A FORTIORI LOGIC


CHAPTER 6 –
A fortiori in Greece
and Rome

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 (datedc. 350 BCE) seem relevant to our research.[3]

In hisRhetoric2:23 (i.e. book II, chapter 23), in §4, Aristotle writes:

“Another line of proof is thea 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 inRhetoric2:23, Aristotle adds a number of examples of allegedlya paria 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 hisTopics2:10 (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 ofa crescendoargument, i.e. of argument resembling a fortiori but having a ‘proportional’ conclusion. He is clearlynotsaying, for instance, that if pleasure is goodit follows deductively thatmore pleasure is better – he is only saying that the question should be asked and that the answer is to be soughtby induction; he explicitly conceives the possibility that it maynotfollow. 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, andnotthe 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, byusinga 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 ofTopics, 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 analyzeTopics3:6. Still inTopics2:10, 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 myJudaic Logicaccount called such argument, in which the major premise is a statement of equality, egalitarian a fortiori; another name for it isa 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 theRhetoricchapter above quoted: “This argument [i.e. a fortiori] might also be used in a case of parity.” He again implies as much inTopics2:11: “You can argue, then, from greater or lessor likedegrees of truth in the aforesaid number of ways” (italics mine) and elsewhere.

It should be noted that, though Aristotle, as we have seen inRhetoric2:23 and in the second passage ofTopics2:10, formulates a fortiori argument primarily in logical-epistemic terms, looking at the third passage ofTopics2:10 it appears that he also conceives of purely ontical a fortiori argument, since he speaks repeatedly of a predicatebelongingto a subject, as against beingsupposedto belong. This is confirmed by some of his a fortiori pronouncements in other contexts; for example inTopics3:6 (see below), or again in theHistory of Animals5:14, 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 theHistory of Animalsshow 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 theRhetoricpassage: “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 rabbinicaldayoprinciple (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 hisTopics3:6 (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 doesnotfollow 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 isa 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 moodinvalid, 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 doesnotfollow 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 isa 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 haveenoughof 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’sTopicsconcerning a fortiori argument, namely: “ii. 10 (114b37); iii. 6 (119b17); iv. 5 (127b18); v. 8 (137b14); vi. 7 (145b34); vii. 1 (152b6); vii. 3 (154b4)”[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:

Topics4:6 – 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, anda 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.

Topics7:3 – 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 fromTopics5:8:

“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 universallyinfera 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 fromTopics4:6:

“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 ofTopics2:10), 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.

Abouta contrario. In this context, I could additionally point to some of Aristotle’s remarks in hisRhetoric, 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 ofRhetoricwill 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, thennonot-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 ‘somenot-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 theTopics, 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]. SeeAppendix 4for 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’sRhetoricandTopicsfound 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 inOn the Soul, 1 case inParva Naturalia, 1 case inHistory of Animals, 1 case inMetaphysics, 2 cases in thePosterior Analytics, and 2 cases inRhetoric. 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 fortioriit 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 inHistory 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, andall the moreas 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 notsufficefor 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 byRhetoric2:23 andTopics2:10 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 inTopics2:10– 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 earlierRhetoricquotation, a different picture emerges.

As regards the suggestion that the two forms of argument are different, note that Wiseman does notquoteAristotle 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 inRhetoric2:23: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.Whyis a man more likely to strike his neighbors than his own father? Because it is generally easier, psychologically, socially and ethicallyto 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 ofRhetoricif 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 hisTopics[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’sRhetoric(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) isnota 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 fora pariargument 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 twoa pariarguments, 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 theira pariversions. 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 ofthe 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 19thcentury 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 willnotavail 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.

Regardinga pariargument, 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 samplea pariargument! 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 ana pariargument 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 rabbinicaldayo(sufficiency) principle, being first mentioned in the MishnaBaba 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’sTopics. 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 hisTopicsor 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 thea 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, anda 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 3rdcentury 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 ana fortioriargument 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 argumentsκατἀ ποιὀτηταwas probably an attempt to systematize what Aristotle says ofa fortioriarguments in various passages of hisTopics” (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 “ana fortioriargument 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 forsomethingis 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 wordlogikain 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 AnalyticsandPosterior Analytics. Alexander also inaugurated the termOrganonto 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 argumentsκατἀ ποιὀτηταwas probably an attempt to systematize what Aristotle says ofa fortioriarguments in various passages of hisTopics.” 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 beused, 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 8thcentury BCE to about the 5thcentury CE, in the Greco-Roman world, mainly in the Greek and Latin languages. Apart from actual occurrences of a fortiori argument, abstractdiscussionsrelating 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’sRhetoricandTopicsare available online at the Internet Classics Archive:classics.mit.edu/Aristotle/rhetoric.htmlandclassics.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 lessexists, the more alsoexists” to the wording above “if the lesslikelything is true, the morelikelything 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 thedayoprinciple in the MishnaBaba Qama2:5.

[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 theRhetoricandTopicstext of the words “degree” or “greater,” and you will find many cases.

[15]In a pdf copy ofThe 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 thePrior 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, thedayoprinciple 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 inRhetoric(2:23), 8 inTopics(2:10, 7:3), 3 inPosterior Analytics(1:1, 1:3, 1:10), and 1 inMetaphysics(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 thedayoprinciple, 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 thedayoprinciple 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 theCategories,OnInterpretation, theTopicsandOn Sophistical Refutations. At one time, theRhetoricand thePoeticswere also (with some justification) included in theOrganon, but later dropped out. Parts of theMetaphysicscould 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.

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.