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© Avi Sion
All rights reserved
LOGIC OF CAUSATION ©
Avi Sion, 1999, 2003. All rights reserved. Phase One: Macroanalysis Chapter
9 - Squeezing
Out More Information
THE LOGIC OF CAUSATION
© Avi Sion, 1999, 2003. All rights reserved.
Phase One: Macroanalysis
9 - Squeezing
Out More Information
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:
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
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 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/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 so-called fourth figure), 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,
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.
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.
Obviously, the significance of not-p or not-q 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/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 propositions not-p can be read as not-pS. 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 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
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.
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 ‘non-conclusions’ (labeled nil), which they are not. We shall consider the significance of negative conclusions now.
In this context, it is important to keep in mind the distinction between a mood not implying a certain conclusion (which is therefore a non-sequitur, 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" (not-p and not-q). 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 ps or qs), 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 not-p and n implies not-q. 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 not-m by itself does not imply p,
not-m + n = np (moods 221, 312). Likewise, although not-n by
itself does not imply q, 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, 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 not-pS or not-qS. 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 = pq or no-causation.
But, not-m + not-n + not-pS + not-qS
may not offhand be interpreted as no-causation, since pq remains
conceivable as pnotSqnotS
or relative to some other complement T. Note well that p+
does not imply pnotS
and likewise q+ not-qS
does not imply qnotS.
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 the relative to S “lone determinations” m-alonerel and n-alonerel, 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 not-imply 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.
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 not-m and/or not-n in Table 9.4, we obtain the following:
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 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, 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 ps and qs, 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 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 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 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 pabs “P is a partial cause of Q” (irrespective of complement) is implied by either of those ‘relative’ propositions pR or pnotR (that specify the complement). It follows of course that the negation of the absolute implies the negation of both the relatives. Also, pabs may be true while only one of pR or pnotR is true and the other is false. That is, the conjunctions ‘pabs + not-pR’ or ‘pabs + not-pnotR’ 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:
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.
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.
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).
 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.
 The expression is Aristotelian in origin.