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A FORTIORI LOGIC

© Avi Sion, 2013 All rights reserved.

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A FORTIORI LOGIC

CHAPTER 6 – A fortiori in Greece and Rome

1. Aristotle’s observations

2. The Kneales’ list

3. Aristotle in practice

4. Relation to syllogism

5. Cicero

6. Alexander of Aphrodisias

7. Historical questions

1. Aristotle’s observations

Looking at the sayings or writings of ancient Greek philosophers – Thales, Anaximander, Anaximenes, Heraclitus, Pythogoras, Philolaus, Xenophanes, Parmenides, Zeno, Empedocles, Leucippus, Democritus, Anaxagoras, Socrates, Plato, and Aristotle, and their successors – one cannot but be awed by the extraordinary breadth and profundity of their thinking, and their anticipation of many ideas considered important today. For example, I recently realized that Empedocles[1] could be regarded as the precursor of the phenomenological approach, on the basis of his statement: “Think on each thing in the way in which it is manifest.”

It is not surprising, therefore, to find some discussion of a fortiori argument in the works of Aristotle (Greece, 384-322 BCE)[2]. The following quotations from his works (dated c. 350 BCE) seem relevant to our research.[3]

In his Rhetoric 2:23 (i.e. book II, chapter 23), in §4, Aristotle writes:

“Another line of proof is the a fortiori[4]. Thus it may be argued that if even the gods are not omniscient, certainly human beings are not. The principle here is that, if a quality does not in fact exist where it is more likely to exist, it clearly does not exist where it is less likely. Again, the argument that a man who strikes his father also strikes his neighbors follows from the principle that, if the less likely thing is true, the more likely thing is true also; for a man is less likely to strike his father than to strike his neighbors. The argument, then, may run thus. Or it may be urged that, if a thing is not true where it is more likely, it is not true where it is less likely; or that, if it is true where it is less likely, it is true where it is more likely: according as we have to show that a thing is or is not true.”

In this passage, Aristotle shows he considers a fortiori argument as a “line of proof” – by which he presumably means that it is a deductive argument. He marks his understanding of a fortiori argument as going from denial of the ‘more’ to denial of the ‘less’, or from affirmation of the ‘less’ to affirmation of the ‘more’. On this basis, we can say that Aristotle was aware of at least two valid moods: positive argument “from minor to major,” and negative argument “from major to minor,” though he does not use such terminology, but only says: “according as we have to show that a thing is or is not true”[5]. Clearly, therefore, what he has in mind here are positive and negative subjectal arguments. His arguments can be reworded as follows to clarify their standard formats (with the symbols P, Q, R, and S, denoting the major, minor, middle and subsidiary terms, respectively):

His first example is negative subjectal: that the gods are omniscient (P) is more credible (R) than that human beings are so (Q); therefore if the gods’ omniscience is not credible enough to be assumed (S), the omniscience of human beings is not credible enough to be assumed. This illustrates the principle: if a quality in a certain place (P) is more likely to be found (R) than the same quality in another place (Q) is, then if the quality in the first place is not sufficiently likely to be found to be considered as existing in fact (S), it follows that the quality in the second place is not sufficiently likely to be found to be considered as existing in fact (S).

His second example is positive subjectal: a man striking his neighbors (P) is a more likely event (R) than the man striking his father (Q); therefore, if a man striking his father is likely enough to be expected (S), then the man striking his neighbors is likely enough to be expected. This illustrates the principle: if something somewhere (P) is more likely (R) than the same thing elsewhere (Q), then if the latter is likely enough to be declared true (S), it follows that the former is likely enough to be declared true. (To which he adds the negative mood: if the former is not likely enough to be declared true, it follows that the latter is not likely enough to be declared true.[6])

Noteworthy here is Aristotle’s formulation of these a fortiori arguments in logical-epistemic terms, i.e. using a logical middle term (such as ‘likely’) and an epistemic subsidiary term (such as ‘believed’)[7]. His above two examples could of course have been formulated in purely ontical terms, as follows. The gods (P) are more well-endowed (R) than human beings (Q) are; therefore, if the gods are not well-endowed enough to be omniscient (S), then human beings are not well-endowed enough to be omniscient. Or again: striking one’s neighbors (P) generally seems more natural (R) than striking one’s father (Q); therefore, if striking his father seems natural enough to a certain man for him to actually do it (S), then striking his neighbors seems natural enough to him for him to actually do it.[8]

Still in Rhetoric 2:23, Aristotle adds a number of examples of allegedly a pari a fortiori argument. I say allegedly, because the proposed arguments are not complete enough to judge the matter. Note that five of the examples have negative form, while two have positive form. In any case, this serves to show us his awareness of such argument:

“This argument might also be used in a case of parity, as in the lines: Thou hast pity for thy sire, who has lost his sons: Hast none for Oeneus, whose brave son is dead? And, again, ‘if Theseus did no wrong, neither did Paris’; or ‘the sons of Tyndareus did no wrong, neither did Paris’; or ‘if Hector did well to slay Patroclus, Paris did well to slay Achilles’. And ‘if other followers of an art are not bad men, neither are philosophers’. And ‘if generals are not bad men because it often happens that they are condemned to death, neither are sophists’. And the remark that ‘if each individual among you ought to think of his own city’s reputation, you ought all to think of the reputation of Greece as a whole’.”

In his Topics 2:10 (book II, chapter 10), where Aristotle begins with: “Moreover, argue from greater and less degrees…,” I will divide what he thereafter says in three parts for purposes of analysis:

“See whether a greater degree of the predicate follows a greater degree of the subject: e.g. if pleasure be good, see whether also a greater pleasure be a greater good: and if to do a wrong be evil, see whether also to do a greater wrong is a greater evil. Now this rule is of use for both purposes: for if an increase of the accident follows an increase of the subject, as we have said, clearly the accident belongs; while if it does not follow, the accident does not belong. You should establish this by induction.”

This first paragraph, if it is at all related to a fortiori argument, makes clear by implication that Aristotle does not universally approve of a crescendo argument, i.e. of argument resembling a fortiori but having a ‘proportional’ conclusion. He is clearly not saying, for instance, that if pleasure is good it follows deductively that more pleasure is better – he is only saying that the question should be asked and that the answer is to be sought by induction; he explicitly conceives the possibility that it may not follow. This is an important finding concerning Aristotle, considering that (as we shall see) many people who historically came after him did not likewise realize the invalidity of ‘proportional’ a fortiori argument. He goes on:

“If one predicate be attributed to two subjects; then supposing it does not belong to the subject to which it is the more likely to belong, neither does it belong where it is less likely to belong; while if it does belong where it is less likely to belong, then it belongs as well where it is more likely. Again: If two predicates be attributed to one subject, then if the one which is more generally thought to belong does not belong, neither does the one that is less generally thought to belong; or, if the one that is less generally thought to belong does belong, so also does the other. Moreover: If two predicates be attributed to two subjects, then if the one which is more usually thought to belong to the one subject does not belong, neither does the remaining predicate belong to the remaining subject; or, if the one which is less usually thought to belong to the one subject does belong, so too does the remaining predicate to the remaining subject.”

Aristotle here details the positive and negative moods of three seemingly distinct a fortiori arguments. The first concerns two subjects (A, B) with a common predicate (C), and its major premise is: ‘A is C’ (P) is more likely (R) than ‘B is C’ (Q). The second concerns one subject (A) with two predicates (B, C), and its major premise is: ‘A is B’ (P) is more generally thought (R) than ‘A is C’ (Q). The third concerns two subjects (A, B) with two predicates (C, D), and its major premise is: ‘A is B’ (P) is more usually thought (R) than ‘C is D’ (Q). Although the middle term (R) is differently worded in each case, no great significance should be attached to this variation: all three may be taken to mean about the same, say ‘likely’. The subsidiary term (S) may in all cases be regarded as ‘believed’ (or ‘adopted’ or any similarly convenient qualification). In each case, the said major premise is followed by the minor premises and conclusions in the standard forms below:

Given something (P) is more likely (R) than another thing (Q) is, it follows that:

if Q is R enough to be believed (S), then P is R enough to be S;

and if P is R not enough to be S, then Q is R not enough to be S.



Clearly, the three sets of argument of positive and negative forms are effectively one and the same set. They illustrate subjectal a fortiori argument with a logical middle term (e.g. ‘likely’) and an epistemic subsidiary term (e.g. ‘believed’). Note well, though these arguments concern whole propositions (labeled P and Q by me), they are not to be regarded as antecedental since no implication between propositions is suggested. Though the major and minor terms, P and Q, are propositions, each stands in this context as a unitary term, a subject of which may be predicated the said logical and epistemic qualifications. The four terms of the a fortiori argument as such are the two effective subjects P and Q, and the two predicates ‘likely’ (R) and ‘believed’ (S). Aristotle does not seem aware of all that.

The three or four terms mentioned by Aristotle as subjects and predicates (labeled A, B, C and D by me) are terms within the propositions P and Q, and not the terms of the a fortiori argument as such, note well. These terms (A, B, C, D) are thus quite incidental to the argument, which are used to illustrate possible uses of such argument. Aristotle could well have mentioned only one such illustration if his intention was to abstractly describe a fortiori argument as such. It appears, then, that it was not his primary intention to do that here. His primary intention was probably to concretely describe different ways a predicate may be found to belong or not belong to a subject, by using a fortiori argument.

Nonetheless, judging by this second paragraph, which clearly concerns a fortiori argument, we can again say that Aristotle was well aware of the positive and negative subjectal moods. Thus far, however, there is still no evidence of his being aware of predicatal arguments. We might, on a superficial reading, have thought that Aristotle here marks the difference between subjectal and predicatal a fortiori, when he speaks of one predicate for two subjects or two predicates for one subject. He might have been referring, in the first case to the subsidiary term being predicated of the major and minor terms (the subjectal mood), and in the second case to the major and minor terms being predicated of the subsidiary term (the predicatal mood). But when we actually set out his arguments in standard forms, we see clearly that they are all subjectal.

I should also stress that though Aristotle’s arguments in the above paragraph of Topics, as well as in the Rhetoric passage earlier considered, can be cast in standard forms using qualifications like ‘likely’ as middle term and ‘believed’ as subsidiary term, it is obvious that Aristotle himself does not formulate his arguments as clearly. He is not sharply aware of the distinct functions of these two terms (R and S) in his arguments. In fact, he tends to lump them together, i.e. treat them as one and the same. This observation will be further confirmed further on, when we analyze Topics 3:6. Still in Topics 2:10, he goes on:

“Moreover, you can argue from the fact that an attribute belongs, or is generally supposed to belong, in a like degree, in three ways, viz. those described in the last three rules given in regard to a greater degree. For supposing that one predicate belongs, or is supposed to belong, to two subjects in a like degree, then if it does not belong to the one, neither does it belong to the other; while if it belongs to the one, it belongs to the remaining one as well. Or, supposing two predicates to belong in a like degree to the same subject, then, if the one does not belong, neither does the remaining one; while if the one does belong, the remaining one belongs as well. The case is the same also if two predicates belong in a like degree to two subjects; for if the one predicate does not belong to the one subject, neither does the remaining predicate belong to the remaining subject, while if the one predicate does belong to the one subject, the remaining predicate belongs to the remaining subject as well.”

Looking at this third paragraph, we can also say that Aristotle realized that a fortiori inference is also possible between equals, not just from the more to the less or vice versa. And as he points out, in such case the argument can function either way, i.e. from minor to major or from major to minor, whether it is positive or negative. I have in my Judaic Logic account called such argument, in which the major premise is a statement of equality, egalitarian a fortiori; another name for it is a pari.

Apart from that, there is nothing new in this paragraph – it still concerns only subjectal moods. There is still no mention of equivalent predicatal moods (which involve quite different arrangements of terms). Even so, this insight of his has some importance. He also says further on in the Rhetoric chapter above quoted: “This argument [i.e. a fortiori] might also be used in a case of parity.” He again implies as much in Topics 2:11: “You can argue, then, from greater or less or like degrees of truth in the aforesaid number of ways” (italics mine) and elsewhere.

It should be noted that, though Aristotle, as we have seen in Rhetoric 2:23 and in the second passage of Topics 2:10, formulates a fortiori argument primarily in logical-epistemic terms, looking at the third passage of Topics 2:10 it appears that he also conceives of purely ontical a fortiori argument, since he speaks repeatedly of a predicate belonging to a subject, as against being supposed to belong. This is confirmed by some of his a fortiori pronouncements in other contexts; for example in Topics 3:6 (see below), or again in the History of Animals 5:14, where he says: “If a sow be highly fed, it is all the more eager for sexual commerce, whether old or young,” implying that being well-fed physically causes (and not merely implies) a sow to want sex.

It is reasonable to suppose that, though Aristotle only mentions the distinction between a property and a supposed property in the said third passage, which deals with terms of “like degree,” he does not consider this distinction as exclusive to egalitarian a fortiori arguments. For a start, he makes no mention of such exclusiveness; and besides, examples like the one just cited from the History of Animals show that he does not intend it. Thus, this distinction between a property and a supposed property can be fairly applied to non-egalitarian a fortiori arguments too.

In other words, we may say that Aristotle is somewhat aware of purely ontical argument, and does not limit on principle a fortiori to the logical-epistemic variety, even if he appeals to the latter more often (so far). In my theory of a fortiori argument, note, the emphasis is rather on the ontological variety. This does not of course exclude the epistemological variety, which Aristotle seemingly emphasized, nor for that matter the ethical and legalistic variety, which the Rabbis and others have emphasized; I view (and have from the start viewed) all these other varieties as special cases of the primary, ontological variety.

A further thing to notice is the uncharacteristic lack of formalization in Aristotle’s treatment of a fortiori argument. This is no doubt because he mentions such argument in passing, without focusing on it particularly or very deeply. Although he does discuss the argument in relatively abstract terms, as when he says in the Rhetoric passage: “if a thing is not true where it is more likely, it is not true where it is less likely; or … if it is true where it is less likely, it is true where it is more likely,” and not merely through concrete examples (like the man striking his father or neighbors), he does not go one step further as he did with syllogistic reasoning and use symbols (A, B, Γ, Δ) in lieu of terms to list all possible moods of the argument and, most importantly, to formally validate or invalidate them. My theory of a fortiori argument does this crucial job.

In this regard, we should note too that Aristotle does not here (or elsewhere, to my knowledge) formulate any rule of reasoning comparable to the rabbinical dayo principle (which appears on the stage of documented history perhaps some four and a half centuries later[9]) – or more precisely, to the principle of deduction as it applies specifically to a fortiori argument, namely the rule that the subsidiary term (which is a predicate in subjectal argument) must be identical in the minor premise (where it concerns the minor term) and in the conclusion (where it is applied to the major term). Aristotle may well in practice reason correctly in accord with this principle, but he does not explicitly express theoretical awareness of it – unless we count the already mentioned passage: “See whether a greater degree of the predicate follows a greater degree of the subject,” which we interpreted as an effective rejection of a crescendo argument, as intended by him to be an admonishment by him not to always reason proportionately.

Let us now move on and examine a passage of his Topics 3:6 (book III, chapter 6), which again I split up as convenient:

“Moreover you should judge by means of greater or smaller or like degrees: for if some member of another genus exhibit such and such a character in a more marked degree than your object, while no member of that genus exhibits that character at all, then you may take it that neither does the object in question exhibit it; e.g. if some form of knowledge be good in a greater degree than pleasure, while no form of knowledge is good, then you may take it that pleasure is not good either.”

In this first paragraph, Aristotle shows stronger awareness of the middle term of a fortiori argument, namely the “such and such a character” (R) which the “other genus” (i.e. the major term, P) exhibits in a more marked degree than “your object” (i.e. the minor term, Q). Notice, too, that this middle term (R) is definitely ontological, rather than as before epistemological. However, his argument is not very well formulated, in that his major premise states that “some members” of P “exhibit this character,” whereas his minor premise states contradictorily that “no members” of P “exhibit this character.” This confusion is not due to his insertion of quantification issues into the equation, but to his conflation between the middle term (in the major premise) and the subsidiary term (in the minor premise and conclusion). The latter is a not uncommon error of formulation[10]. He goes on:

“Also, you should judge by a smaller or like degree in the same way: for so you will find it possible both to demolish and to establish a view, except that whereas both are possible by means of like degrees, by means of a smaller degree it is possible only to establish, not to overthrow. For if a certain form of capacity be good in a like degree to knowledge, and a certain form of capacity be good, then so also is knowledge; while if no form of capacity be good, then neither is knowledge. If, too, a certain form of capacity be good in a less degree than knowledge, and a certain form of capacity be good, then so also is knowledge; but if no form of capacity be good, there is no necessity that no form of knowledge either should be good. Clearly, then, it is only possible to establish a view by means of a less degree.”

This second paragraph serves to show (only by means of example, but clearly enough) that Aristotle is aware that, even though one may argue positively, from predication of the subsidiary term to the minor term to predication of the subsidiary term to the major term, it does not follow that one may argue negatively, from denial of predication of the subsidiary term to the minor term to denial of predication of the subsidiary term to the major term – except, of course, where the argument is a pari. He here obviously refers specifically to subjectal argument, since in fact (although he makes no remark to that effect) the opposite rule would hold for predicatal argument. His statement of this rule is significant, since he thereby declares a mood invalid, whereas previously he only declared moods valid.

Note however that he does not similarly point out that, though (in subjectal argument) one may argue negatively, from denial of predication of the subsidiary term to the major term to denial of predication of the subsidiary term to the minor term, it does not follow that one may likewise argue positively, from predication of the subsidiary term to the major term to predication of the subsidiary term to the minor term – except, of course, where the argument is a pari. That is, even though he has previously mentioned both positive and negative moods for validation purposes, in the present remark he only mentions a negative mood for invalidation purposes and omits to mention the corresponding positive mood for invalidation purposes.

Moreover, Aristotle’s “invalidation” of a mood of a fortiori argument here is merely intuitive, i.e. a raw rational insight – he does not explain or formally prove the invalidity of the mood in question. He tells us that it is wrong reasoning, but he does not tell us why it is so.

Furthermore, in the example he gives, the major term is “knowledge” and the minor term is an unspecified “capacity,” while the middle and subsidiary terms are “good.” In this passage, then, he again confuses the issue somewhat by contradicting elements of his major premise, viz. “if a certain form of capacity be good [middle term, R] in a like degree to knowledge,” in his minor premise and conclusion, viz. “if no form of capacity be good [subsidiary term, S], then neither is knowledge.”

The error here, as already pointed out, is to use one and the same term (viz. “good,” in this example) both as middle and as subsidiary. For the argument to be consistent and valid, these two must be distinct (the middle term might, say, be “valuable” and the subsidiary term “pursued,” so that the argument reads: if a certain capacity is as valuable as knowledge, it follows that if no capacity is valuable enough to be pursued, then knowledge is not valuable enough to be pursued). Aristotle, then, is apparently not aware of this important rule, i.e. of the need to distinguish the middle and subsidiary terms.

We might more generously see, in Aristotle’s affirmation of something in one premise and negation of it in the other, as a recognition by him of the possibility of using a term so abstractly that both its position (e.g. “good”) and its negation (“not good”) are included in it, as different degrees of it (above zero and zero or less, respectively). Looking at a term R in this way, we can both claim that P is more R than Q, and claim that P and Q are not R at all, without self-contradiction. This seems to be the thought in Aristotle’s head, though he does not (here at least) make any explicit remark to that effect. To be sure, knowing that Aristotle is not prone to self-contradiction, this is a credible hypothesis.

It is worth noting too in this context that, although Aristotle associates a fortiori argument with the idea of greater, lesser or equal degrees, there is no evidence in the above cited passages of any notion of “sufficiency,” i.e. of there being a threshold as of which predication occurs and before which it does not occur. This is an important deficiency in his treatment (if indeed, as I presume, he nowhere else mentions this feature of a fortiori argument). Had he been aware of the “sufficiency” issue (i.e. the need to have enough of the middle term for predication) in a fortiori inference, he would have quickly realized that the middle term mentioned in the major premise cannot reasonably be identical with the predicate inferred from the minor premise to the conclusion.

As we shall see further on, all but one of Aristotle’s many a fortiori arguments in practice are formulated without the crucial feature of “sufficiency” of the middle term for predication. The one exception shows that Aristotle was slightly aware of this feature, but not enough to make it explicit in all his a fortiori discourse, and not enough to take it into consideration in his theorizing.

2. The Kneales’ list

In their historical opus, The Development of Logic[11], William and Martha Kneale give seven references in Aristotle’s Topics concerning a fortiori argument, namely: “ii. 10 (114b37); iii. 6 (119 b17); iv. 5 (127 b18); v. 8 (137 b14); vi. 7 (145 b34); vii. 1 (152 b6); vii. 3 (154 b4)”[12]. I have above dealt in detail with the first two of these passages (namely, 2:10 and 3:6), which are the most interesting, in that Aristotle is in them effectively teaching us something about a fortiori argument. The remaining passages are less interesting: Aristotle uses rather than discusses a fortiori argument in two of them (namely, 4:6 and 7:3), while the rest (namely, 5:8, 6:7 and 7:1) have nothing to do with such argument but were only apparently listed because they contain a reference to degrees. Only the following two remaining passages, then, concern a fortiori argument:

Topics 4:6 – This chapter contains an a fortiori argument of positive subjectal form:

“On the other hand, the comparison of the genera and of the species one with another is of use: e.g. supposing A and B to have a like claim to be genus, then if one be a genus, so also is the other. Likewise, also, if what has less claim be a genus, so also is what has more claim: e.g. if ‘capacity’ have more claim than ‘virtue’ to be the genus of self-control, and virtue be the genus, so also is capacity. The same observations will apply also in the case of the species. For instance, supposing A and B to have a like claim to be a species of the genus in question, then if the one be a species, so also is the other: and if that which is less generally thought to be so be a species, so also is that which is more generally thought to be so.”

The reasoning here is: Given that A seems more fitting to be a genus (or a species) than B is, it follows that: if B seems so fitting that it may be declared a genus (or a species), then A must also be fitting enough for that; if A and B are equally fitting (parity), then the inference goes both ways. We can distinguish two moods (from minor to major, and a pari), each with two alternative middle terms (one for genus and one for species); but all four arguments have really one and the same thrust.

Topics 7:3 – This chapter contains a very similar a fortiori argument:

“Moreover, look at it from the point of [sic][13] and like degrees, in all the ways in which it is possible to establish a result by comparing two and two together. Thus if A defines a better than B defines [b?] and B is a definition of [b?] so too is A of a. Further, if A’s claim to define a is like B’s to define b, and B defines b, then A too defines a. This examination from the point of view of greater degrees is of no use when a single definition is compared with two things, or two definitions with one thing; for there cannot possibly be one definition of two things or two of the same thing.”

The reasoning here is: Given that A defines ‘a’ more fittingly than B does ‘b’, it follows that if B defines ‘b’ so fittingly that it may be declared the definition, then A defines ‘a’ must also be fitting enough for that; if Aa and Bb are equally fitting (parity), then the inference from Bb to Aa is also valid (more significantly, the reverse inference is also possible now: though Aristotle does not say so, he probably intended it). Here again, note, there is only really one argument, though it is worded in two ways.

We do not learn anything new about a fortiori argument from these two passages; they each give an example of a fortiori argument, rather than a discussion of it. I should perhaps, after all, say a bit more about the three passages listed by the Kneales that do not contain a fortiori arguments. Aristotle seems there and elsewhere[14] to have some beliefs about the degrees of things that I do not entirely agree with.

Consider for instance the following comment drawn from Topics 5:8:

“Next look from the point of view of greater and less degrees… See, for destructive purposes, if P simply fails to be a property of S simply; for then neither will more-P be a property of more-S, nor less-P of less-S, nor most-P of most-S, nor least-P of least-S. … For constructive purposes, on the other hand, see if P simply is a property of S simply: for then more-P also will be a property of more-S, and less-P of less-S, and least-P of least-S, and most-P of most-S.”

What this, and more of the same (which I have left out, for brevity’s sake), suggests is that Aristotle considers concomitant variation to be a universal law. According to him, if S is P, then to every degree of S there corresponds a comparable degree of P, and if such parallel increase and decrease in magnitude does not occur, then S is not P. This is highly to be doubted, in my view. In some cases, the same value of a predicate P is applicable to all values of a subject S. In some cases, a constant subject S has (over time) different degrees of a predicate P. The variations may be inverted, with increase on one side and decrease on the other, or vice versa. Many other complications are conceivable and occur in practice.

As an example of such inference that Aristotle gives us is: “Thus, inasmuch as a higher degree of sensation is a property of a higher degree of life, a lower degree of sensation also would be a property of a lower degree of life, and the highest of the highest and the lowest of the lowest degree, and sensation simply of life simply.” Well, it may be true that degrees of sensation are proportional to degrees of life (whatever that means: presumably complexity of organization?), but I very much doubt that we can universally infer a concordance of lesser degrees from one of a higher degrees, and so on, as he apparently recommends. Perhaps he only means that such concomitant variation is a good working hypothesis, a probability to be verified empirically.

Again, consider the following comment drawn from Topics 4:6:

“Moreover, judge by means of greater and less degrees: in overthrowing a view, see whether the genus admits of a greater degree, whereas neither the species itself does so, nor any term that is called after it… If, therefore, the genus rendered admits of a greater degree, whereas neither the species does so itself nor yet any term called after it, then what has been rendered could not be the genus.”

Let G be a genus and S be a species, or a species of a species. The question here posed is whether G is or is not indeed a genus of S; or conversely, whether S is or is not indeed a species of G. The answer is sought through comparison of changes in magnitude; actually, only increase in magnitude is mentioned, not decrease (no explanation is given for this unreasonable stipulation). It is not clarified what is here increased – it seems to be the degree of G or S itself, rather than of some property thereof. The changes in degree seem to refer to comparisons of instances (extensional mode), rather than to changes over time (temporal mode).

Aristotle reasons syllogistically that if the genus is variable then the species must be variable too. But to my mind this is an error of logic. Surely a variable is a set of constants, in which case a genus may be variable and yet composed of species some or all of which are (different) constants. The error is to treat the predicate ‘variable’ as distributive, whereas it is here intended as collective – it applies to the class as a whole, not necessarily to any of its parts.

Such comments by Aristotle, though not directly relevant to a fortiori argument, have indirect relevance, since belief in the universality of concomitant variation would lead us to automatically draw an a crescendo conclusion from a fortiori premises, whereas in fact an appropriate pro rata argument is a formally required intermediary for such deduction. But as we have earlier seen (in the previous section, in the first passage of Topics 2:10), Aristotle explicitly (though without naming it) presents argument pro rata as inductive rather than deductive. It follows that he cannot (without self-contradiction) have here intended to suggest that pro rata argument always possible, i.e. formally universal for any terms. Thus, a fortiori argument must be distinguished from a crescendo.

About a contrario. In this context, I could additionally point to some of Aristotle’s remarks in his Rhetoric, which give the impression that he advocates a contrario argument, which has some resemblance to a fortiori argument but is really very different. The following passage, drawn from the already mentioned chapter of Rhetoric will illustrate what I mean:

“One line of positive proof is based upon consideration of the opposite of the thing in question. Observe whether that opposite has the opposite quality. If it has not, you refute the original proposition; if it has, you establish it. E.g. ‘Temperance is beneficial; for licentiousness is hurtful’. Or…: ‘If war is the cause of our present troubles, peace is what we need to put things right again’.”

If we read this literally, we would suppose that ‘If all X are Y, then no not-X is Y’. But such inversion, as Aristotle surely well knew, is not universally valid. We can only educe from ‘all X are Y’ (via: ‘all not-Y are not-X’) that ‘some not-X are not Y’; it remains possible that ‘some not-X are Y’. So, we should view his remarks on such arguments as mere observations. They are presented as forms of rhetoric, rather than of logic, so as to point out noncommittally that people do use them, without intent to imply them to be necessarily valid.

The examples he gives seem credible enough, being particular causative arguments. Since licentiousness hurts, we should try temperance to diminish if not remove our pain. Since war causes troubles, we should try peace to diminish if not stop our malaise. These are only probable arguments, however, which do not guarantee that the desired change will occur. They are not, of course, a fortiori arguments, although they have some resemblance.

Compare the commonly used formulation of a fortiori argument: ‘If Q, which is not R, is S, then, all the more, P, which is R, is S’, with the following a contrario statement (which for the sake our present demonstration involves the vaguer term ‘something’ in the places of P and Q): ‘If something which is not-R is S, then something which is R is not-S’ – and it is easy to see the difference. In the former case (i.e. a fortiori), the predicate is S in both the antecedent and consequent, whereas in the latter case (i.e. a contrario), the predicate is S in the premise and not-S in the conclusion. The resemblance is thus quite superficial.

A contrario argument, like a fortiori, can be copulative or implicational. In the former case, it has the form: ‘If X is Y, then not-X is not-Y’; and in the latter case, it has the form: ‘If X implies Y, then not-X implies not-Y’. While such reasoning is sometimes applicable, it is not – to repeat – universally valid.

Finally, let me quote the Kneales’ sole remark about Aristotle in relation to a fortiori argument:

“…the theory of arguments a fortiori, or, as Aristotle says, ‘from the more and the less’. This is a topic to which he refers many times and always in a way which suggests that he thinks of it as a well-recognized theme. It was natural, therefore, that he should wish to incorporate his views on the subject into his later work on logic, and it seems probable that this is what he had in mind when he spoke later of his intention to write on arguments ‘according to quality’ (κατἀ ποιὀτητα).” (Pp. 42-43.)

This comment suggests that Aristotle was rather interested in a fortiori argument and seemingly intended to treat the subject in more detail eventually. The Kneales do not specifically cite the passages in Aristotle’s works they base these remarks on. As already mentioned, they do give a number of references in the Topics, but I do not see that these passages justify the above claims. Not that it matters greatly, but I would have liked to know what the Kneales meant more precisely. Because, judging by the texts analyzed above, Aristotle’s involvement in theoretical a fortiori logic was not very intense.

3. Aristotle in practice

Let us now take a closer look at Aristotle’s practice of a fortiori argument, which differs considerably from his theoretical treatment. For this purpose, I looked into all instances I could find of Aristotle’s use of the argument[15]. See Appendix 4 for a detailed list of citations[16]. These included 40 occurrences of the fifteen key phrases most often used to signal a fortiori discourse, namely: a fortiori (12), all the more (22), how much more (2), how much less (0), so much more (1), so much less (0), much more (2), much less (1), (how/so) much the more (0), (how/so) much the less (0). Plus 3 occurrences of more widely used character strings, namely: more so (1), less so (0), even more (2), even less (0). Additionally, I referred to the passages in Aristotle’s Rhetoric and Topics found by the Kneales (see previous two sections), which contain numerous a fortiori arguments without use of the key phrases (except once), and found another 37 occurences.

Altogether, I found in Aristotle’s works, 80 cases of a fortiori argument, of which at least 11 were a pari (i.e. involved a major premise with equal major and minor terms). As could be expected, most cases, 48 to be exact, were positive subjectal in form; and indeed, of these 8 could be said to be a crescendo. Without surprise, another 22 cases were found to be negative subjectal. The interesting findings were that 5 cases were positive predicatal and 3 cases were negative predicatal; and moreover that 2 cases were antecedental. What these findings teach us is that, although Aristotle reasoned often enough in subjectal formats, which he mentions in his more theoretical exposés, he also occasionally reasoned in other formats, which he does not consciously distinguish in theoretical contexts.

Aristotle, as everyone knows, was Plato’s star student. Examining the latter’s main works, I found at least 15 instances of a fortiori discourse, 9 of them spoken (if we are to believe Plato) by Socrates, and the rest by others. Of these instances, 9 are positive subjectal in form (and of those, 4 seem to have an a crescendo intent), 1 is negative subjectal, 4 are negative predicatal, and 1 is negative consequental in form. These findings are based on computer searches for specific strings; more cases, involving other wording, may conceivably yet be found. These figures on Plato are also significant, assuming that Aristotle read these works (a fair assumption), since they are additional evidence that Aristotle did not closely examine all the data he had on hand when analyzing a fortiori argument. The corresponding findings for Aristotle are as follows:

Mood of
a fortiori argument

Orientation

Number found

Of which
a pari

Of which
crescendo

Copulative

Positive subjectal {+s}

from minor to major (Q-P)

48

7

8

Negative subjectal (–s)

from major to minor (P-Q)

22

4

Positive predicatal {+p}

from major to minor (P-Q)

5

Negative predicatal (–p)

from minor to major (Q-P)

3

Implicational

Positive antecedental (+a)

from minor to major (Q-P)

2

Negative antecedental (–a)

from major to minor (P-Q)

0

Positive consequental (+c)

from major to minor (P-Q)

0

Negative consequental (–c)

from minor to major (Q-P)

0

Totals

80

11

8

Table 6.1

Needless to say, the arguments are here classified on the basis of their apparent forms, without regard to the truth or falsehood of their contents.

As regards Aristotle’s own use of predicatal argument, 1 case occurs in On the Soul, 1 case in Parva Naturalia, 1 case in History of Animals, 1 case in Metaphysics, 2 cases in the Posterior Analytics, and 2 cases in Rhetoric. For example: “But if the Soul does not, in the way suggested [i.e. with different parts of itself acting simultaneously], perceive in one and the same individual time sensibles of the same sense, a fortiori it is not thus that it perceives sensibles of different senses” (Parva Naturalia, 7). This has to be read as a predicatal argument[17], since the subjects of the minor premise and conclusion are one and the same (viz. “the soul”) and their predicates are different (viz. “it perceives sensibles of the same sense” and “it perceives sensibles of different senses”).

Aristotle’s two uses of implicational argument (both positive antecedental) occur in History of Animals; notice that there is no use of negative antecedental or of positive or negative consequental argument. An example is: “Now, as the nature of blood and the nature of the veins have all the appearance of being primitive, we must discuss their properties first of all, and all the more as some previous writers have treated them very unsatisfactorily” (3:2). This has to be read as an implicational argument[18], because in the minor premise and conclusion, the antecedents and consequents contain different subjects and predicates, so that these propositions consist of theses implying theses.

Thus, judging by his extant works, Aristotle did not pay close attention to his own uses, or his teacher’s uses, of a fortiori argument, when discussing this form of reasoning. Had he done so, he would have discovered predicatal argument and implicational argument.

Furthermore, as regards his 8 uses of a crescendo argument (all positive subjectal), it may be supposed that Aristotle uttered them in good faith, i.e. that he believed that in these specific cases proportionality was justified. But he apparently nowhere remarks on the important difference between purely a fortiori argument and the more elaborate a crescendo argument, even though he uses both these types of reasoning. That is to say, he does not formulate a rule comparable to the much later rabbinical “sufficiency (dayo) principle,” according to which (in the simplest reading of it[19]) the conclusion of an a fortiori argument should exactly mirror its minor premise, and not indulge in proportionality (to which we should add: unless, of course, an appropriate pro rata argument can be additionally put forward to justify such proportionality).

It is noteworthy that, in all the instances of a fortiori argument I found in Plato and Aristotle works, only one instance contains the word ‘enough’ or ‘sufficient’. The instance is found in Aristotle’s work and reads: “But since even water by itself alone, that is, when unmixed, will not suffice for food – for anything which is to form a consistency must be corporeal – , it is still much less conceivable that air should be so corporealized [and thus fitted to be food]” (On Sense and the Sensible, 5). This shows that Plato was unaware of this crucial feature of a fortiori argument, and Aristotle was a bit more but still barely aware of it.

Finally, it is interesting to note the following statistics: of the a fortiori arguments used by Aristotle, only 16 are logical-epistemic[20], the remaining 57 being ontical. What this tells us is that the impression given by Rhetoric 2:23 and Topics 2:10 that he regards a fortiori argument as essentially logical-epistemic is belied by his actual practice.

4. Relation to syllogism

One more important question to ask regarding Aristotle’s theoretical treatment of a fortiori is whether he regarded such argument as capable of identification with syllogism. Wiseman[21] suggests that Aristotle did not make such an equation, saying:

“Interestingly, Aristotle did not consider the a fortiori to be the same as his categorical syllogism; rather, he understands it as an analogic[al] device, unlike what we have encountered in some definitions so far that meant to show it as deductively valid. Perhaps Aristotle was the first to view the a fortiori as an inductive analogy.”

As regards Wiseman’s claim that Aristotle viewed a fortiori as a mere analogical device, I tend not to agree. Wiseman is basing this assumption, I take it, on the first of the above quoted paragraphs in Topics 2:10– which, as already pointed out, is not clearly about a fortiori argument (even though the next paragraph indeed is about it). Aristotle is here neither proposing a necessary deduction (a fortiori or other) nor suggesting a weaker argument by analogy – on the contrary, he is saying one cannot predict which way things will go (“See whether a greater degree of the predicate follows a greater degree of the subject…”) and must resort to induction for the answer. Moreover, if we look at the earlier Rhetoric quotation, a different picture emerges.

As regards the suggestion that the two forms of argument are different, note that Wiseman does not quote Aristotle as saying so; he only theorizes it is so, based on the information available to him. I would certainly lean towards the same assumption, however. It would seem (given his extant works) that Aristotle did not ask himself or try to answer that specific question, about whether a fortiori argument is or is not a sort of syllogistic argument; had he done so, he would surely have stressed the fact explicitly, one way or the other. On the other hand, it could be argued that Aristotle tended to consider syllogism as the essential form of all argument (certainly many people after him seem to have thought he did so) – in which case he would not necessarily think he needed to specifically subsume a fortiori for us.

Consider now an example of a fortiori argument given by Aristotle in Rhetoric 2:23: a man is less likely to strike his father than to strike his neighbors; therefore, if a man strikes his father, he is likely to strike his neighbors too. We see here that Aristotle is aware of the major premise[22], as well as of the minor premise and conclusion. However, he does not discuss the real middle term, which tacitly underlies and would explain and justify the apparent middle term ‘likely’ that he takes for granted. Why is a man more likely to strike his neighbors than his own father? Because it is generally easier, psychologically, socially and ethically to strike one’s neighbors than one’s father. The apparent middle term ‘likely’ is based on an emotional and cultural fact (or at least, the assumption of such a fact).

A fortiori argument usually appears as essentially deductive – in the sense that given the premises we can confidently infer the conclusion – yet in the present example there is clearly a sense that the conclusion is at best probable. Why is that? Because it so happens that the example under scrutiny is about human volition, i.e. something that by nature cannot be predicted with certainty. A man may well generally find it easier to hit neighbors than his own father; but in truth, a man may consider the latter action as more legally permissible, being a private as against public matter, or again, he may out of cowardice hit on his weak old father more readily than he assaults his strong young neighbors.

Such actions are based on personal perceptions or belief systems, and depend on personal inclinations and conscience, and they are ultimately produced by freewill. For this reason, Aristotle indeed had to qualify things as only “likely” throughout his example. But such approximation is not inherent to a fortiori, but a function of the content in this particular sample. If we look at the other example Aristotle gives in the same passage of Rhetoricif even the gods are not omniscient, certainly human beings are not – it is clear that he sees the conclusion as certain[23], and not as a mere rough analogy[24].

We can thus, to conclude, say that since – as far as we know – Aristotle did not fully analyze a fortiori argument, he is not likely to have made a pronouncement as to whether it was the essentially same as syllogism or not; or, for that matter, as to whether it is deductive or merely analogical. The truth is, Aristotle was a genius who ranged far and wide in logic, philosophy and the special sciences, and touched upon a great many subjects, some of which he took time to look into more deeply and systematically, and some of which he only briefly considered in passing. Regarding a fortiori, the latter seems to be applicable. Moreover, of course, Aristotle was human, and however authoritative his viewpoints on many issues, he was not omniscient (as he readily admits in one of the said examples).

Whatever Aristotle may have or not have privately thought on the issue, my own formalization of a fortiori, presented in the preceding chapters, justifies our henceforth definitively adopting the position that Aristotle’s categorical syllogism (and also for that matter hypothetical syllogism, which is very similar in overall form) is very different from copulative (or implicational, as the case may be) a fortiori argument, though the latter is also a form of deduction. Moreover, although we can correlate these two forms of argument in various ways, we cannot formally reduce either of them to the other; they are distinct and relatively independent movements of thought.

5. Cicero

Marcus Tullius Cicero (Rome, 106-43 BCE), who was an influential philosopher and jurist among many other things, left us some interesting reflections on a fortiori argument in his Topics[25]. Cicero there tells us (this was a year before his death) he composed the book as a commentary to Aristotle’s work with the same name, from memory; but his treatment is distinctive. It seems to have been equally influenced by Aristotle’s Rhetoric (II, 23) and by some later, Stoic texts[26]. Concerning argumentation in general, Cicero has this to say:

“6. Every systematic treatment of argumentation has two branches, one is concerned with invention of arguments and the other with judgment of their validity; Aristotle was the founder of both in my opinion.”

By “invention of arguments” he apparently means formulation of arguments. From his mention here of validation, we see that Cicero’s interest was in logic, and not merely in rhetoric. He discusses in some detail all the arguments he lists, giving examples from Roman law practices. Arguments by comparison (i.e. a fortiori) are classified as arguments “from the things which are in some way closely connected with the subject,” which in turn fall under the heading of arguments “inherent in the nature of the subject.” This teaches us that Cicero looked upon a fortiori argument as essentially ontical, rather than as logical-epistemic. He introduces a fortiori argument in §23 as follows:

“23. All arguments from comparison are valid if they are of the following character: what is valid in the greater should be valid in the less (Quod in re maiore valet, valeat in minori), as for example… Likewise the reverse: what is valid in the less should be valid in the greater (Quod in minori valet, valeat in maiore); the same example may be used if reversed. Likewise, what is valid in one of two equal cases should be valid in the other (Quod in re pari valet valeat in hac quae par est); for example… Equity should prevail, which requires equal laws in equal cases.”

Cicero here apparently lists three varieties of the argument: from major to minor; from minor to major; and from equal to equal. Let us look at the examples here proposes for them. The first example concerns reasoning from major to minor: “since there is no action for regulating boundaries, there should be no action for excluding water in the city.” This argument seems to be a negative subjectal; we can formalize it as follows:

Regulating boundaries (P) is more serious a matter (R) than excluding water in the city (Q) is,

yet, regulating boundaries (P) is not a serious matter (R) enough to justify an action (S);

therefore, excluding water in the city (Q) is not a serious matter (R) enough to justify an action (S).[27]



For reasoning from minor to major, Cicero unfortunately gives no example here, but only says “the same example may be used if reversed.” It is not clear what “reversed” (convertere) here means. It surely does not mean simple conversion, for such argument would obviously be logically invalid[28]. That is, we can reasonably assume he is not suggesting that “since there is no action for excluding water in the city, there should be no action for regulating boundaries” follows from the preceding case. Therefore, he presumably intends a hypothetical contraposition of it: “if there was a possibility of action for excluding water in the city, there would be a possibility of action for regulating boundaries,” which signifies: positive subjectal argument.

The example Cicero adduces for a pari argument is: “since use and warranty run for two years in the case of a farm, the same should be true of a (city) house. But a (city) house is not mentioned in the law, and is included with the other things use of which runs for one year”[29]. It is not clear to me what the intended conclusion is, here. The first sentence seems to conclude with equality; but the second sentence denies the equality. I think that the solution to that problem is simply that Cicero here proposes two a pari arguments, one positive and one negative. The first says hypothetically: “if farm and city house were equal, the law of the former would apply to the latter.” The second says factually: “but since they are not equal, the law of the former does not apply to the latter.”

Thus, to summarize, Cicero seems to have pointed to positive and negative subjectal a fortiori argument, including their a pari versions. What about the positive and negative predicatal moods? I do not think that we can judge on the basis of the examples he gives that Cicero consciously limited a fortiori to the subjectal moods, to the exclusion of the predicatal ones; or for that matter, that he intended to limit it to copulative forms, to the exclusion of implicational ones. He obviously simply stated three directions “from major to minor,” “from minor to major,” and “from equal to equal” – unaware of the distinctions between positive and negative, subjectal and predicatal, or copulative and implicational. In other words, let us not misinterpret his vagueness as an exclusive (or even inclusive) intent.

It seems that some of this ambiguity was corrected by later writers, judging by a maxim claimed by Mielziner to have been in use in 19th century jurisprudence[30]: “Quod in minor valet, valebit in majori; et quod in majori non valet, nec valet in minori” – meaning: “what avails in the less, will avail in the greater; and what will not avail in the greater, will not avail in the less.” The similarity of this statement to Cicero’s is striking, but so is the difference. Here, the minor to major case is consciously positive, since the major to minor case is explicitly negative. The trouble with this more precise later statement, however, is that (if it was intended as exhaustive) it effectively limits a fortiori reasoning to the subjectal mode, to the exclusion of the predicatal mode. But such exclusiveness may have been, and probably was, unintentional.

In fact, Cicero further expounds “the topic of comparison” in §68-71.

“68… a definition and example were given above. Now, I must explain more fully how it is used. To begin with, comparison is made between things which are greater, or less or equal. And in this connexion, the following points are considered: quantity, quality, value, and also a particular relation to certain things.”

He then goes on to clarify each of these considerations with many examples. I will reproduce here one example for each. For “quantity”: “more ‘goods’ are preferred to fewer;” for “quality”: “we prefer… the easy task to the difficult;” for “value”: “we prefer… the stable to the uncertain;” for “relation to other things”: “the interests of leading citizens are of more importance than those of the rest.” Clearly, these considerations refer to possible contents of a fortiori argument: the examples he proposes are sample major premises.

The uniform ‘X is preferred to Y,’ format of his proposed major premises suggests to me that Cicero was only consciously aware of subjectal a fortiori argument; he did not consciously notice (though he might have in practice used) predicatal a fortiori argument. Granting this, it follows that when earlier Cicero referred to inference from major to minor, he did have in mind negative subjectal argument; and therefore for him inference from minor to major meant positive subjectal. Note also that the format is also always copulative, never implicational and the middle term is always ‘preference’ – one thing is preferable to another. This is a limitation which we might excuse by saying that Cicero had in mind disputes between people in front of a court.

We can thus guess the forms of argument Cicero had in mind to have been: given ‘X is better than Y,’ it follows that ‘if Y is good enough for Z, then so is X’ and ‘if X is not good enough for Z, then neither is Y.’ He also says: “70… And just as these are the things which in a comparison are regarded as the better, so the opposites of these are regarded as worse.” What he had in mind here is: since ‘X is better than Y’ is convertible to ‘Y is worse than X,’ it also follows that ‘if X is bad enough for Z, then so is Y’ and ‘if Y is not bad enough for Z, then neither is X.’ Cicero does not say this explicitly, but that is evidently what he means. Note that these alternate arguments are formally the same, i.e. just as subjectal.

Regarding a pari argument, he adds: “71. When equals are compared, there is no superiority or inferiority; everything is on the same plane.” He gives a new example of it: “If helping one’s fellow-citizens with advice and giving them active assistance are to be regarded as equally praiseworthy, then those who give advice and those who defend ought to receive equal glory. But the first statement is true, therefore the conclusion is also.” Now, my impression here is that Cicero is having trouble formulating a sample a pari argument! What he has just put forward is not a fortiori argument, but simply apodosis: ‘If A, then B; but A, therefore B.’

The correct formulation of an a pari argument would be, according to me: ‘X is as good as Y, therefore: if X is good enough for Z, so is Y; and if Y is good enough for Z, so is X; and if either is not good enough for Z, neither is the other.’ Or, to use Cicero’s sample terms: ‘Giving advice and actively assisting are equally praiseworthy, therefore: if either is praiseworthy enough to deserve glory, so is the other; and if either is not praiseworthy enough to deserve glory, neither is the other.’ It seems that Cicero did not fully grasp this form.

Finally, we should note that Cicero does not mention anywhere the principle of deduction for purely a fortiori argument, according to which the subsidiary term should be identical in the conclusion to what it is in the minor premise, and not made ‘proportional’ (in an attempt to reflect the proportion between the major and minor terms). There is accordingly no mention by him of the a crescendo argument, where a ‘proportional’ conclusion is indeed allowed, being made possible by means of an additional premise about concomitant variation.

The rabbinical dayo (sufficiency) principle, being first mentioned in the Mishna Baba Qama 2:5, may be said to have appeared in Jewish legal discourse sometime in 70-135 CE at the latest, this being the period when R. Tarfon (who is mentioned in the said Mishna) was active. This principle, as we shall see, prohibits lawmakers from inferring a greater penalty for a greater crime from a lesser penalty for a lesser crime given in the Torah. I have not found evidence of a similar restriction in Cicero’s Topics. However, Roman law does seem to have generated an apparently similar principle, which reads: “In poenis bensignior est interpretatio facienda,” meaning: in penalties, the more benign interpretation is to be applied[31].

I do not know when this principle first appeared in Roman law. If it was developed before or during Cicero’s time, he would surely have mentioned it somewhere (in his Topics or elsewhere), being an expert in Roman law. If it emerged later, it might still have done so before it made its appearance in Jewish jurisprudence – or it may have come after. This historical question must be resolved by competent historians. In any case, it cannot be said with certainty that the law system where the principle appeared first influenced the law system where it appeared second. There could have been a common inspiration, or an inspiration from one to the other, or the two cultures could have arrived at the same idea independently[32].

To summarize, what is evident is that though Cicero had some knowledge of a fortiori argument, he was not conscious of all its forms (namely, predicatal and implicational forms); also, some of the forms he was conscious of (namely the a pari) he did not quite master. Moreover, the issue of ‘proportionality’ apparently eluded him. Another important observation we must make is there is no evidence of formalization or validation in Cicero’s treatment of the subject, though he mentions the issue of “validity” at the beginning of his book. Thus, we must say that on the whole Cicero did not go much further than Aristotle as regards a fortiori logic. Still, he enriches the field a bit through his more conscious distinction between three variants of a fortiori argument (viz. major to minor, minor to major, and a pari) and his listing of various possible contents (quantity, quality, value and importance).

All this is certainly interesting historically, in that it gives us an idea of the state of knowledge and skill regarding a fortiori argument in Cicero’s lifetime in Rome. Because Cicero was one of the foremost legal thinkers, lawyers and orators of his generation, we can reasonably consider his level as the ‘state of the art’ for his time and place, that is about three centuries after Aristotle in the Greco-Roman world. Needless to say, this is said on the basis of a spot check, and not on the basis of a thorough study of all the relevant literature in that region and period. There may well have been other logicians or rhetoricians who said more on a fortiori argument than we have discovered thus far.

6. Alexander of Aphrodisias

The Kneales’ account makes no mention of any discussion of a fortiori argument in the Hellenistic world in the centuries between Aristotle and Alexander of Aphrodisias, who was a 3rd century CE Peripatetic philosopher and commentator of Aristotle’s works. In particular, they do not mention Cicero’s contribution to the subject, which we presented in the previous section, even though they do examine his work on other topics. Obviously, then, their silence regarding a fortiori argument should not be interpreted to mean that there was no discussion of the subject; it could well just mean that they did not consider it important enough to mention. Anyway, as regards the said Alexander, the Kneales tell us the following, further to their earlier comments regarding the treatment of a fortiori argument by Aristotle:

“From Alexander’s explanation it appears that an argument of type (5), i.e. κατἀ ποιὀτητα, is an a fortiori argument with a general conditional premiss[33]. His example is:

If that which appears to be more sufficient for happiness is not in fact sufficient, neither is that which appears to be less sufficient.

Health appears to be more sufficient for happiness than wealth and yet is not sufficient.

Therefore wealth is not sufficient for happiness.

The theory of arguments κατἀ ποιὀτητα was probably an attempt to systematize what Aristotle says of a fortiori arguments in various passages of his Topics” (p. 111).

I do not see that this remark tells us much more about Aristotle or about a fortiori argument, but I quote it to be exhaustive. As regards Alexander’s example, I would rephrase it in standard format as follows:

Health (P) is apparently more conducive to happiness (R) than wealth (Q) is.

Health (P) is not conducive to happiness (R) sufficiently to actually produce happiness (S).

Therefore, wealth (Q) is not conducive for happiness (R) sufficiently to actually produce happiness (S).



In this format, it is seen to be a valid negative subjectal (major to minor). Let us analyze Alexander’s statement in detail, now. The Kneales’ remark about this being “an a fortiori argument with a general conditional premiss” refers to the first proposition: “If that which appears to be more sufficient for happiness is not in fact sufficient, neither is that which appears to be less sufficient.” If we look at this proposition, we see that it is only general regarding the major and minor terms P and Q (respectively, “more sufficient for happiness” and “less so”), but not general as regards the middle term R (which is specified as “sufficient for happiness”). Thus, it is only partly general. To be fully general, i.e. effectively a formal statement, the middle term should have been “something.” That is to say, the proposition should have read: “If that which appears to be more sufficient for something is not in fact sufficient, neither is that which appears to be less sufficient.”

In fact, therefore, since it is not “general” enough to be formal, this first proposition is redundant. Alexander’s second and third propositions contain, without need of the initial not-quite-abstract statement, the whole concrete a fortiori argument. The second proposition, “Health appears to be more sufficient for happiness than wealth and yet is not sufficient,” lists both the operative major premise (“Health appears to be more sufficient for happiness than wealth”) and minor premise (“and yet [health] is not sufficient [for happiness]”); and the third proposition (“Therefore wealth is not sufficient for happiness”) concludes the argument. Now, this is a well-constructed a fortiori argument, because it has an explicit middle term (“sufficient for happiness” – meaning, rather, conducive to happiness), relative to which the major and minor terms are compared, and it has two premises and a conclusion, and its minor premise and conclusion contain the idea of sufficiency (in negative form) for a certain result (actual happiness, in this case).

So this is on the whole a good effort by Alexander, although not perfect. The imperfections are (a) the first proposition, which is not general enough to count as a formal statement and therefore redundant (since the next proposition does the job just as well without it); (b) the lumping together of the operative major and minor premises into an apparently single statement (so that the different roles of the conjuncts in it are blurred); and (c) the use of the term “sufficient” in two senses: as ‘conducive ’ and ‘enough (to actualize)’. The latter equivocation causes some confusion in the reading of Alexander’s a fortiori argument, and is indicative of some confusion within him. It is indicative of a commonplace error, which we have already spotted in Aristotle’s treatment – namely, the conflation of the middle and subsidiary terms, the failure to clearly distinguish them in view of their quite distinct roles in the argument.

Thus, all things considered, Alexander’s statement is a well-constructed example of (subjectal) a fortiori argument, showing considerable implicit understanding of the form of inference – but it is not a successful explicit formalization, showing complete understanding. And of course, so far as we can tell from the Kneales’ account, there is no effort at validation. This is all a bit surprising, since Alexander was an Aristotelian, and so presumably well acquainted with Aristotle’s formal methods. We could regard Alexander’s first, “general” proposition as his attempt at validation. He perhaps viewed this statement as justifying the inference from the second proposition to the third (much like in syllogism the general major premise justifies the inference from the minor premise to the conclusion). But though such application of a wider generality gives an impression of validation, it does not in fact constitute validation, since the wider generality remains unproved.

Still, Alexander’s work is an improvement. He places more emphasis than Aristotle seems to have done on ontical a fortiori. He is also more advanced in his clear focus on sufficiency in the example quoted, whereas Aristotle does not use the word in the present context. Of course, several centuries separate the two. Note in passing that in Alexander’s case we are already in Talmudic times (not that I suggest a causal relation between his thought and that of the rabbis – but the parallelism is interesting).

It is (according to Ventura[34]), be it said in passing, to this Alexander that we owe the Greek word logika in the sense of the modern term ‘logic’. Previously, the word had rather the sense of ‘dialectic’ (e.g. as used by Cicero). Aristotle’s word for what we call logic was ‘analytic’; whence the titles of two of his works: Prior Analytics and Posterior Analytics. Alexander also inaugurated the term Organon to refer to a collection of Aristotle’s logical works[35].

As for the Kneales, their failure to analyze the “general conditional premiss” sufficiently to realize its relative informality shows that they did not have an entirely clear idea of what constitutes formalization. For this reason, and because I have in the past found errors in their analyses in other contexts, I do not take for granted their following statement: “The theory of arguments κατἀ ποιὀτητα was probably an attempt to systematize what Aristotle says of a fortiori arguments in various passages of his Topics.” They do not specify which passages. I would want to see these passages for myself before accepting that there is significant “systematization” in them. All we are shown here is a negative subjectal argument; there is no positive subjectal and there are no predicatal forms on display, to convince us that Alexander indeed achieved a systematic understanding. He made a valuable contribution, but I reserve judgment as to its full scope.

7. Historical questions

What is the precise history of a fortiori argument in ancient Greek, Roman and Hellenistic literature, whether philosophical, religious or secular? This question is always answered briefly and rather vaguely by historians of logic, if at all, because no one has apparently ever systematically researched the answers to it. In fact, this question should be asked for every type of argument, in every culture, if we want to be able to eventually trace the development of reasoning by human beings. But historical research into the a fortiori argument would be a good start, a good model, as it is a rather distinctive form of argument which is used and discussed in the said ancient Western civilizations though not so frequently as to be overwhelming. This is a scientific task, akin to biological research into a particular species of life in a particular environment, and it should be carried out with appropriate rigor and exhaustiveness.

The first step in such research would be collection of all relevant data. This means identifying the precise locations in various extant texts where such argument appears (in full or in part) to be used, and of course registering the argument made there in a data base so that it becomes henceforth readily available for future discussions. The literature[36] to be looked into dates from about the 8th century BCE to about the 5th century CE, in the Greco-Roman world, mainly in the Greek and Latin languages. Apart from actual occurrences of a fortiori argument, abstract discussions relating to the use of such argument must be identified and collected. Discussion of a form of argument signifies a higher degree of logical awareness than mere usage; and any attempts at theory, i.e. to formalize it, to find its varieties and to validate it, signify a higher level still. All these stages in logical awareness should obviously be distinguished, assuming instances of all of them are found.

Once the said raw data is collected, logicians can begin to sift through it and analyze its full significance. We can find out when and where the argument first and subsequently appeared within the period and region studied, and what form it took in each case. We can follow the flowering of varieties of the argument over time and in different places, as practice becomes more sophisticated. We can distinguish the different contexts of usage: poetic, business, legal, philosophical, scientific. We can compare the frequencies of use of such argument in different cultures[37]. We can perhaps trace the travels of the argument from one culture or subculture to another, as it is passed on from one people or social group to another, along trade routes or through various kinds of intellectual influence (for examples, through a philosophical author or a religious holy book). We can hopefully perceive the dawning self-awareness of those using the argument, as they begin to marvel at it, discern its parts and try to understand how it functions.

Clearly, we have here a sketch of a very interesting and enriching research project that someone or some people could and should take up. Similar research should of course also be carried out for other periods of history and regions of the world.


[1] Of Acragas, a Greek colony in Sicily, ca. 482 – ca. 432 BCE. Cited by Freely, p. 18.

[2] When I researched a fortiori argument, back in 1991-92, although my main interest was Judaic logic, I wondered – as a big fan of Aristotle, the undoubted founder of formal logic – whether he had noticed and discussed a fortiori argument. But I lacked the research tools and free time to find out (the Internet did not exist, for a start, and I had little access to reference books). Just recently, looking at Allen Wiseman’s new study of the subject, I was pleased to see that he had found use and mention of a fortiori argument in Aristotle and other ancients (p. 7). Apparently, he did so at least in part thanks to the Kneales’ historical study, to which he refers at length (p. 25). I have since then done some research in the works of Aristotle, and his predecessor Plato, and determined more accurately the extent of use of a fortiori discourse in these authors. The results are given here.

[3] The full texts of Aristotle’s Rhetoric and Topics are available online at the Internet Classics Archive: classics.mit.edu/Aristotle/rhetoric.html and classics.mit.edu/Aristotle/topics.html.

[4] ‘A fortiori’ is of course a Latin expression meaning ‘all the more strongly’. Aristotle’s words in Greek are “ἄλλος ἐκ τοῦ μᾶλλον καὶ ἧττον” – meaning, literally: “Another topic is derived from the more and less.” www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0059%3Abook%3D2%3Achapter%3D23%3Asection%3D4.

[5] Or, in a more literal translation: “according as it is necessary to prove either that a predicate is affirmable or that it is not.” (See Perseus Digital Library reference mentioned earlier.)

[6] However, note that this further remark is not found in all extant versions of the text. (See Perseus Digital Library reference mentioned earlier.)

[7] Actually, judging by another, more literal translation, it is not sure that Aristotle intended the logical-epistemic interpretation in the second example (concerning a man striking his father): “And to say that a man who beats his father also beats his neighbors, is an instance of the rule that, if the less exists, the more also exists.” Compare the wording here “if the less exists, the more also exists” to the wording above “if the less likely thing is true, the more likely thing is true also.” (See Perseus Digital Library reference mentioned earlier.)

[8] Needless to say, I am only here discussing the formal aspect of these arguments; I am not endorsing their content.

[9] Counting from 350 BCE, the approximate date when Aristotle’s works treating a fortiori argument were written, to say 100 CE, presumably roughly when R. Tarfon and the Sages had their famous clash on the dayo principle in the Mishna Baba Qama 2:5.

[10] See for instance a similar fault in the sample argument given by Alexander of Aphrodisias, further on.

[11] Oxford, London: Clarendon, 1962. This is available (in part) online at Google Books: books.google.com/books?id=FtXAwgy1w9cC&printsec=frontcover&dq=Kneale&hl=en&ei=RV7ZTOONOZCbOpnd_fAI&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCUQ6AEwAA#v=onepage&q&f=false. This is a great piece of work. Pity, though, that it contains so much material in the Greek or Latin original without English translation. Someone should remedy this and prepare a new edition.

[12] On p. 42, fn. 4. Note that I assume that there was a typing error with regard to “iv. 5,” and that the intent was really “iv. 6,” since the former chapter has nothing of relevance in it, whereas the latter does. The text here reproduced is drawn from the Internet Classics Archive; translation by W. A. Pickard-Cambridge.

[13] Presumably the original text said “from the point of view of greater, lesser and like degrees.”

[14] Just search for all occurrences in the Rhetoric and Topics text of the words “degree” or “greater,” and you will find many cases.

[15] In a pdf copy of The Works of Aristotle. (Ed. William David Ross. Chicago: Encyclopædia Britannica, 1952.) Presumably, this contains all his extant works; as for works which may have been lost, nothing can be said, obviously.

[16] It is interesting to note that I did not find (using the main key phrases) use, mention or discussion of a fortiori argument in the Prior Analytics.

[17] I read the argument as: If the soul (S) is not versatile (R) enough to perceive simultaneously sensibles of the same sense (Q), then the soul (S) is not versatile (R) enough to perceive simultaneously sensibles of different senses (P). The required major premise is obviously: More versatility (R) is required for P than for Q.

[18] I read the argument as: If the primitiveness of the properties of blood and veins (Q) implies urgency (R) enough for us to discuss them first (S), then their having been unsatisfactorily treated by past writers (P) implies urgency (R) enough for us to discuss them first (S). The required major premise is obviously: P implies more urgency (R) than Q.

[19] In truth, as discussed elsewhere, the dayo principle is more complex and more specifically religious than here suggested, and we should rather refer to a larger ‘principle of deduction’.

[20] These are distributed as follows: 4 in Rhetoric (2:23), 8 in Topics (2:10, 7:3), 3 in Posterior Analytics (1:1, 1:3, 1:10), and 1 in Metaphysics (3:4). Note in passing that none of the a fortiori arguments used by Plato are logical-epistemic.

[21] A Contemporary Examination of the A Fortiori Argument Involving Jewish Traditions, p. 25.

[22] Even though the other example here given, about the non-omniscience of gods and humans, does not likewise mention a major premise (namely that gods are more qualified to be omniscient than humans) at all. Needless to say, not verbalizing the major premise does not mean it is not mentally present in the background. This is true not only in a fortiori argument but in all reasoning (and is called abridged argument, or enthymeme). Much of our thought remains tacit, even if it has a logical impact on what we do verbalize.

[23] Though of course we might contend that, since gods do not exist and are figments of the imagination, his certainty was in fact unjustified. But our concern here is with inference – given the truth of the premises, does the conclusion’s truth follow or not? This issue applies to all inference, not just to a fortiori.

[24] Even if Aristotle goes on to abstract from this example a principle stated in terms of likelihood, the fact remains that the example itself is distinctively stated in terms of certainty.

[25] Topica. Trans. H. M. Hubble. Cambridge, Mass. Harvard UP, 1949. The full text of this book in Latin, with an English translation, may be read online at: www.scribd.com/doc/45159491/Cicero-Topica.

[26] See the Introduction, presumably written by the translator, H. M. Hubbell.

[27] This matter is a bit obscure to us; a footnote explains that “boundaries” refers to five foot strips no man’s land between estates, and “excluding water” refers to water diverted by one neighbor into another’s property.

[29] It is explained in a footnote that a farm owner would sell the warranted use of his land for two years, after which the purchaser would acquire title by “adverse possession”.

[30] On p. 131, footnote 1. Mielziner gives as reference: “quoted by Coke on Littleton, 260.”

[31] This is cited by Wiseman (p. 165). The reference he gives is: Digest of Justinian, no 49, in Albert Gautier, Introduction to Roman Law for Studies in Canon Law, (Rome: Faculty of Canon Law, St. Thomas University, 1994), page 154. I cannot compare and contrast this principle more precisely to the dayo principle, because I have not so far seen examples of just how it was used in practice.

[32] Thus, Maccoby’s suggestion, in his essays on the subject, that the dayo principle was an independent rabbinical production may turn out to be true, or false – it is not possible to tell which without more thorough research.

[33] In Aristotelis An. Pr. Lib. I Commentarium, ed. Wallies, C.I.A.G. ii (i), p. 265. (Footnote by the Kneales.)

[34] In his Introduction to Maimonides’ Terminologie Logique (p. 14).

[35] Ventura, p. 12, footnote 17. The term was later extended to include not only the said purely logical works, but related works like the Categories, On Interpretation, the Topics and On Sophistical Refutations. At one time, the Rhetoric and the Poetics were also (with some justification) included in the Organon, but later dropped out. Parts of the Metaphysics could have been included but were not.

[36] Literature in whatever form, of course – including archaeological fragments, epigraphy and the like. Obviously, too, when dealing with second-hand information, distinction must be made between the date of a report and the alleged date of what is reported. Remember, too, that a lot of the early literature was oral for a long time before it was put in writing. Also, even written material changes a bit over time, during transcription or by deliberate editing or amplification. All such factors must of course be taken into consideration and specified when estimating historical dates.

[37] This certainly exists. There is no doubt that a fortiori argument plays a larger role in Jewish law deliberations than in those of any other culture, for instance. I would also suggest, as another example, comparison between colloquial use of a fortiori discourse in French and English; the French seem to me to use it much more often.

2016-08-23T09:50:37+00:00