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 avalidity rateof 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 ofcompounds(16) orsubalterns(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 premisesmandmyield the conclusionm. 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/mplus1/n/n/n.
Such equivalencies are due to the fact thatthe 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/mqhad concludedmq, say, instead of justm), 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 oftransposition of determinationsin the premises. Moods can be paired-off 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/pand 1/p/pq/p; the conclusion has the same determinationp, whether the determinations of the premises arepq/porp/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 withtransposition of the premisesthemselves. 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 so-called 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 aboutthe 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 conclusionm; if we gradually increase the strength of the major premise (tomqormn), 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 tomn, the conclusion increases in strength tomn. 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 fromptomor fromqton(and likewise to the joint determinationsmqornp). For instance, compare moods 1/mn/p/pand 1/mn/m/m; here, keeping the major premise constant (mn), as we upgrade the minor premise fromptom, we find the conclusion upgraded fromptom. Similarly with moods 1/p/mn/pand 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:pandm, though defined as mutually exclusive, are meant as differentdegreesof positive causation; similarly for the negative aspects of causation,qandn. Thus, some such transitions were to be expected.
We can in this way interpret our list of valid moods asa 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, additionalvalid moods involving a negative premiseand, consequently, a negative conclusion.
This is made possible by using Aristotle’smethod of indirect reduction, or reductionad 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 reducead absurdumto 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 | not-m | not-m | 265 |
m | not-n | not-n | 256 |
mn | not-p | not-p | 217 |
mn | not-q | not-q | 218 |
Figure 1 from Figure 3 – keep same minor | |||
not-m | n | not-m | 356 |
not-n | m | not-n | 365 |
not-p | mn | not-p | 371 |
not-q | mn | not-q | 381 |
Figure 2 from Figure 1 – keep same major | |||
m | not-m | not-m | 155 |
n | not-n | not-n | 166 |
mn | not-p | not-p | 117 |
np | not-p | not-p | 137 |
pq | not-p | not-p | 147 |
mn | not-q | not-q | 118 |
mq | not-q | not-q | 128 |
pq | not-q | not-q | 148 |
p | not-p | not(mn) | 171 |
q | not-q | not(mn) | 181 |
p | not-p | not(pq) | 174 |
q | not-q | not(pq) | 184 |
Figure 2 from Figure 3 – keep the minor as a major | |||
not-n | n | not-m | 365 |
not-m | m | not-n | 356 |
not-p | p | not(mn) | 371 |
not-q | q | not(mn) | 381 |
Figure 3 from Figure 2 – keep the major as a minor | |||
n | not-n | not-m | 256 |
m | not-m | not-n | 265 |
p | not-p | not(mn) | 217 |
q | not-q | not(mn) | 218 |
Figure 3 from Figure 1 – keep same minor | |||
not-m | m | not-m | 155 |
not-n | n | not-n | 166 |
not-p | mn | not-p | 171 |
not-p | pq | not-p | 174 |
not-q | mn | not-q | 181 |
not-q | pq | not-q | 184 |
not-p | p | not(mn) | 117 |
not-q | q | not(mn) | 118 |
not-q | q | not(mq) | 128 |
not-p | p | not(np) | 137 |
not-p | p | not(pq) | 147 |
not-q | q | not(pq) | 148 |
Obviously, the significance ofnot-pornot-qin 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/not-p/not-p, 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 propositionsnot-pcan be read asnot-p_{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/not-p/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 tableinvolve a negative generic premise. Until now, we have only managed to validate moods with positive premises, i.e. positive moods. These are the firstnegative moodswe 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 conclusionsnot(mn),not(mq),not(np), ornot(pq)can be interpreted as disjunctive propositions involving all remaining (i.e. not negated) formal possibilities. Thus, for instance,not(mn)means “eithermqornporpqornon-causation”.
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 conceivablepositiveconclusions from positive moods, i.e. from moods both of whose premises are positive (generic and/or joint) causative propositions. But we have virtually ignorednegative conclusionsfrom these (positive) moods, effectively lumping them with ‘non-conclusions’ (labelednil), 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 moodnot implyinga certain conclusion (which is therefore anon-sequitur, an ‘it does not follow’, which is invalid, but whose contradictory may yet be a valid or invalid conclusion), and a moodimplying the negation of(i.e. denying) a certain conclusion (which is therefore more specifically anantinomy, 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 isnota partial andnota contingent cause of Q” (not-pandnot-q). We thus haveat least as manyvalid negative conclusions as we have valid positive ones. Such syllogisms with negative conclusions are, of course, meresubalternsof 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 bydenying, i.e. implying the negation of, some clause(s). Specifically, this occurred in the 14 cases listed in the following table (where ‘+’ means implied, ‘-’ meansdenied, and ‘?’ means neither implied nor denied).
Notice that this table concerns negations ofporqrelative to the complement S(whence my use here of the notationp_{s}orq_{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 ofmorn, they are absolute anyway since there are no complements for them. Also note that:
·Wheremornis affirmed (as in moods 221, 231, 312, 313), thenporq(respectively) may be denied absolutely, i.e. whatever complement (S, notS or any other) be considered forporq. That is,mimpliesnot-pandnimpliesnot-q. This can also be stated asm=mnormqandn=mnornp, wherein the complement is unspecified (possibly but not necessarily S, or notS, or any other).
·Althoughnot-mby itself does not implyp,not-m + n = np(moods 221, 312). Likewise, althoughnot-nby itself does not implyq,not-n + 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, wheremand/ornis/are denied (as occurs in all 14 cases to some extent), the additional denialif anyofpand/orq(as in 221, 231, 241, and the six moods of Figure 3) has to initially be understood as a restricted negation, i.e. asnot-p_{S}ornot-q_{S}. Additional work is required to prove radical negation of the weak determinations.
·Since causation is by joint determination or not at all,not-m+not-n = pqorno-causation. But,not-m+not-n +not-p_{S}+not-q_{S}may not offhand be interpreted asno-causation, sincepqremains conceivable asp_{notS}q_{notS}orrelative to some other complement T. Note well thatp+not-p_{S}doesnotimplyp_{notS}and likewiseq+not-q_{S}doesnotimplyq_{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 | + | – | – | – | mqandnot-q_{s} | Sincem + not-n = mq |
221 | mq | mn | – | + | – | – | npandnot-p_{s} | Sincen + not-m = np |
233 | np | np | – | ? | ? | ? | not-m | Many outcomes possible |
222 | mq | mq | ? | – | ? | ? | not-n | Many outcomes possible |
224 | mq | pq | – | – | ? | ? | pqorno-causation | Sincenot-m + not-n |
234 | np | pq | – | – | ? | ? | pqorno-causation | Sincenot-m + not-n |
244 | pq | pq | – | – | ? | ? | pqorno-causation | Sincenot-m + not-n |
241 | pq | mn | – | – | – | – | pqbutnot-p_{s}+not-q_{s} orno-causation | Sinceifcausation, then not-m + not-n =pq |
Figure 3 | ||||||||
313 | mn | np | + | – | – | – | mqandnot-q_{s} | Sincem + not-n = mq |
312 | mn | mq | – | + | – | – | npandnot-p_{s} | Sincen + not-m = np |
324 | mq | pq | ? | – | – | – | not-nandnot-p_{s}+not-q_{s} | Various outcomes possible |
334 | np | pq | – | ? | – | – | not-mandnot-p_{s}+not-q_{s} | Various outcomes possible |
314 | mn | pq | – | – | – | – | pqbutnot-p_{s}+not-q_{s} orno-causation | Sinceifcausation, then not-m + not-n =pq |
344 | pq | pq | – | – | – | – | pqbutnot-p_{s}+not-q_{s} orno-causation | Sinceifcausation, then not-m + not-n =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 self-contradictory, as already seen.
Note in passing that the conclusions of moods 231, 313 and 221, 312 may be read as therelative to S“lone determinations”m-alone_{rel}andn-alone_{rel}, respectively; but it of course does not follow from this that absolute lone determinations exist – indeed we see here that inabsoluteterms the respective conclusion ismqornp. The latter imply that relative to some item other than S, be it notS or some other item T,qorp(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),exhaustnegative conclusions drawable from positive moods. The answer seems to beyes, 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 tonot-implysome 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 formallyimply 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 ofyielding no positive conclusions; but here they have been declared ‘valid’with regard tocertain 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 reductionad absurdum. If we focus, for the purpose of illustration, on the negative conclusionsnot-mand/ornot-nin 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 | nand/orm | not(mn) | 241 |
mq | m | not(mn) | 221 |
np | n | not(mn) | 231 |
mq | n | not(mq) | 222 |
np | m | not(np) | 233 |
mqornporpq | nand/orm | not(pq) | 224, 234, 244 |
Figure 1 from Figure 3 – keep same minor | |||
nand/orm | pq | not(mn)andnot(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 |
nand/orm | mnorpq | 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 | nand/orm | not(mq)andnot(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 | nand/orm | not(pq) | 241, 244 |
We can analyze these results as follows, for examples.
With regard to the Figure 1 moods in the above table derivedad absurdumfrom Nos. 222 and 233, namelymq/n/not(mq)andnp/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 no-causation).
The same moods in Figure 3, correspond to the moods 326 and 335. In their case, however, we had positive conclusions from them, namelymfrommq/nandnfromnp/m. The additional negative conclusions obtained from them here, namelynot(mq) andnot(np), respectively, constitute further information extraction, since they are not formally implied by the previous conclusions.
Note well thatpandqin these four cases meanp_{s}andq_{s}, respectively, since we are in subfigure (c). Therefore, in Figure 3, we shouldnotgo on to infer thatm + not(mq) = mn, or thatn + not(np) = mn, i.e. that both moods 326 and 335 yield the full conclusionmn! They only in fact yieldmandnin 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 “PpreventsQ” may be explicated as “P causesnotQ”. 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 additionallycorrelate 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 P-Q relations whatever we like, or nothing at all.
b)A particularly interesting negative term is when a partial or contingent causative proposition involves anegative 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’ propositionp_{abs}“P is a partial cause of Q” (irrespective of complement) is implied byeitherof those ‘relative’ propositionsp_{R}orp_{notR}(that specify the complement). It follows of course that the negation of the absolute implies the negation ofboththe relatives. Also,p_{abs}may be true while only one ofp_{R}orp_{notR}is true and the other is false. That is, the conjunctions ‘p_{abs}+not-p_{R}’ or ‘p_{abs}+not-p_{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 bydirect 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 inFL, p. 38). I have nevertheless disregarded it in the present treatise, to avoid excessive detail.
[2]That is to say, more precisely, conclusions that denygenericdeterminations. As we shall see further on, there are additionally (and derivatively from the present investigation) positive moods yielding negations ofjointdeterminations, 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. 221-231, 222-233, 224-234, 312-313, 324-334), others (241, 244, 314, 344) have no mirrors.
[4]The expression is Aristotelian in origin.