A FORTIORI LOGIC
CHAPTER 5 – COMPARISONS AND CORRELATIONS
2. Is a fortiori argument syllogism?
5. From syllogism to a fortiori argument
We need to be very clear about the differences between a fortiori argument and other forms of argument, with which a fortiori argument is often compared and even confused. Many commentators have wrongly characterized a fortiori argument as analogical argument[1], and many more as syllogism, and they need to be corrected once and for all. We shall begin our comparison and correlation of argument forms with regard to argument by analogy, and then deal with syllogism.
1. Analogical argument
Just what form does argument by analogy have, and how does it differ from a fortiori argument? Can either of these forms be reduced to the other?
The forms of analogy. Our first task is to formalize analogical argument and identify the conditions of its validity. Qualitative analogical argument, like pure a fortiori argument, consists of four terms, which we may label P, Q, R, S, and refer to as the major, minor, middle and subsidiary terms as before, although here without implying that the major term is greater in any way than the minor. The argument may then take the following four copulative forms:
a. The positive subjectal mood. Given that subject P is similar to subject Q with respect to predicate R, and that Q is S, it follows that P is S. We may analyze this argument step by step as follows:
Major premise: P and Q are alike in that both of them have R.
This implies both ‘P is R’ and ‘Q is R’, and is implied by them together.
Minor premise: Q is S.
The term S may of course be any predicate; although in legalistic reasoning, it is usually a legal predicate, like ‘imperative’, ‘forbidden’, ‘permitted’, or ‘exempted’.
Intermediate conclusion and further premise: All R are S.
This proposition is obtained from the preceding two as follows. Given that Q is S and Q is R, it follows by a substitutive third figure syllogism that there is an R which is S, i.e. that ‘some R are S’. This particular conclusion is then generalized to ‘All R are S’, provided of course we have no counter-evidence. If we can, from whatever source, adduce evidence that some R (other than Q) are not S, then of course we cannot logically claim that all R are S. Thus, this stage of the argument by analogy is partly deductive and partly inductive.
Final conclusion: P is S.
This conclusion is derived syllogistically from All R are S and P is R.
If the middle term R is known and specified, the analogy between P and Q will be characterized as ‘complex’; if R is unknown, or vaguely known but unspecified, the analogy between P and Q will be characterized as ‘simple’. In complex analogy, the middle term R is clearly present; but in simple analogy, it is tacit. In complex analogy, the similarity between P and Q is indirectly established, being manifestly due to their having some known feature R in common; whereas in simple analogy, the similarity between them is effectively directly intuited, and R is merely some indefinite thing assumed to underlie it, so that in the absence of additional information we are content define it as ‘whatever it is that P and Q have in common’.
Needless to say, the above argument would be equally valid going from P to Q. I have here presented it as going from Q to P to facilitate comparison and contrast to a fortiori argument, which topic will be dealt with further on.[2]
Quantification. Let us next consider the issue of quantity of the terms, which is not dealt with in the above prototype.
In the singular version of this argument, the major premise is ‘This P is R and this Q is R’, where ‘this’ refers to two different individuals. The minor premise is ‘This Q is S’, where ‘this Q’ refers to the same individual as ‘this Q’ in the major premise does. From the minor premise and part of the major premise we infer (by syllogism 3/RRI[3]) that there is an R which is S, i.e. that some R are S – and this is generalized to all R are S, assuming (unless or until evidence to the contrary is found) there is no R which is not S. From the generality thus obtained and the rest of the major premise, viz. this P is R, we infer (by syllogism 1/ARR) the conclusion ‘This P is S’, where ‘this P’ refers to the same individual as ‘this P’ in the major premise does.
In the corresponding general version of the argument, the major premise is ‘All P are R and all Q are R’ and the minor premise is ‘All Q are S’. From the minor premise and part of the major premise we infer (by syllogism 3/AAI) that some R are S – and this is generalized to all R are S, assuming (unless or until evidence to the contrary is found) there is no R which is not S. From the generality thus obtained and the rest of the major premise, viz. All P are R, we infer (by syllogism 1/AAA) the conclusion ‘All P are S’. Note that the minor premise must here be general, because if only some Q are S, i.e. if some Q are not S, then, if all Q are R, it follows that some R are not S (by 3/OAO), and we cannot generalize to all R are S; and if only some Q are R, we have no valid syllogism to infer even that some R are S.
As regards the quantity of P and Q, there is much leeway. It suffices for the major premise to specify only that some Q are R; because, even if some Q are not R, we can still with all Q are S infer that some R are S (3/AII), and proceed with the same generalization and conclusion. Likewise, the major premise may be particular with respect to P, provided the conclusion follows suit; for, even if some P are not R, we can still from some P are R and all R are S conclude with some P are S (1/AII). Needless to say, we can substitute negative terms (e.g. not-S for S) throughout the argument, without affecting its validity.
It is inductive argument. Thus, more briefly put, the said analogical argument has the following form: Given that P and Q are alike in having R, and that Q is S, it follows that P is S. The validation of this argument is given in our above analysis of it. What we see there is that the argument as a whole is not entirely deductive, but partly inductive, since the general proposition ‘All R are S’ that it depends on is obtained by generalization.
Thus, it may well happen that, given the same major premise, we find (empirically or through some other reasoning process) that Q is S but P is not S. This just tells us that the generalization to ‘All R are S’ was in this case not appropriate – it does not put analogical argument as such in doubt. Such cases might be characterized as ‘denials of analogy’ or ‘non-analogies’. Note also that if ‘All R are S’ is already given, so that the said generalization is not needed, then the argument as a whole is not analogical, but entirely syllogistic; i.e. it is: All R are S and P is R, therefore P is S. Thus, analogy as such is inherently inductive. And obviously, simple analogy is more inductive than complex, since less is clearly known and sure in the former than in the latter.
It is interesting in passing to relate this argument form to the rabbinical hermeneutic principles. The second rule of R. Ishmael, the principle of gezerah shavah, which is based on the terms having some Biblical wording or intent in common, may be said to constitute simple analogy. This is because (evident) same wording, or (assumed) same ‘intent’ of different wordings, do not provide a sufficiently explicit predicate (R) in common to the subjects compared (P and Q). Words are explicit, but they are incidental to what they verbalize; therefore, the assumption that the Torah intends them as significant enough to justify an inference is open to debate[4].
The same can be said of the twelfth rule of R. Ishmael, which refers to contextual inferences (meinyano, misofo, and the like): such reasoning is simple analogy. However, the third rule of R. Ishmael, the principle of binyan av, falls squarely under the heading of complex analogy. In fact, our above description of complex analogy is an exact description of binyan av reasoning. When the rabbis want to extend the scope of a Torah law (S), they show that some new subject (P) has some feature (R) in common with the Torah-given subject (Q), and assuming that this feature is the reason for the law (this assumption constitutes a generalization, even if it superficially may seem to be a direct insight), they carry the law over from the given case to the unspecified case.
Other moods. The above, prototypical mood was positive subjectal. Let us now consider the other possible forms of analogical argument.
b. The negative subjectal mood. Given that subject P is similar to subject Q with respect to predicate R, and that P is not S, it follows that Q is not S. This mood follows from the positive mood by reductio ad absurdum: given the major premise, if Q were S, then P would be S; but P is not S is a given; therefore, Q is not S. This argument is of course just as inductive as the one it is derived from; it is not deductive.
c. The positive predicatal mood. Given that predicate P is similar to predicate Q in relation to subject R, and that S is P, it follows that S is Q. We may analyze this argument step by step as follows:
Major premise: P and Q are alike in that R has both of them.
This implies both ‘R is P’ and ‘R is Q’, and is implied by them together.
Minor premise: S is P.
Intermediate conclusion and further premise: S is R.
This proposition is obtained from the preceding two as follows. Given that R is P, it follows by conversion that there is a P which is R, i.e. that ‘some P are R’, which is then generalized to ‘all P are R’, provided of course we have no counter-evidence. If we can, from whatever source, adduce evidence that some P are not R, then of course we cannot logically claim that all P are R. Next, using this generality, i.e. ‘all P are R’, coupled with the minor premise ‘S is P’, we infer through first figure syllogism that ‘S is R’. Clearly, here again, this stage of the argument by analogy is partly deductive and partly inductive.
Final conclusion: S is Q.
This conclusion is derived syllogistically from R is Q and S is R.
Note that the generalized proposition here concerns the major and middle terms, whereas in the preceding case it concerned the middle and subsidiary terms. Needless to say, this argument would be equally valid going from Q to P. I have here presented it as going from P to Q to facilitate comparison and contrast to a fortiori argument, which topic will be dealt with further on.
Let us now quantify the argument. In the singular version, the major premise is: this R is both P and Q, and in the general version it is: all R are both P and Q. The accompanying minor premise and conclusion are, in either case: and a certain S is P (or some or all S are P, for that matter); therefore, that S is Q (or some or all S are Q, as the case may be). We could also validate the argument if the major premise is: some R are P and all R are Q; but if only some R are Q, i.e. if some R are not Q, we cannot do so for then the final syllogistic inference would be made impossible[5]. Such argument is clearly inductive, since it relies on generalization. No need for us to further belabor this topic.
d. The negative predicatal mood. Given that predicate P is similar to predicate Q in relation to subject R, and that S is not Q, it follows that S is not P. This mood follows from the positive mood by reductio ad absurdum: given the major premise, if S were P, then S would be Q; but S is not Q is a given; therefore, S is not P. This argument is of course just as inductive as the one it is derived from; it is not deductive.
We can similarly develop four implicational moods of analogical argument, where P, Q, R, S, symbolize theses instead of terms and they are related through implications rather than through the copula ‘is’. The positive antecedental would read: Given that antecedent P is similar to antecedent Q with respect to consequent R, and that Q implies S, it follows that P implies S. The negative antecedental would read: Given the same major premise, and that P does not imply S, it follows that Q does not imply S. The positive predicatal mood would read: Given that consequent P is similar to consequent Q in relation to antecedent R, and that S implies P, it follows that S implies Q. The negative predicatal mood would read: Given the same major premise, and that S does not imply Q, it follows that S does not imply P. These are, of course, partly inductive arguments since they involve generalizations. Validation of these four moods should proceed in much the same way as that of the four copulative moods.
Quantitative analogy. Analogy may be qualitative or quantitative. The four (or eight) moods of analogical argument above described are the qualitative. In special cases, given the appropriate additional information, they become quantitative.
a. The positive subjectal mood in such case would read: Given that subject P is greater than subject Q with respect to predicate R, and that Q is S (Sq), it follows that P is proportionately more S (Sp). Obviously, this reasoning depends on an additional (though often tacit) premise that the ratio of Sp to Sq is the same as the ratio of P to Q (with respect to R).
Very often in practice the ratios are not exactly the same, but only roughly the same. Also, the reference to the ratio of P to Q (with respect to R) should perhaps be more precisely expressed as the ratio of Rp to Rq. Note that this argument effectively has five terms instead of only four (since term S splits off into two terms, Sp and Sq). Of course, the additional premise about proportionality is usually known by inductive means. It might initially be assumed, and thereafter found to be untrue or open to doubt. In such event, the argument would cease to be quantitative analogy and would revert to being merely qualitative analogy. Thus, quantitative analogy is inherently even more inductive than qualitative analogy.
Note that the argument here is, briefly put: ‘just as P > Q, so Sp > Sq’. We can similarly argue ‘just as P < Q, so Sp < Sq’, or ‘just as P = Q, so Sp = Sq’. In other words, positive subjectal quantitative analogy may as well be from the inferior to the superior (as in the initial case), from the superior to the inferior, or from equal to equal; it is not restrictive with regard to direction. In this respect, it differs radically from positive subjectal a fortiori argument, which only allows for inference from the inferior to the superior, or from equal to equal, and excludes inference from the superior to the inferior. All this seems obvious intuitively; having validated the qualitative analogy as already shown, all we have left to validate here is the idea of ratios, and that is a function of mathematics.[6]
Similar comments can be made with regard to the other three copulative moods of quantitative analogy, namely:
b. The negative subjectal mood: Given that subject P is greater than subject Q with respect to predicate R, and that P is not S (Sp), and that the ratio of Sp to Sq is the same as the ratio of P to Q (with respect to R), it follows that Q is not proportionately less S (Sq).
This mood can be validated by reductio ad absurdum to the positive one. Both the major premise (viz. that P > Q, with respect to R) and the additional premise about proportionality (viz. that Sp:Sq = Rp:Rq) remain unchanged. What has ‘changed’ is that the minor premise of the negative mood is the denial of the conclusion of the positive mood, and the conclusion of the negative mood is the denial of the minor premise of the positive mood. Note that here instead of ‘not more S (Sp)’ and ‘not S (Sq)’, I have put ‘not S (Sp)’ and ‘not less S (Sq)’; this is done only to preserve the normal order of thought – it does not affect the argument as such. Here again, needless to say, though the mood shown is based on P > Q, it can easily be reformulated with P < Q or P = Q; this only affects the conclusion’s magnitude (making Sq mean ‘more S’ or ‘equally S’ as appropriate).
c. The positive predicatal mood: given that predicate P is greater than predicate Q in relation to subject R, and that a certain amount of S (Sp) is P, and that the ratio of Sp to Sq is the same as the ratio of P to Q (in relation to R), it follows that a proportionately lesser amount of S (Sq) is Q.
Here, the argument is essentially that ‘just as P > Q, so Sp > Sq’, i.e. that the amounts of subject S (viz. Sp and Sq) in the minor premise and conclusion differ in accord with the amounts of predicates P and Q (in relation to R). Or maybe we should say that subject R differs in magnitude or degree when its predicate is P (Rp) and when its predicate is Q (Rq), and that subject S differs accordingly (i.e. Sp and Sq differ in the same ratio as Rp to Rq). This is again an inductive argument, and would be equally valid in the forms ‘just as P < Q, so Sp < Sq’, or ‘just as P = Q, so Sp = Sq’.
d. The negative predicatal mood: given that predicate P is greater than predicate Q in relation to subject R, and that a certain amount of S (Sq) is not Q, and that the ratio of Sp to Sq is the same as the ratio of P to Q (in relation to R), it follows that a proportionately greater amount of S (Sp) is not P.
This mood can be validated by reductio ad absurdum to the positive one. That is, given the same major premise and additional premise about proportionality, we would say: since the lesser amount of S (Sq) is not Q, it must be that the greater amount of S (Sp) is not P. Here again, if the major premise has P < Q or P = Q instead of P > Q, the conclusion follows suit (i.e. Sp < or = instead of > Sq).
We can similarly develop four implicational moods of quantitative analogy. Thus, all eight moods of qualitative analogical argument can be turned into quantitative ones, provided we add additional information attesting to ‘proportionality’.
Face-off with a fortiori. Clearly, while qualitative analogy is somewhat comparable to purely a fortiori argument, quantitative analogy is somewhat comparable to a crescendo argument; but they are still far from the same. Let us first compare and contrast qualitative analogical argument to pure a fortiori argument. For this purpose, let us first focus on the positive subjectal mood, viz.:
P is more R than (or as much R as) Q, |
and Q is R enough to be S; |
therefore, P is R enough to be S. |
Here, as in analogy, the major premise implies that both P and Q are R, but unlike in analogy, it additionally implies that Rp ≥ Rq, i.e. that the quantity of R in P is greater than (or equal to) that in Q. Thus, though we can deduce the major premise of analogical argument from that of a fortiori argument, we cannot reconstruct the major premise of a fortiori argument only from that of analogical argument. Similarly, though the minor premise of a fortiori argument implies that Q is S, and therefore implies the minor premise of analogical argument, the reverse is not true. The difference between the two minor premises is that in a fortiori argument there is the element of sufficiency of R to be S, which is clearly lacking in argument by analogy. For the same reason, although the conclusion of a fortiori argument implies that of analogy, the latter does not by itself enable us to reconstruct the former.
Moreover, even though each of the propositions (the major and minor premises and the conclusion) involved in a fortiori argument implies the corresponding proposition of analogical argument, this does not mean that an a fortiori argument implies an analogical one. For, the a fortiori argument is deductive, i.e. its conclusion follow necessarily from its two premises; whereas, as we have just shown, the argument by analogy, even in its complex form, is inherently inductive, i.e. it requires a generalization of its minor premise to enable us to draw its conclusion. Therefore, even if both arguments may be said to yield a common conclusion, namely ‘P is S’, that conclusion has a very different logical status in the one and in the other.
It follows that we can neither reduce a fortiori argument to argument by analogy, since the latter’s conclusion does not imply the former’s (even though the premises of the former do imply those of the latter), nor can we do the reverse, since the premises of the latter do not imply those of the former (even though the conclusion of the former does imply that of the latter). It does happen that we know enough to form the major premise needed for a fortiori argument, but we do not know enough for its minor premise; or we know enough to form the minor premise needed for a fortiori argument, but we do not know enough for its major premise – in such cases we might have enough information to at least formulate an analogical argument. Thus, sometimes we have more information than we need for an analogy, but not enough for an a fortiori argument – in such cases we can only formulate an analogy.
Therefore, though we can say that a fortiori argument and argument by analogy have some features in common, we must admit that they are logically very distinct forms of argument. This is a formal and undeniable demonstration, once and for all. To repeat: neither argument can be reduced to the other. However, every valid a fortiori argument implies a corresponding argument by analogy involving less information and certainty. The premises of the latter, as we have just seen, lose the quantitative and/or sufficiency factors involved in the former; and the conclusion of the analogical argument is, as a result, both less informative and less sure (being now inductive instead of deductive). But of course, except for the present theoretical clarification, there is in practice no point in resorting to such implication, since the given a fortiori argument is better in all respects.
As regards the opposite direction, it cannot be said that every analogical argument implies a corresponding a fortiori argument. All we can say is that we can, sometimes, when the facts of the case permit it, construct an a fortiori argument which implies the given analogical argument. This is possible if the latter argument has a middle term (R), or an appropriate middle term can be found for it, which can both be used as a continuum of comparison (which, I think, is always possible in practice, although we cannot tell a priori which term is greater than the other) and at the same time serve as the sufficient condition for the subject (Q) to access the predicate (S) in the minor premise (and this is, of course, not always possible in practice). Thus, the construction of a corresponding a fortiori argument from a given analogical argument is not a mechanical matter and cannot always be performed. In effect, when it is found possible, it just means that we should in the first place have resorted to the stronger a fortiori argument yet foolishly opted for the weaker analogical argument.
All that we have said here applies equally well, mutatis mutandis, to the negative subjectal forms of these arguments, and to positive and negative predicatal forms, and again to the four implicational forms. These jobs are left to interested readers.
As regards comparison and contrast between quantitative analogy and a crescendo argument, i.e. ‘proportional’ a fortiori argument, the following need be said. The major premises are the same in both. But the minor premises and conclusions obviously differ, insofar as in quantitative analogy there is no idea of a threshold value of the middle term as there is in a fortiori argument. This explains why the ‘proportionality’ is essentially non-directional in quantitative analogical argument (inference is always possible both from minor to major and from major to minor); whereas it is clearly directional in a fortiori argument (inference is only possible from minor to major in positive subjectal and negative predicatal argument, and from major to minor in negative subjectal and positive predicatal argument).
Note in passing that although quantitative analogy and mere pro rata argument (i.e. used alone, outside of a crescendo argument) are not formally identical the two are effectively the same. Compare for example the following two formulas; clearly, the provisos in them are essentially the same (a concomitant variation between the values of S and the values of R) even if the terms are differently laid out.
Given that P is greater than Q with respect to R, and that Q is S (Sq), it follows that P is proportionately more S (Sp), provided that the ratio of Sp to Sq is the same as the ratio of P to Q (quantitative analogy).
Given that if R has value Rq then S has value Sq, it follows that if R has value more than Rq (say Rp), then S has value more than Sq (say Sp), provided that the values of S vary in proportion to the values of R (pro rata argument).
To conclude: there is, to be sure, an element of ‘analogy’ in all human thinking, including in syllogism and in a fortiori argument, since all abstraction is based on mental acts of comparison and contrast; but to say this loosely is not the same as equating syllogism or a fortiori argument to argument by analogy. When we look into the exact forms of these arguments, we clearly see their significant differences.
2. Is a fortiori argument syllogism?
The relationship(s) between a fortiori argument and syllogism have been a subject of debate for a long time, with some logicians and commentators equating the two or at least assimilating one to the other, and others denying such correlations. Ignoring for now the historical narrative, let us first focus on the formal issues involved and develop an independent judgment on them. I considered them very briefly in my Judaic Logic[7], saying:
“It could be said that there is an a-fortiori movement of thought inherent in syllogism, inasmuch as we pass from a larger quantity (all) to a lesser (some). But in syllogism, the transition is made possible by means of the relatively incidental extension of the middle term, whereas, as we have seen, in a-fortiori proper, it is the range of values inherent to the middle term which make it possible.”
Let us here look more deeply into the matter, without prejudice. First, it is well to realize that there are variant versions of the thesis of identification between these forms of argument. The most extreme position is of course that syllogism and a fortiori argument are one and the same thing. At the other extreme, all comparison and correlation between the two forms of reasoning might be rejected. But most logicians and commentators, myself included, adopt an median stance.
It should be made clear at the outset that we are here using the word ‘syllogism’ in its strict, Aristotelian sense, which is etymologically composed of ‘syn’ = together and ‘logos’ = thought, and which refers to an argument involving three items disposed in two premises and a conclusion in certain ways. We are not by this word referring more loosely (as some people do) to any form of argument in which an item serves as intermediary, i.e. to ‘mediate inference’. The latter, more generic expression is equally applicable, for instance, to apodosis (i.e. modus ponens or modus tollens), and obviously equally to a fortiori argument.
Quite often, those who try to explain a fortiori argument do so by suggesting that a fortiori is a sort of syllogistic reasoning. As a fortiori has been less studied than syllogism, it is less widely known and understood; so it is natural for people to try to refer it to a more familiar form of reasoning. The way they proceed to do this, however, is (funnily enough) not very logical, since instead of showing that a fortiori argument is or can be reduced to syllogism, they do the opposite – they usually try to show that syllogism can be formulated as a fortiori argument.
Usually, this is done by means of an example (usually, a syllogism in ‘Barbara’ format, 1/AAA). Often, they try to buttress the demonstration by using the words ‘a fortiori’ instead of ‘therefore’ to introduce the syllogistic conclusion, effectively saying: since ‘a fortiori’ means ‘perforce’, and the conclusion of a syllogism follows perforce from its premises, syllogism is comparable to a fortiori argument; this is obviously silly. Rather, it seems to me, what they need to try and do is recast a fortiori argument into syllogistic form. But this is of course more difficult, as it requires a prior clear awareness of the formalities of a fortiori argument, which most people lack.
What are the proponents of the identification thesis claiming, exactly? Some seem to think that all a fortiori argument is syllogism and all syllogistic argument is a fortiori argument. Others seem only to claim that all syllogistic argument is a fortiori. Still others seem to claim the reverse, i.e. that all a fortiori argument is syllogistic. Yet others seem to regard that there are some convergences between the two forms of argument, without going so far as to claim that all cases of the one fall under the other or vice versa. They may speak of possible reduction or analogy, instead of outright equation.
An important issue here is what exactly is meant by equation between two forms distinct enough to be called two. It may be that these forms are logically equivalent, irrespective of superficial verbal differences; i.e. they are poetically different ways to express the exact same logical thought. Alternatively, perhaps, one form can be fully transformed into or wholly reduced to the other, but not vice versa, so that the latter is logically prior to the former; or such transformation or reduction can occur in both directions. This may occur with or without loss of information, i.e. reversibly or not. Or again, the two forms may share some characteristics, i.e. be analogous in some respects, but differ sufficiently to require separate logical treatment.
That is to say, there is a big difference between saying that a fortiori argument ‘is’ or ‘is a special case of’ syllogism and saying that it ‘can be expressed as’ or ‘is reducible to’ syllogism; or vice versa; or both. Thus, these are different sorts and degrees of equation between two kinds of reasoning. To give a familiar example, even though all second and third figure syllogisms can be directly or indirectly reduced to first figure syllogisms, the former remain distinct forms of reasoning and significant in their own right. On the other hand, all cognoscenti agree, many fourth figure syllogisms have no real existence apart from first figure syllogisms, being distinguished only by the order of presentation of the premises and the order of terms in the conclusion.
Before proposing a theory of the precise correlation between the two forms of argument, let us first clarify what we mean by correlation in more formal terms and endow ourselves with the required vocabulary.
3. Correlating arguments
Let us investigate the possible relations between any two arguments, whatever their forms. An argument consists of one or more premise(s), say p, and a conclusion, say q; if the argument is valid, then p implies q. The given premise(s) p may of course yield more than one conclusion, i.e. q need not be their only conclusion; however, if there are other conclusions, they together with the same premise(s) constitute and may be regarded as other arguments. We will symbolize the two arguments being correlated as p1q1 and p2q2. In all cases, to repeat, p1 implies q1 and p2 implies q2. (I recommend that the reader draw flow charts to illustrate what is described below.)
The two arguments may be said to be ‘implicants’ if their premises mutually imply each other or are identical, i.e. if p1 implies p2 and p2 implies p1; for it follows that p1 also has the conclusion q2, and that p2 also has the conclusion q1. In such case, either the two conclusions, q1 and q2, imply each other, or only one implies the other, or neither implies the other. If q1 and q2 do not mutually imply each other, then though the arguments are implicants they obviously remain logically distinct by virtue of this difference.
- If neither of q1, q2 implies the other, we have two effectively ‘independent’ arguments with identical or logically similar premises (i.e. premises that imply each other). Neither argument can be logically reduced to the other.
- If q1 implies q2 (but not vice versa), then the argument p2q2 may be said to be a ‘subaltern’ of p1q1; if the reverse is the case, the result is of course reversed. A subaltern argument is reducible to, i.e. can be validated by, the argument it is subaltern to. For example, the syllogism 1/AAI is a subaltern mood of 1/AAA, since the premises are the same and the conclusion I is a subaltern of the conclusion A.
- If q1 and q2 imply each other, we may call the two arguments ‘intertwined’. Each is technically reducible to the other. For example, 1/AII and 3/AII are intertwined, being identical except that the minor premise of each is the converse of that of the other. If p1 is formally identical to p2 and q1 to q2, the two arguments can be characterized as ‘same’ or ‘identical’. But if any premise and/or the conclusion is not formally identical in the two arguments, we may not thus fully equate them, even though they are indeed logically closely related. For in such case not only is there some formal difference between them, but that difference may together with some other common premise(s) result in some difference in conclusion.
Now, consider cases where p1 implies p2 but p2 does not imply p1. We may in such cases say that the argument p1q1 ‘implies’ p2q2, but not vice versa. We can infer (via p2) that p1 also has the conclusion q2, but we cannot likewise infer (via p1) that p2 has the conclusion q1 (although that may happen in some cases). As regards the relationships between the conclusions, here again either they imply each other, or only one implies the other, or neither implies the other.
- If neither of q1, q2 implies the other, the two arguments are clearly independent, even though their premises are somewhat logically related. Neither argument is reducible to the other.
- If q2 implies q1 (but not vice versa), then the argument p1q1 may be said to be a subaltern of p2q2, because q1 may be viewed as following from p1 through the intermediary of p2 and q2, i.e. p1q1 is directly reducible to p2q2. For example, the syllogism 3/AAI is a subaltern mood of 3/AII or likewise of 3/IAI.
- If q1 implies q2 (but not vice versa), then the argument p2q2 may be said to be a ‘corollary’ of p1q1. Note well: this does not mean that p2q2 is a subaltern of p1q1 or vice versa; nor can the relation between the arguments be characterized as independent. We cannot here claim to logically reduce either argument to the other. We will encounter many examples of this relationship in the present context.
- If q1 and q2 imply each other, then the argument p1q1 is a subaltern of p2q2, whereas p2q2 is a corollary of p1q1. The relation between the arguments is not symmetrical, note well, because p1 implies p2 but p2 does not imply p1.
If p2 implies p1 but p1 does not imply p2, the same can be said of their possible relationships in reverse. If neither of p1 and p2 implies the other, they are ‘unrelated’ arguments; of course, they have to be compatible to occur together in a given context or body of knowledge. So much, then, for all the possible correlations between a pair of arguments, we now have a typology and terminology to work with. Note especially the applications of the term ‘corollary’ here introduced.
4. Structural comparisons
Let us next compare and contrast the structures of syllogism and a fortiori argument. Consider the following typical samples:
a) A categorical syllogism | A copulative a fortiori argument |
All Y are Z, and | P is more R than Q is, |
All X are Y, | and, Q is R enough to be S; |
therefore: All X are Z. | therefore, all the more, P is R enough to be S. |
(Here, the items X, Y, Z and P, Q, R, S are terms, note.)
b) A hypothetical syllogism | An implicational a fortiori argument |
Y implies Z, | P implies more R than Q does, |
X implies Y, | and, Q implies enough R to imply S; |
therefore: X implies Z | therefore, all the more, P implies enough R to imply S. |
(Here, the items X, Y, Z and P, Q, R, S are theses, note.)
Syllogism refers to inference from one term or thesis (X) to a second (Z) via a third (Y). The third item, known as the middle item, is the intermediary through which the inference is made; and it is found in the two premises, but not in the conclusion. The other two items are found one in each premise and both in the conclusion; they are traditionally called the minor and major item, because valid positive moods of the first figure serve to include a narrower item in a larger (or equal) one – but this initial scenario does not apply to all valid moods (it does not apply to negative moods of the first figure or to moods of the second and third figures), so the words major and minor should not in all cases be taken literally, they are conventional labels. Do not be misled, either, by my adoption of similar terminology for a fortiori argument: the words here differ in meaning.
A fortiori argument also involves three propositions. But the first, called the major premise, differs significantly in form from the second, called the minor premise, and from the conclusion, while the latter two propositions are of the same form. Moreover, these three propositions involve, not just three terms or theses, but four of them. The middle item (R), here, is present (implicitly if not explicitly) in the conclusion as well as in the two premises; it is labeled ‘middle’ because it interrelates the three other items. The major and minor items (P and Q), here, are so named because they refer respectively to a larger and smaller quantity of the middle item; in the limiting cases of equality between them, their quantitative relations become interchangeable, of course. These two items are both present, together with the middle item, in the major premise.
Note well that the minor premise and conclusion do not always contain the same items (as they do in syllogism). In ‘major to minor’ moods the minor premise contains the major item and the conclusion contains the minor item, whereas in ‘minor to major’ moods the minor premise contains the minor item and the conclusion contains the major item. The middle item is, to repeat, found in both propositions. Finally, we here have a fourth item (S), called the subsidiary item, which is present in the minor premise and conclusion, but absent in the major premise.
Syllogism occurs through the inclusion or exclusion between the three classes or theses concerned; whereas a fortiori compares the measures or degrees of the quality or qualification signified by middle item in its relation to the three other items. Both forms of argument, it is true, involve quantity – but they do so with different emphasis and effect.
The quantities (like ‘all’ or ‘some’) involved in categorical syllogism are applicable to the classes concerned as conceptual groupings and not to the individual members they subsume; similar quantifications are involved in hypothetical syllogism with reference to the conditions underlying the theses. The quantities (like ‘more’ or ‘enough’) involved in copulative a fortiori argument relate to a common property of the individual instances of its other terms; or, in implicational a fortiori, of the conditions underlying its other theses. The focus of syllogism is not primarily on quantity, but on essentially qualitative information; in categorical syllogism, it relates to classification; while in hypothetical syllogism, it relates to sequencing. The focus of a fortiori argument is primarily on quantity – it is quantitative ordering of thoughts, entities, qualities or events.
As just explained, the two forms of argument are structurally very different. As I will presently demonstrate, we can – though sometimes in a rather forced manner – derive a fortiori arguments from syllogisms, and vice versa. Nevertheless, these two species of reasoning cannot be considered the same, or one as a subspecies of the other, because, though some of the original meaning is retained when we attempt to transform syllogism into a fortiori argument or vice versa, some important information is lost.
5. From syllogism to a fortiori argument
Let us now investigate whether syllogism can be reworded as a fortiori argument (yes, it can) and whether such rewording entails any loss of information (yes, it does). We shall begin with a detailed analysis relative to the most typical valid syllogisms, namely the positive first figure moods 1/AAA and 1/AII of categorical syllogism (and their singular equivalent), and later extend our consideration to all other forms. The following translation of such syllogism into a fortiori is proposed. The major premise of the latter is derived from that of the former; the minor premise of the former becomes that of the latter; and the original conclusion is reworded as shown.
a) Categorical syllogism (given) | A fortiori argument (derived) |
All Y are Z, and | The class Z is bigger than (or as big as) the class Y, and |
All (or some) X are Y, | The class Y is big enough to include all (or certain) members of the class X, |
therefore: | therefore: |
We could view this transition from ordinary (Aristotelian) syllogism to a fortiori argument as made through the intermediary of a class-logic syllogism. That is, the major premise “All Y are Z” is first interpreted as “The class of Y is entirely subsumed by the class of Z,” the minor premise likewise as “The class of X is entirely (or partly) subsumed by the class of Y,” and the conclusion as “The class of X is entirely (or partly) subsumed by the class of Z.” Then these three propositions are used to produce the three of the resultant a fortiori argument. This is said in passing.
The resulting a fortiori argument, note, is positive subjectal (minor to major). An example often given to illustrate and justify this operation is: “All men (Y) are mortal (Z) and Socrates (X) is a man, therefore Socrates is mortal” becomes “Since the class of mortals is more extensive than that of men (which is included in it), it follows that if the latter includes Socrates as a member, the former is bound to include him as a member as well.” (People don’t normally think like that, admittedly, but logicians have to sometimes!) But as we shall presently explain, this example is not fully accurate and so a bit misleading.
Note that, in this instance (though not always, as we shall later see), the major term (Z) of the syllogism gives rise to the major term (the class Z) of the a fortiori argument. However, the middle term (Y) of the syllogism gives rise to the minor term (the class Y) of the a fortiori, and the minor term of the syllogism (X) gives rise to the subsidiary term (the class X) of the a fortiori. Thus, do not confuse the appellations of the terms in the two arguments. The middle term of the a fortiori argument is “bigness;” although it is not explicitly given in the syllogism, it is read out of it. Here, “bigness” refers to the extension of each class concerned, i.e. the number of members it includes. We could as well have used the more technical term extensiveness.
Now, the major premise of the a fortiori argument tells us that the term Z is more extensive than the term Y (or only as extensive as Y, if it so happens that All Z are Y is also true). It does not and cannot tell us whether or not Z includes Y, note well. The minor premise and conclusion of the a fortiori argument likewise only tell us that the terms Y and Z respectively are more extensive than (or at least as extensive as) the term X. They do not and cannot tell us that Y and (therefore) Z actually include X; they only affirm in the infinitive that they are big enough to do so, i.e. their inclusion is a logical possibility but not an inferred certainty.
That is to say, the two premises of the derived a fortiori argument do not fully reproduce the information in the given syllogism – they only focus on the relative extensions implied by them, but do not carry over the information about precise inclusions. Consequently, the conclusion of the a fortiori – which likewise contains no actual (only ‘potential’) information about inclusions – cannot be used to infer the conclusion of the syllogism. This means that the given syllogism logically implies the proposed a fortiori argument – but the latter does not logically imply the former! So the derived a fortiori argument is a corollary of the given syllogism.
This correlation becomes more evident if we realize that the a fortiori argument shown here would remain true even if the classes Y and Z were mutually exclusive (!) – provided we knew somehow that their relative sizes (extensions) were as here stated (i.e. such that Z is greater or equal to Y). Even in such case (i.e. that of mutual exclusion), the said minor premise would imply the said conclusion, for neither of these two propositions formally means that the term X is (wholly or partly) actually included by the terms Y and Z – they only tell us that they could be.
As far as the a fortiori argument is concerned, all information about inclusion found in the two premises of the given syllogism is redundant – it is simply not carried over in it; it is effectively lost. The propositional forms used in the a fortiori argument are simply incapable of containing this significant classificatory information, and they have no need of it to deduce the conclusion they yield. Similarly, the latter conclusion (of the a fortiori) has nothing definite to tell us about the inclusion (i.e. of X in Z) that the original syllogistic conclusion refers to, and therefore is unable to reconstruct it.
We could of course have formulated the major premise as “The class Z is bigger than (or as big as) its subclass Y,” and similarly the minor premise as “The class Y is big enough to include all (or certain) members of its subclass X” – but the additional information so tagged on would still play no role in the inference made possible by the a fortiori reasoning as such. It would of course suggest to us that X is indeed included in Y and therefore in Z – but that suggestion would be emerging not from the a fortiori argument itself, but from the syllogism mentally underlying it. The a fortiori argument as such could logically still only yield the said conclusion “The class Z is big enough to include all (or certain) members of the class X,” without specifying that the Xs are indeed included in Z.
It follows that we cannot say that the derived a fortiori argument “is” or “is identical to” the given syllogism. The most we can say is that the former is implicit in the latter; i.e. that it is a part of it or an aspect of it. The derived format cannot logically replace the given format in all respects. Important information is lost in the transition. To return to the above example about Socrates, we now see that the clause about the class of men being included in that of mortals, and the suggestion that Socrates is included in both these classes, are not in fact constituents of the a fortiori argument as such.
As already mentioned, many people affirm that syllogisms can be reworded in a fortiori form, and they give an example or two to prove it, usually a positive first figure mood. But I asked myself two original questions: (a) can all valid moods of the syllogism be likewise recast in a fortiori format, and (b) are all valid forms of a fortiori argument generated by such translations?
To my surprise, the answers to both these questions were found to be: yes, even if some of the processes do seem rather artificial. At the risk of boring the reader stiff (skip it all if you take my word for it), I will now show step by step that all valid moods of syllogism can be recast in a fortiori form; and I will also show that such translations or reinterpretations generate all the varieties of a fortiori argument. Though most of this work was easy enough, some of it was a bit difficult – so it was well worth doing.
We have already dealt with wholly positive first figure syllogism (moods 1/AAA, 1/AII). We can similarly propose a translation of first figure syllogism with a negative major premise and conclusion (moods 1/EAE, 1/EIO), by simply using the term nonZ in place of Z. as follows:
b) Syllogistic format | A fortiori format |
No Y are Z (= All Y are nonZ), and | The class nonZ is bigger than the class Y, and |
All (or some) X are Y, | The class Y is big enough to include all (or certain) members of the class X, |
therefore: | therefore: |
Clearly, however, this is still positive a fortiori argument (albeit with a negative term nonZ), and does not correspond to negative a fortiori argument. The question arises, can we generate a negative a fortiori argument from some other syllogism? Yes, but to do so we need to move over to the second figure of syllogism, as follows:
c) Syllogistic format | A fortiori format |
All Z are Y, and | The class Y is bigger than the class Z, and |
All (or some) X are not Y, | The class Y is not big enough to include all (or certain) members of the class X, |
therefore: | therefore: |
The derivation might be considered roughly adequate, if we accept that X is not Z (or Y) means that Z (or Y) is not “big enough to include” X. But even if we do so, it is very reluctantly, as we are clearly indulging in an unnatural way of thinking. The above translation concerns the syllogistic moods 2/AEE, 2/AOO; we can do the same for the moods 2/EAE, 2/EIO simply by replacing Z with nonZ, as shown next:
d) Syllogistic format | A fortiori format |
No Z are Y (= All Z are nonY), and | The class nonY is bigger than the class Z, and |
All (or some) X are not Y (= are nonY), | The class nonY is not big enough to include all (or certain) members of the class X, |
therefore: | therefore: |
Of course, such a fortiori argument is even more awkward. Still, let us say we have now managed to translate all first and second figure syllogism to positive and negative a fortiori arguments, respectively. What of third figure syllogism – to what would we translate them? Moreover, all the a fortiori forms encountered so far have been subjectal – what of predicatal a fortiori? The answers to these two questions are the same. We can propose the translations of positive and negative moods of third figure syllogism into positive and negative predicatal a fortiori arguments, as follows:
e) Syllogistic format | A fortiori format |
All Y are Z, and | A bigger class is required to subsume the class Z than to subsume the class Y, and |
Some (or all) Y are X, | The class X is big enough to include some (or even all) members of the class Y, |
therefore: | therefore: |
The above is appropriate for the translation of the mood 3/AII (and likewise its subaltern 3/AAI). For the mood 3/IAI (and its same subaltern 3/AAI), we would have to transpose the premises and convert the particular conclusion.
f) Syllogistic format | A fortiori format |
Some (or all) Y are Z, and | A bigger class is required to subsume the class X than to subsume the class Y, and |
All Y are X, | The class Z is big enough to include some (or even all) members of the class Y, |
therefore: | therefore: |
The mood 3/OAO (and its subaltern 3/EAO) can be translated as shown next; note the transposition of premises here too.
g) Syllogistic format | A fortiori format |
Some (or all) Y are not Z, and | A bigger class is required to subsume the class X than to subsume the class Y, and |
All Y are X, | The class Z is not big enough to include some (or even all) members of the class Y, |
therefore: | therefore: |
And similarly for the mood 3/EIO (and its subaltern 3/EAO) shown next.
h) Syllogistic format | A fortiori format |
No Y are Z (= All Y are nonZ), and | A bigger class is required to subsume the class nonZ than to subsume the class Y, and |
Some (or all) Y are X (= are not nonX), | The class nonX is not big enough to include some (or even all) members of the class Y, |
therefore: Some X are not Z | therefore: |
Note in passing that we could similarly process modal syllogism, the modalities simply passing over from the syllogism to the derived a fortiori argument. We have thus shown that all valid moods of the syllogism can be rephrased (however awkwardly) as a fortiori argument of various sorts. The first figure moods yielded positive subjectal a fortiori; the second figure moods yielded negative subjectal a fortiori; and the third figure moods yielded positive and negative predicatal a fortiori. We have thus also shown that all four forms of a fortiori argument are produced by these processes.
Needless to say (but I will add it anyway, to be foolproof), the middle term of these a fortiori arguments, viz. “big,” is just one possible middle term – most a fortiori arguments we encounter in practice do not use this particular middle term but may use any of countless middle terms. That is to say, although we have generated instances of all sorts of a fortiori, we have certainly not generated all particular instances of a fortiori! Thus, ‘a fortiori argument in general’ must be admitted to be a larger class than ‘a fortiori argument generated from syllogism’ is. Keeping that in mind, we will not overestimate the import of the preceding demonstration.
Our demonstration was made with reference to categorical syllogism and copulative a fortiori – but obviously the same can be done with reference to hypothetical (and likewise, ‘de re’ conditional) syllogism. We can I think take that for granted without worry, so I won’t bore you further with a repetition of all the above work to prove the point. However, it is worth our while looking at just one sample of translation of hypothetical syllogism into a fortiori argument, to highlight and examine the different concepts and language involved in the latter:
Hypothetical syllogism (given) | A fortiori argument (derived) |
If Y, then Z, and | Thesis Z is bigger than thesis Y, and |
If X, then Y | Thesis Y is big enough to include all (or certain) conditions of thesis X, |
therefore: If X, then Z | therefore: |
The given propositions can also be written/read respectively as “Y implies Z,” “X implies Y” (or in the weaker case, “X does not imply not Y”), and “X implies Z” (or in the weaker case, “X does not imply not Z”), note. The derived a fortiori argument here is still copulative, note, and not implicational. Its language is clearly similar here to that we used in translating categorical syllogism, except that we say “thesis Z” instead of “class Z,” etc., and we speak of “conditions” instead of “members.”
This change of wording reflects the nature of implication: necessary implication (the stronger variant) means that the consequent occurs under all the conditions applicable to the antecedent, whereas possible implication (the weaker variant) refers to some of these conditions. In logical conditioning, the conditions referred to are internal contexts of knowledge; in ‘de re’ modes of conditioning – i.e. the natural, temporal and spatial modes – they are external circumstances, times or places.
Note well that when we say that “thesis Z is bigger than thesis Y,” we do not mean that the subject of Z is bigger than the subject of Y (taking Z and Y as single categorical propositions here, for the sake of argument). We are not referring to the extensions of the terms involved within the theses, but to the conditions underlying the theses. The proposition “Y implies Z” does not formally exclude the possibility that Y may be singular and Z universal, or vice versa, provided that under all the conditions concerned Y is accompanied by Z. Similarly with the other forms mentioned in the above arguments.
To conclude, it is now formally proved true that syllogism can always be recast in a fortiori form, although it was also evident in the course of our demonstration that such translations are, to varying degrees, very contrived. Maybe logicians occasionally have such convoluted thoughts; but we do not ordinarily think in such awkward ways. Syllogism is quite thinkable and commonly thought by itself, i.e. without need to refer to a fortiori argument as above done. Indeed, syllogism is a simpler and more primitive movement of thought; more easily and widely comprehensible. So we would not in practice rephrase a syllogism as an a fortiori argument. Still, we have answered half the question initially asked.
6. From a fortiori argument to syllogism
We need now turn to the second part of our initial question: can a fortiori argument be said to be syllogism or at least be expressed as or reduced to syllogism? We can put it more technically, and ask: how can a four-term argument (a fortiori) be reworded as a three-term one (syllogism), with minimal loss of meaning and conviction?
One possible answer is: by ignoring or concealing one of the terms – namely the middle term. We very commonly do leave tacit the middle term in our a fortiori arguments. An argument used by Aristotle that does this is: if even the gods are not omniscient, certainly human beings are not. Here, the middle term is tacitly present in the words ‘even’ and ‘certainly’; often, it is signaled by expressions like ‘enough’ and ‘all the more’. Put in formal terms, this would read: if Q is S enough, then all the more P is so – the a fortiori middle term R being left out entirely. This is a three-term argument – but is it syllogism? Clearly not, since we have no operative syllogistic middle term (S cannot be said to play that role here).
We can in this context, incidentally, pinpoint the usual error of the logicians and commentators who seek to explain a fortiori argument as a sort of syllogism – they wrongly assume that the three terms of such derived syllogism would be the minor, major and subsidiary (P, Q, S), whereas in fact they are the different quantities of the middle term (R) occurring relative to these three terms. This is a subtle distinction that they could not see, because they had not sufficiently analyzed a fortiori argument as such as gravitating around a fourth term, which is often in practice left unstated or so intertwined with the other three that it is almost imperceptible.
To understand how we can transmute a fortiori argument into syllogism, we need to go back to the way a fortiori argument is formally validated – i.e. we need to decorticate its propositions and find out whether the constituents of the given premises justify the constituents of the putative conclusion. To do this we need to look again at Table 1.1 and Diagram 1.1, given in the first chapter.
Once the centrality of the middle term of a fortiori argument is grasped, it is easy for us to formulate its various moods in syllogistic form. We shall now do this with reference to copulative a fortiori argument. The major premise of all four a fortiori arguments can be expressed in the hypothetical form “Rp implies Rq,” which means “the quantity of R corresponding to P implies the quantity of R corresponding to Q.” Rp may be said to imply Rq because a larger quantity implies all lesser quantities; as for example 5 equally implies, 4, 3, 2 or 1 – i.e. you cannot reach the larger amount without passing through the smaller amounts. If you have $5, it is also true that you have $4, $3, etc.
The minor premises and conclusions can similarly, by consideration of their quantitative significances, be expressed as if–then propositions or negations of such, as shown below. Note well that the middle items of the syllogisms produced vary: in cases (a) and (d) the middle theses is Rq, while in cases (b) and (c) it is Rp. Note too that in cases (a) and (b) the premises must be switched to get the stated conclusion; i.e. the major premise becomes the minor and vice versa. Notice, finally, that we obtain valid moods in all three figures of the syllogism, viz. 1/AAA (twice), 2/AOO and 3/OAO, although (predictably, in view of the numbers) we do not produce all the valid moods of syllogism out of those of a fortiori.
a) Positive subjectal copulative a fortiori | Syllogism (1/AAA) |
P is more R than (or as much R as) Q is, | Rp implies Rq |
and, Q is R enough to be S; therefore, | Rq implies Rs |
all the more (or equally), P is R enough to be S. | So, Rp implies Rs |
b) Negative subjectal copulative a fortiori | Syllogism (3/OAO) |
P is more R than (or as much R as) Q is, | Rp implies Rq |
yet, P is R not enough to be S; therefore, | Rp does not imply Rs |
all the more (or equally), Q is R not enough to be S. | So, Rq does not imply Rs |
c) Positive predicatal copulative a fortiori | Syllogism (1/AAA) |
More (or as much) R is required to be P than to be Q, | Rp implies Rq |
and, S is R enough to be P; therefore, | Rs implies Rp |
all the more (or equally), S is R enough to be Q. | So, Rs implies Rq |
d) Negative predicatal copulative a fortiori | Syllogism (2/AOO) |
More (or as much) R is required to be P than to be Q, | Rp implies Rq |
yet, S is R not enough to be Q; therefore, | Rs does not imply Rq |
all the more (or equally), S is R not enough to be P. | So, Rs does not imply Rp |
To give an example[8]: “It is psychologically more difficult (R) for a man to strike his father (P) than to strike his neighbors (Q); so, if John (S) was able to strike his father, he is all the more capable of striking his neighbors” (major to minor, positive predicatal) becomes “John’s psychological state is such that he could strike his father, and striking one’s father is more difficult than striking one’s neighbors, therefore John’s psychological state is such that he could strike his neighbors.”
Now, just as we found that in the attempted transformation of syllogism into a fortiori argument there was a loss of information (about inclusions between the classes concerned), so it is with regard to the attempted transformation of a fortiori argument into syllogism some information is inevitably lost in the process. What information is lost? I will now explain, with reference to our earlier analysis of the propositional forms involved in a fortiori argument. Consider for instance process (a), the translation of positive subjectal a fortiori argument into syllogism.
The form “P is more R than Q” tells us more than just “Rp implies Rq” – it also tells us that “P implies Rp” and “Q implies Rq” and “Rp is greater than Rq.” Likewise, the form “Q is R enough to be S” tells us more than just “Rq implies Rs” – it also tells us that “Rs implies S” and “Q implies Rq” and “Rs includes Rq.” Thus, the two premises of the derived syllogism do not carry over all the information that was in the given a fortiori argument, but only parts of it. The conclusion of the derived syllogism, viz. “Rp implies Rs,” is similarly less informative than the conclusion of the given a fortiori argument, which tells us that “Rs implies S” and “P implies Rp” and “Rs includes Rp.” From “Rp implies Rs,” we can infer the element “Rs includes Rp,” but not the elements “Rs implies S” and “P implies Rp.”
So we are unable to logically reconstitute the conclusion of the original a fortiori argument from the conclusion of the derived syllogism – we only have part of its discourse leftover, some of it having been left out on the way. It follows that, though a significant syllogism is formally discernible within the a fortiori argument, that derived syllogism does not store enough information to in turn produce the original a fortiori argument.
It follows that the a fortiori argument logically implies but is not implied by the syllogism shown. That is to say, we cannot claim that the given a fortiori argument “is” or “is identical to” the derived syllogism. The most we can say is that the latter is a corollary of the former: it is implicit in it, a part of it or an aspect of it. But evidently, the latter cannot logically replace the former in all respects. Important information is lost in the transition.
The same can be said with regard to the translation of other forms of a fortiori argument to syllogism, i.e. the processes labeled (b), (c) and (d) above. Though these processes are obviously valid, they do not produce a syllogistic replica of a fortiori argument – only at best a corollary. We could do and say the same for implicational a fortiori argument, but there is no need to repeat ourselves. Suffice it to show and comment on one sample:
Implicational a fortiori argument | Syllogism |
P implies more R than (or as much R as) Q does, | Rp implies Rq |
and, Q implies enough R to imply S; therefore, | Rq implies Rs |
all the more (or equally), P implies enough R to imply S. | So, Rp implies Rs |
Note that the language of the derived syllogism here is exactly the same as that for copulative a fortiori, since we are still concerned with various degrees of R implying each other. The syllogism derived here – as that derived from copulative a fortiori – is formally hypothetical, not categorical. The items involved in all such derived syllogisms, viz. Rp, Rq and Rs, are the terms or theses “the value of R that P is or implies,” “the value of R that Q is or implies,” and “the value of R that S is or implies,” respectively (using “is” for copulative sources and “implies” for implicational ones). No more need be said.
We have thus demonstrated using formal means that the essence of a fortiori argument can be expressed through syllogism. This is equally true of positive or negative, subjectal or predicatal, copulative or implicational a fortiori. This does not mean that a fortiori argument is the same as syllogism, but it does mean that syllogistic movements of thought are involved in a fortiori reasoning, or (in other words) that a fortiori argument can be partly reduced to and validated by means of syllogism. Even so, as just explained, some information is invariably lost on the way.
7. Reiterating translations
To complete our analysis of the relationships between a fortiori argument and syllogism we need to examine one more issue. Having found that we can generate a fortiori arguments from syllogisms, and syllogisms from a fortiori arguments – we need to ask the question of reiteration. This does not mean reversibility, since we have already shown that such translations are not reversible – i.e. the arguments generated by such translations are always mere corollaries because some information is always lost in the process. Rather, the question to ask is this: having extracted an a fortiori argument from a syllogism, what syllogism can we in turn extract from that derived a fortiori argument? And conversely, having extracted a syllogism from an a fortiori argument, what a fortiori argument can we in turn extract from that derived syllogism?
Every syllogism implies a corresponding a fortiori argument which does not in turn imply it back – but it does go on to imply some syllogism. Similarly, every a fortiori argument implies a corresponding syllogism which does not in turn imply it back – but it does go on to imply some a fortiori argument. It should be obvious that these pairs of statements are not in contradiction, but I will explain why anyway. Consider the following samples:
a) Syllogism | A fortiori argument |
All Y are Z, and | The class Z is bigger than (or as big as) the class Y, and |
All (or some) X are Y; | The class Y is big enough to include all (or certain) members of the class X, |
therefore: | therefore: |
b) A fortiori argument | Syllogism |
P is more R than (or as much R as) Q is, and, | Rp implies Rq, and |
Q is R enough to be S; therefore, | Rq implies Rs; |
all the more (or equally), P is R enough to be S. | So, Rp implies Rs. |
Consider first process (a): if we wanted to draw a syllogism in accord with process (b) out of the a fortiori argument derived from the given syllogism, we would obtain the syllogism shown next, which is considerably different from the original one:
The size of class Z implies the size of class Y, |
the size of class Y implies the size of class X; |
therefore, the size of class Z implies the size of class X. |
I use the word “size” here to avoid using the more barbaric “bigness;” we are of course concerned with extensions, meaning that a bigger size implies a smaller one, as already explained. Note that this new syllogism is not categorical (with “is” relating terms) like the original one, but hypothetical (with “implies” relating theses[9]). If now we compare the information in this derived syllogism to that in the original syllogism, we note that valuable information was lost in the transition: the inclusions of Y in Z and X is Y (and therefore of X in Z) are long gone; instead, all we now discuss are the relative sizes of these classes (irrespective of any subsumption between them). Thus, though we have indeed obtained a new syllogism, it is certainly not the same as the original syllogism but a watered-down corollary of it.
Consider now process (b): if we wanted to draw an a fortiori argument in accord with process (a) out of the syllogism derived from the given a fortiori argument, we would obtain the a fortiori argument shown next, which is considerably different from the original one:
Thesis Rp is bigger than (or as big as) thesis Rq, and |
Thesis Rq is big enough to include all the conditions of thesis Rs; therefore, |
all the more (or equally), Thesis Rp is big enough to include all the conditions of thesis Rs. |
Here, as already explained, sizes (as in “bigger,” “big enough”) refer to numbers of conditions underlying theses, not to numbers of members within classes. Note that the new a fortiori argument is copulative, since it uses “is” rather than “implies” to relate items. Here again, if we compare the information in this derived a fortiori argument to that in the original a fortiori argument, we note that valuable information was lost in the transition: for instance, information we had initially on Q actually being (R enough to be) S is long gone – all we still know about them now is that the value of R corresponding to Q occurs in numerically more conditions than the value of R corresponding to S does. Thus, though we have indeed obtained a new a fortiori argument, it is certainly not the same as the original a fortiori argument but a watered down corollary of it.
Clearly, such reiterations do not produce very interesting results. No doubt if we tried reiterating further, i.e. translating the syllogism and a fortiori argument produced by the above reiterations into new derivatives, we would likewise not produce anything very interesting. But the main point we wished to make – viz. that reiteration should not be confused with reversing – has been convincingly established.
8. Lessons learned
We can here conclude our research into the relationships between a fortiori argument and syllogism. We looked into this topic because it is one often and hotly debated in literature on a fortiori logic. We decided to carry out a more formal and systematic investigation than past logicians and commentators have done, so as to be able to judge the matter more objectively and definitively. We first looked into the conceivable correlations between any two arguments, and in particular formally defined what we mean by a corollary. Then we compared the structures of syllogism and a fortiori argument, showing the various respects in which they differ significantly.
We then demonstrated that every syllogism contains an implicit a fortiori argument – but we also found that the latter is a mere corollary of the former, information being lost in transition, so that the process of derivation cannot be reversed. We then demonstrated the converse, i.e. that every a fortiori argument contains an implicit syllogism – but we also found that the latter is a mere corollary of the former, information being lost in transition, so that the process of derivation cannot be reversed. Finally, we showed that reiteration of these processes, though logically possible, is not very interesting.
We are now in a position to formulate the following overall lessons learned. Syllogism and a fortiori argument are very different movements of thought. They are structurally different, and each serves a different rational purpose; so they are not equivalent or interchangeable. Although they can be formally interrelated in various ways, they remain logically distinct and irreplaceable in important ways. Syllogism orders terms or theses by reference to the inclusions (or exclusions) between them, while a fortiori argument orders them by reference to the measures or degrees of some property they have (or do not have) in common. Neither function can be substituted for the other. We can (however awkwardly) reword one form of argument into the other; but such translations are not exactly transformations, because significant information cannot be passed on from one form to the other; therefore, neither form of argument can be fully reduced to the other. Thus, both forms are needed by reason to pursue its business; they are complementary instruments of reasoning.
In spite of all that, it should be remembered that a fortiori argument, whether copulative or implicational, is inextricably dependent on hypothetical (rather than categorical) syllogism, as we have shown in detail in the first chapter of the present volume, in the section dealing with validation (1.3). Hypothetical syllogisms (arguments such as if a, then b and if b then c, therefore if a then c) are not by themselves sufficient for validation of a fortiori arguments, since we also need comparative arguments (such as a > b and b > c, therefore a > c); but without them the validations would be impossible. Thus, a fortiori argument is a sort of composite argument, whose components are brought to the surface during the process of validation. Nevertheless, it is a form of argument in its own right, insofar as we are able to reason through it directly, without always having to resort to validation – i.e. it is intuitively credible anyway.
[1] Notably, Islamic commentators (such as al-Ghazali) seem to have tended in this direction, although this tendency has been implicit rather than explicit.
[2] Note in passing that we could similarly validate an argument with a negative major premise. Given that ‘P is dissimilar to Q with respect to R’ (i.e. say, P is R but Q is not R), then since ‘Q is S’, there is an S which is not R, whence by generalization No S is R, and this together with P is R implies that ‘P is not S’.
[3] Here, the symbol R refers to a singular affiRmative proposition, as against G for a singular neGative one. I introduced these symbols in my Future Logic, but singular syllogism is not something new. The Kneales (p. 67) point out that Aristotle gives an example of syllogism with a singular premise in his Prior Analytics, 2: 27. The example they mean is supposedly: “Pittacus is generous, since ambitious men are generous and Pittacus is ambitious” (1/ARR). Actually, there is another example in the same passage, viz.: “wise men [i.e. at least some of them] are good, since Pittacus is not only good but wise” (3/RRI). Note that the reason I did not choose the symbol F for aFfirmative was probably simply to avoid confusion with the symbol F for False. In any case, some symbols were clearly needed for singular propositions, since the traditional symbols A, E, I, O only concern plural propositions.
[4] In other words, the traditional Judaic belief (or dogma) that names are part of the nature of the things they name, if not their very essence, is – as far as formal logic is concerned – only a theory. There is nothing obvious or axiomatic about it. It is a hypothesis that must remain open to scrutiny and testing like any other. Modern linguistics would deny this hypothesis in view of the demonstrable fact that all languages, including Hebrew, have evolved over time.
[5] However, if we know that some R are Q, and do not know that some R are not Q, we can generalize the positive particular to obtain the ‘all R are Q’ proposition needed to infer the final conclusion. In that case, the argument as a whole would be doubly inductive, since involving two generalizations.
[6] As regards negation of the major premise, here, we can deal with it very simply as follows. ‘P is not greater than Q with respect to R’ can be restated as ‘P is either lesser than or equal to Q with respect to R’; therefore, given that Q is Sq and that Sp:Sq = P:Q (or Rp:Rq), it follows that P is Sp, where Sp < or = Sq. In other words, when the major premise is negative, we resort to two positive quantitative analogies in its stead.
[7] In a footnote to chapter 4.
[8] This is one of Aristotle’s examples reworked a bit.
[9] The propositions involved, written more fully, would be: ‘If a class has the number of members in class Z, then it has the number of members in class Y’ (meaning that the first number is greater or equal to the second number, as e.g. 5 implies 3), and so forth.