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A FORTIORI LOGIC

© Avi Sion, 2013 All rights reserved.

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A FORTIORI LOGIC

CHAPTER 8 – In the Talmud, continued

1. Natural, conventional or revealed?

2. Measure for measure

3. The dayo principle in formal terms

4. The human element

5. Qal vachomer without dayo

6. Three additional Gemara arguments

7. Assessment of the Talmud’s logic

8. The syllogistic Midot

9. Historical questions

The present chapter is a continuation of the preceding, aimed at further clarifying some details.

1. Natural, conventional or revealed?

Our above critique of the Gemara was based to some extent on the assumption that it considers dayo as a principle, which the Sages regard as a hard and fast rule and R. Tarfon views as a conditional rule, depending on whether or not its application “defeats the purpose of the qal vachomer.” But in truth, the idea of dayo as a “principle” may be an interpolation, because the original Aramaic text (viz. “ורט לית ליה דיו והא דיו דאורייתא הוא”) does not use the word “principle” in conjunction with the word “dayo.”

The translation given in the Soncino Babylonian Talmud (viz. “Does R. Tarfon really ignore the principle of dayo? Is not dayo of Biblical origin?”) does of course use this word. But if we look at the Talmud Bavli translation (with their running commentary here put in square brackets), viz. “And does R’ Tarfon not subscribe to [the principle of] ‘It is sufficient…’ – Why, [the principle of] ‘It is sufficient…’ is contained in the [Written] Torah, [and R’ Tarfon must therefore certainly accept it!]” – it becomes evident that the word “principle” is an add-on. This of course does not mean that it is unjustified, but it opens possibilities.

If we do accept the translations, it is clear that the word “principle” is here equivocal, anyway – granting that for the Sages it means a universal proposition whereas for R. Tarfon it means a merely conditional one. This equivocation implies that the positions of the two parties are not as harmonious as the Gemara tries to suggest. They do not agree on principle and merely differ on matters of detail, as it were. On one side, there is a hard and fast rule; and on the other, one that is subject to adaptation in different situations. This is a radical difference, which is hardly diminished by assuming the “principle” to be of Biblical origin.

In view of this, it is difficult to guess what might be the Gemara’s purpose in positing that the dayo principle is deoraita (of Biblical origin – as against derabbanan, of rabbinic origin) and is known and essentially accepted by R. Tarfon. Moreover, as we have exposed, the Gemara’s scenario for R. Tarfon’s thesis is forced and untenable, being based on doubtful readings of the Torah and Mishna texts it refers to and, worst of all, on a parody of logic. Certainly, the Gemara’s scenario does not prove the claim of Biblical origin. If anything, that claim is weakened by virtue of having been supported by such rhetoric. But is the claim now disproved, or can it be supported by other means?

The Gemara is, of course, correct is in linking the issue of Biblical origin with that of R. Tarfon’s knowledge and acceptance. If the principle is of Biblical origin – i.e. is given in the Written Torah, or (since it is not manifest in the Pentateuch) at least the Oral Torah – it must be assumed to be known and accepted by him, as well as by the Sages. If he did not know and accept it, but only the Sages did, it cannot be of Biblical origin. However, I do not see how the Gemara can claim a different understanding of the dayo principle of Biblical origin for R. Tarfon than for the Sages. What would be the common factor between their views, which would be a “principle” of Biblical origin? The difference between universal and only-conditional applicability is too radical; these two theses are logically contrary. Their only possible intersection is that valid dayo objections may occur. This is hardly enough to constitute a “principle,” although we might in the limit grant it such status.

On the other hand, it would be quite consistent to say that the Sages and R. Tarfon both believe in a dayo principle of Biblical origin that is only conditionally applicable, but only differ with regard to the precise conditions of its application. Thus, the Biblical origin hypothesis remains conceivable, provided the word “principle” is understood in its softer sense, in such a way that debate is logically possible in particular cases, so that R. Tarfon might win in some cases and the Sages in other cases. The dayo principle would then consist in the bare fact that “some dayo objections are justifiable, though some are not;” and its being of Biblical origin would mean that this vague, contingent prediction was given at Sinai. Such conceivability does not of course prove that this much-reduced dayo principle was indeed of Biblical origin. Nor does it explain why the Gemara tried so hard to establish it as such. But it at least leaves the hypothesis in the running, so long as no other plausible reasons are found to discard it.

As mentioned at the end of our analysis of the Mishna, there are yet other equally viable hypotheses. We can still uphold the conflict between the Sages and R. Tarfon to be one between a hard and fast view of the dayo principle and an only-conditional view of it, provided we do not claim this principle to be of Biblical origin, but only of rabbinic origin (derabbanan). In the latter case, the Sages are collectively in the process of legislating the dayo principle in our Mishna, and though R. Tarfon initially tries to argue against this innovation by means of his two arguments, at the end he is forced to accept the majority decision. This scenario is equally consistent, and to my knowledge the Gemara offers no reason for dismissing it.

In this context, we could suggest that the dayo principle being “of Biblical origin” means, not that is was explicitly mentioned in or logically deduced from the Torah, but simply that something to be found in the Torah inspired the rabbis to formulate and adopt this principle. We might even propose (this is pure speculation on my part) the inspiration to have come specifically from Deuteronomy 4:2[1], which reads: “Ye shall not add unto the word which I command you, neither shall ye diminish from it.” It could well be that the rabbis, consciously or otherwise, saw in this warning of the Torah a justification for the cautiousness called for by their dayo principle. In that event, both R. Tarfon and the Sages obviously agreed regarding the truth of the inspiring Torah passage, but they differed as to how far the inspiration should be allowed to go. The dayo principle is not, in either case, precisely deducible from the said Torah passage, but a relation of sorts between the two can be claimed. The rabbinical principle, however broadly understood, is not in ‘the letter of the law’, but it is surely in ‘the spirit of the law’.

Another possibility is that there is no dayo principle, whether universal or conditional, at all, but each recorded dayo objection stands on its own as an individual rabbinical decree, for whatever reason the rabbis consider fit. This too can be used to explain the disagreements between R. Tarfon and the Sages in a consistent manner. This hypothesis logically differs very little from the above mentioned one of a conditional dayo principle, except in that the conditional dayo principle scenario implies an explicit Divine prediction at Sinai, whereas the no dayo principle scenario assumes no specific Sinaitic transmission on this topic (even if the general authority of the rabbis to judge and maybe innovate may have there been explicitly established). Here again, then, we have a consistent alternative hypothesis that the Gemara did not take into consideration and eliminate, before affirming its own thesis.

The methodology of the Talmud is of course essentially dogmatic. It engages in discussions and arguments, usually genuinely logical; but it does not go all the way with logic, systematically applying its techniques and referring to its results. It accepts some arbitrary ideas. This here seems to be a case in point, where the Gemara seeks to prove some preconceived notion and does everything it can to give the impression that it has. But we must always consider alternatives and evaluate them fairly.

The issue we will explore now is whether the dayo principle is to be regarded as natural, conventional or revealed. By ‘natural’ I mean that it is a law of nature, i.e. more specifically of logic or perhaps of natural ethics. By ‘conventional’ I mean that it is a collective decision of the rabbis, or more generally of human authorities, for whatever motive. And by ‘revealed’ I mean here that it is Divinely-decreed, handed down to us through prophecy or other supernatural means; i.e. more specifically, primarily at the Sinai revelation through Moses, and then written in the Torah or passed on orally through an unbroken tradition.

We have, I believe, definitely established in our above treatment that the dayo principle is not a law of logic. Many people have thought of it – and for a long time, I must confess, I too did so – as signifying that the (predicate of the) conclusion of (purely) a fortiori argument cannot quantitatively surpass the (predicate of the) minor premise. The dayo principle, in that view, corresponds to the principle of deduction, i.e. to a reminder that you cannot get more out of it than you put into it. In that perspective, I used to think the rabbis collectively instituted the dayo principle in order to prevent other people from erroneously drawing a ‘proportional’ conclusion from purely a fortiori premises. I was misled into this belief, perhaps, by the fact that rabbinical a fortiori reasoning is in practice usually correct, and also by the fact that the mentions of qal vachomer in the lists of Hillel and R. Ishmael do not mention the dayo principle as a separate hermeneutic rule, and therefore apparently consider the latter as an integral part of the former’s structure, which though it can be distinguished from it cannot correctly be dissociated from it.[2]

But as we have demonstrated in the present study the dayo principle is something much more complex than that. However, although this principle is not a natural principle in the sense of a law of logic, it might still be considered as a natural principle in the sense of a truth of ethics in a secular perspective. If we were to consider it as such, we would have to say that when the rabbis apply it, they are merely expressing their moral sensibilities as ordinary human beings. In that event, we would have to say that the dayo principle is applicable not only in legal contexts peculiar to the Jewish religion, but in all legal contexts, whether Jewish or non-Jewish, religious or secular. But the latter does not seem true – certainly, if we look at legal rulings in other traditions, the idea of dayo hardly if at all arises. So this idea seems to be a particularly Jewish (indeed, rabbinical) sensibility.

Thus, the dayo principle should rather be viewed as either conventional or revealed. As we have seen, contrary to what the Gemara insists, there is no incontrovertible proof that it is revealed. It may be “of Torah origin” in a broad sense, in the sense of “of Sinaitic origin.” But it is clearly (for any honest observer) not explicitly stated in the Written Torah; so it must be assumed to be part of the Oral Torah. Of course, the Gemara does seem to be claiming this principle to be logically derived from Num. 12:14-15 – but as we have seen, this ‘proof’ is unfortunately circular: it is read into the text rather than out of it. This means that the only way we know that the principle is “of Torah origin” is because the rabbis (led by the Gemara) tell us that it is. Such assertion is considered by the rabbis as sufficient proof that the alleged tradition is indeed Sinaitic. But scientifically it is surely not sufficient, as all sorts of things could have happened in the millennia in between.

Thus, while in the first instance (lehatchila) the rabbis would affirm the principle as derived from the Written Torah, if they are pressed hard enough they would probably as a last resort (bedieved) opt instead for the Oral Torah explanation. But, to my mind at least, this is logically equivalent to saying that the rabbis are the effective source of the principle. That is, it is derabbanan, and not at all deoraita. For we only have their say-so as proof of their assertion. Of course, it is still conceivable that the principle was indeed handed down at Sinai – we have not disproved that, and have no way to do so. But, as there is no way (short of a new revelation) to prove it, either, this conceivable scenario remains a mere speculation. So that the logical status of the principle is pretty much exactly the same as if the rabbis had simply conventionally decided to adopt it. This is the conclusion I adopt as a result of the present study: the dayo principle is of rabbinical origin.

To conclude, it is not clear why the Gemara makes such a big thing about the “Biblical origin” of the dayo principle, even going so far as to construct fictitious inference rules and arguments to prove its point. Did the Gemara have some halakhic purposes in mind, or was it just engaging in idle chatter (pilpul)? As we have seen, the Mishna can well be understood – indeed, in a number of ways – without pressing need to resolve the issue of the origin of the dayo principle. Why then is the Gemara’s commentary so focused on this specific issue, ignoring all other aspects? Perhaps it needs the proposition that the dayo principle is “of Biblical origin” for some other purpose(s), elsewhere. Not being a Talmudic scholar, I cannot answer this question. But in any event, to my mind, whatever the Gemara’s motives may have been, it failed miserably in this particular discourse.

Moreover – let us not forget this fact – when the Gemara refers to the dayo principle, it means just the first expression of that principle, as it is applicable to R. Tarfon’s first argument. The Gemara has not shown any awareness of the existence and significance of R. Tarfon’s second argument, and therefore of the difference in the Sages’ dayo objection to it. Thus, even if it had succeeded to prove somehow that the Sages’ first dayo objection was “of Biblical origin,” it would not have proven that their second objection was of equally elevated origin. This, too, is a disappointment concerning the Gemara: its powers of observation and analytic powers were here also less acute than they ought to have been.

We have thus far considered the issue of the origin of the dayo principle, but now let us look into that of qal vachomer. It is worth noting for a start that qal vachomer and the dayo principle are viewed by the Gemara as two distinct thought processes. The dayo principle is applied ex post facto, to the conclusion of a preexisting qal vachomer. The dayo principle (presumably) cannot be invoked until and unless a qal vachomer is formulated. If the dayo principle is not applied (as is possible in R. Tarfon’s view, according to the Gemara), the qal vachomer stands on its own. Thus, qal vachomer inference is independent of the dayo principle, even if the latter process is not independent of the former. Therefore, claiming that the dayo principle is “of Biblical origin” does not necessarily imply a claim that qal vachomer inference is also so justified. It may thus well be a natural process, if not a rabbinical convention.

In this context it is interesting to note that, in the lists of hermeneutic principles of Hillel and R. Ishmael, the dayo principle is nowhere mentioned, but only qal vachomer is mentioned. Since qal vachomer can occur, according to the Gemara, without the dayo principle, why is the latter not mentioned also as a separate hermeneutic principle? And if the dayo principle is “of Biblical origin,” as the Gemara has it, should it not all the more be mentioned in such lists? Conversely, if qal vachomer is a natural thought process, why does it need to be mentioned is such lists? Perhaps the answer to these questions is simply that the term “qal vachomer” in these lists is intended as an all-inclusive title, meaning “anything to do with qal vachomer, including on occasion application of the dayo principle.” Since, whatever the source of qal vachomer, whenever it is mentioned the question arises as to whether or not the dayo principle is applicable to it, the former always brings to mind the latter. Moreover, the traditional view seems to be that the dayo principle is only applicable to qal vachomer, so this question will not arise in other contexts.

In the Mishna, there is no explicit reference to the issue of the origin of the inference processes used. No explicit claim is made by anyone there that the dayo principle is “of Biblical origin” or any other origin; and nothing of this sort is said of qal vachomer. If we look at R. Tarfon’s wording, we are tempted to say that he regards his reasoning as natural. When he says: “I infer horn from foot” and “does it not stand to reason that we should apply the same strictness to horn?” – he seems to be appealing to logic rather than to some dogmatic given; and furthermore, by saying “I” and “we,” he seems to suggest that the decision process is in human hands. The Sages do not in their replies reprove him for this naturalistic approach; but they merely, it seems, say what they for their part consider to be a wiser ruling.

For the Gemara (i.e. the particular Gemara commentary that concerns us here, and not necessarily the Gemara in general), as we have seen, “qal vachomer” is understood as referring specifically to a crescendo argument, i.e. to a fortiori argument with a ‘proportional’ conclusion. The Gemara bases this understanding on the baraita it quotes. It does not mention purely a fortiori argument, which suggests that it is not aware of such form of argument. This is of course an important error on its part, because without awareness of the difference between purely a fortiori argument and a crescendo argument it cannot realize the logical skill of R. Tarfon’s second argument and the challenge it posed to the Sages’ first formulation of the dayo principle. The Gemara’s blindness to purely a fortiori argument explains its blindness to R. Tarfon’s second argument.

Even so, it is safe to say that the Gemara considers qal vachomer as natural in origin. Certainly, it does not explicitly state it to be “of Biblical origin,” as it does for the dayo principle. Although the Gemara’s assumption that Num. 12:14-15 contains an example of qal vachomer is reasonable, this Torah passage certainly does not use any verbal expression indicative of it, like “qal vachomer” or “all the more;” so, human insight is needed to see the implicit qal vachomer. The Gemara cannot be said to regard qal vachomer as a conventional construct by the rabbis, since the argument is in its view already found in the Torah. Since the Gemara does not even raise the issue (though it could and should have), it may be supposed to regard qal vachomer as ordinary human reasoning.

We might, however, suppose that the Gemara considers that the Miriam example is also given in the Torah to teach us that the correct conclusion of qal vachomer is ‘proportional’ – i.e. that this rule of inference was Divinely-ordained together with the dayo principle. But such a supposition is objectively nonsensical, since a fortiori argument is in fact not universally ‘proportional’. It would suggest that God, well after the Creation, may tell us to disregard logic and judge contrary to its laws. Yet, the laws of logic are not arbitrary dictates that can be discarded at will – even at Divine will – they are inextricably tied to the world as it is and our rational cognition of it. Therefore, to attribute such opinion to the Gemara would be to its discredit.

If we look at the three other a fortiori arguments in the Pentateuch listed in Genesis Rabbah, there is as in the Miriam instance no explicit ‘proportionality’, but we could in two of them at least similarly assume implicit ‘proportionality’, namely Ex. 6:12 and Deut. 31:27. Moreover, there is one passage in the Pentateuch that is explicitly ‘proportional’, namely Gen. 4.24: “If Cain shall be avenged sevenfold, truly Lamekh seventy and seven-fold”[3] –– but the speaker of this statement being Lamekh, someone apparently not regarded as exemplary, it can hardly be considered as halakhically authoritative. There are also many passages in the rest of the Bible that seem either explicitly or implicitly ‘proportional’, and so could be brought to bear in the present context. But the Gemara does not (at least, not here) find it necessary to mention any of them.

Thus, it is reasonable to suppose that the Gemara views qal vachomer (or at least its ‘proportional’ version) as natural argumentation – i.e. as not needing a special Divine dispensation to be credible. In other words, it is purely logical. In Talmudic terminology, this would qualify qal vachomer as a sort of svara, an inference naturally obvious to human reason. This seems to be the way most rabbis throughout history would characterize the argument. Certainly, most of the exceptional rules and dispensations they have enacted in relation to this argument form suggest it; although the fact that some have tried to interdict its free use suggests a doubt in their mind in this regard.

But even though svara refers to natural and universal logical insight, qal vachomer is always counted as one of the “midot,” i.e. of the rabbinical hermeneutic principles. There is a difficulty in this fact, because a hermeneutic principle is thought of as a discursive tool (ordained directly by God or indirectly by rabbinical decision) for use specifically in Torah interpretation. Such principles being essentially non-natural, they may well be not rationally evident or even perhaps contrary to logic. Not so in the case of qal vachomer. So there is a problem with its inclusion in the lists of midot. The solution of this paradox, I would say, is simply that the rabbis themselves did not make such fine distinctions between natural and conventional logic. Or equally well: they could lump qal vachomer with more uncommon forms of reasoning, because in their minds all are “logical.” This is indeed suggested in many rabbinical texts in English, where the word “midot” is translated as “principles of logic.”

2. Measure for measure

The Gemara perhaps sought to justify the dayo principle by claiming it to be “of Biblical origin” – but there was no pressing need for it to do so, since other explanations were readily available and perhaps less problematic. It seems that the Gemara, not having previously analyzed qal vachomer reasoning in formal terms, was unable to precisely perceive its constituent premises, and under what conditions they resulted in this or that conclusion; and thence, how such an argument could be rebutted. In the Gemara author’s mind, therefore, apparently, the status of a Divine decree (“Biblical origin”) was necessary for the dayo principle to have the power to rebut the qal vachomer argument (as he saw it).

As we have shown, the two arguments proposed by R. Tarfon and the dayo objections to them put forward by the Sages can be interpreted in a number of ways. R. Tarfon’s two arguments could have been (1) intended as two mere arguments by analogy (more precisely, pro rata); or (2) the first one may have been pro rata, while the second was (purely) a fortiori; or (3) they could (as the Gemara did) both be construed as having been a crescendo. The Sages’ dayo statements, could be viewed as (a) particular ad hoc objections, decided by the rabbis collegially; or (b) as general objections, either (i) clearly given in the Written Torah or deduced from it (as the Gemara wrongly claims); or (ii) inductively or rhetorically derived from it (as the Gemara actually attempted); or (iii) known from the Oral Torah (i.e. by unbroken tradition since the Sinai revelation); or again (iv) decided by the rabbis.

If we said that R. Tarfon’s first argument was purely a fortiori, we would thereby imply that he did not know how to reason correctly in the a fortiori mode; nevertheless, if he did so reason incorrectly, the Sages’ dayo objection to his argument would in that event be equivalent to the principle of deduction, interdicting a ‘proportional’ conclusion from the given premises. Many commentators have so interpreted the debate, but in truth they did so without paying attention to R. Tarfon’s second argument, which could also be considered as purely a fortiori and yet be free of the Sages’ same objection. So this hypothesis is farfetched and unconvincing, and best brushed aside.

More probably, R. Tarfon put forward his first argument in pro rata or a crescendo form; and the Sages objected “dayo” to it in particular or in general, as already said. The purpose of this objection was to annul the premise of ‘proportionality’ inherent in R. Tarfon first argument. R. Tarfon, being an intelligent man, got the message and proposed instead a neat second argument, which was not subject to the same rebuttal, for the simple reason that whatever its form (pro rata, a crescendo or purely a fortiori) it yielded one and the same seemingly ‘proportional’ conclusion. Nevertheless, the Sages again objected “dayo” to it, in particular or in general, in exactly the same terms. By so doing, the Sages enlarged the meaning of their dayo objection, since it could here only refer to the generalization process preceding the deduction, since annulling the premise of ‘proportionality’ was useless.

As earlier explained, the principle of deduction is that the putative conclusion of any deductive argument whatsoever must in its entirety follow necessarily from (i.e. be logically implied by) the given premise(s), and therefore cannot contain any information not found explicitly or implicitly in the said premise(s). If a putative conclusion contains additional information and yet seems true, that information must be proved or corroborated from some other deductive or inductive source(s). This principle is true not only of valid a fortiori argument, but of all other valid forms of deductive argument, such as for instances syllogism or dilemma. Inference in accord with this principle is truly deductive. Inference not in accord with this principle may still be inductively valid, but is certainly not deductively valid.

It seems evident that when the Gemara says “a fortiori” (qal vachomer) it means a crescendo. Yet the Gemara does not clearly acknowledge the implications of such an assumption (at least not in the sugya under scrutiny). To be fully credible, the Gemara should have demonstrated its understanding that the arguments it characterized as a fortiori were not purely so, but involved an additional premise, one which establishes a pro rata relationship between the subsidiary and middle items. The issue is not merely verbal, note well, but depends on acknowledging a logical precondition for validity. Unfortunately, (to my knowledge) the Gemara nowhere explicitly acknowledges this crucial precondition. Nevertheless, we can generously suppose that the Gemara unconsciously or tacitly intends it, and move on. Our inquiry must now turn to the question: What is the required additional premise, in more concrete terms?

The tacit premise. It is a principle of justice (perhaps even the essence of it) that: on the positive side, the reward ought to fit the good deed and be commensurate with it; and on the negative side, the punishment ought to fit the wrongdoing and be commensurate with it. If these conditions are not fulfilled, justice has not been entirely served. This principle is in accord with our natural human ‘sense of justice’. It is an insight which cannot be proved, but which expresses (at least in part, if not wholly) what we commonly mean by ‘justice’. It is the basis of many laws legislated by mankind and guides many courts of law (namely, those that are characterized as ‘just’) in their deliberations and their rulings. For examples, a greater penalty is incurred by armed bank robbery than by shoplifting; or by premeditated murder than by murder in a moment of passion. In this negative guise, the principle of justice is known (in Latin) as the lex talionis, or law of retaliation.

Of course, the ‘sense of justice’ is not something literally ‘sensory’, but rather something ‘intuitive’, an insight of sorts. We know from within ourselves what is just and what is not. Of course, such knowledge is mere opinion that has to be confirmed over time using inductive techniques. We individually may see things differently at different times; and different people may see things differently. The sense of justice may be honed by use or blunted by disuse. It may be influenced by surrounding culture, whether incidentally or by deliberate propaganda. All the same, even though this faculty can be put to sleep or smothered, swayed or manipulated, each of us (as a being capable of personally suffering in a similar situation) does have an underlying sense of justice.

Of course, it is not always easy to intuit, much less demonstrate indubitably, what is ‘fitting’ and ‘commensurate’ reward or punishment. Justice is not an exact science. In Judaism, where this principle is known as midah keneged midah (meaning: measure for measure), the right measure is determined either by Divine fiat or by rabbinical decision; in the latter case the wisdom of the rabbis being assumed to be above average. I have not seriously researched the issue as to when this principle began to play an explicit role in rabbinical decision making, but I assume it was very early in view of its implicit presence in many stories and commandments of the Jewish Bible (Torah and Nakh).

The story of Miriam’s punishment for criticizing Moses, which the Gemara focuses on so insistently, is a case in point. In the Mishna debate, it is obvious that R. Tarfon’s two arguments are motivated by the measure for measure principle, even though not in so many words, but in the background, pre-verbally. Some commentators see the statement by God in Gen. 9:6, “Whoso sheddeth man’s blood, by man shall his blood be shed,” as the Biblical precursor of the measure for measure principle, even though it is more specific, in view of its symmetrical format (shed blood justifies blood shedding). The value and importance of justice in Judaism may be seen, for instance, in the Deut. 16:20 injunction: “Justice, justice shalt thou pursue.”

As regards stories, an illustration often appealed to, of God’s practice of ‘measure for measure’, is the correspondence between the crimes of the Egyptians against the Israelites and the punishments that later befell them; for example: they wanted to drown the babies (Ex. 1:22) – their army was drowned in the sea (Ex. 14:28). In Joshua 7:25, “Why hast thou troubled us? Hashem shall trouble thee this day,” a ‘tit for tat’ is clearly implied. The principle is well-nigh explicit in 2 Samuel 22:24-28; for instance in v. 26, David says: “With the merciful Thou dost show Thyself merciful, with the upright man Thou dost show Thyself upright.” Or compare Proverbs 1:11 and 1:18. Many examples of such reciprocity can also be found in the Talmud; see for instance Sotah 8b-11b. The concept is certainly older than the name attached to it.

I have not to date managed to find out when and where the exact Hebrew phrase “midah keneged midah” first appears. But I found a Mishna (Sotah 1:7) with very similar words: “By the measure that a man measures, so is he measured (במדה שאדם מודד בה מודדין לו, bemidah sheadam moded bah, modedin lo)”[4]. The meaning is admittedly not literally identical, since ‘measure for measure’ is understood to mean more broadly that the way a man behaves determines his recompense. However, if we understand “the measure that a man measures” as signifying the thoughts which determine his behavior, and “so is he measured” as referring to the Divine judgment in consequence of his actions, which determines his recompense, the two ideas may be pretty well equated.[5]

On the basis of this equity principle, it appears reasonable to us (for instance) that someone who has offended God deserves more punishment than someone who has merely offended a human being even if the latter be one’s own father. On this basis, then, it appears reasonable to us that, in the episode narrated in Num. 12:14-15, Miriam should indeed, as the Gemara suggests, theoretically deserve a penalty of (say) fourteen days isolation instead of just seven days. The fourteen is perhaps just an illustrative number, because surely offending God deserves more than double the punishment due for offending one’s father. Indeed, even the seven days penalty in the latter case is an arbitrary number – in this case, a Divine decree – so the fourteen days penalty is bound to be so too.[6]

Clearly, the Sages’ dayo principle is not a redundant restatement of the principle of deduction for a fortiori argument, as it might sometimes appear to be; nor does it have any other purely logical purpose. Rather, it serves an important additional, more moral purpose. We could imagine that the Gemara tacitly agrees that, in the Miriam example, the qal vachomer by itself (per se) can only logically yield the conclusion of seven days. But in the present case, even though this is not explicitly said anywhere, the qal vachomer is not ‘by itself’: it happens (per accidens) to be accompanied by an expectation of fourteen days based, not on formal grounds relating to purely a fortiori inference, but on the principle of justice that we have just now enunciated.

The dayo principle then comes to teach us: even in a case like this, where a greater penalty is expected due to implications of the principle of justice, the rabbinical conclusion (i.e. the law, the halakha) should not diverge from the quantity given in the Torah-based premises, whether such premises are used to draw a conclusion by mere analogy or by a fortiori argument or any other inductive or deductive means. The use of inference should not end up concealing and exceeding the penalty amounts mentioned in the premises given by Scripture. Such quantities should be understood as davka (as is), and not used for extrapolations however just those might seem based on human reasoning. The dayo principle is then, as the Gemara suggests, “Biblical,” if only in the sense that it advocates strict adherence to Biblical givens whenever penalties are to be inferred, whether by deduction or by induction.

The motive of the Sages seems obvious enough: the dayo principle is essentially a precautionary measure, enacted to avoid human errors of judgment in processes of inference in legal contexts. When a human court condemns an accused to some penalty, it is taking on a very serious responsibility. If that penalty is Divinely-ordained, i.e. explicitly written in the Torah, the responsibility of the human judges is limited to whether or not they correctly subsumed the case at hand to a given set of laws. Whereas, if the judges add something to the given penalty, on the basis of some ‘proportional’ reasoning, they are taking an additional risk of committing an injustice. So, it is best for them to stick to the Torah-given penalty.

It is interesting to note the comment by R. Obadiah Sforno (Italy, 1475-1550), regarding the principle of “an eye for eye” in Exodus 21:23-25, that “strict justice demanded the principle of measure for measure, but Jewish tradition mitigated it to [monetary] compensation to avoid the possibility of exceeding the exact measure.”[7] This suggests that the idea of compensation was instituted in that context to prevent eventual excess in the application of physical retribution – which, of course, would not be justice, but injustice[8]. We may refer to this idea to perhaps better understand and justify the dayo principle. In instituting this principle, the rabbis were not merely “tempering justice with mercy,” but also making sure that there would not be occasional occurrences of injustice, by mistake or due to excessive zeal. It was, at least in part, a precautionary measure.[9]

Viewed as a restraint on ‘proportional’ inference, the Sages’ dayo principle is not a principle of logic, but a merely hermeneutic principle inclining rabbinical judgment to mercy. It is not intended to regulate the qal vachomer inference as such, but rather to restrict a parallel application of the principle of justice – or perhaps more accurately put, a parallel intuition from our ‘sense of justice’. The Sages are telling us: although our human sense of justice produces in us an expectation that (to take the Gemara’s example) Miriam deserves (say) a fourteen days penalty, nevertheless God mercifully decreed (in the Torah) only seven days penalty for her. On the basis of this exemplary decree in the Biblical story of Miriam, Jewish legislators and law courts must henceforth always judge with the same restraint and limit the concluding penalty to the penalty given in the premise, even when the principle of justice would suggest a more severe punishment.

This is surely the real sense of the Sages’ dayo principle: they were not reiterating any law of logic, but setting a limitation on the principle of justice. And now, having perceived this, we can understand many things in this Talmudic sugya. We can understand why the Gemara would wish to establish that the dayo principle is Divinely-decreed. For it might seem unjust to restrict application of the principle of justice; it might be argued that the conclusion of a strict deduction is as reliable as its premises. Moreover, we can see how it is conceivable that, as the Gemara has it, R. Tarfon can differ from the Sages’ view and ignore the dayo principle in some situations. For no law of logic is being ignored or breached thereby, but only a moral principle; and a moral principle is logically more flexible, i.e. it may apply differently to different situations.[10]

Other angles. The dayo principle as above presented is designed to prevent the rabbis from ruling too severely. What of rulings that are too lenient, we might ask? Surely, ruling too leniently can conceivably be a problem. Justice is not served if criminals are not punished as they deserve (as indeed unfortunately often happens in practice in present day society). Too much leniency can be a bad thing for society, just as too much severity often is. So the dayo principle ought conceivably to forbid excessive mercy, as well as excessive justice.

If we think about it, measure for measure is essentially a principle of justice rather than one of mercy. By definition, mercy is intended to temper strict justice. It is not measure for measure, but beyond measurement. Justice is logical, while mercy is humane. Logically, the judgment should be so and so; mercy mitigates the conclusion. Mercy is surely desirable; but excessive mercy would obviously constitute injustice. Overdoing it would be negation of measure for measure! Thus, the right balance is needed. Arguing thus, we might easily advocate that the dayo principle is applicable to inferences that increase leniency, as well as to those that increase severity.

But my impression from rabbinic discourse generally is that the dayo principle is always intended as a principle of justice, and not occasionally as a principle of mercy. The rabbis are not so worried about irrational bursts of magnanimity; they are worried about inflicting undeserved punishment.

There is another objection that can be raised to our moral interpretation of the dayo principle. It seems reasonable enough in the present negative legal context, where the qal vachomer has as its conclusion a punishment for a wrongdoing. But what of equivalent positive legal contexts, where the qal vachomer has as its conclusion a reward for a good deed? Surely, the rabbis cannot here say that it is merciful to diminish the reward’s proportionality. Also, what of non-legal contexts, when the qal vachomer is constructed in pursuit of a factual conclusion – do the rabbis simply ignore the dayo principle in such cases? The question is, then: how general is the Sages’ dayo principle, or rather: what are the limits of its application?

The answers to these questions are, I think, broadly speaking, as follows. Jewish law, like most law systems, is essentially concerned with sanctions for wrongdoing rather than with rewarding good deeds. For this reason, only the negative side of the measure for measure principle is relevant to the rabbinical legislative process, and applications of the dayo principle occur only in relation to penalties. I doubt that any legalistic a fortiori argument with a conclusion of reward occurs in Jewish law; but if any indeed does, and the principle of measure for measure seems applicable, I very much doubt that the rabbis would block, on the basis of the dayo principle, the inference of increased or decreased rewards.

As regards a fortiori arguments in homiletic and other non-legal contexts, I do believe the dayo principle is indeed ignored in practice. It is admittedly sometimes apparently used – but such use is rhetorical. In other contexts, maintaining the a crescendo conclusion may be preferred. Since the principle has no binding legal impact either way, the decision to use or not-use it depends entirely on what the speaker wishes to communicate.

All the above comments circumscribing use of the dayo principle are of course mere personal impressions and educated guesses; they are open to discussion. They would have to be justified empirically, by thorough systematic research through the whole Talmud and indeed all Jewish law literature. Until such data is gathered by scholars, and fully analyzed by competent logicians, we cannot answer the said questions with much greater precision and certainty than just done. Nevertheless, by asking questions and proposing answers, we have at least raised issues and sketched possible results. It would, of course, be interesting and valuable to find rabbinical statements that clearly justify what has been said.

3. The dayo principle in formal terms

We shall here review our new interpretation of the dayo principle in more formal terms. This is done with reference to Mishna Baba Qama 2:5, where the principle is traditionally given pride of place, first dealing with the Sages’ objection to R. Tarfon’s first argument, and then with their objection to his second argument. As already seen, these are two distinct expressions of the dayo principle, although they have a common motive. The corresponding Gemara in Baba Qama 25a, as we saw, only seems to have noticed the first version of the dayo principle; but later commentators (notably, it seems, Rashi and Tosafot) did notice the second[11]. We shall show here more precisely why the Gemara’s view is inadequate.

A further reason why we wish to now investigate the dayo principle in more formal terms is because both formulations in the Mishna relate specifically to the positive subjectal form of a crescendo argument. Nothing is there said of eventual applications to the negative subjectal form, or to the positive or negative predicatal forms. Our purpose here is to consider theoretically what such other applications would look like. Whether such other applications actually occur or not in the Talmud (or other rabbinic literature) is not the main issue, here; but it is abstractly conceivable that they might occur. In any case, we are sure to clarify our concept of the dayo principle by this enlarged research.

Let us to begin with deal with the Sages’ dayo objection to the first argument of R. Tarfon. Here, R. Tarfon tried to infer a liability of full payment for damage by horn on private property (conclusion), from a liability of half payment for damage by horn on public property (minor premise). He was thus presumably using a crescendo argument, of positive subjectal form, as follows:

Action P is a more serious breach of a certain law (R) than another action Q is.

Action Q is a breach of that law (R) enough to merit a certain penalty (S).

The magnitude of penalty S is ‘proportional’ to the seriousness of the breach of law R.

Therefore, action P is a breach of that law (R) enough to merit a greater penalty (S+).



The Sages’ dayo objection to this attempt can be stated as: if the minor premise predicates a certain penalty (S) for a certain action (Q), then the conclusion cannot predicate a greater penalty (S+) for a more illegal action (P). This objection can be perceived as neutralizing the additional premise concerning ‘proportionality’. The Sages are saying: although by commonsense such ‘proportionality’ seems just, by Jewish law it is not to be applied, and we can only predicate the same penalty (S) in the conclusion as was previously given (in the minor premise).

What the dayo objection does here is to block, or switch off, as it were, the operation of the additional premise regarding ‘proportionality’: though that moral premise might usually be granted credibility, it is rendered inoperative in the present context, to avoid any possible excess of penalization (as earlier explained). This means that the a crescendo argument is effectively abolished and replaced with a purely a fortiori argument. Evidently, then, the Gemara’s view, according to which the a crescendo argument is allowed to proceed, and then the dayo principle reverses its action[12], is technically incorrect. The action of dayo is preventive, rather than curative; it takes place before the ‘proportional’ conclusion is drawn, and not after.

We can easily, by formal analogy, extend this principle to other forms of a crescendo argument, if only out of theoretical curiosity. The analogous positive predicatal argument would have the following form:

A more serious breach of a certain law (R) is required to merit penalty P than to merit another penalty Q.
Action S is a breach of that law (R) enough to merit penalty P.
The seriousness of the breach of law R is ‘proportional’ to the magnitude of action S.

Therefore, a lesser action (S) is a breach of that law (R) enough to merit penalty Q.



Notice that the additional premise about ‘proportionality’ is different in subjectal and predicatal arguments. The order is reversed. In the former, the subsidiary term S, being a predicate, is proportional to the middle term R; whereas in the latter, it is the middle term R that is proportional to the subsidiary term S, which is a subject. This is due to the order of things in the minor premise, which the conclusion naturally reflects, where predication is made possible only if the value of R for the subject matches or exceeds the minimum value of R necessary for the predicate.

In this context, the Sages’ dayo objection would be stated as: if the minor premise predicates a certain penalty (P) for a certain action (S), then the conclusion cannot predicate a lesser penalty (Q) for a less illegal action (S–). This objection can be perceived as a denial of the additional premise concerning ‘proportionality’. Here, the Sages might say: although by commonsense such ‘proportionality’ seems just, by Jewish law it is not to be applied, and we can only address the same action S in the conclusion as was given (in the minor premise). This statement, to repeat, is formulated by analogy, merely for theoretical purposes; it is not given in the original Mishna debate.

There is admittedly a difficulty in the latter extension of the dayo principle. For whereas applying dayo to a positive subjectal argument results in preventing potentially excessive justice, by mechanically attributing a greater penalty to a more serious breach of law, the application of dayo to a positive predicatal argument results in the prevention of increasing leniency, which is what attributing a lesser penalty to a less serious breach of law would constitute. We shall return to this issue further on.

As regards the corresponding negative arguments, they can easily be determined using the method of ad absurdum. In each case, the major premise and the additional premise about ‘proportionality’ remain the same, while the negation of the conclusion becomes the new minor premise and the negation of the minor premise becomes the new conclusion. Application of the (first) dayo principle to them would have the effect of inhibiting the deduction of the putative negative a crescendo conclusion from the given negative minor premise, through rejection of the additional premise.

As for implicational arguments, they can be dealt with in comparable ways.

Let us now deal with the Sages’ dayo objection to the second argument of R. Tarfon. Here, R. Tarfon tried to infer a liability of full payment for damage by horn on private property (conclusion), from a liability of full payment for damage by tooth & foot on private property (minor premise). He was thus using an argument, again of positive subjectal form, that yields the same conclusion whether construed as a crescendo argument or as purely a fortiori. This means that the first version of the Sages’ dayo principle would be useless in this second case, for the minor premise and conclusion naturally have the exact same predicate (full payment). Therefore, since the Sages nevertheless declared dayo applicable, they must have been referring to some other feature of the argument.

The only other logical operation they could have been referring to is the inductive formation of the major premise, by generalization from the liability of half payment for damage by horn on public property and the liability of no payment for damage by tooth & foot on public property. That is, the major premise that ‘liability for damage by horn is generally greater than liability for damage by tooth & foot’ was derived from the same given concerning horn as before, namely that ‘liability for damage by horn on public property is half payment’. Here, then, the dayo principle must be stated in such a way as to interdict this preliminary generalization.

The Sages apparently hint at this solution to the problem by restating their second objection in exactly the same terms as the first. There is no other explanation for their using the exact same words. In this context, then, the Sages’ dayo objection would be stated as: if the major premise is inductively based on information about a certain action (P) meriting a certain penalty (S), in one set of circumstances, then the conclusion drawn from it cannot be that the same action (P) in another set of circumstances merits a greater penalty (S+). That is, under the dayo principle, we can only conclude that ‘P is S’, not that ‘P is S+’. Note well how this second version of the dayo principle is very different from the previous.

It is important to realize that, unlike the preceding one, this dayo objection cannot be perceived as neutralizing the additional premise concerning ‘proportionality’. For here, a crescendo and purely a fortiori argument have the exact same conclusion; so that whether or not we ‘switch off’ this third premise makes no difference whatever to the result. This means that, in the present case, the argument is necessarily purely a fortiori, i.e. devoid of an additional premise. No a crescendo argument can usefully be proposed here, since the conclusion is already maximal through purely a fortiori argument. Therefore, in such case, we must prevent the unwanted conclusion further upstream in the reasoning process; that is, at the stage where the major premise is getting formed by means of a generalization.

We can easily, by formal analogy, formulate a similar principle with regard to positive predicatal argument. In this context, the Sages’ dayo objection would be stated as: if the major premise is inductively based on information about a certain action (S) meriting a certain penalty (Q), in certain circumstances, then the conclusion drawn from it cannot be that a lesser action (S–) in whatever other circumstances merits the same penalty (Q). That is, under the dayo principle, we can only conclude that ‘S is Q’, not that ‘S– is Q’. This statement, to repeat, is formulated by analogy, merely for theoretical purposes; it is not given in the original Mishna debate.

Admittedly, our formal extension of the second dayo principle from positive subjectal argument to positive predicatal argument is open to debate. For whereas in the former case dayo serves to prevent increased severity, in the latter case it seems to have the opposite effect of preventing increased leniency. This issue will have to be addressed, further on.

Returning now to the Gemara, we can see from the above formal treatment, that it was wrong in considering the dayo principle as concerned essentially with a crescendo argument. In the first case, which the Gemara did try to analyze, the Sages’ dayo objection effectively advocated a purely a fortiori argument instead of R. Tarfon’s apparent attempt at a crescendo argument. But in the second case, which was unfortunately ignored by the Gemara, the Sages’ dayo objection couldn’t function in a like manner, by blocking the usual velleity of ‘proportionality’, since this would be without effect on the conclusion. It had to apply to a presupposition of R. Tarfon’s argument, however construed – namely the generalization earlier used to construct its major premise.[13]

Let us now return to the issue glimpsed above, as to whether or not the dayo principle is only meaningful in relation to positive subjectal a crescendo argument, which proceeds from a lesser penalty for a lesser infraction to a greater penalty for a greater infraction. We have seen that we can formally enlarge the idea of preventing proportionality implied in dayo application to positive subjectal argument, to negative subjectal, and to positive and negative predicatal arguments – but is such analogy meaningful when more concretely examined? We shall here try to answer this question.

Remember our earlier determination that the dayo principle is not a logical principle, but a “moral” one, i.e. it has to do with ethics or law in the context of the Jewish religion. It is not logically necessitated by the principle of deduction or by the use of a fortiori argument or any other purely logical consideration; no contradiction would arise if we simply ignored it. It is, rather, something Divinely or rabbinically prescribed, to lawmakers and courts of law, for cases where a qal vachomer is being attempted in order to infer a greater penalty for some wrongdoing. It is an artificial injection into the Jewish legislative process apparently motivated by mercy, i.e. to temper justice. There is no reason to apply it in contexts other than the sort just specified, or for that matter in other religions or outside religion.

We could eventually expect the same idea to be extended from penalties to duties. Such conceptual extrapolation might well be found exemplified in the Talmud or other Jewish literature (I have not looked for examples). That is conceivable if we think of penalties and duties as having in common the character of burdens on the individual or community subjected to them. If we look on increased duties (mitzvoth) as positive rewards, in the way that a servant might rejoice at receiving increased responsibilities, the analogy of course fails. But if we look on duties as burdens, an analogy is possible. It that case, the dayo principle could be taken to mean more broadly that burdens in general must not be increased on the basis of a qal vachomer argument from the Torah.

Granting the above clarifications of the dayo principle, the first question to ask is: is its function limited to contexts of positive subjectal qal vachomer – or can this definition be extended to other a fortiori argument formats? The format focused on by the rabbis is, to repeat, positive subjectal, which means that it is minor to major (miqal lechomer), whence the appropriateness of the name qal vachomer. Let us now consider what dayo application to the negative subjectal format would mean. Such argument is, of course, major to minor (michomer leqal) in orientation. It would look as follows:

Action P is a more serious breach of a certain law (R) than another action Q is.

Action P is a breach of that law (R) not enough to merit a certain penalty (S).

The magnitude of penalty S is ‘proportional’ to the seriousness of the breach of law R.

Whence, action Q is a breach of that law (R) not enough to merit a lesser penalty (S).



The major premise and the additional premise about ‘proportionality’, which (as we saw earlier) is in practice derived from the principle of midah keneged midah (measure for measure), both remain the same, here. What changes is that the minor premise and conclusion are now negative propositions and the major term (P) appears in the former and the minor term (Q) appears in the latter. It remains true that the value of S associated with P is greater than that associated with Q; however, note that here the greater value appears in the minor premise and the lesser in the conclusion.

Our question is: what would be the significance of the dayo principle, in either of its senses, in such negative subjectal context? Note that above argument is formally valid. The question is thus not whether its conclusion follows from its said premises. The question is whether to reject its additional premise (first type of dayo application) or its major premise (second type of dayo application).

At first sight the answer is that the dayo principle would not be called for – because there is no velleity in such a context to use the principle of measure for measure, and dayo is intended as a restraint on such velleities. Since the minor premise and conclusion are negative, we can say that no actual penalty, small or large, is claimed in either of these propositions; in that case, we are not naturally inclined to engage in measure-for-measure reasoning, and therefore no dayo principle is needed to block such reasoning. It would appear, then, that the dayo principle is not useable in such negative context.

However, we could also look upon such negative argument as tacitly positive. Assuming that all law-breaking merits some penalty, we could argue that where an illegal action is not sufficiently illegal to merit a certain penalty we may infer it to positively merit a lesser penalty, though we cannot predict how much less. In that case, the negative subjectal argument would be interpreted as saying that P is illegal enough to positively merit a penalty of magnitude ‘somewhat less than S’, and therefore Q is illegal enough to positively merit a penalty of magnitude even smaller than ‘somewhat less than S’. This thought clearly involves measure-for-measure reasoning; so the dayo principle ought to now be applicable.

But of course it is not in fact applicable, because this new argument infers a decrease in penalty, whereas the dayo principle is essentially aimed at preventing inferences of increase in penalty. It is intended as a principle of mercy, pushing towards leniency rather severity of judgment; therefore, its application here would be inappropriate. In other words, we would not normally try to interdict the conclusion of a negative subjectal argument (even one recast in more positive form), whether by denial of the additional premise or of the major premise, for the simple reason that such reaction would not be in accord with the spirit and intent of the dayo principle.

We can argue in much the same way with respect to positive predicatal a crescendo argument:

A more serious breach of a certain law (R) is required to merit penalty P than to merit another penalty Q.
Action S is a breach of that law (R) enough to merit penalty P.
The seriousness of the breach of law R is ‘proportional’ to the magnitude of action S.

Therefore, a lesser action (S) is a breach of that law (R) enough to merit penalty Q.



Here again, we have reasoning from major to minor – specifically, from a more illegal action (S) with a greater penalty (P) to a less illegal action (S) with a smaller penalty (Q) – so, there would be no sense in applying (in either way) the dayo principle to it. Such an argument would, if our analysis of the moral motives of this principle has been correct, be allowed to proceed unhindered.

However, things get more complicated when we turn to negative predicatal argument, since the orientation is again from minor to major, while the minor premise and conclusion are negative in polarity:

A more serious breach of a certain law (R) is required to merit penalty P than to merit another penalty Q.
Action S is a breach of that law (R) not enough to merit penalty Q.
The seriousness of the breach of law R is ‘proportional’ to the magnitude of action S.
Therefore, a greater action (S+) is a breach of that law (R) not enough to merit penalty P.


In view of the negative polarities involved, we are tempted to say that there is no call for the dayo principle since no actual penalties are claimed. However, if we recast the argument in more positive form, following the idea that all law-breaking merits some penalty, we could say that the minor premise concerns some positive penalty of magnitude ‘somewhat less than Q’ (for action S) and likewise the conclusion concerns some positive penalty of magnitude ‘somewhat less than P’ (for action S+). Assuming that ‘somewhat less than P’ is greater than ‘somewhat less than Q’, which seems reasonable granting the additional premise, we can say that this argument is indeed from minor to major in a positive sense. In that case, the dayo principle ought to be applied to it, to prevent justification of the increased penalty advocated by the conclusion. Thus, either the additional premise about ‘proportionality’ or the generalization leading to the major premise will be interdicted.

Thus, to sum up, whereas when we think in bare formalities the four forms of a crescendo argument might seem liable to dayo principle interference, upon reflection it is only the positive subjectal and negative predicatal forms which are concerned, because they go from minor to major. The other two forms, the negative subjectal and the positive predicatal, are not concerned, because they go from major to minor. So the issue is not so much the polarity of the argument as its orientation. All the above can be repeated regarding implicational arguments, of course.

What we have said here, of course, refers to arguments that predicate penalties[14]. Arguments that predicate rewards are not to be treated in an analogous manner, because (as we have seen earlier) the dayo principle is only aimed at preventing increased punishment, not increased reward. But, one might ask, what of decreased rewards? Is not a decrease in reward comparable to an increase in punishment? The answer to that I would suggest is again practical rather than formal: Jewish law is not concerned with rewarding good deeds, but in penalizing bad ones. Furthermore, it does not address all bad deeds, but only some of them – namely, those subject to judgment by rabbinical courts. The purpose of Jewish law, as indeed most law systems, is to ensure at least social peace; it is not to control everything. Accordingly, the dayo principle is not intended to deal with changes in magnitude relating to rewards. It will simply not be invoked in such contexts; and indeed, such contexts are not expected to arise.

This is all assuming, of course, that my understanding of the matter is correct. It is not unthinkable that the empirical truth is a bit different from what I have assumed; and for instance, there are in fact occasional applications of the dayo principle in situations where I have just said it is logically inapplicable. In that event, needless to say, the above account would have to be modified in accord with actual facts. This should not be too difficult, since the formal issues are already transparent. It is not unthinkable that over time the original intent of the Sages’ dayo (given in Mishna Baba Qama 2:5) has been misunderstood, forgotten or intentionally ignored, and the concept of dayo was eventually used more broadly. This is in fact suggested by the broad or vague way that the dayo principle is usually presented in rabbinical literature.

Judging by the study of Mishnaic qal vachomer presented in Appendix 2, we cannot resolve the empirical issue with reference to the Mishna. For, surprisingly, of the 46 arguments found there, only the famous two in Mishna Baba Qama 2:5 involve the dayo principle! This is an important finding. There are nine other arguments which are possibly a crescendo, and therefore could be subject to dayo; but there is no mention of dayo in relation to them – either because they are not really a crescendo or because they do not serve to infer a penalty from the Torah.

Therefore, we must look to the Gemara (and indeed, later rabbinic literature), to find out whether the dayo principle is consistently applied in practice as here postulated. Only after all a fortiori arguments in the whole rabbinic corpus have been identified and properly analyzed will this question be scientifically answered. In Appendix 3, I try to at least partly answer the question, using the Rodkinson English edition of the Talmud. My finding in this pilot study is that there are only six Talmudic contexts where the dayo principle is explicitly appealed to! In five of these cases, the dayo principle may be said to be used as I have predicted, i.e. to prevent increase in legal responsibility through a fortiori argument. In the remaining case, this is partly true (see fuller explanation there).

Considering the prime position given to qal vachomer in the rabbinic lists of middot (hermeneutic principles), and the great attention accorded by rabbinical commentators to the Mishna Baba Qama 2:5 which introduces the dayo principle, one would expect the Tannaim (the rabbis of the Mishnaic period) to resort to dayo objections quite often. That this is statistically not the case is, to repeat, quite surprising. It may well be that more instances of dayo use by Tannaim will be found in some baraitot (statements attributed by Tannaim not included in the Mishna), many (maybe most) of which are quoted by Amoraim (the rabbis of the Gemara period) in different passages of the Talmud. This matter deserves systematic research, if we want to get a realistic idea of the quantity of dayo use by the Tannaim[15].

Besides that, we of course need to further research independent dayo use by the later rabbis, i.e. the Amoraim and their successors, respectively. Its use also in the early and late Midrashic literature deserves close study too. As regards the Amoraim, it is also quite surprising how little they appeal to the principle, at least explicitly, at least in the Rodkinson edition. However, my expectation is that, though some more use of the dayo principle by the Tannaim and the Amoraim may well be found, it will not be significantly much more.

I would like now to deal with a couple of further details, before closing this topic.

To begin with, let us reflect on the fact that rabbinical formulations (apparently of more recent vintage historically) usually describe a fortiori argument as an instrument of legal reasoning that can proceed in both directions, i.e. both from minor to major and from major to minor. For instance, consider the following formulation by R. Feigenbaum:

“Any stringent ruling with regard to the lenient issue must be true of the stringent issue as well; [and] any lenient ruling regarding the stringent issue must be true with regard to the lenient matter as well.”[16]

According to this statement, given that a stringent ruling (S) applies to the lenient issue (Q), it must also apply to the stringent issue (P); and given that a lenient ruling (S) applies to the stringent issue (P), it must also apply to the lenient issue (Q). The first part of that statement matches positive subjectal a fortiori (minor to major). The second part of it presumably refers to the negative subjectal form, since it is major to minor (and obviously not predicatal). Indeed, that is how I interpreted it in my Judaic Logic[17]. My thinking there was that: Given that there has been some breach of law (R), then some penalty is deserved; in that event, “not-deserving a stringent penalty” implies “deserving a lenient penalty”! The terms stringent and lenient being understood as relative to each other, not as absolute.

Thus, a formulation such as R. Feigenbaum’s tacitly assumes that “all law-breaking merits some penalty.” It is only on this basis that we can indeed logically transfer a lenient ruling from a stringent issue to a lenient matter, as he and others postulate. Although his above formula is stated entirely in positive terms, it in fact refers to both positive and negative arguments. Note in passing that the dayo principle is not mentioned in that writer’s formula. That is because he is here thinking in purely a fortiori terms, and not a crescendo like the Gemara. He is not saying that the inferred ruling is to be more stringent or more lenient, but only as much so. The same stringency or leniency is passed on.

Not having R. Feigenbaum’s book in my possession any longer, I do not know what, if anything, he said in it about the dayo principle. I doubt offhand that he distinguished between purely a fortiori and a crescendo argument, and that he related that principle exclusively to the latter form and limited dayo use to increased stringencies. But, using at the language of his above statement, I would say it ought to be amplified as follows. In cases where purely a fortiori inference is appropriate, the same degree of stringency or leniency is concluded, and the dayo principle is irrelevant. But in cases where a crescendo inference is appropriate, the natural conclusion would be more stringency or more leniency. In such cases, if the conclusion is a more stringent penalty than the one proposed in the Torah, dayo should be applied; whereas if it is more lenient it need not be.

Another point I would like to clarify is the idea emitted above that in predicatal a crescendo argument the subsidiary term (the subject of the minor premise and conclusion) is decreased (in the positive mood) or increased (in the negative mood). What does it mean to say, as we did, that an action is lesser or greater? This is best clarified by giving an example. We might, for instance, conceive two kinds of killing: intentional killing and unintentional killing, and argue thus: More badness (middle term, R) is required to merit a more severe penalty (major term, P) than to merit a less severe penalty (minor term, Q); so if, under the law relating to killing, intentional killing (S1) is bad enough to merit a more severe penalty, then unintentional killing (S2) is bad enough to merit a less severe penalty. This is a positive predicatal a crescendo argument.

Formal application of the dayo principle to this reasoning would mean that it is forbidden to here follow the principle of measure for measure and infer a lesser penalty for the less serious crime. Intuitively, such interdiction is obviously contrary to reason: we would rather let the ‘proportional’ conclusion stand since it is more indulgent. Neither justice nor mercy would be well served by applying the dayo principle to such cases. To punish a less serious crime the same way as a more serious one would be contrary to both justice and mercy. To punish a less serious crime less severely than a more serious one is in accord with both our sense of justice and our sense of mercy.

Clearly, then, the dayo principle should remain inoperative in cases of positive predicatal a crescendo argument concerning retribution for crime. Similar reasoning, as we have seen, applies to negative subjectal a crescendo argument. It is only with regard to positive subjectal or negative predicatal a crescendo arguments that the dayo principle makes sense and has relevance, for only in their case may there be an over-enthusiastic upsurge of justice, so that mercy requires a more cautious and temperate approach. In other words, dayo is potentially relevant only to a crescendo arguments that go from minor to major; it plays no role in such arguments that go from major to minor. Dayo is also, of course, irrelevant to purely a fortiori arguments (whether a minori or a majori), since the subsidiary term (whether it is a subject or a predicate) remains unchanged in them.[18]

This is spoken entirely from a theoretical perspective. It does not mean that the rabbis have all always been as conscious as that of the various possibilities. But I suspect they at least subconsciously have indeed reasoned in this way and limited dayo in the ways above described. Exceptions might conceivably be found in the mass of Talmudic and other rabbinic literature. This is an empirical question that must be answered empirically. If examples of upside down application of dayo are found, they would need to be rationalized somehow ad hoc – or, alternatively, they could be viewed as occasional errors of reasoning.

To conclude our formal exposition, we can say that the dayo principle is much leaner than what we may have originally imagined. It is not a formal law of a fortiori logic, but a very specific religiously-inspired rule for Jewish legislators and judges. Moreover, it is not a rule to be applied indiscriminately, but specifically with regard to attempts at increasing penalties on the basis of proportional qal vachomer reasoning. I should add: since a crescendo argument as such, i.e. as distinct from the dayo principle used to freeze its conclusions as just explained, is purely logical – it is inaccurate to call qal vachomer a hermeneutic rule! The first hermeneutic rule in Hillel’s list or in R. Ishmael’s list is, strictly speaking, not the qal vachomer argument, but the dayo principle applied in the context of such argument. We may nevertheless maintain the use of “qal vachomer” as the title of the first rule on the basis that the dayo principle is called for solely in that specific context, because it is only in such context that a quantitative increase (in penalty) might be inferred.

One might unthinkingly assume that the dayo principle might equally well be used in conjunction with other forms of analogical reasoning (e.g. gezerah shavah or binyan av). Indeed, one might argue that if dayo is applicable in such a maximally deductive context as qal vachomer, then it should all the more be applicable in more inductive contexts like gezerah shavah or binyan av. But further reflection should convince that what distinguishes qal vachomer is that it deals with quantities and the dayo principle is a restriction of increase in quantity (of the subsidiary term, to be exact) when inferring a penalty from the Torah. Since gezerah shavah, binyan av and other hermeneutic principles do not prescribe quantitative changes, the dayo principle does not concern them.

It remains conceivable, however, that yet other forms of reasoning could result in quantitative changes that would call for application of dayo. Come to think of it, it does seem like the rabbis “temper justice with mercy” even in situations that do not involve qal vachomer or any other hermeneutic principle. But of course such judgments might not be characterized as based on the dayo principle, since they are made more directly. What I am referring to here is the rabbinical interpretation of the lex talionis (the law of retaliation) found in Exodus 21:23–25 and Leviticus 24:19–21 – the famous “an eye for an eye, a tooth for a tooth” principle. The rabbis do not read this Torah law literally, but as a call for monetary compensation in cases of injury; this is shown using various arguments, including a qal vachomer.[19]

4. The human element

Looking at rabbinical practices and principles, we can safely say that the rabbis were very careful to acknowledge the human element in reasoning a fortiori, or by means of any other of the listed hermeneutic principles (and by extension, even unlisted thought processes).

This is evident, first of all, in their practice of teshuvah (Heb.) or pirka (Aram.) – usually rendered in English as ‘objection’ or ‘challenge’ – consisting in retorting to or rebutting an argument, and in particular an a fortiori argument, by showing or at least pointing out that one (or more) of its premises is (wholly or partly) open to doubt or false, or that the putative conclusion cannot in fact be drawn from the given premises. This demonstrated their awareness, if only pre-verbally in some instances, of the inductive sources of many of the propositions used in their reasoning. In some cases, as well, such practice on their part demonstrated awareness of the relative artificiality of certain forms of argumentation they used and thence the tenuousness of their conclusions.

Such awareness of the human element in apparently deductive inference is also made evident in their setting a number of explicit restrictions on the use of a fortiori argument. Such argument could only be used for inferring laws by qualified rabbis involved with their peers in the development of Jewish law (meaning in principle members of the Sanhedrin, though in practice some participants were probably not officially members). Inferences made had to be accepted unanimously or by ruling of a majority. Inferences could be made only from written Torah laws, and not from oral Torah traditions, even if they were reputed to go all the way back to Moses, and all the more so if they were considered to be of more recent vintage. One could not infer a new ruling from a previously inferred ruling, i.e. use the conclusion of one a fortiori argument as a premise in the next.

I would additionally suggest, an a fortiori inference from a Torah law would be considered questionable if it was found to conflict with another Torah law. This seems reasonable on the general understanding that written Torah law carries more weight in Judaism than any human inference. An example is apparently given by Louis Jacobs in his The Jewish Religion: A Companion[20] with reference to a responsum of the Radbaz (Spain-Israel, R. David ben Zimra, 1479-1573) to the question why the Torah does not forbid a man’s marriage to his own grandmother, and yet forbids him his wife’s grandmother (who is a more remote relative), although we would expect by a fortiori argument from the prohibition in the latter case that the former case would also be prohibited. Jacobs explains: “Typical of Radbaz’s attitude to the limited role of human reasoning in Judaism is his reply that the a fortiori argument is based on human reasoning, whereas the forbidden degrees of marriage are a divine decree, so that human reasoning is inoperative there. All we can say is that God has so ordained. One degree of relationship is forbidden, the other permitted.”

The a fortiori argument here is: a man’s own grandmother (P) is more closely related (R) to him than his wife’s grandmother (Q); if his wife’s grandmother (Q) is closely related (R) enough to be forbidden in marriage to him (S), then a man’s own grandmother (P) is closely related (R) enough to be forbidden in marriage to him (S). The difficulty is that, although the former is forbidden, the latter is not forbidden. However, I do not see why the rabbis do not accept this a fortiori argument, as they do many others, and simply prohibit marriage to one’s own grandma, since there is no written permission to contend with. The answer given by the Radbaz, and before him by Menahem Meiri (France, 1249-1316), is that there is no need for the inferred prohibition as no one would be likely to do such a thing anyway in view of age differences. That is, more precisely put, while a man might be attracted to his wife’s grandmother (e.g. if his wife is thirteen years old, and her mother twenty-six and her grandmother thirty-nine, and he is forty), he is unlikely to be attracted to his own grandmother (who would be in her mid-sixties at least). But this argument may seem a bit weak, as some men are attracted by much older women, even if rarely.[21]

Another restriction was that a ruling based on a fortiori argument could not take precedence over a Torah law from which it was inferred, if the two happened to come into conflict. For example, it is inferable from the Torah law (Ex. 23:4) that one should return one’s enemy’s lost ox or ass that one should likewise, a fortiori, return one’s friend’s lost ox or ass. One might think that, having thus made a deductive inference, it would follow that when simultaneously encountering two lost animals, one from each of these people, one could legally prefer to return that belonging to one’s friend rather than (or at least before) returning that belonging to one’s enemy. But no: the premise remains more binding than the conclusion, and one must therefore give precedence to the enemy’s animal[22]. Yet another important restriction was that a rabbinical law court could not sentence someone to corporeal punishment on the basis of a legal ruling derived by a fortiori argument. Meaning that, however reliable the justifying deduction might well have been, there was still a drop of doubt in it sufficient to preclude such drastic penalties.

Some of these restrictions were perhaps more theoretical than practical, because if we look at Talmudic discussions (Mishna, Gemara and later commentaries and super-commentaries all included) one is struck by the ease and frequency with which the rabbis engaged in a fortiori argument if only rhetorically. One would have to examine all rabbinic literature in great detail to determine whether these theoretical restrictions have all in fact been consistently adhered to in practice (this is certainly a worthwhile research project for someone). Nevertheless, on the whole, these restrictions show the rabbis’ acute awareness of the natural limits of the human powers of experience and reason.[23]

The dayo principle as I have above described it falls right into this pattern of restricting excessive reliance on logical means. A ruling based on qal vachomer argumentation remains somewhat doubtful, even though the conclusion (if correct) follows the premises with absolute certainty, because there is inevitably some human element in the induction of the premises. These premises may be in part or even largely Torah-based, but still some part(s) of them were inevitably based on human insight or convention, so it is wise to remain a bit open-minded concerning their conclusion[24]. But this is nothing to do with the dayo principle, as we have latterly discovered. This principle is not designed to throw doubt on qal vachomer argumentation as such, but to prevent extrapolation from Torah-based premises by means of the principle of justice.

A question we could ask is: why is the dayo (sufficiency) principle not directly and always applied to the midah keneged midah (measure for measure) principle? In my above treatment of these principles, I have identified the latter as inserting an additional premise of ‘proportionality’ between the minor premise and conclusion, and the former as either blocking the operation of this additional premise or preventing the formation of the major premise through generalization. Thus, we may view the measure for measure principle as tending to turn a purely a fortiori conclusion into an a crescendo one, and the sufficiency principle as on the contrary tending to restrain (in one way or another) such proportionality. The two balance each other out, and the result is that the purely a fortiori conclusion stands unchanged.

The question is: could we not say, more generally: whenever we encounter a midah keneged midah, we must apply dayo? Why does the qal vachomer need to be mentioned at all? Obviously, if such a general rule was promulgated, the two said principles would effectively cancel each other out and cease to exist! Obviously, too, this is not the intent of the dayo principle; i.e. it is not meant to altogether neutralize the midah keneged midah principle. So it is reasonable to suppose the dayo principle to be intended for a specific context; namely, for when a qal vachomer is formulated and we are tempted to extrapolate its conclusion by a thought of measure for measure. And more specifically still, for when the speaker (like R. Tarfon in mBQ 2:5) attempts to infer a larger penalty from a lesser penalty prescribed in the Torah.

If there were no qal vachomer, or other deductive inference, the measure for measure principle might conceivably have been applied without restriction. Why then, we might well ask, was the dayo principle needed in the context of qal vachomer? Perhaps the answer to that important question is that if the measure for measure extrapolation occurs in a non-deductive context, we naturally remain aware of the human element in it and maintain a healthy measure of skepticism. Whereas in a deductive context, especially where the powerful logic of qal vachomer is used, since we have already proved part of the quantity, we are more likely to view its measure for measure extrapolation as also ‘proved’. The dayo principle comes to remind us that the proposed extrapolation does not have the same degree of reliability as the more limited conclusion of the qal vachomer has. Indeed, the dayo principle precludes any temptation to extrapolate rather than let us run the risk wrongful extrapolation.

This may conceivably have been the justification of the dayo principle in the rabbis’ minds. Even if they did not fully realize that it concerned a thought of midah keneged midah accompanying a qal vachomer, rather than the latter argument per se, they would have sensed the danger of unbridled extrapolation. And according to the Gemara, as we have seen, the preemptive measure against such extrapolation (viz. the dayo principle) was not a mere rabbinical ruling (by the Sages), but a Divine decree (through Num. 12:14-15). It perhaps had to be a Torah-based hermeneutic rule, so that it could not in turn be open to doubt as a human construct. Even so, as we have seen, R. Tarfon and others did (according to the Gemara) claim the dayo principle could in some situations be bypassed or even ignored. But, for the most part, the Sages’ posture has prevailed.

It is worth noting lastly that, according to later authorities (at least some of them), qal vachomer argument (or more precisely the dayo principle associated with it) could only be used in the Talmudic law making process. After the closure of this process, it was considered illegal to use this hermeneutic principle, or any other of the thirteen rules of R. Ishmael for that matter, to interpret the written Torah for legislative purposes. The references for this sweeping ruling are given by R. Bergman[25] as: “Maharik Shoresh 139; Ra’ah to Ketubos cited in Yad Malachi 144.” This limitation in time is additional evidence that Judaism does not view the dayo principle as a law of logic but as a revealed ad hoc religious law. Laws of logic cannot be abrogated; decrees can. Similarly for the other hermeneutic principles.

Why this limitation in time? Because, I presume, the hermeneutic rules were a prerogative of the Sanhedrin, the Jewish Supreme Court; when its deliberations were interrupted due to foreign conquest and rule, rabbis were no longer empowered to use these interpretative principles. An implication of this explanation is that if – or when – the Sanhedrin is reinstituted (presumably by the Messiah) the dayo principle and other such guidelines will again be useable by its members. This is a neat answer to the question, except that most of the Babylonian Talmud’s deliberations took place in Babylon, far from the traditional seat of the Sanhedrin in the Land of Israel. Presumably, the Babylonian rabbis involved were considered to be worthy successors to the Sanhedrin. The reason for the time limitation would then simply be that the Talmud was ‘closed’ in about 500 CE (say), and subsequent rabbis were considered as at a lower spiritual level than their teachers.

5. Qal vachomer without dayo

It should be pointed out that Talmudic use of qal vachomer does not always require application of the dayo principle, for the simple reason that the conclusion sometimes naturally lacks the required quantitative aspect, i.e. there is no propensity to ‘proportionality’ that needs to be interdicted. In other words, the argument is purely a fortiori rather than a crescendo. Consider the following argument:

“All these things they [the rabbis] prescribed [as culpable] on a Festival, how much more [are they culpable] on Sabbath. The Festival differs from the Sabbath only in respect of the preparation of food.” (Mishna Beitzah, 5:2.)[26]

There is, surprisingly, no remark in the corresponding Gemara (Yom Tov, 37a) on this significantly different use of a fortiori reasoning. Here, unlike in the Miriam example and cognate cases, there is no appeal to the dayo principle. Does the Talmud notice and discuss this difference anywhere else? I do not know. In any case, this example is very interesting and worth analyzing further.

The Mishna here clearly teaches that: what is forbidden (assur) on a Festival is, a fortiori, also forbidden on the Sabbath. We can express this in a standard form of a fortiori argument (namely, the positive subjectal, from minor to major) as follows:

The Sabbath (P) is more religiously important (R) than any Festival (Q); whence:

if a certain action on a Festival (Q) is important (R) enough to be forbidden (S),

it follows that the same action on the Sabbath (P) is important (R) enough to be forbidden (S).



This is a passable representation of the argument. However, if we ask what we mean here by more “religiously important,” we might reply that the Sabbath is more “demanding” (or strictly regulated) than any Festival. In that perspective, the argument would seem to be, though still ‘minor to major’, more precisely negative predicatal in form, and we should preferably formulate it as follows:

More holiness (R) is required to observe the Sabbath (P) than to observe any Festival (Q).

If some action[27] (S) is not sufficiently holy (R) to be compatible with observance of a Festival (and thus must[28] be forbidden on it) (Q),

then that action (S) is not sufficiently holy (R) to be compatible with observance of the Sabbath (and thus must be forbidden on it) (P).



Note that I have inserted “holiness” (of an action) as this argument’s operative middle term (R) on the basis of rabbinical explanatory statements in the present context that the holiness of the Sabbath is greater than that of any Festival day. The way I have used this word is a bit awkward, I’ll admit; but it does the job anyway.

More fully expressed the argument has three components: (a) Given that (in the minor premise) S implies not-Q, it follows by contraposition that if Q is prescribed, S must be forbidden. (b) And given that S implies not-Q, it follows by a fortiori that S implies not-P. Finally, (c) since (in the conclusion) S implies not-P, it follows by contraposition that if P is prescribed, S must be forbidden. The two ‘contrapositions’ used are simple ethical logic: anything that interferes with achievement of a set goal is obviously to be prohibited; the means must be compatible with the ends.

We can present the corresponding positive predicatal (major to minor) as follows:

More holiness (R) is required to observe the Sabbath (P) than to observe any Festival (Q).

If some action (S) is sufficiently holy (R) to be compatible with observance of the Sabbath (and thus may be permitted on it) (P).

then that action (S) is sufficiently holy (R) to be compatible with observance of a Festival (and thus may be permitted on it) (Q),



This follows from the negative form by reductio ad absurdum, of course. The meaning of this new argument is: what is permitted (i.e. not forbidden) (mutar) on the Sabbath is, a fortiori, also permitted on a Festival. That is, the argument could as well be put in negative subjectal form, as follows:

The Sabbath (P) is more religiously important (R) than any Festival (Q); whence:

if a certain action on the Sabbath (P) is important (R) not enough to be forbidden (S),

it follows that the same action on a Festival (Q) is important (R) not enough to be forbidden (S).



The expression “not enough to be forbidden” may be taken to imply that the action in in fact “permitted.”

Obviously, we cannot reverse these two statements, viz. that what is forbidden on a Festival must be forbidden on the Sabbath, and what is permitted on the latter must be permitted on the former. Obviously, something forbidden on the Sabbath (e.g. cooking food) is not necessarily also forbidden on a Festival. Something permitted on a Festival (e.g. cooking food) is not necessarily also permitted on the Sabbath. Reasoning of the latter sort would be fallacious by the ordinary rules of a fortiori logic.

Note also: although I have above classified the two arguments as predicatal (i.e. copulative), it might be more accurate to call them consequental (i.e. implicational). For, what the negative form tells us is that a certain action (S) by a Jew causes some deficiency of, let us say, holiness (R) in him and thus causes him to fail to observe a Festival (Q) or the Sabbath (P); similarly for the positive form, mutatis mutandis. In other words, while it is true that P, Q, R, S are terms, there is an unstated underlying subject (a Jewish man, or woman) in relation to which they are all predicates, so that theses (rather than terms) are in fact tacitly intended here.

Furthermore, according to formal logic, if the above two arguments are true, the following two (in which the negative term not-S replaces the positive term S) must also be true.

More holiness (R) is required to observe the Sabbath (P) than to observe any Festival (Q).

If some inaction[29] (not-S) is not sufficiently holy (R) to be compatible with observance of a Festival (and thus must be forbidden on it) (Q),

then that inaction (not-S) is not sufficiently holy (R) to be compatible with observance of the Sabbath (and thus must be forbidden on it) (P).



This is a negative predicatal (minor to major) argument. The meaning of this new argument is, clearly: what is imperative (chayav) on a Festival is, a fortiori, also imperative on the Sabbath. In this form, it is positive subjectal.

More fully expressed the argument has three components: (a) Given that (in the minor premise) not-S implies not-Q, it follows by contraposition that if Q is prescribed, S must be prescribed. (b) And given that not-S implies not-Q, it follows by a fortiori that not-S implies not-P. (c) Finally, since (in the conclusion) not-S implies not-P, it follows by contraposition that if P is prescribed, S must be prescribed. The two ‘contrapositions’ used are simple ethical logic: anything without which a set goal cannot be achieved is obviously to be prescribed; the means necessary for an end are indispensable.

We can present the corresponding positive predicatal (major to minor) as follows:

More holiness (R) is required to observe the Sabbath (P) than to observe any Festival (Q).

If some inaction (not-S) is sufficiently holy (R) to be compatible with observance of the Sabbath (and thus may be permitted on it) (P).

then that inaction (not-S) is sufficiently holy (R) to be compatible with observance of a Festival (and thus may be permitted on it) (Q),



This follows from the negative form by reductio ad absurdum, of course. The meaning of this new argument is: what is exempted (i.e. not prescribed) (patur) on the Sabbath is, a fortiori, also exempted on a Festival. In this form, it is negative subjectal.

Obviously, here again, we cannot reverse these two statements, viz. that what is imperative on a Festival must be imperative on the Sabbath, and what is exempted on the latter must be exempted on the former. Something imperative on the Sabbath (e.g. the additional sacrifices on it) is not necessarily also imperative on a Festival. Something exempted on a Festival (e.g. the said additional sacrifices) is not necessarily also exempted on the Sabbath. Reasoning of the latter sort would be fallacious by the ordinary rules of a fortiori logic.

Clearly, the Sabbath and the Festivals involve some distinctive practices; and Festivals are not all identical. The Festivals are not merely lighter forms of Sabbath, and the Sabbath is not merely a heavier form of Festival; and the various Festivals involve different rituals. We cannot deductively predict all features of one holy day from the other, or vice versa, but must refer to Biblical injunctions or hints for the special features of each. The above a fortiori arguments do not provide a complete set of relationships, which mechanically exclude innovations from the Biblical proof-text.

What can be inferred from the Sabbath to Festivals or vice versa is a product of two forces: (a) the major premise, which relates these two kinds of holy day through a middle term that we took to be ‘holiness’; and (b) the minor premise, which links one of these holy days to a certain subsidiary term through the same middle term. This limits the possibilities of inference, insofar as the middle term does not have unlimited scope. For a start, ‘holiness’ is a vague abstraction, difficult to establish objectively; moreover, it does not provide links to any and all subsidiary terms, but only at best to a specified few.

Thus, much in these arguments depends on traditional understanding of the terms involved. That is to say, the arguments are descriptive propositions as much as deductive processes. They give verbal expression to pre-existing traditions or traditions taking shape, as well as assist in the inference of information. They are formulas designed to enshrine traditional principles and facilitate logical access to them.

It is perhaps historically in this way, by development from the Beitzah 5:2 example of a fortiori argument, that the more general rabbinic definition of qal vachomer emerged (presumably later)[30]. To take a modern statement, R. Chavel defines the argument as follows:

“A form of reasoning by which a certain stricture applying to a minor matter is established as applying all the more to a major matter. Conversely, if a certain leniency applies to a major matter, it must apply all the more to the minor matter.”[31]

This seems to refer primarily to the first two of our above examples, where the “minor matter” is a Festival day and the “major matter” is the Sabbath, and the “stricture” is the proscribing of some action and the “leniency” is its permission. Stricture, of course, suggests restriction, a negative; but it can here be taken to mean more broadly strictness or stringency and thus also refer to a prescription, just as leniency can also refer to an exemption. This is evident in the similar but more accurately worded description of a fortiori reasoning by R. Feigenbaum:

“Any stringent ruling with regard to the lenient issue must be true of the stringent issue as well; [and] any lenient ruling regarding the stringent issue must be true with regard to the lenient matter as well.”[32]

A similar description may also be found in Steinsaltz’s Reference Guide and many other books. What this tells us is that although the examples traditionally drawn from Beitzah 5:2 initially refer to qal vachomer inferences from prohibition to prohibition and from permission to permission, the rabbis also eventually admit the inferences from imperative to imperative and from exemption to exemption that we have just logically demonstrated.

Mielziner, by the way, shows explicit awareness of all four moods, to the extent that where the conclusion is “assur” (forbidden) he adds in brackets the alternative of “chayav” (imperative), and where the conclusion “eino din sheassur” (permitted) he adds in brackets the alternative of “[eino din] shechayav” (exempt). That is, he makes allowance for both the negative and the positive interpretations. He additionally gives us Talmudic examples of an imperative implying an imperative by such qal vachomer: in Baba Metzia 95a, it is inferred that the borrower must restore what was stolen (from him the borrower by some third party) to the lender; or again, in Baba Metzia 94b, that the borrower must restore what he (the borrower) lost to the lender.[33]

However, I am not sure exactly when, in documented history, the transition occurred from the principle specifically concerning Festivals and Sabbaths given in Mishna Beitzah 5:2, and perhaps other passages of the Mishna with a similar thrust, to the general formulations that authors like Mielziner, Chavel, Feigenbaum or Steinsaltz, give nowadays. I suspect the general formulations are not that modern, and may be found in the Talmud or other early literature. It would be very interesting to discover exactly how the progression from material principle to formal principle occurred, i.e. thanks to whom and on what dates.

To conclude this section, what we need to note well is that no application of the dayo principle is needed or even possible in cases of the sort here considered, since obviously an action is either forbidden or permitted, either imperative or exempted, and there are no degrees in between. Admittedly, as regards permitted actions, some may be more ‘desirable’ or ‘to be preferred’ or ‘recommended’ than others, but these are not degrees of permission as such. Observe that we have no inclination, in the above inference from permission on the Sabbath to permission on a Festival, to regard the latter permission as of a lesser (or greater) degree than the former. Similarly with regard to exemption: it has in itself no degrees. Very often, the conclusion of a fortiori argument is like that – without degree. This is clearly purely a fortiori inference, and not to be confused with a crescendo inference.

I do not know if the rabbis explicitly made this distinction, between qal vachomer use with appeal to dayo principle and qal vachomer without relevance of dayo. As I have explained, the dayo principle is needed to block reasoning through the midah keneged midah (measure for measure) principle or similar ‘proportional’ propositions. It is not directly related to a fortiori argument as such; it is only indirectly related, to prevent a common penchant for ‘proportionality’ in special cases. In many cases, if not in most, there is no such propensity, because there is no parallel principle like midah keneged midah pressing us towards ‘proportionality’, and therefore the issue of dayo does not even arise. In truth, a fortiori reasoning is always the same, irrespective of whether there is ‘proportionality’ or not and whether dayo is thereafter used or not.

In view of all this, it is hard to understand why the Gemara commentary in Baba Qama 25a is so categorical in its treatment, giving the impression that a fortiori argument is necessarily a crescendo, and failing to explicitly note that the dayo principle, whether it is applied to all a crescendo arguments (as the Sages apparently hold, in the Gemara’s view) or only to some (as R. Tarfon holds, according to the Gemara), is not applicable to purely a fortiori arguments, i.e. those which do not involve (explicitly or implicitly) an additional premise about ‘proportionality’. Surely, if the author of this Gemara was aware of the full sweep of Talmudic discourse, he would have noticed these distinctions and taken them into consideration in his commentary.[34]

6. Three additional Gemara arguments

Further on in tractate Baba Qama, on pp. 25b-26a, the Gemara proposes three a fortiori arguments in which the previously used propositions, about damage by horn and by tooth & foot on public and private grounds, are recycled and reshuffled in various ways, and the resulting conclusions are tested. For this reason, I have dubbed them “experimental” arguments. It is not immediately clear what the purpose(s) of these additional arguments might be. At first sight, their insertion here looks like a process of consistency checking. Possibly, the Gemara is using them to settle some legal matter specified in the larger context. Alternatively, it is merely exploring theoretical possibilities, trying different permutations and seeing where they lead. Or again, perhaps the Gemara is simply engaged in intellectual exercise for its own sake. In any case, we shall here try to throw some light on these arguments by means of logical analysis.

Before we do so, however, let us briefly recall here the original Mishna (BQ 2:5) arguments to which they refer, for this will facilitate our work. The first Mishna argument can be presented in several ways. Its premises and conclusion can be laid out as a set of if-then propositions spelling out the legal liability for damage by different causes in different domains, as follows:

If tooth & foot and public, then no liability (by extreme inversion of Ex. 22:4).

If tooth & foot and private, then full liability (Ex. 22:4).

If horn and public, then half liability (Ex. 21:35).

If horn and private, then full liability (R. Tarfon’s putative conclusion).

If horn and private, then half liability (the Sages’ conclusion, after application of dayo type I).

As we saw in our earlier detailed treatment, this basic argument can be recast in analogical, pro rata, a crescendo or purely a fortiori forms, as follows:

Analogy

Just as, in the case of tooth & foot, damage in the private domain implies more legal liability than damage in the public domain (since the former implies full liability and the latter none).

Likewise, in the case of horn, damage in the private domain implies more legal liability than damage in the public domain (i.e. given half liability in the latter, conclude with full in the former).



Pro rata

The degree of legal liability for damage is ‘proportional’ to the status of the property the damage is made on, with damage in the private domain implying more legal liability than damage in the public domain.

This is true of tooth and foot damage, for which liability is known to be nil in the public domain and full in the private domain.

Therefore, with regard to horn damage, for which liability is known to be half in the public domain, liability may be inferred to be full in the private domain.



A crescendo

Private domain damage (P) is more important (R) than public domain damage (Q) [as we infer by extrapolation from tooth & foot damage (where liability is respectively full and half in the two domains) to all causes of damage, including horn].

Horn damage in the public domain (Q) is important (Rq) enough to make the payment half (Sq).

The payment due (S) is ‘proportional’ to the degree of legal liability (R).

Therefore, horn damage in the private domain (P) is important (Rp) enough to make the payment full (Sp = more than Sq).



Pure a fortiori

Private domain damage (P) is more important (R) than public domain damage (Q) [as we infer by extrapolation from tooth & foot damage, to repeat].

Horn damage in the public domain (Q) is important (R) enough to make the payment half (S).

Therefore, horn damage in the private domain (P) is important (R) enough to make the payment half (S).



As we learned previously, the above analogical, pro rata or a crescendo arguments correspond to R. Tarfon’s reasoning. The Mishna Sages reject his reasoning by means of a dayo objection of the first type, i.e. which denies the ‘proportionality’ assumed by their colleague. Effectively, then, the Sages advocate the purely a fortiori argument exclusively. The second Mishna argument can likewise be presented in several ways. As a set of if-then propositions, it looks as follows:

If tooth & foot and public, then no liability (by extreme inversion of Ex. 22:4).

If horn and public, then half liability (Ex. 21:35).

If tooth & foot and private, then full liability (Ex. 22:4).

If horn and private, then full liability (R. Tarfon’s same putative conclusion).

If horn and private, then half liability (the Sages’ conclusion, after application of dayo type II).

And here again, the basic argument can be recast in analogical, pro rata, a crescendo or purely a fortiori forms, as follows:

Analogy

Just as, in the public domain, damage by horn implies more legal liability than damage by tooth & foot (since the former implies half liability and the latter none).

Likewise, in the private domain, damage by horn implies more legal liability than damage by tooth & foot (i.e. given full liability in the latter, conclude with full in the former).



Pro rata

The degree of legal liability for damage is ‘proportional’ to the intentionality of the cause of damage, with damage by horn implying more legal liability than damage by tooth & foot.

This is true of the public domain, for which liability is known to be nil for damage by tooth and foot and half for damage by horn.

Therefore, with regard to the private domain, for which liability is known to be full for damage by tooth and foot, liability may be inferred to be full for damage by horn.



A crescendo

Horn damage (P) is more important (R) than tooth & foot damage (Q) [as we infer by extrapolation from the public domain (where liability is respectively half and nil in the two cases) to all domains, including the private].

Tooth & foot damage in the private domain, (Q) is important (R) enough to make the payment full (S).

The payment due (S) is ‘proportional’ to the degree of legal liability (R).

Therefore, horn damage in the private domain (P) is important (R) enough to make the payment full (S).



Pure a fortiori

Horn damage (P) is more important (R) than tooth & foot damage (Q) [as we infer by extrapolation from the public domain, to repeat].

Tooth & foot damage in the private domain, (Q) is important (R) enough to make the payment full (S).

Therefore, horn damage in the private domain (P) is important (R) enough to make the payment full (S).



As we found out previously, this time all of the above argument forms, including the purely a fortiori one, match R. Tarfon’s reasoning. So, the Mishna Sages cannot reject his reasoning by means of a dayo objection of the first type, since ‘proportionality’ is not essential to its stringent conclusion of full liability. Nevertheless, they maintain their dayo objection, and again advocate a moderate conclusion of only half liability. Therefore, the latter dayo objection must be of a second type. It is indeed, interdicting the inductive process of generalization through which the major premise of such argument is produced. We need not say more than that here, having already dealt with the issues involved at length.

Now, what is interesting is the way the Gemara takes the final conclusion of the Mishna Sages, namely that horn damage in the private domain implies half liability, and uses it as a constant premise in each of its three experimental arguments. This proposition is of course implied by Ex. 21:35, which specifies half liability for horn damage, without specifying a domain; but the Sages have effectively ruled that it is not a minimum but a maximum[35], i.e. it is to be read as davka half. Nevertheless, the Gemara here additionally uses a watered down version of Ex. 21:35 in two of its arguments (the first two).

Another proposition relevant to all three Gemara arguments is Ex. 22:4, which specifies full liability for tooth & foot damage in the private domain[36]. This proposition is repeated in two of the Gemara arguments (the first and last). In the Mishna, the liability for tooth & foot damage in the public domain is taken to be the extreme inverse of Ex. 22:4, i.e. no liability. And this is also assumed in two Gemara arguments (the last two); however, at the end of one Gemara argument (the first one), a moderate inversion is attempted, i.e. “not full” is taken to mean “half” rather than “nil.”

Let us now examine the three new arguments in the Gemara more closely.

First experiment. The Gemara states: “But should we not let Tooth and Foot involve liability for damage done [even] on public ground because of the following a fortiori:

If in the case of Horn, where [even] for damage done on the plaintiff’s premises only half payment is involved, there is yet liability to pay for damage done on public ground,

does it not necessarily follow that in the case of Tooth and Foot, where for damage done on the plaintiff’s premises the payment is in full, there should be liability for damage done on public ground?

— Scripture, however, says: And it shall feed in another man’s field, excluding thus [damage done on] public ground. But have we ever suggested payment in full? It was only half payment that we were arguing for![37]

Note at the outset the sources of the premises in the Gemara’s argument. One is the earlier conclusion of the Mishna Sages (via their dayo objections to R. Tarfon’s claims) that for damage by horn on private property the ox owner’s liability is half. The other two premises are more directly derived from the Torah (Ex. 22:4 and Ex. 21:35). The conclusion concerns damage by tooth & foot on public property.

Expressed as a set of brief if-then statements, this Gemara argument looks as follows. Note that the first two have in common the factor of private property.

If horn and private, then half liability (ruling of the Mishna Sages).

If tooth & foot and private, then full liability (Ex. 22:4).

If horn and public, then some liability (from Ex. 21:35).

If tooth & foot and public, then some liability (putative conclusion).

Or in analogical format, as follows:

Just as, in the private domain, damage by tooth & foot implies more legal liability than damage by horn, since the former implies full and the latter half.

Likewise, in the public domain, damage by tooth & foot implies more legal liability than damage by horn; whence given that the latter implies some liability (note that although Ex. 21:35 implies a specific amount, the Gemara here deliberately avoids mentioning it in its premise), then the former implies some liability.



Or again, in purely a fortiori format, of positive antecedental form (minor to major), as follows[38]:

Tooth & foot damage (P) is more important (R) than horn damage (Q) [as we infer by extrapolation from their liabilities for damage in the private domain, respectively full and half, to all domains, including the public].

Horn damage in the public domain (Q) is important (R) enough to imply some liability (S).

Therefore, tooth & foot damage in the public domain (P) is important (R) enough to imply some liability (S).



The Gemara is thus justified in describing its argument here as qal vachomer (מקל וחומר), although this must be taken to refer to purely a fortiori argument and not a crescendo. We see clearly from the a fortiori formulation that the major premise is produced by a generalization, from the particular case of private property to all property, and its application to the particular case of public property. On this basis, the minor premise about unspecified liability for horn leads to the conclusion about unspecified liability for tooth & foot.

Now, the main question to ask here is: why is the Gemara opting for such vague language? There are actually two separate questions, here: (a) Why is its premise is deliberately vague, saying “there is yet liability” (חייבת), i.e. some liability, without specifying just how much liability even though the amount is already known from Ex. 21:35 to be precisely half? And (b) Why is its conclusion also vague, saying “there should be liability” (חייב), i.e. some liability, although the amount of this liability may be assumed by partial instead of full denial of Ex. 22:4 to be half? We shall now propose our answers.

The way to answer our question about the vagueness of the minor premise is to consider what would happen if more explicit language were to be used. To start with, had the Gemara used half liability as the consequent of the minor premise, and argued a crescendo instead of purely a fortiori, its conclusion would have been full liability for tooth & foot damage in the public domain, and thus contrary to Ex. 22:4, according to which full liability is reserved for tooth & foot damage in the private domain. This is evident in the following lines:

Tooth & foot damage (P) is more important (R) than horn damage (Q) [as we infer by extrapolation, as before].

Horn damage in the public domain (Q) is important (R) enough to imply half liability (S) (as specified in Ex. 21:35).

The payment due (S) is ‘proportional’ to the degree of legal liability (R).

Therefore, tooth & foot damage in the public domain (P) is important (R) enough to imply full liability (S) (contrary to the davka reading of Ex. 22:4).



Thus, the Gemara’s thinking (consciously or otherwise) in this respect was effectively as follows. Since the full liability conclusion is contrary to a Scriptural given (namely Ex. 22:4, which specifies full liability to be applicable only to private property) the argument must be rejected somehow. Since the major and minor premises are already accepted, and the inference process is clearly valid, the only way to reject the argument is by denying the additional premise about ‘proportionality’ – or, in other words, by applying a dayo objection of type I. That is to say, the a crescendo argument is to be discarded, leaving only the underlying purely a fortiori argument. This leftover argument is similar to the Gemara’s (previously mentioned), except that it infers half liability from half liability[39], instead of some liability from some liability.

Another route the Gemara may have tried is the following. As we learned from R. Tarfon, we can by judicious reshuffling of the premises obtain an alternative a fortiori argument. In the present case, this would be done as shown next. In terms of if-then statements, our competing argument would be as follows. Note that the first two statements, which we use to form our major premise, are both about horn damage.

If horn and private, then half liability (ruling of the Mishna Sages).

If horn and public, then half liability (Ex. 21:35).

If tooth & foot and private, then full liability (Ex. 22:4).

If tooth & foot and public, then full liability (putative conclusion, contrary to Ex. 22:4).

This can be recast in analogical form thusly:

Just as, in the case of horn, damage in the public domain implies as much legal liability as in the private domain (since both imply half liability).

Likewise, in the case of tooth & foot, damage in the public domain implies as much legal liability as in the private domain; whence given that the latter implies full liability, then the former implies full liability (contrary to Ex. 22:4, which specifies full liability to be applicable only to private property).



More to the point, we can formulate it in purely a fortiori format as follows. Note that this argument is positive antecedental and a pari (i.e. egalitarian).

Public domain damage (P) is as important (R) as private domain damage (Q) [as we infer by extrapolation from horn damage (where liability is half in both domains) to all causes of damage, including tooth & foot].

Tooth & foot damage in the private domain (Q) is important (R) enough to imply full liability (S).

Therefore, tooth & foot damage in the public domain (P) is important (R) enough to imply full liability (S) (contrary to Ex. 22:4).



Now, observe why this argument seems more secure than the preceding a crescendo. It also goes from minor to major; but since the minor premise predicates what is already the maximum amount allowable (namely, full liability), the conclusion has to predicate the same maximum amount (i.e. full liability). Yet here again the conclusion is contrary to a Scriptural given (Ex. 22:4, which specifies full liability to be applicable only to private property). Therefore, it must be rejected. The only way to do this is through a dayo objection of type II, i.e. by preventing the generalization that gave rise to its major premise from proceeding. The final conclusion will then again be half liability.

What the above suggests, then, is that the Gemara opted for vague language in the minor premise, speaking of liability indefinitely, because it knew or at least sensed that specifying half liability would in any event lead to a conclusion of full liability, contrary to Scripture; which conclusion would have to be prevented by application of dayo objections of both types. In the Judaic frame of reference, a conclusion contrary to what the Torah teaches is a conclusion contrary to ‘fact’, which must be prevented to avoid inconsistency. Apparently, then, rather than get involved in that long discussion, or pilpul, it opted for a vaguer statement of the minor premise, to arrive at its desired conclusion more directly.

As regards its vague conclusion, a minimum of reflection shows that the liability implied, though stated indefinitely, can only be half liability. This is evident already in the above two arguments from the minor premise of half liability, since their conclusion of full liability is unacceptable because contrary to Scripture. However, we could arrive at the same result by working on the vague conclusion of the Gemara’s own purely a fortiori argument (from some to some liability). Given the conclusion that tooth & foot damage on public property implies some liability, i.e. denies no liability, this can only mean half liability, since full liability is excluded by Ex. 22:4. This seemed so obvious to the Gemara that it did not even see any necessity to say it out loud.

As we have seen, according to the rabbis, based on Biblical practice, the variable “liability” allows in the present context for only three possible values; namely, no liability, half liability and full liability. Therefore, an indefinite amount of liability, i.e. some liability, which is the negation of no liability, means “half or full” liability. Therefore, to say “there is liability,” meaning some liability, is not as open a statement as it might seem – it allows for only two possibilities, viz. half or full liability. So, if one of these is known to be false (in this case, with reference to the Torah), the other must be true. The latter argument is a disjunctive apodosis: “either this or that, but not this, therefore that.”

Note well that the Gemara here proposes an alternative judgment on damage by tooth & foot on public property to that previously accepted (in the debate between R. Tarfon and the Sages). Previously, the Mishna and the Gemara interpreted Ex. 22:4 (“If a man… shall let his beast loose, and it feed in another man’s field, etc.”), which imposes full liability for tooth & foot damage on specifically private grounds, as implying that there is no liability for tooth & foot damage on public grounds. Here, the Gemara (logically enough) proposes an alternative reading for the latter case, such that “not full” is taken to mean “half” instead of the more extreme “nil,” and it backs up this moderate reading by reasoning that so concludes.

Thus, the Gemara’s use of vague language in its first argument was not some subterfuge relying on half-truths; it was just intended as a shortcut to a result that was in any case logically inevitable. The Gemara achieved its objective here, which was to establish that Ex. 22:4, which imposes full liability for tooth & foot damage on private grounds, need not be taken to imply (as it was in the Mishna) that there is no liability for tooth & foot damage on public grounds; for the alternative of half liability is logically equally cogent. That the Gemara was consciously doing this is evident from its statement: “It was only half payment that we were arguing for!” At worst, the Gemara can be criticized for being too laconic; but its reasoning is sound.

Second experiment. The Gemara states: “But should we not let Tooth and Foot doing damage on the plaintiff’s premises involve the liability for half damages only because of the following a fortiori:

If in the case of Horn, where there is liability for damage done even on public ground, there is yet no more than half payment for damage done on the plaintiff’s premises,

does it not follow that in the case of Tooth and Foot, where there is exemption for damage done on public ground, the liability regarding damage done on the plaintiff’s premises should be for half compensation [only]?[40]

— Scripture says: He shall make restitution, meaning full compensation.

We should here again at the outset note that the Gemara’s argument uses as a premise the earlier conclusion of the Mishna Sages (via their dayo objections to R. Tarfon’s claims) that for damage by horn on private property the ox owner’s liability is half. The other two premises are derived from the Torah as follows: one directly, from Ex. 21:35; and the other indirectly, by extreme inversion of Ex. 22:4 (by which I mean that “not full” is here taken to mean “nil” as in the Mishna, instead of “half” as proposed in the preceding experimental argument of the Gemara). The conclusion concerns damage by tooth & foot on private property. The Gemara demonstrates that a conclusion of half liability, contrary to the full liability given in Ex. 22:4, would follow from the said premises.

Expressed as a set of brief if-then statements, this Gemara argument looks as follows. Note that the first two have in common the factor of public property.

If horn and public, then some liability (from Ex. 21:35).

If tooth & foot and public, then no liability (by extreme inversion of Ex. 22:4).

If horn and private, then only half liability (ruling of the Mishna Sages).

If tooth & foot and private, then [only] half liability (putative conclusion, contrary to Ex. 22:4).

This can be expressed in analogical form, as follows. Note that I here use the term “exemption” in the sense of “freedom of liability,” allowing for degrees of zero, half and total exemption; the term is thus intended as the reverse of the range of “liability.”

Just as, in the public domain, damage by tooth & foot implies more legal exemption than damage by horn, since the former implies no liability and the latter some liability (note that although we can infer from Ex. 21:35 the amount to be half, the Gemara here deliberately avoids specifying it in its premise).

Likewise, in the private domain, damage by tooth & foot implies more legal exemption than damage by horn; whence given that the latter implies only half liability, then the former implies only half liability (contrary to Ex. 22:4, which imposes full liability for this).



We can represent the same argument in purely a fortiori form, as follows. Note the negative polarity of the middle term (R) used; this is necessary to ensure that tooth & foot damage emerge as the major term (P) and horn damage as the minor term (Q). The resulting argument is thus minor to major, positive antecedental.

Tooth & foot damage (P) is more unimportant (R) than horn damage (Q) [as we infer by extrapolation from their liabilities for damage in the public domain (respectively none and some) to all domains, including the private].

Horn damage in the private domain (Q) is unimportant (R) enough to imply only half liability (S).

Therefore, tooth & foot damage in the private domain (P) is unimportant (R) enough to imply only half liability (S) (contrary to Ex. 22:4, which imposes full liability for this).



The Gemara is thus justified in describing its argument here as qal vachomer (מקו), although again this should be understood to refer to purely a fortiori argument rather than a crescendo. We see clearly from the a fortiori formulation that the major premise is produced by a generalization, from the particular case of public property to all property, and its application to the particular case of private property. On this basis, the minor premise about half liability for horn leads to the conclusion about half liability for tooth & foot.

Thus, whether we reason analogically or purely a fortiori, we obtain a conclusion contrary to Scripture. Since the processes used are faultless, what this means is that one or more of the premises must be wrong. In order to try and understand where the problem lies, let us look again at the Gemara’s formulation. The first question to ask (in view of what we learned in the previous case) is why does the Gemara say vaguely “there is liability” (חייבת) for damage by horn in the public domain, when it is known from Ex. 21:35 that the amount of liability is precisely half? Looking at the major premise of the above a fortiori argument, which is generalized from this information, it is clear that it would have made no difference to it if the Gemara had specified half liability. The argument by analogy would similarly be unaffected. So there seems to be no reason for the Gemara not to have said half[41].

Another question is why does the Gemara find it necessary to say “no more than” (אלא) half regarding the liability for damage by horn on private property? Until now, “half” has always meant precisely half, without need to specify that only half is intended. If more than half liability was possibly included in the term half, the meaning of it would have been “half or full,” and this could be stated as before as indefinite “liability.” Perhaps the answer is that if the liability for damage by horn on private property had been full, as R. Tarfon advocated, then the conclusion here would be full liability for damage by tooth and foot on private property. So the Gemara is specifying “no more than half” merely to indicate that it is abiding by the ruling of the Mishna Sages, and not adopting the contrary opinion of R. Tarfon.

In fact, we could represent almost the same argument in a crescendo form, as follows. Note the similarities to the preceding purely a fortiori formulation, but also the totally different conclusion. Instead of half liability, the conclusion here is no liability. But the effect is the same, in that this is contrary to Ex. 22:4.

Tooth & foot damage (P) is more unimportant (R) than horn damage (Q) [as we infer by extrapolation from their liabilities for damage in the public domain (respectively none and some (or more precisely half)) to all domains, including the private].

Horn damage in the private domain (Q) is unimportant (R) enough to imply half liability (S).

The payment due (S) is ‘proportional’ to the degree of legal liability (R).

Therefore, tooth & foot damage in the private domain (P) is unimportant (R) enough to imply no liability (S) (contrary to Ex. 22:4, which imposes full liability for this).



If, in view of the conflict of this conclusion with Ex. 22:4, we interdicted the premise about ‘proportionality’ by means of a dayo objection of type I, we would obtain the same conclusion as the pure a fortiori argument above; namely, half liability. This would of course still leave us with a conclusion contrary to the Scriptural given of Ex. 22:4. Although the Gemara originally does not express this conclusion, however obtained, as “only half,” it is interesting to note that the translator does add on the qualification of exclusion. This is no doubt to exclude “full” liability (rather than to exclude “no” liability), because this is the crux of the issue in this Gemara argument.

Now, this conclusion of half (i.e. not full) liability is especially troubling because the premises that give rise to it were previously regarded as quite acceptable. The major premise is based on Ex. 21:35 (whether we read it as half liability or more vaguely as some) and on the extreme inversion of Ex. 22:4 (i.e. reading not-full as nil, to the exclusion of half) taken for granted by all participants in the Mishna. And the minor premise is the ruling of the Sages in the Mishna, which is in any case implied in Ex. 21:35 (since this verse does not make an explicit distinction between public and private property). How then can these givens result in a conclusion contrary to Scripture, i.e. to Ex. 22:4? This is the difficulty.

Obviously, the problem must lie with the major premise of the a fortiori argument (whether non-proportional or proportional). The extrapolation of “Tooth & foot damage is more unimportant than horn damage” from public property to private property has to be interdicted by a dayo objection of type II, so as to avoid the antinomic conclusion. This could be considered as the intent of the final statement “Scripture says: He shall make restitution, meaning full compensation,” although there is no explicit mention of dayo here. The Gemara is effectively saying: the conclusion cannot be right, therefore block it from happening. This is regular reductio ad absurdum reasoning.

We could also, by the way, obtain the conclusion of no liability by purely a fortiori argument (instead of a crescendo, as just shown), by imitating the Mishna’s R. Tarfon and using another direction of generalization, as shown next. First, let us reshuffle the initial if-then statements, so that the ones we use to form our major premise are both about horn damage, as follows:

If horn and public, then half liability (Ex. 21:35).

If horn and private, then half liability (ruling of the Mishna Sages).

If tooth & foot and public, then no liability (by extreme inversion of Ex. 22:4).

If tooth & foot and private, then no liability (putative conclusion, contrary to Ex. 22:4).

Next, let us formulate the argument in analogical form, keeping to the language of exemption for symmetry with the previous formulation, as follows:

Just as, in the case of horn, damage in the private domain implies as much legal exemption as in the public domain (since both imply half liability):

So, in the case of tooth & foot, damage in the private domain implies as much legal exemption as damage in the public domain; whence given the latter implies no liability, then the former implies no liability (contrary to Ex. 22:4, which imposes full liability for this).



Lastly, we formulate the argument as a purely a fortiori one, of positive antecedental form (minor to major), as follows:

Damage in the private domain (P) is as unimportant (R) as damage in the public domain (Q) [as we infer by extrapolation from horn damage (where liability is half in both domains) to all causes of damage, including tooth & foot].

Tooth & foot damage in the public domain (Q) is unimportant (R) enough to imply no liability (S).

Therefore, tooth & foot damage in the private domain (P) is unimportant (R) enough to imply no liability (S) (contrary to Ex. 22:4, which imposes full liability for this).



This argument seems more solid than the preceding a crescendo argument because it argues from no liability to no liability, rather than from half to none. So it cannot be prevented by means of a dayo objection of type I. And yet its conclusion is the same, viz. no liability. Which poses a problem, since it is inconsistent with the Scriptural imposition of full liability (in Ex. 22:4). Here, then, we must resort to a dayo objection of type II, interdicting the generalization that led to the major premise. We might then be tempted to accept the next amount of half liability as the final result – but no, this is still contrary to Ex. 22:4, and so must be avoided too.

To sum up, the initial premises used in different ways in the various arguments we considered representing the Gemara’s second experiment cannot readily be rejected, yet they lead to a conclusion contrary to Scripture. To prevent such paradoxical result, we had to again resort to dayo objections of both types. This means that the initial premises are together viable provided we do not indulge in proportional thinking or in generalizations in relation to them. Our room for maneuver with them is severely limited; we must proceed with caution.

Third experiment. The Gemara states: “But should we not [on the other hand] let Horn doing damage on public ground involve no liability at all, because of the following a fortiori:

If in the case of Tooth and Foot, where the payment for damage done on the plaintiff’s premises is in full there is exemption for damage done on public ground.

does it not follow that, in the case of Horn, where the payment for damage done on the plaintiff’s premises is [only][42] half, there should be exemption for damage done on public ground?

— Said R. Johanan: Scripture says. [And the dead also] they shall divide, to emphasise that in respect of half payment there is no distinction between public ground and private premises.

We can here again at the outset note that the Gemara’s argument uses as a premise the earlier conclusion of the Mishna Sages (via their dayo objections to R. Tarfon’s claims) that for damage by horn on private property the ox owner’s liability is half. The other two premises are derived from the Torah as follows: one directly, from Ex. 22:4; and the other indirectly, by extreme inversion of Ex. 22:4 (by which I mean that “not full” is here taken to mean “nil” as in the Mishna, instead of “half” as proposed in the first experimental argument of the Gemara). The conclusion concerns damage by horn on public property. The Gemara demonstrates that a conclusion of no liability, contrary to the half liability given in Ex. 21:35, would follow from the said premises.

Expressed as a set of brief if-then statements, this Gemara argument looks as follows. Note that the first two have in common the factor of private property.

If tooth & foot and private, then full liability (Ex. 22:4).

If horn and private, then [only] half liability (ruling of the Mishna Sages).

If tooth & foot and public, then no liability (by extreme inversion of Ex. 22:4).

If horn and public, then no liability (putative conclusion, contrary to Ex. 21:35).

This can be expressed in analogical form, as follows. Note that I here use the term “exemption” in the sense of “freedom of liability,” allowing for degrees of zero, half and total exemption; the term is thus intended as the reverse of the range of “liability.”

Just as, in the private domain, damage by horn implies more legal exemption than damage by tooth & foot, since the former implies [only] half liability and the latter full liability.

Likewise, in the public domain, damage by horn implies more legal exemption than damage by tooth & foot; whence given that the latter implies no liability, then the former implies no liability (contrary to Ex. 21:35, which imposes half liability).



We can represent the same argument in purely a fortiori form, as follows. Note the negative polarity of the middle term (R) used; this is necessary to ensure that horn damage emerge as the major term (P) and tooth & foot damage as the minor term (Q). The resulting argument is thus minor to major, positive antecedental.

Horn damage (P) is more unimportant (R) than tooth & foot damage (Q) [as we infer by extrapolation from private domain damage (for which the liabilities are half and full respectively) to all domains, including the public].

Tooth & foot damage in the public domain (Q) is unimportant (R) enough to imply no liability (S).

Therefore, horn damage in the public domain (P) is unimportant (R) enough to imply no liability (S) (contrary to Ex. 21:35, which imposes half liability).



The Gemara is thus justified in describing its argument here as qal vachomer (מקו), although again this should be understood to refer to purely a fortiori argument rather than a crescendo. We see clearly from the a fortiori formulation that the major premise is produced by a generalization, from the particular case of private property to all property, and its application to the particular case of public property. On this basis, the minor premise about no liability for tooth & foot leads to the conclusion about no liability for horn.

No ‘proportionality’ can be presumed here, for the simple reason that the minor premise and conclusion are already an extreme value (namely, no liability). Thus, an a crescendo argument with the same terms would be identical with the above purely a fortiori argument.

Manifestly, whether we reason analogically or purely a fortiori, we obtain a conclusion contrary to Scripture. Since the processes used are faultless, what this means is that one or more of the premises must be wrong. Examining the Gemara’s formulation, we see that in the present case, unlike the preceding two, there is no ambiguous language. The word exemption (פטורה) is clearly intended here, in both its occurrences, in the sense of full exemption, i.e. zero liability.

It is noteworthy that, although the Gemara originally does not express the liability for damage by horn as “only half,” the translator adds on the qualification of exclusion. But this is no doubt simply to exclude the “full” liability here due according to the dissenting opinion of R. Tarfon; and it does not seriously affect the argument, since if full were adopted instead of half, the major premise would become egalitarian, but the minor premise and conclusion would remain the same.

Now, this conclusion of no liability (instead of half) is obviously problematic, since the premises that give rise to it were previously regarded as quite acceptable. The major premise is based on Ex. 22:4 and on the ruling of the Sages in the Mishna, which is in any case implied in Ex. 21:35 (since this verse does not make an explicit distinction between public and private property, as R. Johanan reminds us[43]). And the minor premise is based on the extreme inversion of Ex. 22:4 (i.e. reading not-full as nil, to the exclusion of half) taken for granted by all participants in the Mishna. How then can these givens result in a conclusion contrary to Scripture, i.e. to Ex. 21:35? This is the difficulty.

Obviously, the problem must lie with the major premise of the a fortiori argument. The extrapolation of “Horn damage is more unimportant than tooth & foot damage” from private property to public property has to be interdicted by a dayo objection of type II, so as to avoid the antinomic conclusion. This could be considered as the intent of the final statement concerning damage by horn that “in respect of half payment there is no distinction between public ground and private premises,” although there is no explicit mention of dayo here. The Gemara is effectively saying: the conclusion cannot be right, therefore block it from happening. This is regular reductio ad absurdum reasoning.

Our next obvious move would be to investigate if a conclusion consistent with Scripture would be obtained by imitating the Mishna’s R. Tarfon, and judiciously reshuffling the given information so as to attempt another direction of generalization. This would proceed as follows. First, we reshuffle the initial if-then statements, so that the ones we use to form our major premise are both about tooth & foot damage, as follows:

If tooth & foot and private, then full liability (Ex. 22:4).

If tooth & foot and public, then no liability (by extreme inversion of Ex. 22:4).

If horn and private, then [only] half liability (ruling of the Mishna Sages).

If horn and public, then [only] half liability (conclusion in accord with Ex. 21:35).

Next, we formulate the argument in analogical form, keeping to the language of exemption for symmetry with the previous formulation, as follows:

Just as, in the case of tooth & foot, damage in the public domain implies more legal exemption than in the private domain (since these respectively imply no and full liability):

So, in the case of horn, damage in the public domain implies more legal exemption than damage in the private domain; whence given the latter implies only half liability, then the former implies only half liability (in accord with Ex. 21:35).



Lastly, we formulate the argument as a purely a fortiori one, of positive antecedental form (minor to major), as follows:

Damage in the public domain (P) is more unimportant (R) than damage in the private domain (Q) [as we infer by extrapolation from tooth & foot damage (for which liability is respectively nil and full) to all causes of damage, including horn].

Horn damage in the private domain (Q) is unimportant (R) enough to imply only half liability (S).

Therefore, horn damage in the public domain (P) is unimportant (R) enough to imply only half liability (S) (in accord with Ex. 21:35).



However, before we can adopt this purely a fortiori argument we must look into the corresponding a crescendo argument. The latter is as follows:

Damage in the public domain (P) is more unimportant (R) than damage in the private domain (Q) [as we infer by extrapolation from tooth & foot damage (for which liability is respectively nil and full) to all causes of damage, including horn].

Horn damage in the private domain (Q) is unimportant (R) enough to imply half liability (S).

The payment due (S) is ‘proportional’ to the degree of legal liability (R).

Therefore, horn damage in the public domain (P) is unimportant (R) enough to imply no liability (S) (contrary to Ex. 21:35, which imposes half liability).



Evidently, arguing a crescendo with these premise results in the undesirable conclusion of no liability for horn damage in the public domain, which is contrary to Scripture (Ex. 21:35). This being the case, such a crescendo argument has to be interdicted by means of a dayo objection of type I. So doing, we return to the purely a fortiori argument formulated just before, which yields the conclusion of half liability. Since the latter conclusion is consistent with Scripture (Ex. 31.35), we have no need to interdict it by means of a dayo objection of type II. We can therefore adopt the said a fortiori argument as a viable alternative to the third one proposed by the Gemara, which yielded an unacceptable conclusion.

From this we see that, while the Gemara’s third experiment is in many ways similar to its second, they are ultimately quite different, in that while the second experiment leaves us without a viable a fortiori counter-argument, the third one does have a viable a fortiori counter-argument. It is surprising that the Gemara did not remark on this significant difference, but remained content with simply listing two arguments with conclusions inconsistent with Scriptural givens.

To sum up. The Gemara’s three experimental arguments have in common as a premise the conclusion of the Sages in the Mishna that damage by horn in the private domain implies half liability. The arguments then seek to determine what conclusion can be drawn from that constant premise about the other situations, viz. tooth & foot damage in the public and private domains, and horn damage in the public domain, respectively. The purpose of the exercise is apparently to compare such conclusions to, respectively, an assumption in the Mishna (viz. that tooth & foot damage on public property implies no liability, based on extreme inversion of Ex. 22:4) and to certain Scriptural givens (viz. Ex. 22:4, which imposes full liability for tooth & foot damage on private property, and Ex. 21:35, which imposes half liability for horn damage on public property).

The Gemara’s logical virtuosity in proposing these three arguments is rather impressive, considering its lack of formal tools. Although the above proposed explicit logical analyses of the three arguments are absent in the Gemara, similar analyses may be reasonably be supposed to have consciously or subconsciously colored the Gemara’s thinking, for otherwise it would be difficult to explain its intent in presenting these arguments. Note in particular that though the dayo principle is nowhere here mentioned by the Gemara, both versions of it are very present in the background of its discourse.[44]

7. Assessment of the Talmud’s logic

We have in the preceding pages examined in great detail, using up to date methods of formal logic, the a fortiori reasoning of both the Mishna and the Gemara, or at least their reasoning in the immediate vicinity of the present sugya[45] (i.e. mBQ 2:5 and bBQ 25a). We judged these texts on their own merits, note well, and not through the prism of later commentaries. Our general conclusion may well be that both the earlier and later Talmudic sages, the Tannaim and the Amoraim, were amazingly powerful logic practitioners, even if they were not great theoreticians. Judging by the Talmudic material we have looked at here, their reasoning seems on the whole sound, even if too often much is left unstated.

What is amazing is precisely that, albeit the brevity of their statements, the people involved were able to reason with such accuracy. I am amazed because, with my pedestrian mind, without reference to formal methods and without full exposition of all implicit discourse, I would be unable to arrive at similar results with equal aplomb. Nevertheless, it must be said and admitted that self-assurance, however esthetically impressive, is not enough. Logic is not just an art; it is first of all a science. To reason correctly is good; but to know just why one’s reasoning is correct is much better. To reason correctly based only on intuition, i.e. on immediate logical insight, is not as convincing as to do so based on broad theoretical understanding, i.e. on abstract study of the exact conditions for correct reasoning (even if, to be sure, such study is also based on the same faculty of logical insight). In the former case, there is some reliance on luck; in the latter, nothing is left to chance.

Comparing now the logic in the Mishna to that in the Gemara, certain trends are evident. The Mishna’s thinking is more straightforward; the Gemara’s thinking is more tortuous. In the Mishna, R. Tarfon puts forward an argument in support of his contention that the legal liability for damage by an ox on private property ought to be full compensation. This argument is not accepted by his colleagues, the Sages, apparently because it relies on proportionality. R. Tarfon then very skillfully proposes an alternative argument, which is not open to such objection. The Sages nevertheless reject the latter argument, apparently by resorting to another kind of objection.

R. Tarfon’s two arguments are traditionally presumed to be qal vachomer, i.e. a fortiori arguments, although just what that means (besides the descriptive name) is nowhere defined. In fact, looking at these arguments very objectively, they could be interpreted as arguments by analogy or more precisely as arguments pro rata, or as arguments a crescendo (i.e. proportional a fortiori) or as purely a fortiori arguments. Moreover, there is no attempt to theoretically validate these arguments. But in any event, they are intuitively quite reasonable; and it seems from the text that it is on this logical basis that R. Tarfon advocates them.

The Sages’ objections, labeled dayo (from their opening word, which means “it is enough”) are not likewise justified by any theoretical discussion. What is clear after our detailed analysis is that they are not essentially logical objections; they are not indicative of breaches of deductive logic, though they might be postulated to signify some inductive restraint. They should rather be viewed as arbitrary decisions (I here use the term ‘arbitrary’ non-pejoratively, in the sense of ‘resorting to arbitration’) by the Sages themselves, based on certain ethical considerations. It can reasonably be doubted that the Sages are here evoking some ancient tradition, perhaps a teaching dating back from Sinai, because R. Tarfon, their colleague and equal, evidently does not preemptively take it into consideration in his two arguments.

Turning now to the Gemara, i.e. the later Talmudic commentary on this passage of the Mishna, we find a very different frame of mind. One would expect the Gemara to initiate a thorough theoretical reflection on R. Tarfon’s two lines of reasoning and the difference in the Sages’ dayo objections to them. But no; the Gemara ignores these burning issues and goes off on a tangent, focusing on the relatively not very relevant issue of the distance between R. Tarfon’s and the Sages’ positions. Apparently, the Gemara’s only concern here is whether R. Tarfon knew and agreed with the Sages’ dayo considerations. Obviously, he could not have fully agreed with them, since his conclusions differ from theirs; so the question is how far their views on the dayo principle differ.

In pursuit of the answer to that question, the Gemara engages is a very complicated scenario of its own, according to which R. Tarfon advocated a more conditional dayo principle than the Sages did. Briefly put, it proposes a distinction (which it attributes to R. Tarfon ex post facto) between applications of the dayo principle that “would defeat the purpose of” the qal vachomer and those that “would not defeat” it. In the former case, the ‘proportional’ gain made possible by an a fortiori argument (taken by the Gemara, on the authority of a baraita, to mean a crescendo argument) would be wiped out by dayo, so it should not be applied; whereas in the latter case, it would not by wiped out by dayo, so it may be applied.

In defense of this fanciful scenario, the Gemara proposes different readings of a Torah text, viz. Num. 12:14-15, by R. Tarfon and the Sages. However, both these readings are far removed from the plain meaning of the text, in that they do not take all of it into consideration. Most important, the view attributed by the Gemara to R. Tarfon assumes an a fortiori argument to be intended in the text while discarding the verses that would justify such assumption! It thus mendaciously infers an a crescendo conclusion of fourteen days ex nihilo, instead of with reference to the textual given of seven days. This means that the Gemara’s whole idea, of a distinction between applications of the dayo principle that “would defeat the purpose of” the qal vachomer and those that “would not defeat” it, is an outright deception. The bottom line is that the Gemara in fact fails to achieve its stated goal of harmonizing the opinions of the Mishna contestants.

Now, this is a bit of a shock, but not too astonishing. Anyone who has studied the Gemara to any extent can see for himself that its thinking, though based on the Mishna to some extent, is often more convoluted and open to doubt. Of course, more fundamentalist readers would never agree with such an assessment, but instead insist that in such cases the Gemara has intellectual intentions and ways too sublime for us ornery folk to grasp. But we, while making no claim to infallibility or omniscience, do claim to be honest and lucid, and stand by what is evident to the senses and to reason. In the present case, the Gemara’s ideas must obviously not be confused with the discourses found in the Mishna. With regard to this, the following general comment of Louis Jacobs in Rabbinic Thought in the Talmud (pp. 17-18) is apropos:

“A much discussed question is whether the interpretations of the Mishnah found in the Gemara are really a reading of ideas into the Mishnah or whether they are authentic accounts of what the Mishnah itself intended. Now students of the Mishnah in the Middle Ages noted that some, at least, of the Gemara’s interpretations of the Mishnah are so far-fetched and artificial that they cannot possibly be accepted as real interpretations of what the Mishnah intends to say, which is why Maimonides and other early commentators were prepared to disregard the Gemara to interpret the Mishnah on its own terms. To conclude from this that the Gemara has, at times, ‘misunderstood’ the Mishnah is precarious. It is possible that the Gemara, at times, consciously departs from the plain meaning of the Mishnah in order to produce its own original work….”

There is no harm, in our view, in producing original work, provided it is openly acknowledged as such. Unfortunately, the traditionalist’s way of thinking is that what he reads into a text must have been intended in the original; to him, interpretation is a sort of deduction. This is applicable at all levels – from the Mishna reading meanings into the Torah, to the Gemara reading them into the Mishna, to later commentators reading them into the Talmud[46]. And this applies to both halakhic and haggadic material. What is sorely needed to cure this serious intellectual malady is to understand the inductive nature of interpretation. An interpretation is a theory designed to fit the ‘facts’ that the given text constitutes. Its logical status is that of a hypothesis, which may and probably does have competing hypotheses. Rarely is an explanation the only conceivable hypothesis, though this happens occasionally. Therefore, a reading should always be acknowledged to be one possible interpretation, even if it fits the given data.

But in the case under consideration, as we have definitely shown, the Gemara’s proposed interpretation of the Mishna simply does not convince. It is not a credible theory, because it is built on illusion, on make-believe. Furthermore, the Gemara does not demonstrate its having noticed and understood the differences between R. Tarfon’s two arguments and between the Sages’ two dayo objections to them. That later commentators have projected such understanding into the Gemara does not prove that the Gemara in fact had it, only at best that it might have. Such ex post facto attribution of knowledge to the Gemara is only evidence of the faith later commentators had in it. The Gemara itself does not explicitly remark on these crucial issues, nor even implicitly suggest them.

It is interesting to note that, whereas the Mishna participants are involved in a purely legal debate, without stepping aside and reflecting on the methodological issues it implies, the Gemara does, in an attempt to clarify the primary, legal discussion, initiate a secondary, more methodological reflection. The latter discourse is intended as an accessory to the former, in that the legal conclusions that might be drawn depend on the methodological lessons learned. Thus, we can say that there is in the Gemara a sugya within a sugya, or there are two intertwined layers of discussion – the main one being legalistic in content, whereas the accessory one is methodological in content. The problem is that, although the Gemara could have used this opportunity to develop a deep reflection on the methodological issues involved, it disappointingly engaged in a very tangential and artificial discourse, driven by quite ideological considerations.

The present work being a treatise on logic, with an emphasis on a fortiori logic, our concern is naturally with the methodological topic of conversation; we are not really interested in the legal topic except possibly as an example. I personally have no legal axe to grind; I am not out to modify or overturn any halakhah. I certainly have no desire to put down anyone, either. Our interest in this research is relatively abstract, and certainly impartial. Our present study is aimed at logic theory and history; it is not essentially Talmudic in orientation, in contradistinction to the rabbis, whose main interest is always legal rather than logical. Nevertheless, we had to consider the legal debate in some detail, since it houses information we needed for historical purposes and to empirically judge the level of rabbinical understanding of a fortiori reasoning.

The author of the Gemara commentary we have studied is obviously someone with an intelligent, imaginative and logically sharp mind. But it is not an entirely scientific mind, which frankly considers all alternatives, lays out all the pros and cons, and judges the matter fairly in accord with objective standards. It is an authoritarian mind, which therefore functions to some extent manipulatively. The Gemara’s author does not derive a conclusion from given premises in an unbiased manner; he starts with a desired conclusion and proceeds to give the impression of having proven it by intricate argument. He is satisfied with the result, even though he in fact did not prove it, either because he fools himself or because he assumes no one would notice the logical trickery involved in his argument and dare cry foul. In the latter event, he relies on the psychology denounced in H. C. Anderson’s The Emperor’s New Clothes.

What is evident looking at Baba Qama 25a is that the Mishna’s narrative and that of the Gemara are quite distinct. The Gemara presents itself as a mere conduit, authoritatively clarifying and explaining the Mishna – but it speaks for itself alone. There is no evidence that it truly represents the views of R. Tarfon and the Sages. When the Gemara speaks in their names, it is just telling us what it thinks they said or meant. The thesis the Gemara presents must be treated as just a hypothesis, even as a mere speculation, since there is no way to establish its historicity. The dialogues it puts forward are imaginary. Its argument is rhetorical and not logical.

The Gemara’s doctrinal goal seems to be to reconcile the seemingly antagonistic positions of the participants in the Mishna, i.e. R. Tarfon and the Sages. This is in accord, we may remark in passing, with the general rabbinical dogma that everything a Talmudic rabbi (or indeed an important later rabbi) says is essentially right, even if it seems to conflict with what others say. This doctrine that “the Torah has seventy facets” presumably arose ex post facto, first implicitly and then explicitly, perhaps somewhere mid-course in the Talmud, maybe only in the Gemara (when exactly, I do not know[47]). In any case, it clearly plays an active role in the present portion of the Gemara, and this is important to keep in mind.

To repeat, the Gemara’s treatment of the Mishna is quite superficial, failing to spot and take into consideration important details in the proof-text. The Gemara cheerfully refers to the Miriam story as its model for understanding the Mishna, failing to notice that though this passage of the Torah can be used to throw light on the first argument of the Mishna, it is useless with regard to the second. Moreover, the Gemara not having even tried to make a preparatory theoretical analysis of qal vachomer, fails to realize the different possibilities of interpretation inherent in the Mishna. It takes for granted without reflection that the qal vachomer inferences in the Mishna are all ‘proportional’, and does not see the possibility in it of purely a fortiori arguments or even non a fortiori arguments.

The Gemara then embarks on a quite abstruse theory of qal vachomer, which it attributes to R. Tarfon, according to which (in positive subjectal a fortiori argument) a conclusion can only be drawn from a minor premise with the same subject. The Gemara does not notice that this imagined narrative is not in accord with what is explicitly given in the Mishna, let alone realize that it has no basis in formal logic. It does this to justify making a distinction between the argument implied in the Miriam story and the argument (it only perceives one, the first) given in the Mishna, so as to explain the difference of opinion there between R. Tarfon and the Sages in relation to a presumed dayo principle. Furthermore, in attempting to depict this theory, the Gemara has R. Tarfon drawing an alleged qal vachomer conclusion from no premises at all when he applies it to the Miriam story.

The Gemara’s general idea that the Miriam story contains an a crescendo conclusion of fourteen days restricted by a dayo principle to seven days (rather than a straight a fortiori argument with a conclusion of seven days) is still not unthinkable, note well. Even though the Gemara does not admit v. 14a (about offending one’s father) as a premise of this argument, and takes v. 14b (about seven days quarantine) as its final conclusion, the argument can be imagined as occurring in between these verses. It must however be stressed that, contrary to what the Gemara claims, there is no actual concrete hint of this scenario in the Biblical text. Even a purely (i.e. non-proportional) a fortiori reading is open to debate; all the more so an a crescendo one. Consequently, any claim that the passage points to a dayo principle is also open to debate.

The a fortiori reading is not inevitable; but it is a reasonable assumption, provided it is made to explain the connection between the first and second part of v. 14. Note well that the a fortiori argument is not just used to infer a number of days, but especially the punishment of isolation away from the community. The seven days prescribed are only a qualification of this predicate, serving to quantify the penalty; they are not the main issue of the argument. Thus, the final exchange in the Gemara between R. Tarfon and the Sages, regarding where in the Biblical text the alleged two sets of “seven days” come into play focuses on a side issue, diverting attention from the main one. The Gemara gives the impression that the qal vachomer is all about numbers of days; this is misleading.

Our wisest course is to blame the Gemara alone for these various rationalizations. The Gemara is plainly indulging in sophistry, masquerading as rational discourse. Its narrative is an obvious and absurd invention, which has little to do with the Mishna’s. If we accept the scenario the Gemara advocates, we would be unfairly imputing the errors of reasoning it commits to R. Tarfon and the Sages. We cannot justify lumping together the players in the Mishna with those perhaps two or three centuries later (and some five hundred miles away) in the Gemara, just out of some ideological desire to make them appear to all speak with one voice. It is better to blame the author of this Gemara in particular for them than to insist they are true and embarrass everyone else! These very critical remarks of mine are sure to revolt traditional Talmudists, but they are unavoidable.[48]

I am, of course, well aware that such statements undermine rabbinical authority. We can say, having found such casuistry, that the rabbis are not always right, i.e. that their logic is not infallible. But I knew this already, having uncovered much problematic reasoning by them in the course of my earlier research on rabbinic hermeneutics, as detailed in my Judaic Logic[49]. Many more instances are uncovered in the present work; see for examples the fallacies discussed in chapters 3.4, 9.7, and especially 18.2.

The issue is how often do they reason incorrectly? This question cannot be answered offhand but requires systematic and thorough research throughout Talmudic literature – by competent researchers, I might add (for someone who does not know logic much better than them cannot judge theirs). If errors are only occasional, that is surely not too serious, since we are all human beings with limitations; if they are very frequent, that is certainly quite serious, since some inexcusable negligence is involved. It might be possible to lay the blame for all or most errors found on some specific rabbis. This would somewhat improve the logical credibility of the rabbis collectively, although we could still wonder why the errors were not spotted and corrected by other rabbis.

In any case, our approach as logicians must be objective and impartial, and not swayed by any imagined or actual threat of hostility and rejection. From a metaphysical point of view, if God is the ultimate reality of the world we experience, and the meaning of human life is to tend towards Him, then truth is a paramount value and honesty is an indispensable virtue. There is no rational excuse for evading or stonewalling, let alone opposing and denigrating, just criticism. It would be unrealistic to expect utter perfection from any human being, even if he is an important rabbi. When we come across logical faults, we should not deny them, but humbly admit them and try to correct them. While some might consider criticism of rabbinical arguments as cause for condemnation, we should rather view such events as welcome opportunities for improvement.

By this I mean that once we realize and admit that Talmudic and more broadly rabbinic logic is not inerrant – but sometimes debatable, contrived or erroneous – we open a safe door to halakhic review and revision. This of course cannot be taken as a blanket license for general change in Judaism as convenient; but there may be circumscribed opportunities for evolution based on ad hoc logical analysis. For the law must surely be in accord not only with empirical scientific knowledge of nature and history, but also with logic. Just as ignorance of the former is bound to lead to error in law, so is faulty logic also bound to lead to such error.

One of the major rabbinical authorities of modern times, R. Moses Sofer (Germany, 1762-1839) wrote this about logic (higayon): “whoever mixes words of logic with matters of Torah offends against the law of: ‘Thou shalt not plough with an ox and an ass together’ [Deut. 22:10].”[50] But logic is not, as this farfetched statement suggests, something arbitrary that we have a choice about using or not. Mentally, we are of course able disregard it; but intellectually, if we are honest, we cannot do that, because logic is our main means for verifying and certifying the truthfulness and consistency of our judgments. If any verse of the Torah is to be brought to bear in this matter, it is rather this: “Thou shalt not have in thy bag diverse weights” (Deut. 25:13). But there is no need for that; it is obvious.

Apologists for religion reproach secular scientific knowledge of nature and history for varying in time. They suggest that such variation is proof of its unreliability. But this is of course a spurious argument. Scientific knowledge varies because it is essentially inductive, freely and dynamically adapting to new empirical discovery and rational review. This is not a fault or weakness – it is the very virtue of science. The truly scientific view[51] at any point in time is comparatively the best hypothesis human beings as a group have to offer. That it may later change does not make it any the less ‘the best’ at the time concerned. Certainly, it is always better than a static hypothesis based on religious dogma that is out of touch with empirical fact and rational scrutiny.

Browbeating is not a form of proof. Religion must learn to humbly adapt to scientific change. This would certainly not be the end of religion, because religion is a necessary expression and instrument of human spirituality. See how those who lost it suffer, from the lack of direction in their lives. Just as science makes possible the accumulation and transmission of human knowledge of nature and history, so religion makes possible the accumulation and transmission of human knowledge of spirituality. Of course, the latter tends to be more plural than the former, because spirituality allows for many paths. But in any case, whatever the chosen path, empirical science and logic must be taken into consideration to ensure its full truth.

8. The syllogistic Midot

As regards syllogism, it is also naturally found in rabbinic thinking and even within many of their hermeneutic techniques (midot). This is said to contradict the claim of many commentators that none of the rabbinical hermeneutic techniques are syllogistic.

This position, for instance, is to some extent found in Louis Jacobs’ treatment, insofar as the only rabbinic argument he sees as syllogistic is the one referred to as ha-kol. In his Studies (1961), he says: “There is a form of Talmudic reasoning which has no connection with the qal wa-chomer but bears a remarkable affinity to the Syllogism;” and he goes further in a footnote, saying: “the ‘ha-kol’ formula… is identical with the Syllogism,” and giving as example the following argument implied in Avot 6:3:

“He who learns from his fellow has to pay him honour;

I have learned from my fellow;

Therefore I am obliged to pay him honour.”

Michael Avraham, for his part, asserts categorically (in the English abstract of a 1992 Hebrew paper) that none of the 13 principles of R. Ishmael are syllogistic; as he puts it: “the Kal Vachomer – like the rest of the 13 ‘Middot’ – is not a syllogism” (my italics). This opinion is apparently not new, judging by a statement made by Aviram Ravitsky (my italics):

“Maimonides viewed most of the halakhic world as conventional, and this view enabled him to treat the halakhic arguments as dialectical ones, although he did not think that halakhic arguments could be reduced to syllogistic figures.” [52]

But in my Judaic Logic (1995), I show that many of the thirteen midot of R. Ishmael involve syllogistic thought processes. For a start, a fortiori argument is in part based on hypothetical syllogism. Syllogistic reasoning is implicit in the midot dealing with the scope of terms, collectively called klalim uphratim (rules 4-7), insofar as these have to do with subsumption and exclusion of cases in classes[53]. But more to the point, most of the midot dealing with harmonization (specifically the rules 8-11) are clearly syllogistic, so much so that they can be represented and resolved diagrammatically. While my work on a fortiori argument has attracted some attention, my work on these harmonization midot has apparently not been noticed. For this reason, I think it useful to reiterate some of these findings in the present context, to show how a lot of rabbinic thinking is syllogistic.

The first three (actually, four) of the principles of R. Ishmael concerned with harmonization begin with the phrase kol davar shehayah bikhlal veyatsa…, meaning literally “anything which was in a generality and came out…”. Broadly put, in formal terms, these rules are concerned with the following exegetic situation:

Given the three premises (#s 1, 2, 3), common to the four harmonization rules 8a, 8b, 9, 10:

All S1 are P1 (common major premise, #1),

and All S2 are P2 (common minor premise, #2),

where All S2 are S1, but not all S1 are S2 (common subjectal premise, #3),

and the fourth premise (#4), as applicable in each of these rules:

P1 and P2 are in some relation f{P1, P2} (d) (distinctive predicatal premise, #4):

§ In rule No. 8a, P2 implies but is not implied by P1; that is:

All P2 are P1, but not all P1 are P2.

§ In rule No. 8b, P1 implies P2 (and P2 may or not imply P1); that is:

All P1 are P2 (whether All P2 are P1 or some P2 are not P1).

§ In rule No. 9, P1 and P2 are otherwise compatible; that is:

Some P1 are P2 and some P1 are not P2; some P2 are P1 and some P2 are not P1.[54]

§ In rule No. 10, P1 and P2 are incompatible; that is:

No P1 is P2 and No P2 is P1.[55]

What, other than the above given, are resulting relations (conclusions)?

Between S1 and P2 (this is the primary issue, #5);

and (secondarily) between S2 and P1, and between S1 and P1, and between S2 and P2.

We can for a start, by means of syllogism, draw the following conclusions, common to all four rules, from the first three premises, without reference to the fourth premise:

· From the minor and subjectal premises, Some S1 are P2 (mood 3/AAI).

· From the major and subjectal premises, All S2 are P1 (mood 1/AAA).

· From the major and subjectal premises, Some P1 are not S2 (mood 3/OAO).

What this means is that, no matter which predicatal premise is used, it cannot logically yield a conclusion incompatible with ‘Some S1 are P2’. The following specifies what can additionally be said in each of the four rules under scrutiny (the sources and discussion of the examples here proposed are given ad loc. in my Judaic Logic):

In rule No. 8a, nothing further about S1 and P2 can be deductively inferred; yet R. Ishmael apparently claims ‘All S1 are P2’ (which is too much). For example: A sorceress (or by extension, a sorcerer) is liable to the death penalty (#1); a male or female medium or necromancer is liable to death by stoning (#2); a male or female medium or necromancer is a sorcerer or sorceress (#3); death by stoning is a species of death penalty (#4); therefore, all sorts of sorcerers or sorceresses are liable to be stoned (#5).

In rule No. 8b, we can syllogistically infer (mood 1/AAA) that ‘All S1 are P2’; yet R. Ishmael apparently claims ‘Some S1 are not P2’ (which is inconsistent). For example: whoever approaches holy offerings while impure is liable to the penalty of excision (#1); anyone who eats peace-offerings while impure is liable to the penalty of excision (#2); peace-offerings are holy offerings (#3); the penalty is the same in both cases, viz. excision (#4); therefore, the consumption of offerings of lesser holiness than peace-offerings is not subject to the penalty of excision (#5).

In rule No. 9, we can syllogistically infer (mood 2/AOO) that ‘Some P2 are not S1’; it is not clear how R. Ishmael’s proposed conclusion here should be presented in formal terms (such lack of clarity being of course a deficiency). I have not found a sufficiently informative example of application of this rule[56].

In rule No. 10, the predicatal premise is logically incompatible with the other three premises, so no syllogistic inference is possible; R. Ishmael apparently resolved the conflict by modifying the major premise to read ‘Some, but not all, S1 are P1’ (which is logically acceptable, though not the only option open to us). For example: the release of a Hebrew slave is subject to a certain set of laws (#1); the release of a daughter sold as maid-servant is subject to another set of laws (#2); a daughter sold as maid-servant is nominally a subcategory of Hebrew slave (#3); yet, the laws for the maid-servant and those for the Hebrew slave in general are very different (#4); therefore, the category of Hebrew slave intended here is in fact not so broad as to subsume such maid-servants (#5).

From these reflections, we learn that at least four of the rules of R. Ishmael (as I have tentatively interpreted them, based on a small number of examples) are syllogistic in form. These four all include at least the syllogism: ‘All S2 are S1 and All S2 are P2, therefore Some S1 are P2’ (3/AAI). Two of the four involve an additional syllogism (of form 1/AAA in rule 8b, and of form 2/AOO in rule 9); one rule involves no additional syllogism (rule 8a); and the fourth rule involves inconsistent premises. It is interesting to note that R. Ishmael’s apparent solutions to these four syllogistic problems are in some way or other deficient. Nevertheless, it does not change the fact that these four rules are essentially of a syllogistic nature.[57]

I have also demonstrated, in an earlier chapter of the present volume, in the section on analogical argument (5.1), the presence of syllogistic thinking in rabbinic analogical arguments, namely in rule 2 (gezerah shavah), rule 3 (binyan av) and rule 12 (meinyano and misofo) in R. Ishmael’s list. These arguments are not solely syllogistic – they involve inductive processes too – but they definitely do include syllogism. These findings are indubitable, and they put to rest once and for all the rather widespread notion in some quarters that the rabbinic hermeneutic principles do not depend on syllogistic reasoning.

Syllogism can, I suspect, be discerned in yet other midot, if we examine them closely enough. I would go much further than that, and assert that these examples drawn from the 13 midot are only the tip of the iceberg. The midot are by far not a full listing of the reasoning processes actually used by the rabbis; it is certain many of their actual reasoning processes are not included in their listings. The listings only bring together certain forms of thought which the rabbis considered worthy of notice and emphasis for some reason. But like all human beings, they used many thought processes unconsciously – including the process of syllogism. It is impossible for anyone to reason without certain basic thought forms; and the syllogism is definitely one of these unavoidable thought forms, since it is required for all mental acts of inclusion or exclusion.

9. Historical questions

There is, I would say, a significant difference between a fortiori use in Talmudic contexts its use in other ancient literature, such as in Platonic or Aristotelian texts[58]. In the latter, it is probably more accurate to speak of a fortiori discourse rather than a fortiori argument, because it is used more as a rhetorical device than as a form of reasoning. The author in such cases could well have rephrased his text in such a way as to pass the same message without using a fortiori language. Whereas, in Talmudic contexts, the use of a fortiori is definitely argumentative; it is necessary to prove something that has a legal impact and that could not be arrived at by other means. So when we speak of a fortiori use in the Talmud, we are referring to something much more serious.

When we speak of Talmudic and rabbinic logic, we must have in mind and look for both explicit theories and implicit practices by people concerned. Theorizing has different levels: just being aware that one is engaged in an argument is one level; the next higher level is awareness that the argument is of a certain peculiar type, and a name is assigned to it (such as ‘a fortiori argument’); the third level consists in attempting to give form to such argument, using symbols in the place of terms; and the highest level is wondering at the argument’s validity and seeking to establish it once and for all. Study of the history of an argument is also theory, though of a more intellectual-cultural sort. As regards practice, it may be far ahead of theory. Theory can improve further practice, but is generally based on prior ‘intuitive’ practice. Therefore any investigation that aims to understand the logic of some group of people or humanity in general must focus strongly on actual practices.

Even if much conscious research has been carried out on Talmudic and rabbinic logic (including hermeneutics), I wager that there is still a lot to discover in this field. We shall never arrive at an accurate, scientific history – or indeed, theory – of Talmudic logic, and in particular of Talmudic a fortiori logic, without a thorough, systematic listing and competent analysis of all the arguments in the Talmud and related texts. Someone has to do this major work some day; or else we shall always be dealing in rough hypotheses based on limited samples.

Take for instance a fortiori argument, which is our object of study here. What we need, for a start, is a table listing all the apparent cases of a fortiori argument. In each case, we should note its location in the Talmud, and who (named or unnamed) is apparently formulating it, so that the best estimate of its date can later to be put forward. We must distinguish the person(s) formulating it from the person(s) commenting upon it in subsequent developments of the Talmud (by which I mean here, the two Talmuds, that of the Land of Israel and the Babylonian, and related contemporary literature).

Each argument must be analyzed, first by classifying it, i.e. identifying which of the eight standard moods it fits into. Is it copulative or implicational? Label the major term (or thesis) P, the minor term (or thesis) Q, the middle term (or thesis) R and the subsidiary term (or thesis) S. Is the argument subjectal (or antecedental) or predicatal (or consequental)? Is it positive or negative? Moreover, is the argument purely a fortiori or a crescendo? Was dayo applied? Very often, some creative work will be necessary, insofar as the a fortiori argument is not entirely explicit. It may, for instance, be necessary to construct an appropriate major premise, and the operative middle term (or thesis) may have to be suggested by the researcher. Obviously, any such contributions to the argument made by the researcher must be noted as such and not confused with the raw data. That is to say, the fact that the original argument is in some way incompletely formulated is a significant detail of the analysis.

Once we have such an exhaustive database of the a fortiori arguments in the Talmud, we can begin to develop a truly scientific account of this argument form in that document. We can say with certainty what moods (if not all) of the argument were known to the named participants and anonymous redactors, and how well they understood them. We can compare the logical skills in this domain of the different players involved. We can find out more precisely what their theoretical understanding of a fortiori forms were and what terminology they used for them. We must not forget that the Talmud is a document built-up over centuries, by hundreds of people. The Talmud is not a monolith, but has many temporal and geographical layers[59]. Therefore, research must also try to trace the development of skills and understanding of a fortiori and other argumentation across time and place.

We can also more accurately compare Talmudic use and knowledge of the a fortiori argument to use and knowledge in surrounding cultures – notably the Greek and Roman as regards the Mishna and the Jerusalem Talmud, and possibly further afield as regards the Gemara of the Babylonian Talmud, since it was developed in Babylon where perhaps some Indian influences might have occurred. This too, of course, has a long timeline. Take for example the distinction between miqal le chomer (from minor to major) and michomer leqal (from major to minor). This distinction is taken for granted today – but it surely has a rich history. Does it appear anywhere in the Talmud, or is it a later discovery? It is found in later rabbinic literature – but the question is when and where did it first appear? Was this an independent Jewish discovery, or can Greek or Roman or other ancient influence, or later on Christian or Moslem influence, be traced? If the distinction is made in the Talmud, just when and by whom, in what context(s)?

This distinction, note well, signifies some level of conceptual analysis of a fortiori reasoning. But it is still relatively vague or equivocal, insofar as ‘minor to major’ can signify either positive subjectal or negative predicatal argument, while ‘major to minor’ can signify either negative subjectal or positive predicatal argument[60]. A question to ask is, therefore: what was the original intent of this distinction – was it meant as a distinction between positive and negative moods of subjectal a fortiori argument, or was it a distinction between positive subjectal and positive predicatal arguments, or was there awareness of all four possibilities, or did it remain vague? Indeed, granting that positive subjectal argument is the most obvious and widespread form, when were negative subjectal and positive and negative predicatal argument forms first realized in Judaic logic (or elsewhere, for that matter)?

And so on. There are evidently many questions worth asking and the answers cannot be settled till we have a thorough database, as already said. It should be noted that today, with the digitalization of most ancient texts well on the way if not already completed, the job is immensely facilitated, since exhaustive searches of different verbal strings are possible in a jiffy and information can be cut and pasted without difficulty! The historical work and the logical analysis involved may or may not be done by the same person(s). The ideal scholar would be a good wide-ranging historical researcher, knowledgeable and at ease in the Talmud and other significant texts in the original languages, and a good logician to boot. These qualities are not necessarily all found in the same person, but a multi-disciplinary team might be constituted by a university. I do hope some people someday realize the need and value of such research and organize a determined effort in that direction.



[1] Likewise, Deut. 13:1.

[2] To tell the truth, I had inexcusably, at the time I wrote Judaic Logic, not actually studied this Talmudic sugya, but instead took for accurate what other commentators said about it. I was at the time much more naïvely trusting than I am today!

[3] This passage is not included in the Genesis Rabbah listing of ten cases of qal vachomer, but is mentioned in Rashi’s commentary. According to Jacobs in his Rabbinic Thought in the Talmud (p. 116) this instance is mentioned in much earlier rabbinic texts: “Avot de-Rabbi Nathan (version B) 44; Gen. Rabbah 4:24 (ed. Theodor-Albeck, p. 225) and the Jerusalem Talmud Sanh. 10:1 (27d).”

[4] This concerns a suspected adulteress. The Mishna goes on, giving examples: “She adorned herself for a transgression; the Holy One, blessed be He, made her repulsive;” etc. And the corresponding Gemara starts with: “R. Joseph said: although the measure has ceased, [the principle] in the measure has not ceased.” I found this passage thanks to Jacobs, who quotes it in his Rabbinic Thought in the Talmud (p. 78, fn. 1). I had previously by chance found this maxim in the Mekhilta de Rabbi Ishmael (Beshallach, 1); but this Midrash is a later document, thought to date from the 3rd century.

[5] In Oriental religions, of course, the ‘measure for measure’ principle is expressed as the ‘law of karma’. This is a more mechanical version than the Judaic principle, which clearly involves Divine intervention and thus distinctively allows for eventual exceptions, i.e. reduced punishment or increased reward out of Divine love.

[6] Actually, there are explanations of these specific numbers in later commentaries, but I won’t go into them here, so as not to complicate matters unduly. (E.g. one explanation refers to the fact that 7 days is the minimum period of quarantine in the event of leprosy, so that another 7 days is the least possible additional period of quarantine.)

[7] I am here quoting the paraphrase of Sforno’s comment given in The Soncino Chumash (ad loc.), not Sforno directly.

[8] It is interesting that, in Shakespeare’s The Merchant of Venice (Act IV, Scene 1), the Italian Jewish protagonist, Shylock, is refused the “pound of flesh” he had contracted for on the basis that he might inadvertently take more than that (namely, some blood with the flesh). So it seems, ironically, that the legal principle Shakespeare appealed to might have been formulated a few decades before him by… an Italian Jew, i.e. Sforno! (Indeed, according to a Wikipedia article, Shakespeare’s play was written in 1596-98, and this and other elements of it are based on a tale by Giovanni Fiorentino called Il Pecorone, published in Milan in 1558. Sforno died in 1550.)

[9] There is of course some tension between what I said a bit higher up, about the dayo principle being “Biblical,” if only in the sense that it advocates strict adherence to Biblical givens etc., and Sforno’s suggestion that “eye for eye” was mitigated to monetary compensation. But, in the latter case, the literal reading of the Biblical law is looked upon as metaphorical and is replaced by a less harsh reading; whereas, in cases of dayo, the literal reading is not discarded, but proportional inferences from it are disallowed, so as to prevent harsher practices. These are two clearly very different treatments of Biblical text by the rabbis.

[10] For my part, I must confess that I originally believed the dayo principle to be a rabbinical statement of the principle of deduction, proposed specifically for qal vachomer only because such argument was for the rabbis the very essence of deductive reasoning. This is essentially the position I took in my Judaic Logic, although I also there considered that proportionality was still possible though a separate act of reasoning (whether deductive or inductive). But now, having realized this more accurate interpretation of the dayo principle, as applicable to any extrapolation attempted on the basis of midah keneged midah on the products or preliminaries of qal vachomer (or any type of reasoning with similar effect), I definitely opt for this latter hypothesis.

[11] These later commentators generously project their insights onto the Gemara; but this is of course anachronism, motivated by their wish to claim a continuity of tradition.

[12] Notice the sequence of events in the following sentence in the Gemara: “nevertheless, by the working of the a fortiori, fourteen days may be suggested: there follows, however, the principle of dayo so that the additional seven days are excluded.” This means that: first, fourteen days are inferred using qal vachomer; and after that (“there follows”), the number of days is reduced by dayo to seven.

[13] It should be said that R. Tarfon’s first argument could conceivably be inhibited by the second type of dayo objection (viz. blocking formation of the major premise by generalization), as well as by the first type (viz. blocking operation of the third premise about proportionality). But this does not seem to be the thrust of the Sages’ rebuttal of the first argument; they seem rather to adopt a purely a fortiori stance in opposition to their colleague’s a crescendo approach.

[14] Or eventually, maybe, duties – viewed as burdens, as earlier explained.

[15] I have read that there are separate collections of baraitot. These would, of course, have to be consulted too to resolve the issue once and for all.

[16] Understanding the Talmud, p. 88-90. Feigenbaum rightly characterizes qal vachomer as “a particular logical structure,” but he introduces the above formula by saying: “it is logical to assume that….” This is a sort of contradiction: if the structure is truly logical, the argument is not a mere assumption, but a thought process that can be validated. Feigenbaum evidently has not attempted to logically validate his formula. He does, however, describe two ways in which the Gemara may “refute” such argument – either by showing that the proposed ruling is found inapplicable in relation to another relatively stringent (or, respectively, relatively lenient) issue, or by showing that the lenient issue is in some respects more stringent (or, respectively, that the stringent issue is in some respects more lenient). But the latter “refutations” are, of course, material rather than formal: they effectively deny the truth of the minor or major premise in a given case, not the validity of the argument properly formulated.

[17] See chapter 4.5 there.

[18] The same can of course be said of the implicational equivalents of those various arguments. Dayo will only apply to positive antecedental or negative consequental a crescendo arguments concerning punishment for illegal acts.

[19] See in the present volume, in the chapter on Moses Mielziner, the section called ‘Concerning the jus talionis’ (13.3). Also see Baba Qama, 83b-84a.

[20] Oxford: OUP, 1995.

[21] In any case, this is not a very good example of the above stated restriction on a fortiori inference, because the conflict here is between an inferred prohibition and a Torah ‘permission’ (presumed merely due to absence of written prohibition, note well), and not between an inferred permission (or exemption) and a written Torah prohibition (or imperative). But, even though I cannot here adduce a fully appropriate example, I think the said restriction does exist and is quite reasonable. Even if I turn out to be wrong, the issue is worth investigating.

[22] This example and its explication are given by R. Schochet in the already cited online video. However, I have not found the Talmudic reference for it (though it is one of the five examples given by Saadia Gaon in his commentary on the 13 midot). Moreover, elsewhere, namely here: www.come-and-hear.com/supplement/so-daat-emet/en_gentiles3.html, it is pointed out that returning a lost animal to a brother is based on Deut. 22:2 – in which case, I do not see the need for a qal vachomer from Ex. 23:4 (unless a ‘friend’ and a ‘brother’ mean different things). Nevertheless, I will not get into a discussion of this concrete issue, nor look for a less controversial example – an illustration of the rabbinic restriction was all that was needed here and this perhaps hypothetical one will suffice.

[23] This is the general point I want to make here. In fact, rabbinic restrictions on use of qal vachomer (and/or the dayo principle) and other hermeneutic principles are far more numerous and intricate than here suggested (indeed, sometimes they seem to me ad hoc, i.e. tailored for the convenience of a particular discussion only). But I do not want to get bogged down in this special field of study. You can find some further details and clarifications in Steinsaltz or Mielziner, for instances.

[24] Francis Bacon, in his The Advancement of Learning, expresses a similar thought: “As in nature, the more you remove yourself from particulars, the greater peril of error you do incur; so much more in divinity, the more you recede from the Scriptures by inferences and consequences, the more weak and dilute are your positions” (2:25:12).

[25] See his chapter 13.

[27] I call the subsidiary term S an “action” to stress that it is something that the people towards whom the law is addressed have a choice to do or not do. No law is possible or meaningful if not addressed to humans with freewill; and no law can be made about something which it is outside their control.

[28] The injunction “must be forbidden” is addressed to the judges who will legislate and implement the law, whereas the law which says that “S is forbidden, etc.” is addressed to the people.

[29] Note that I here call S an action and not-S an inaction merely for convenience – it may be that S is an inaction and not-S is an action. The important thing is that they be contradictories.

[30] This is of course a historical question worth investigating empirically.

[31] Encyclopedia of Torah Thoughts, p. 27, n. 106.

[32] Understanding the Talmud, p. 88.

[33] Introduction to the Talmud, pp. 132-4. “Must restore” is, of course, an imperative, a positive instruction.

[34] I have not here resolved the question as to whether in the Talmud (Mishna and Gemara) the language of purely a fortiori argument is different from that of a crescendo argument. Probably not, but it is worth looking into the matter empirically. I do so with regard to the Mishna in Appendix 2, but only in English translation (not in the original Hebrew).

[35] The Sages opinion is obviously accepted as henceforth binding.

[36] Which is taken to mean to the exclusion of the public domain. Such exclusion is based on davka interpretation of Scripture. That is, what is specified as applicable to private property is taken to include only private property, thus excluding public property. The thinking here is: ‘Otherwise, why specifically mention private property?’ In general, “If A and B, then C” does not formally exclude “If A and not B, then C”; taken together they imply “If A, then C.” However, in the exclusive reading, “If A and B, then C” is taken to imply “If A and not B, then not C.”

[37] Note that I have left out a sentence here, because I do not understand it and do not see its logical significance. This says: “Scripture further says, And they shall divide the money of it [to indicate that this is confined to] ‘the money of it’ [i.e. the goring ox] but does not extend to compensation [for damage caused] by another ox.” What has “another ox” got to do with it?

[38] Note that, to simplify, I here use “is more important” as equivalent to “implies more liability.”

[39] It is perhaps to this implicit a fortiori argument that the Soncino edition refers, when it explains (in a footnote) the Gemara’s conclusion of half liability for tooth & foot damage in the public domain by saying: “On the analogy to Horn where the liability is only for half damages in the case of Tam. The Scriptural text may have been intended to exclude only full compensation.”

[40] I have added the square brackets around this last “only,” because it is not found in the original and therefore seems to be an interpolation by the Soncino edition translators.

[41] Possibly it used vague language here simply to harmonize the language in this experiment with that in the preceding one.

[42] I have added the square brackets around this last “only,” because it is not found in the original and therefore seems to be an interpolation by the Soncino edition translators.

[43] More precisely, R. Johanan, an early authority, interprets the Scriptural verse “[And the dead also] they shall divide,” which is the last sentence of Ex. 21:35, to mean that half liability applies to the public domain as well as to the private domain. Taken literally, of course, this verse does not have exactly that meaning (i.e. another reading is conceivable); but it is reasonable to suppose that Ex. 21:35 as a whole applies to both domains, since neither is explicitly specified or excluded.

[44] It should be noted in passing that all the a fortiori arguments explicitly formulated by the Gemara in the present context are pure; none are a crescendo. This implies that the Gemara does (unconsciously if not consciously) admit that some a fortiori arguments are not a crescendo (unlike the baraita it quotes earlier on, which seems to suggest that a fortiori argument is always a crescendo).

[45] A sugya is a portion of the Talmud dealing with a specific topic.

[46] Not to mention the Bible writer(s) interpreting empirical givens.

[47] The explicit sentence is found in Bamidbar (Numbers) Rabbah 13:15, an 11th or 12th century CE midrash. But the implicit concept is no doubt much earlier. I would guess that it is a viewpoint of the Amoraim with regard to the Tannaim. It is essentially an expression of absolute faith in the tradition as handed down. The philosophic (logical, epistemological, ontological) significance of such a doctrine needs to be reflected on. Can conflicting viewpoints all be true?

[48] It is not unthinkable that the Mishna and Gemara under scrutiny have the same views, but the data at hand phenomenologically does not justify such an extreme judgment. We are duty bound to look at the matter more subtly, and keep track of who said what and when, so that we can pinpoint more precisely who is in error and where, and judge with greater accuracy. This is the scientifically sound approach, and it is also more favorable to the honor of the rabbis. It is preferable by far, and is the policy I have adopted.

[49] See especially chapter 11, on harmonization processes. But in chapter 10, too, there are relevant findings. See for instance my doubts regarding the ‘freedom’ of terms (mufneh) doctrine relative to gezerah shavah; this issue arises in the present Gemara on p. 25b, by the way.

[50] In Hatam Sofer, OH, No. 51. Quoted by Louis Jacobs, in A Tree of Life: Diversity, Flexibility and Creativity in Jewish Law (2nd ed. London: Littman, 2000), p. 8. It is true that earlier authorities like Maimonides or Nachmanides defended the use of logic in Torah related contexts – but they lived in a period when the dangers to religion inherent in strict logical scrutiny were not yet known. R. Sofer was contending with the Haskalah, the modern Jewish Enlightenment critics.

[51] Of course, by scientific, I mean strictly scientific – and not pseudo-scientific pronouncements by scientists (in physics, biology or psychology) about having ‘proved’ the non-existence of God or of soul or of volition or even of consciousness.

[52] See his essay: “Halakhic Arguments as Dialectical Arguments and Exegetical Principles as Aristotelian Topoi in Maimonides’ Philosophy.” In Tarbits, 73 (2004), p. 219.

[53] R. Akiva’s competing principles of ribbui umiut could also be argued to suggest syllogistic reasoning. For interesting examples, see Nissan Mendel, pp. 100-102.

[54] Of course, ‘some P1 are P2’ and ‘some P2 are P1’ imply each other.

[55] Of course,’ no P1 is P2’ and ‘no P2 is P1’ imply each other.

[56] An example is given in Sifra on Lev. 13, but I have not so far found a way to formalize it. See Mendel Nissan, pp. 104-105. There may also be useful examples in Mielziner’s book.

[57] The fifth rule which begins with the phrase kol davar shehayah bikhlal veyatsa… (rule 11) is somewhat different but also syllogistic. See my analysis of it in my Judaic Logic.

[58] For example, in one of Plato’s Dialogues, Socrates says: “I am not a match for one of you; and a fortiori I must run away from two.” In truth, Aristotle does often use a fortiori argument with a more scientific intent; but even then his argument as a whole does not depend so heavily on such argument as it does in Talmudic literature. And of course, Aristotle must be acknowledged for his early reflections on a fortiori argument in the Topics (2:10) and the Rhetoric (2:23), even if such reflections were scant.

[59] I must draw attention, here, to the different theories of Talmudic formation, including those of Abraham Weiss and David Weiss-Halivni. They are of course very relevant to attempts at dating specific passages of the Talmud.

[60] And similarly with implicational moods. I doubt however that the distinction between copulative and implicational moods is to be found anywhere outside my book Judaic Logic – so it is no use asking that question.

2016-08-23T09:50:36+00:00