complement is S everywhere and the negative propositions notp can be read as notp_{S}. We may also generalize this argument to all complements, since whatever the complement happen to be it will return in the conclusion. It follows that if the minor premise is absolute, so is the conclusion.
In some other cases, however, the transition is not so simple. For example, when 2/p/notp/not(mn) is reduced ad absurdum to 1/p/mn/p, we apparently have subfigure 2d (say) derived from subfigure 1c. But the number of complements does not match, so this case is rather artificial in construction. But I will not delve further into such issues here, not wanting to complicate matters unnecessarily. The conscientious reader will find personal investigation of these details a rewarding exercise.
Nevertheless, many of the above results are not without practical interest and value. For a start, they allow us to squeeze a bit more information out of causative propositions, and thus tell us a little more about the topography of the field of determination mentioned earlier. Most importantly, all the moods listed in this table involve a negative generic premise. Until now, we have only managed to validate moods with positive premises, i.e. positive moods. These are the first negative moods we manage to validate, by indirect reduction to (primary) positive moods.
This supplementary class of valid moods yields negative conclusions, whether the negation of a generic determination or that of a joint determination. Remember that the conclusions not(mn), not(mq), not(np), or not(pq) can be interpreted as disjunctive propositions involving all remaining (i.e. not negated) formal possibilities. Thus, for instance, not(mn) means “either mq or np or pq or noncausation”.
Summarizing, we have a total of 20 valid moods with a negative major premise, and 20 with a negative minor premise, making a total of 40 new moods. In Figure 1, the statistics are 4 + 4 = 8; in Figure2, they are 4 + 12 = 16; and in Figure 3, they are 12+ 4 = 16. We could similarly derive additional negative moods, by indirect reduction to compound and subaltern moods: this exercise is left to the reader.
3. Negative Conclusions from Positive Moods.
We have in the preceding chapters evaluated all conceivable positive conclusions from positive moods, i.e. from moods both of whose premises are positive (generic and/or joint) causative propositions. But we have virtually ignored negative conclusions from these (positive) moods, effectively lumping them with ‘nonconclusions’ (labeled nil), which they are not. We shall consider the significance of negative conclusions now[2].
In this context, it is important to keep in mind the distinction between a mood not implying a certain conclusion (which is therefore a nonsequitur, an ‘it does not follow’, which is invalid, but whose contradictory may yet be a valid or invalid conclusion), and a mood implying the negation of (i.e. denying) a certain conclusion (which is therefore more specifically an antinomy, so that not only is it invalid, but moreover it is so because its contradictory is a valid conclusion).
a) For a start, we have to note that wherever a positive mood yields a valid positive conclusion, it also incidentally yields a valid negative conclusion, namely one denying the contrary determination(s). Thus, for example, mood 111 (mn/mn) yields the positive conclusions “P is a complete and necessary cause of Q” (mn); it therefore also yields as negative conclusions “P is not a partial and not a contingent cause of Q” (notp and notq). We thus have at least as many valid negative conclusions as we have valid positive ones. Such syllogisms with negative conclusions are, of course, mere subalterns of those with positive conclusions they are derived from.
b) Moreover, we may notice that some of the crucial matricial analyses developed in the previous chapter invalidated certain conclusions, not merely by leaving one or more of their constituent clauses open, but more radically by denying, i.e. implying the negation of, some clause(s). Specifically, this occurred in the 14 cases listed in the following table (where ‘+’ means implied, ‘’ means denied, and ‘?’ means neither implied nor denied).
Notice that this table concerns negations of p or q relative to the complement S (whence my use here of the notation p_{s} or q_{s}), which is not the same as absolute negation. It is very important to specify the complement, otherwise contradictions might wrongly be thought to appear at later stages. In the case of negations of m or n, they are absolute anyway since there are no complements for them. Also note that:
· Where m or n is affirmed (as in moods 221, 231, 312, 313), then p or q (respectively) may be denied absolutely, i.e. whatever complement (S, notS or any other) be considered for p or q. That is, m implies notp and n implies notq. This can also be stated as m = mn or mq and n = mn or np, wherein the complement is unspecified (possibly but not necessarily S, or notS, or any other).
· Although notm by itself does not imply p, notm + n = np (moods 221, 312). Likewise, although notn by itself does not imply q, notn + m = mq (moods 231, 313). This is evident from the fact that absolute lone determinations are impossible. Here again, note well, the complement concerned is not specified (i.e. it may be, but need not be, S, or notS, or any other, say T).
· Furthermore, where m and/or n is/are denied (as occurs in all 14 cases to some extent), the additional denial if any of p and/or q (as in 221, 231, 241, and the six moods of Figure 3) has to initially be understood as a restricted negation, i.e. as notp_{S} or notq_{S}. Additional work is required to prove radical negation of the weak determinations.
· Since causation is by joint determination or not at all, notm + notn = pq or nocausation. But, notm + notn + notp_{S} + notq_{S} may not offhand be interpreted as nocausation, since pq remains conceivable as p_{notS}q_{notS} or relative to some other complement T. Note well that p+ notp_{S} does not imply p_{notS} and likewise q+ notq_{S} does not imply q_{notS}.
Table 9.4. Positive (generic and/or joint) premises whose conclusion includes additional negative elements.  
No.  Premises  m  n  p_{s}  q_{s}  Full conclusion  Comments  
Figure 1  
None 








Figure 2  
231  np  mn  +  –  –  –  mq and notq_{s}  Since m + notn = mq 
221  mq  mn  –  +  –  –  np and notp_{s}  Since n + notm = np 
233  np  np  –  ?  ?  ?  notm  Many outcomes possible 
222  mq  mq  ?  –  ?  ?  notn  Many outcomes possible 
224  mq  pq  –  –  ?  ?  pq or nocausation  Since notm + notn 
234  np  pq  –  –  ?  ?  pq or nocausation  Since notm + notn 
244  pq  pq  –  –  ?  ?  pq or nocausation  Since notm + notn 
241  pq  mn  –  –  –  –  pq but notp_{s} + notq_{s } or nocausation  Since if causation, then notm + notn = pq 
Figure 3  
313  mn  np  +  –  –  –  mq and notq_{s}  Since m + notn = mq 
312  mn  mq  –  +  –  –  np and notp_{s}  Since n + notm = np 
324  mq  pq  ?  –  –  –  notn and notp_{s} + notq_{s}  Various outcomes possible 
334  np  pq  –  ?  –  –  notm and notp_{s} + notq_{s}  Various outcomes possible 
314  mn  pq  –  –  –  –  pq but notp_{s} + notq_{s } or nocausation  Since if causation, then notm + notn = pq 
344  pq  pq  –  –  –  –  pq but notp_{s} + notq_{s } or nocausation  Since if causation, then notm + notn = pq 
There are thus 8 moods in the second figure and 6 in the third figure with additional negative conclusions (as revealed by matricial analysis in the preceding chapter). The differences between these two figures are simply due to moods 322 and 342 being selfcontradictory, as already seen.
Note in passing that the conclusions of moods 231, 313 and 221, 312 may be read as the relative to S “lone determinations” malone_{rel} and nalone_{rel}, respectively; but it of course does not follow from this that absolute lone determinations exist – indeed we see here that in absolute terms the respective conclusion is mq or np. The latter imply that relative to some item other than S, be it notS or some other item T, q or p (as applicable) is true. That is of course not much information, but better than nothing.
It should be noted that none of these moods is implied by others, so that the negative conclusions implied by them are not repeated in such putative other moods. (See Diagram 1 and Table 7.2, in chapter 7, on reduction.) An issue nevertheless arises, as to whether the moods mentioned, above under (a) and (b), exhaust negative conclusions drawable from positive moods. The answer seems to be yes, we have covered all negative conclusions. This may be demonstrated as follows.
Suppose a mood (i.e. premises) labeled ‘A’ is found by matricial analysis to notimply some positive conclusion ‘C’. Consider another mood ‘B’, such that A implies B. It follows that B does not imply C, since if B implied C, then A would imply C – in contradiction to what was given. But our question is: may B still formally imply notC? Well, suppose B indeed implied notC, then A would imply notC, in conflict with the subalternative result of our matricial analysis that A does not imply C. Granting that matricial analysis yields the maximum result, such conflict is unacceptable. Therefore, it is not logically conceivable that B imply notC as a rule.
We can thus remain confident that the negative conclusions of positive moods mentioned above make up an exhaustive list, provided of course that we remain conscious of the complement under discussion at all times.
In any case, we have in this way succeeded in squeezing some more information out of causative propositions occurring in syllogistic conjunctions. No moods of this sort were found in Figure 1. In Figure 2, two moods (221, 231) were already valid in the sense of yielding positive conclusions; their validity has now been reinforced with additional information; six other moods in this figure (222, 224, 233, 234, 241, 244) were previously classed as ‘invalid,’ in the sense of yielding no positive conclusions; but here they have been declared ‘valid’ with regard to certain negative conclusions. Similarly, in Figure 3, two moods (312, 313) have increased in validity, while another four (314, 324, 334, 344) have acquired some validity. So, in sum, we have four moods with reinforced validity and ten with newly acquired validity.[3]
We can derive additional valid moods from these, as we did before, by use of indirect reduction, or reduction ad absurdum. If we focus, for the purpose of illustration, on the negative conclusions notm and/or notn in Table 9.4, we obtain the following:
Table 9.5. Positive moods with a negative conclusion  
Major  Minor  Conclusion  Source 
Figure 1 from Figure 2 – keep same major  
pq  n and/or m  not(mn)  241 
mq  m  not(mn)  221 
np  n  not(mn)  231 
mq  n  not(mq)  222 
np  m  not(np)  233 
mq or np or pq  n and/or m  not(pq)  224, 234, 244 
Figure 1 from Figure 3 – keep same minor  
n and/or m  pq  not(mn) and not(pq)  314, 344 
n  mq  not(mn)  312 
m  np  not(mn)  313 
n  pq  not(mq)  324 
m  pq  not(np)  334 
Figure 2 from Figure 3 – keep the minor as a major  
m  mn  not(mq)  312 
n  mn  not(np)  313 
n and/or m  mn or pq  not(pq)  314, 344 
n  mq  not(pq)  324 
m  np  not(pq)  334 
Figure 3 from Figure 2 – keep the major as a minor  
pq  n and/or m  not(mq) and not(np)  224, 234 
mn  m  not(mq)  221 
mq  n  not(mq)  222 
mn  n  not(np)  231 
np  m  not(np)  233 
mn or pq  n and/or m  not(pq)  241, 244 
We can analyze these results as follows, for examples.
With regard to the Figure 1 moods in the above table derived ad absurdum from Nos. 222 and 233, namely mq/n/not(mq) and np/m/not(np), they correspond respectively to moods 126 and 135. Until here, these moods were invalid, because we had no positive conclusions from them. But here we have found some very vague conclusions, which negate joint determinations (a relatively indefinite result, since it signifies a disjunction of possible conclusions: i.e. either the remaining joint determinations or nocausation).
The same moods in Figure 3, correspond to the moods 326 and 335. In their case, however, we had positive conclusions from them, namely m from mq/n and n from np/m. The additional negative conclusions obtained from them here, namely not(mq) and not(np), respectively, constitute further information extraction, since they are not formally implied by the previous conclusions.
Note well that p and q in these four cases mean p_{s} and q_{s}, respectively, since we are in subfigure (c). Therefore, in Figure 3, we should not go on to infer that m + not(mq) = mn, or that n + not(np) = mn, i.e. that both moods 326 and 335 yield the full conclusion mn! They only in fact yield m and n in absolute terms, the rest of the conclusions being only relative to S. It would not be reasonable to expect more determination than that, because it would mean we are getting more out of our syllogism than we put in to it, contrary to the rules of inference.
Imperfect moods[4] of causative syllogism are those involving negative items as terms. That is, instead of directly concerning P, Q, R, S, they might relate to notP, notQ, notR and/or notS. We would not expect the investigation of such negative terms to enrich us with any new formal information, but rather to unnecessarily burden us with useless repetition. All the logic of such propositions can be derived quite easily from that of propositions with positive terms. We certainly will not engage in that exercise here (although some logician may be tempted to develop this field once and for all for the record). But we need to point out a couple of interesting facets of this issue.
a) As pointed out in a footnote in the chapter on immediate inferences, we commonly use positive forms with a negative intent, i.e. whose terms are positive on the surface but negative under it. Thus, the expression “P prevents Q” may be explicated as “P causes notQ”. Rather than work out all the logical properties of this new copula called “prevention,” we can simply reduce it to that of causation, by changing all occurrences of Q in causative logic to notQ. We could thus speak of complete or partial prevention, necessary or contingent prevention; and we could correlate such various forms with each other, in oppositions, eductions and syllogisms.
However, we could additionally correlate the forms of prevention in every which way with the forms of causation. It is in the event that we wish to do this, that the need to develop a logic of imperfect moods would arise. Such an enlarged logic would concern not only forms like “P causes Q” (causation) and “P causes notQ” (prevention), but also forms like “notP causes Q” and “notP causes notQ.” I cannot at this time think of any existing verbs that would fit the latter two definitions; we can call the implicit new PQ relations whatever we like, or nothing at all.
b) A particularly interesting negative term is when a partial or contingent causative proposition involves a negative complement. For example, the proposition “P (with complement notR) is a partial cause of Q,” involving the negative complement notR, needs to be investigated to fully comprehend the proposition “P (with complement R) is a partial cause of Q,” involving the positive complement R. Some of this work has been done in the chapter on immediate inferences.
We saw there that the ‘absolute’ proposition p_{abs} “P is a partial cause of Q” (irrespective of complement) is implied by either of those ‘relative’ propositions p_{R} or p_{notR} (that specify the complement). It follows of course that the negation of the absolute implies the negation of both the relatives. Also, p_{abs} may be true while only one of p_{R} or p_{notR} is true and the other is false. That is, the conjunctions ‘p_{abs} + notp_{R}’ or ‘p_{abs} + notp_{notR}’ are logically possible. Similarly with regard to contingent causation, q.
Now, what shall arouse our interest in syllogistic theory are occurrences of a negative minor or subsidiary item. As the reader may recall, in Table 5.2 we identified four ‘subfigures’ (labeled a, b, c, d) for each of the three figures of causative syllogism, according to the presence and position of a positive complement in either premise or in the conclusion. We can here identify five more subfigures (to be labeled e, f, g, h, i) for each of the three figures. These ‘imperfect’ subfigures are clarified in the table below:
Table 9.6. Imperfect subfigures of each figure.  
Subfigures  e  f  g  h  i 
Figure 1  QR  Q(S)R  Q(P)R  Q(notP)R  Q(notP)R 
 P(S)Q  PQ  P(S)Q  P(S)Q  P(S)Q 
 P(notS)R  P(notS)R  P(notS)R  P(S)R  P(notS)R 
Figure 2  RQ  R(S)Q  R(P)Q  R(notP)Q  R(notP)Q 
 P(S)Q  PQ  P(S)Q  P(S)Q  P(S)Q 
 P(notS)R  P(notS)R  P(notS)R  P(S)R  P(notS)R 
Figure 3  QR  Q(S)R  Q(P)R  Q(notP)R  Q(notP)R 
 Q(S)P  QP  Q(S)P  Q(S)P  Q(S)P 
 P(notS)R  P(notS)R  P(notS)R  P(S)R  P(notS)R 
Subfigures ‘e’ and ‘f’ are the most interesting. In both, the complement in the conclusion is negative compared to its origin in one of the premises; the subsidiary term has thus changed polarity. In subfigure ‘e’, the original complement is in the minor premise; in ‘f’, it is in the major premise. Subfigures ‘g,’ ‘h,’ ‘i’ are more complicated, since they involve the minor item or its negation as complement in the major premise. This is a conceivable situation, though one we are not likely to encounter often.
The layouts described by ‘e’ and ‘f’ are relatively common in our causative reasoning, inasmuch as we often have to distinguish between absolute and relative partial or contingent causation, or their negations. To make such distinctions, and decide just how much can be inferred from given premises, we have to refer to these subfigures. Logicians are therefore called upon to develop this particular field further, although the information is already tacit in the results of the subfigures we have already dealt with.
This work will not be pursued further here, except for the following general contribution. The table below predicts how subfigures may be derived from others by direct reduction (i.e. conversion of major or minor premise), i.e. it shows the logical interrelationships between the various subfigures in the different figures. Included in this table are indications for the reduction of perfect as well as imperfect subfigures of Figures 2 and 3 to subfigures of Figure 1. In one case, we reduce a subfigure of Figure 1 to subfigures of Figures 2, 3. This table, obtained by reflection on Tables 5.2 and 9.4, can be viewed as a guide to action for a future logician who may volunteer to finish this job.
Table 9.7. Reductions of Moods between Figures.  
Stages of development of study  If mood is evaluated in subfigure  Then mood is derivable in subfigure 
Firstly,  2a  1a 
perfect  2b  1b 
moods  2c  1f 
 2d  1h 
 3a  1a 
 3b  1e 
 3c  1c 
 3d  1g 
Secondly,  2e  1e 
main  2f  1c 
imperfect  2h  1d 
moods  2i  1g 
 3e  1b 
 3f  1f 
 3g  1d 
 3i  1h 
Thirdly, remaining imperfect moods  1i  2g, 3h 
[1]Aristotle regarded the fourth figure (PQ/QR/PR) as an impractical way of thinking, and so ignored it. My own position is more mitigated (see discussion in FL, p. 38). I have nevertheless disregarded it in the present treatise, to avoid excessive detail.
[2]That is to say, more precisely, conclusions that deny generic determinations. As we shall see further on, there are additionally (and derivatively from the present investigation) positive moods yielding negations of joint determinations, such as mood numbers 126, 135, 326, 335 (see Table 9.5, below).
[3] Note also that some of these are pairs of mirror moods (viz. 221231, 222233, 224234, 312313, 324334), others (241, 244, 314, 344) have no mirrors.
[4] The expression is Aristotelian in origin.