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A FORTIORI LOGIC

© Avi Sion, 2013 All rights reserved.

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A FORTIORI LOGIC

CHAPTER 13 – Moses Mielziner

1. Description of the argument

2. Structural analyses

3. Concerning the jus talionis

4. Restrictions and refutations

Moses Mielziner (Poland-USA, 1828-1903) was an American Reform[1] rabbi and author. His work Introduction to the Talmud is definitely a classic of the genre, one of the best I have seen[2]. It starts with a “historical and literary introduction;” then it deals in considerable formal and material detail with the “legal hermeneutics of the Talmud;” this is followed by an extensive study of “Talmudical terminology and methodology;” and lastly various additional essays and notes. I wish I could take the time to review it all, but here is obviously not the place for this job; such a major digression from our central concern here would be unjustified. I will therefore confine my analysis here mainly to Mielziner’s treatment of “the inference of kal ve-chomer” (part II, chapter I).

1. Description of the argument

Although he states at the outset that this hermeneutic rule “has quite a logical foundation, being a kind of syllogism” (my italics) – this cannot be taken to mean (as some[3] have done) that he equates a fortiori argument with syllogism. That is evident from what comes after. All he means here, I believe, is that a fortiori argument is, like syllogism[4], mediate inference, i.e. inference through a middle term (as distinct from immediate inference). Moreover, he makes clear that though the terms “kal” and “chomer,” referring respectively to things of lesser and greater legal importance or significance, are called the minor and major terms, these words should not be confused with the same ones commonly used in regard to syllogism[5].

Mielziner describes the qal vachomer argument as follows:

“The principle underlying the inference of [kal ve-chomer] is, that the law is assumed to have the tendency to proportionate its effects to the importance of the cases referred to, so as to be more rigorous and restrictive in important [matters], and more lenient and permissive in unimportant matters. Hence, if a certain rigorous restriction of the law is found regarding a matter of minor importance, we may infer that the same restriction is the more applicable to that which is of major importance, though that restriction be not expressly made in the law for this case. And on the other hand, if a certain allowance is made by the law regarding a thing of major importance, we may properly conclude that the same allowance is the more applicable to that which is of comparatively minor importance” (pp. 130-131).

Reading the first sentence, we might well assume that Mielziner identifies a fortiori argument with a crescendo, which yields a ‘proportional’ conclusion. But his next two sentences clearly refer to purely a fortiori argument, which yields an identical conclusion. He gives as his prime example the argument in the Mishna Beitza 5:2 (which we have dealt with in detail in an earlier chapter, viz. 8.5) – namely (briefly put) work permitted on the Sabbath is “more permissible” on a Festival, and vice versa, work forbidden on Festivals is “all the more imperatively to be forbidden” on the Sabbath. Judging by the language used in this example, it appears that Mielziner has not realized the difference between purely a fortiori and a crescendo argument. This is an important lacuna in his analysis.

As we have seen in earlier chapters (1-2) of the present work, despite the misleading suggestions of commonly used phraseology of a fortiori argument like “all the more,” there is in fact no ‘proportionality’ intrinsic to such argument. The conclusion of a purely a fortiori argument logically must needs refer to exactly the same subsidiary term as the minor premise of it does; otherwise, the argument effectively contains five terms instead of four, and the principle of deduction (according to which no more information may be found in a deduced proposition than was explicit or logically implicit in the propositions it was deduced from) is breached. To draw a ‘proportional’ conclusion, an additional premise about ‘proportionality’ must be admitted. This is sometimes readily forthcoming, but not always.

Mielziner rightly perceives a fortiori argument as a sort of argument by analogy. He parenthetically mentions the use of analogous reasoning in modern jurisprudence, quoting the maxim[6]: “Quod in minor valet, valebit in majori; et quod in majori non valet, nec valet in minori” – meaning: “what avails in the less, will avail in the greater; and what will not avail in the greater, will not avail in the less.” This maxim seems to be based on Cicero’s statement in Topica 23, though the wording is more polished here. But though this external analogy is apropos and interesting, Mielziner does not remark on its evident lack of ‘proportionality’, i.e. on there being no difference of degree in “availing” mentioned in it.[7]

Mielziner goes on to give the example of qal vachomer in Numbers 12:14 concerning the isolation of Miriam[8] as the “Biblical prototype” of such argument, like the Gemara does (Baba Qama, 25a). But he does not immediately question the Gemara’s interpretation of the argument, as prescribing a “fourteen days” penalty that is thereafter mitigated to “seven days” by means of the dayo principle (the principle of sufficiency formulated by the Sages in the corresponding Mishna). Only further on does he introduce the latter principle as a “restriction in the application of inferences,” and further analyze the Miriam example. But here again, Mielziner does not show that he realizes that a fortiori argument is not intrinsically proportional. All he says is:

“Thus, in the inference made in Scripture in regard to Miriam, we might have expected that the time of her exclusion from the camp should be more than seven days, since the Lord’s disfavor is of more consequence than a human father’s; nevertheless, Scripture says, ‘Let her be shut out from the camp seven days,’ which is just as long as she would have felt humiliated if her father had treated her with contumely. On this passage the restrictive rule just mentioned is founded” (p. 135).

Notice the claim: “we might have expected… more than seven days.” Whence, it is clear that Mielziner buys the traditional superficial reading of the Gemara, lock stock and barrel, without much reflection on the subject. For him, then, qal vachomer is naturally proportional (as his initial description partly suggests) and it is only due to the restriction imposed by the rabbinical (and ultimately Scriptural, in the rabbis’ view) dayo principle that the conclusion is kept the same in magnitude as the minor premise. Nevertheless, despite this naivety, Mielziner displays considerably more awareness of the structure of a fortiori argument than many other commentators, even later ones.

2. Structural analyses

He clearly analyzes the structure of a fortiori argument as consisting of three propositions. The “first premise” (what I have called the major premise) informs us that “two certain things, A and B, stand to each other in the relation of major and minor importance.” It should be noted that many commentators tend to ignore this crucial information, and Mielziner is rather exceptional is mentioning it explicitly as a premise. However, he fails to mention in what respect the two terms A and B are differently “important” – that is to say, he does not identify the presence of a specific middle term.

The “second premise” (what I have called the minor premise) tells us that “with one of these two things (A) a certain restrictive or permissive law is connected.” The conclusion is that “the same law is more applicable to the other thing (B).” Here, we can congratulate Mielziner on stating that the conclusion is “the same law” as given in the second premise, although his qualifying it as “more applicable” is, as already stated, inaccurate (though a common error). However, here again, there is no mention of the middle term, which is left tacit in both the minor premise and conclusion – and no mention that it is by virtue of having enough of this middle term that the law (i.e. the subsidiary term) is connected.

Note that although A and B are in the first premise described as the items of major and minor importance (apparently respectively), A is always placed in the second premise and B is always placed in the conclusion – so A and B should not be viewed as always symbolizing the same (major or minor) term. This becomes clear when Mielziner goes on to describe the different possibilities (i.e. moods) of qal vachomer argument. He points out that it is “is usually expressed by two compound propositions, one of which is the antecedent and the other the consequent” (his italics), i.e. if the minor premise, then the conclusion, effectively; and he describes two directions of inference.

“Inference from minor to major” goes from a minor thing A being “subject to a certain severity of the law” (which may be, he says it explicitly in Hebrew, a prohibition or an imperative, i.e. a negative or positive severity) to a major thing B being “more subject to the same severity” (again that extraneous and misleading “more” wording!). “Inference from major to minor” goes from a major thing A being subject to “a certain allowance” (which may be, he again says it explicitly in Hebrew, a permission or an exemption, i.e. a negative or positive allowance) to a minor thing B being “more” (grrr!) subject to “the same allowance.” What is good here is Mielziner’s awareness that severity and allowance may each be positive or negative.

What is missing, however, is an explicit statement of the square of opposition between chayav (imperative), assur (forbidden), mutar (permitted) and patur (exempted), which would more fully clarify that the ‘minor to major’ and ‘major to minor’ moods Mielziner lists here are related to each other. “If A which is minor is forbidden (or imperative), then B which is major is forbidden (or imperative)” implies, by contraposition, that “If B which is major is not forbidden (or imperative), i.e. is permitted (or exempted), then A which is minor is not forbidden (or imperative), i.e. is permitted (or exempted).” The two moods are related, they are not two separate phenomena. No doubt Mielziner is tacitly aware of this to some extent, but he does not say it out loud.

More serious than this lack of a distinction between positive and negative moods, is the absence of a distinction between subjectal and predicatal moods, and another between copulative and implicational moods. The forms of a fortiori argument that Mielziner explicitly acknowledges are essentially all copulative, subjectal and positive. He is apparently not aware that the terms A and B may sometimes be predicates instead of subjects; that the propositions they are in may sometimes be negative instead of positive; and that the argument may sometimes be organized around theses instead of terms. Moreover, he does not make any attempt at validation of qal vachomer argument, but takes it as intuitively obvious.

To sum up, although Mielziner gives us a competent basic guide to qal vachomer reasoning in the Talmud, he does not proceed far beyond the traditional approach and systematically develop the field. What I have done here is only a bare sketch of Mielziner’s presentation, for the purpose of determining where he stood with regard to qal vachomer proportionality and the dayo principle, and how far he went in formal analysis of the argument.

3. Concerning the jus talionis

Mielziner gives some interesting illustrations of qal vachomer use in the Talmud and related documents. One of these concerns the so-called jus talionis (the law of ‘talion’, or retaliation), the law given in Exodus 21:23-25: “life for life, eye for an eye, tooth for tooth, etc.” As is well known, the rabbis interpreted this Torah law in an indulgent manner, prescribing monetary compensation rather than a literal application, at least for eyes and teeth (contrary to what detractors of Rabbinic Judaism have claimed, starting with the Sadducees[9]). To justify this ruling, the rabbis made use, among other arguments, of a qal vachomer; this was, as Mielziner puts it (p. 134):

“Referring to the law (Exodus xxi. 29-30), by which, under certain circumstances, the proprietor of a beast which is notably dangerous and which has killed a person, is judged liable to the death penalty; but the capital punishment could be redeemed by money. Now, if the law expressly admits a pecuniary compensation in a case where the guilty person deserved capital punishment, how much the more is a pecuniary compensation admissible in our case where it does not concern capital punishment. (Mechilta to Exodus xxi. 24.)[10]

Mielziner rightly classified this reasoning as “inference from major to minor,” as can be seen by recasting it in standard (positive predicatal) form. Notice how such reformatting clarifies an argument and makes manifest its validity:

More severe penalties (middle term, R) are required for heavier offences (major term, P) than for lighter ones (minor term, Q).

Redemption by pecuniary compensation (subsidiary term, S) is (explicitly according to the Torah) a penalty severe (R) enough in the case detailed in Ex. 21:29-30, which is in principle a capital offence (P).

All the more so, redemption by pecuniary compensation (S) is (implicitly, by way of this inference) a penalty severe (R) enough in the case detailed in Ex. 21:24-25, which is in principle an offence liable to loss of limb (Q).



Several things are worth remarking here. The first is that, though Mielziner did not realize this, the argument used here being positive has to be predicatal[11]; argument from major to minor can be subjectal only if negative in form. Secondly, the major premise of this argument is not mentioned in the Torah, but is intuitively obvious, or can be deemed an application of the general principle of justice known as ‘measure for measure’ (midah keneged midah). Indeed, the law of retaliation (eye for eye, etc.) might be viewed as a special case of this very principle. Its literal application would perhaps be strict justice; but an effort is being made here to mitigate it: it is too extreme, lacking in mercy. Still, paradoxically, the general principle is used as a premise in the argument designed to mitigate it.

Thirdly, the argument is obviously major to minor only in cases of “limb for limb” (Ex. 21:24-25) – but less obviously so in the case of “life for life” (Ex. 21:23)[12]. Is the latter statement intended as very general? It seems doubtful that it includes cases of intentional homicide, which are dealt with elsewhere. Rather it seems to concern unintentional homicide, which presumably does not deserve capital punishment. Perhaps it is even narrower in intent, and concerns only, as the context (viz. Ex. 21:22) suggests, the unintentional killing of an unborn baby; in such case one might well ask: is the court supposed to execute the culprit’s baby? And what if he doesn’t have a baby?[13] It is very doubtful that such drastic retaliation would be even considered. Thus, upon reflection, inferring monetary compensation in cases of “life for life” is also argument a majori ad minus (and not a pari as might superficially seem).

Fourthly, there is no mention of the size of the monetary compensation. There is no attempt at explicit proportionality in that respect; nor at a contrary application of the dayo (sufficiency) principle. At least, Mielziner does not mention the issue here, though it would seem very appropriate to do so. The amount of monetary compensation is obviously to be determined by the courts. Presumably, they would make it proportional, i.e. charge the culprit less for a limb than for a life; in other words, I presume, they would not apply the dayo principle in this case. Why so? Again, I answer the question presumptuously (i.e. I do not know what the rabbis actually say on this subject) – because the inference is from major to minor. If dayo were here literally applied, the monetary compensation for loss of limb would have to be equal to that for loss of life – which is intuitively absurd.

This example seems to confirm what I suggested in my treatment (in an earlier chapter, viz. 8.2-3) of the dayo principle, namely that it can only be reasonably applied in arguments from minor to major. That is to say, it is obviously intended as a principle of mitigation of penalty – not of aggravation of penalty. This example also seems to confirm my contention that any proportionality that is admitted is not due to an operation of the qal vachomer argument as such but to an external application of the principle of midah keneged midah. A fortiori argument cannot be manipulated, being a logical ‘law of nature’. But the measure for measure principle certainly can be adapted to different circumstances; and indeed it ought to adapt, if justice is to be precisely served. The measure for measure principle should be regarded as a general guideline, a rough rule of thumb, a heuristic principle, rather than as a mathematical law to be applied indiscriminately.

Although, as already said, Mielziner briefly discusses the dayo principle, his definition of it as “the law transferred to B (the major), must never surpass in severity the original law in A (the minor), from which the inference was made” (p. 134) is focused on qal vachomer inference from minor to major; he does not ask the question how it might be applied to inference from major to minor. This may well be due to there being no such discussion in the Talmud and other rabbinic literature, either (I do not know if there is any).

Moreover, and most significantly, Mielziner’s definition of the dayo principle corresponds to the first type of dayo, that used by the Mishna Sages in reaction to R. Tarfon’s first argument. He makes no mention of the second type of dayo, used by the Mishna Sages in reaction to R. Tarfon’s second argument, which is significantly different. Here too, he may be excused as having been misled by the Gemara, which also failed to notice the difference. However, since commentators later than the Gemara and before Mielziner did notice the difference, he has less excuse.

4. Restrictions and refutations

Mielziner discusses three Talmudic “restrictions in the application of inferences” of qal vachomer. The first is the dayo principle: “It is sufficient that the result derived from an inference be equivalent to the law from which it is drawn.” This, he informs us, is founded on the Scriptural inference in regard to Miriam (Numbers 12:14) and there is an ample application of it in Mishna Baba Kamma II. 5. He does not remark that the Scriptural source is identified in the Gemara commentary to that Mishna, and there given as a baraita. Also, as we just pointed out, he does not remark that the dayo principle has two distinct senses, corresponding to the two uses of it by the Sages in the Mishna in response to the two arguments by R. Tarfon.

The second restrictive rule is: “The inference from minor to major is not to be applied in the penal law.” Mielziner explains it as follows: “The reason for this rule lies in the possibility that the conclusions drawn by the inference might have been erroneous, so that the infliction of a penalty derived from such a conclusion would not be justified.” He points out that similar provisions can be found in modern legal statutes. He informs us that “an application of [this] rule is made in Talmud Maccoth 5b.” He does not clarify the difference between this second rule and the first one, which also forbids application of proportional penalties based on inference. Presumably, though they overlap somewhat, they differ in scope to some extent[14].

The third restriction he mentions is: “No inferences must be made from traditional laws to establish a new law.” This is “laid down in Mishna Yadaim III. 2.” However, he tells us, R. Akiva did not accept this rule. By “traditional laws” is here meant laws not given in the written Torah but considered to have been orally transmitted since the Sinai revelation, presumably, as well as later legal developments. There are other restrictions relating to inference, but Mielziner does not mention them – at least not here.

Mielziner goes on to describe “refutation of inferences.” As he puts it: “Not every kal ve-chomer offered in Talmudic discussions of the law is correct and valid. We sometimes find there very problematic and even sophistical inferences set forth merely as suppositions or hypotheses; these are, however, finally refuted.” Such refutation is called pirka (meaning: objection). It can be done in one of two ways: either a premise or the conclusion may be challenged. More specifically, the antecedent item “A which was supposed to be of minor importance is in some other respects really of major importance;” or again “the peculiar law connected with A can not be transferred to B[,] as it is not transferred to [some third item] C, which is in certain respects like B.”

He gives Talmudic examples of such refutations, namely: “Chullin 115b; Mechilta to Exodus xxiii. 19.” for the former, and “Mishna Pesachim vi. 1, 2.” for the latter. We shall examine these examples presently. Moreover, Mielziner teaches us: “When an inference has been refuted in one of the two ways just mentioned, the attempt is sometimes made to defend and retain it by removing the objection raised in the refutation. If the arguments proffered for this purpose are found to be correct, the original inference is reinstated; if not, the refutation is sustained and the inference finally rejected.” He gives an example of such discourse, too; but I will not here give further details on that.

The first question that pops into my mind in this context is: how is it possible for any of the rabbis involved in such debates to even momentarily propose an idea or argument that turns out to be wrong? I (and indeed anyone with a very logical mind) would regard such discursive conflicts as concrete evidence from the Talmud itself that the rabbis were neither omniscient nor infallible. But neither traditional commentators, nor for that matter Mielziner (here), reflect on this larger implication. We could, of course, say that the refuted rabbi was playfully putting forward an argument he knew full well would be refuted by a colleague, so as to teach us (future onlookers) that this argument would not stand scrutiny. And indeed this may be true in some cases. But, surely, in some cases such argumentation and counter-argumentation is evidence that the thinking of the rabbis, collectively speaking, was inductive – i.e. ordinary human trial and error. In which event, the rabbis can hardly claim absolute authority for their dictates.

The second question to ask is: how does such refutation proceed in more formal terms?

Let us consider the first example given by Mielziner, illustrating “refutation of a premise.” Regarding orlah fruits (“the fruits of a tree during its first three years”), one rabbi argues: “If those fruits, regarding which no law has been violated, are forbidden to be used in any way, ought not meat and milk, which, in violation of a law have been boiled together, the more forbidden to be used in any way?” Mielziner explains the counterargument as follows: “The premise in this inference is that orlah is of minor importance compared with meat and milk; but this premise is disputed [by another rabbi] by demonstrating that in certain respects it was, in fact, of major importance, since those fruits had at no time before been permitted to be used, while in regard to meat and milk there had been a time (namely, before being boiled together), when the use of each of these components was allowed.”

What is the formal basis of this particular refutation? It is not exactly as Mielziner here suggests (and no doubt many before and after him) that “orlah fruit” is in one “respect” less important, and in another “respect” more important, than “meat and milk.” Rather, I would say, the first rabbi claims that meat-and-milk is always more unlawful than orlah-fruit, and constructs his qal vachomer on that basis, whereas the second rabbi denies his colleague’s major premise, i.e. demonstrates that it is sometimes not true, and so neutralizes the a fortiori reasoning. Their arguments look like this:

First rabbi: (positive subjectal a fortiori argument)

Meat-and-milk (P) is (always) more unlawful (R) than orlah-fruit (Q) is.

So, since orlah-fruit (Q) is unlawful (R) enough to be forbidden to use (S),

it follows that meat-and-milk (P) is unlawful (R) enough to be forbidden to use (S).



Second rabbi: (denies major premise of preceding)

Meat-and-milk is sometimes (before mixing) not more unlawful than orlah-fruit.

So, even though orlah-fruit is unlawful enough to be forbidden to use,

it does not follow that meat-and-milk is unlawful enough to be forbidden to use.



Note that whereas the former tries to prove that meat-and-milk should be forbidden to use, the latter does not try (at least not here) to prove the contradictory conclusion but is content to block his colleague’s attempted proof. The effective middle term, say “degree of unlawfulness,” does not change from one argument to the next; so there is not really a change of “importance” in different “respects.” What changes, rather, in this example at least, is the modality of the proposed major premise, which goes from (tacitly) “always” to (explicitly) “only sometimes.” This modality may be viewed as temporal, i.e. as relating to the life cycle of the conjunctive term “meat and milk.” Or it may be viewed as extensional, i.e. as distinguishing two physical kinds of “meat and milk” conjunctions – namely “meat and milk separately” and “meat and milk together.”

The upshot of this analysis is that the language usually used to describe or explain such refutation is misleading. The impression given is that the initial a fortiori argument is rebutted by an equally cogent or more cogent a fortiori argument (much as one dilemma may be checkmated by another). But this is not the actual scenario in cases of this sort. When the critic shows that the relative importance of the two terms in question can in some circumstances be reversed, he is not proposing a counteracting a fortiori argument – he is simply discrediting the major premise of the initial argument. Note, finally, that the minor premise is not attacked in the above example; but, clearly, this would be an equally viable alternative approach to refutation.

Let us now examine the second example given by Mielziner, illustrating “refutation of a conclusion” (p. 138). It concerns a Passover festival that happens to fall on the eve of a Sabbath, so that the laws of both are somewhat in conflict. R. Eliezer argues: “If slaughtering [the paschal lamb], though a real labor, abrogates the Sabbath, ought not things not regarded as real labor the more abrogate the Sabbath?” To which R. Joshua replies: “A common holiday proves that this conclusion is not admissible, for on such a day some real labors (as cooking, baking, etc.) are permitted, while at the same time certain actions, which fall under the category of [things not regarded as real labor], are positively forbidden.”

The phrase ‘real labor’ used here refers to melachah, while ‘not regarded as real labor’ refers to shevut (these are technical terms in Hebrew for these two categories of action). The expression “abrogates the Sabbath” means that some law normally applicable on the Sabbath is not applicable on the Sabbath under consideration; which means that something normally forbidden is here permitted. The Sabbath discussed here is that coinciding with the eve of Passover.

The arguments can be presented more formally as follows:

First rabbi: (positive subjectal a fortiori argument)

Shevut (P) is generally more lawful (R) than melachah (Q).

So, since some melachah (such as slaughtering the paschal lamb) (Q) are lawful (R) enough be permitted on the Sabbath [coinciding with Pessach eve] (S),

it follows that [all] shevut (such as carrying the lamb to the temple) (P) are lawful (R) enough to be permitted on the Sabbath [coinciding with Pessach eve] (S).



Second rabbi: (denies conclusion of preceding)

Even though shevut is in principle more lawful than melachah,

and indeed, some melachah (such as cooking, baking, etc.) are [lawful enough to be] permitted on a Festival,

nevertheless, some shevut (such as certain private affairs) are not [lawful enough to be] permitted on a Festival.



Let us now analyze these arguments more closely. The first rabbi’s argument is clearly intended to be a fortiori, moving from minor to major. The major and minor premises must be as above shown, since the desired conclusion is permission of shevut on the specified Sabbaths. The middle term “lawful” is chosen by me offhand to fit the bill; it is possible that the rabbi who formulated the argument had another, better one in mind. The second rabbi’s argument is not intended as a rival a fortiori, but as a critique of the first rabbi’s a fortiori. While he apparently admits the former’s major and minor premise, he denies the generality of his conclusion. That is, he points out that, while some melachah (activities forbidden on a Sabbath) are permitted on a Festival, not all shevut are permissible on a Festival (and therefore, a fortiori, not all shevut are permissible on a Sabbath coinciding with a Festival).

From which it follows that the major premise and/or the minor premise must be wrong in some respect(s). The second rabbi’s statement that some melachah are permitted on a Festival seems to be intended as an acknowledgment of the first’s minor premise, even though logically that some melachah are permitted on a Festival does not imply that they would be permitted on a Festival coinciding with a Sabbath. We can suppose that “Festival” here refers to “Festival coinciding with a Sabbath,” since he does not explicitly claim that no melachah is permitted on a Festival coinciding with a Sabbath. The second rabbi’s objection seems rather to be directed at the first’s major premise. That is, his purpose must be to deny that shevut is generally more lawful than melachah, even if he grants that it is in principle (i.e. usually) more so. It is rather by this means that he inhibits the first rabbi’s conclusion.

Thus, while the first rabbi’s conclusion seems to concern all shevut, the second rabbi argues that it can only concern some shevut, and therefore by implication the former’s major premise must also be less general than it seems. For the first rabbi “Shevut is more lawful than melachah” is literally a general proposition, whereas for the second rabbi it is only true ‘in principle’ since exceptions to it exist. In other words, it is not entirely accurate to describe this case as one of “refutation of the conclusion.” It is true that the first rabbi’s conclusion is contradicted by the second rabbi, since the latter puts forward a proposition that conflicts with it. But by doing that the latter is effectively also mitigating the major premise that gave rise to it, and thereby blocking the whole argument.

Thus, in the last analysis, what is touted as ‘refutation of a conclusion’ is really no different from ‘refutation of a premise’. Logically, it is impossible to refute a conclusion if we admit all the premises and, of course, the validity of the deductive process used[15]. What is here (in the above example, least) called ‘refutation of a conclusion’ is denial of a conclusion coupled with denial of the generality of the major premise. We could say that the denial of the conclusion is the means through which the generality of the major premise is denied; and in that sense the denial of the conclusion comes first. In other words, what is involved here is indirect attack on a premise (or both of them), through an attack on the conclusion. The latter clarifies why this process can reasonably be labeled “refutation of a conclusion.”[16]

Lastly, Mielziner draws attention to “some sophistical inferences of kal ve-chomer mentioned in the Talmudic literature, which are simply refuted by an argument ad absurdum,” giving a number of illustrations. The example I found intriguing and amusing was that from Derech Eretz Rabba, chapter I. A layman “tried, in the presence of R. Gamaliel, to ridicule the application of inferences in ritual laws by the following paralogism:

‘If the marriage with one’s own daughter is prohibited, although the marriage with her mother is permitted, how much more unlawful must it be to marry another married woman’s daughter, since the marriage with her mother, a married woman, is positively prohibited?’”

To which Mielziner comments: “the fallacy in this inference is that the conclusion contradicts the premise… But R. Gamaliel answered caustically…” (p. 140). Mielziner is of course right to point out that premise and conclusion were at odds, but I would like to see more precisely how the conclusion was arrived at. The answer has to be that an excessive generalization is involved. We are given that: (1) marriage with one’s wife [one’s daughter’s mother] is permitted, (2) marriage with one’s wife’s [one’s own] daughter is prohibited, and (3) marriage with another man’s wife [a married woman] is prohibited; and the putative conclusion is: (4) marriage with another man’s wife’s [a married woman’s] daughter is prohibited. This argument can be put in qal vachomer form, as follows:

Marriage with a married woman’s daughter (P) is more unlawful (R) than marriage with the married woman herself (Q).

Marriage with another man’s wife (Q) is unlawful (R) enough to be prohibited (S).

Therefore, all the more, marriage with another man’s wife’s daughter (P) is unlawful (R) enough to be prohibited (S).



Clearly, the major premise of this argument is generalized from givens (1) and (2), i.e. from one’s own wife and daughter to all wives and daughters. The a fortiori argument itself is formally valid; i.e. its conclusion does logically follow from its premises. But its conclusion is, as Mielziner points out, an absurdity, since it effectively implies that all marriage is unlawful, and therefore that even one’s own wife, who was naturally another man’s wife’s daughter, is prohibited, even though we are initially given that it is permitted[17]. It follows that one of the premises must be false. It cannot be the minor premise, which is a Scriptural given. Therefore, it must be the major premise. What is wrong with it? Obviously, it was an overgeneralization from the given information. The prohibition to marry one’s daughter is a special case, which cannot be generalized to all people’s daughters without leading to absurdity.

To sum up, then. What is “sophistical” about this example (at least) is simply the false impression it gives of veracity due to its outward form being perfectly logical. But the validity of the explicitly claimed deduction conceals an invalid previous act of induction, which is not explicitly admitted. In other words, the proposed argument can readily be refuted by attacking its unstated major premise. So in fact, this refutation is formally not much different from the one used in an earlier example (concerning meat and milk and orlah fruit). This example again shows that if we know logic well, and we analyze particular cases carefully, we can never be fooled by any sophistry.

In my view, the epithet of sophistry is best reserved for non-sequiturs, i.e. arguments whose conclusions do not follow from their premises. In the case of a fortiori, this would refer a positive subjectal or negative predicatal argument from major to minor, or to a positive predicatal or negative subjectal argument from minor to major; and similarly, of course, with regard to implicational moods. Whether such cases actually occur in the Talmud, I cannot say; offhand, I would suspect they occasionally do. Thus, to refute an a fortiori argument, we may deny either or both of its premises, either directly or by attacking the conclusion, or we may deny that the conclusion logically follows from the premises.



[1] Note that “Reform” Judaism in his day was less radical than it is today; it was perhaps more akin to today’s “Conservative” denomination.

[2] The edition I have in hand is the third, published in 1925. But the original edition dates from 1894. He is thus just a few years ahead of Schwarz, at least with regard to analysis of the qal vachomer argument. This work is certainly worth perpetuating. I wish I had read it while writing my Judaic Logic; the latter work would have been greatly enriched.

[3] Notably Schumann, p.7.

[4] The etymology of syllogism is syn+logism, i.e. confluence of discourses.

[5] In syllogism, the minor term is found in the minor premise and conclusion, and the major term in the major premise and conclusion; as for the middle term, it appears in the two premises but not in the conclusion. In a fortiori argument, the minor and major terms are both found in the major premise, and either one may appear in the minor premise while the other appears in the conclusion; as for the middle term, it appears in both premises and in the conclusion; moreover, here there is a fourth term, the subsidiary term (which is in fact two terms with a common ground, in the special case of a crescendo argument).

[6] On p. 131, footnote 1. Mielziner gives as reference: “quoted by Coke on Littleton, 260.” Concerning Cicero, see the section devoted to him in an earlier chapter of the present volume (6.5). Cicero calls this “argument by comparison;” but he obviously refers to a fortiori argument, since he distinguishes between moods that go from major to minor; from minor to major; and from equal to equal.

[7] Moreover, notice, the Latin statement does not mention the threshold or sufficiency of the middle term that logically enables the inference.

[8] Which we have already analyzed in detail in an earlier chapter, viz. 7.4).

[9] In this respect, I read an interesting comment somewhere that prior to the appearance of the law of “an eye for an eye,” people probably practiced “two eyes for an eye” or other such arbitrarily disproportionate revenge or punishment. The law of talion was thus originally not, as some contend, a law of harsh retaliation, but rather one of kindly mitigation of retaliation. It introduced concepts of ‘rule of law’ and ‘just proportionality’ into the penal system. And this is taking it literally, as it was presumably originally intended. The rabbis, perhaps much later, went further in this humane direction, by taking the principle metaphorically rather than literally, and introducing monetary compensation instead of (as the case may be) capital punishment, blinding eyes or breaking teeth.

[10] See also Baba Qama, 83b-84a.

[11] That is, the major and minor terms in it, P and Q, are predicates.

[12] One might propose an a pari a fortiori argument from pecuniary compensation in lieu of death in relation to Ex. 21:29-30 to same in relation to Ex. 21:23.

[13] Similarly, of course: if he is blind, how can “eye for eye” be applied? If he is toothless, how can “tooth for tooth” be applied? And so on. The law would be difficult to apply literally in some cases, and therefore may well be taken as having been intended metaphorically.

[14] See more detailed comments on this topic in the chapter on Abitbol (21.3).

[15] Sometimes, it is the validity of the deductive process, rather than either premise, which is open to criticism. A case in point is in the Mishna Zebahim 7:4, where R. Yehoshua attacks R. Eliezer’s a fortiori argument by pointing out that the subsidiary term is not really as uniform as it seems in minor premise and conclusion.

[16] It should be noted that the dispute in question does not end where I ended it, but is pursued further with the first rabbi defending his position against the second, and then the second rabbi being in turn defended by a third (R. Akiba), and the rabbis finally opting for rejection of the first rabbi’s conclusion. But these additional details do not impinge on what is being said here.

[17] Maccoby suggests in this context that all marriage would be forbidden except “to the daughters of unmarried [i.e. single] mothers, widows or divorced women.” I am not sure whether this exception is true; I always assumed that, in Jewish law, having sex with a currently unmarried (Jewish) woman is tantamount to marrying her. But in any case, the argument must be revised to precisely adapt it to this interesting issue; I leave the job to others.

2016-06-14T04:51:30+00:00