CHAPTER 9 – Post-Talmudic rabbis

1. Logic and history issues

2. Philo of Alexandria

3. Sifra

4. The Korach arguments

5. Saadia Gaon

6. Rashi and Tosafot

7. Kol zeh assim

8. Maimonides

9. More on medieval authors

10. Moshe Chaim Luzzatto

11. More research is needed


1. Logic and history issues

In the present chapter, our object shall be to discuss and to some extent trace some of the developments in rabbinic and more broadly Jewish thought concerning the a fortiori argument, and to a lesser extent more broadly the hermeneutic principles. This is of course a massive task that we cannot remotely hope to carry out exhaustively in the present study; we can however hopefully reflect on some of the issues involved and give scattered examples of the kind of research and evaluation that are needed in this context.

The first thing to make clear is the distinction between hermeneutic and logical principles. Although the rabbis to some extent regarded their hermeneutic principles as logical principles, the truth is that logic was not a prime interest for them: their primary interest was in justifying the traditional legal system enshrined in the Mishna and expanded on in the Gemara and subsequently. I will not here even try to roughly trace the development of Jewish law from its Biblical beginnings, through the formative period from Ezra to the Mishna, followed by the Gemara and later rabbinic work. I can only recommend to the reader who has not already done so to read works (preferably critical) on the subject, such as Mielziner’s Introduction to the Talmud. The important thing, in the present context, is to take to heart what Mielziner writes regarding the “circumstances that necessitated artificial interpretation”:

“As long as the validity of this oral law had not been questioned, there was no need of founding it on a Scriptural basis. It stood on its own footing, and was shielded by the authority of tradition. From the time however when the Sadducean ideas began to spread, which tended to undermine the authority of the traditional law and reject everything not founded on the Scriptures, the effort was made by the teachers to place the traditions under the shield of the word to the Thora. To accomplish this task, the plain and natural interpretation did not always suffice. More artificial methods had to be devised by which the sphere of the written law could be extended so as to offer a basis and support for every traditional law, and, at the same time, to enrich the substance of this law with new provisions for cases not yet provided for. This artificial interpretation which originated in the urgent desire to ingraft the traditions on the stem of Scripture or harmonize the oral with the written law, could, of course, in many instances not be effected without strained constructions and the exercise of some violence on the biblical text…” (pp. 120-121).[1]

Two ideas should be emphasized in this context. The first is that the hermeneutic principles have a history. They did not come out of the blue all of a sudden, whether at Sinai or later, but were gradually developed in response to specific needs by specific persons, and often against conflicting opinions by other persons. Changes evidently occurred over time. This development can be precisely traced to some extent, even though traditional commentators make every effort to deny significance to the known history. The second idea is that the hermeneutic principles are not necessarily logical. Mielziner rightly refers to “artificial” as against “natural” methods, using exactly the same terms as I did independently fifteen years ago when I wrote my Judaic Logic. In that work, I showed, clearly and by formal means, to what extent the hermeneutic principles could be regarded as logical and to what extent they could not. Mielziner, in his reference to “strained constructions and the exercise of some violence on the biblical text,” had the honesty and courage to admit the limits of rabbinic logic.

In the present work, following detailed logical analyses mainly of the Mishna Baba Qama 2:5 and the related Gemara Baba Qama 25a, I have developed a more precise assessment of Talmudic logic. It appeared from this exploration that the a fortiori logic found in the Mishna is more natural, less artificial, than that found in the Gemara. Judging from the Talmudic passage we examined, the understanding of a fortiori argument by the earlier rabbis was simpler and more straightforward, while that of the later rabbis was more complicated and tortuous. The two groups should not be lumped together. This is as regards their practice; neither group engages in much theoretical reflection (if any) on the subject. So the artificiality that Mielziner speaks of is more centered in the Gemara than in the Mishna (at least as regards a fortiori argument).

What is clear from our research is that it is misleading and futile to try to interpret and justify the rabbinical hermeneutic principles entirely through logic. They undoubtedly have some logical character, and are often thought of and intended as logic, but they are not purely and entirely logical. They are, as Mielziner well described them, ad hoc responses to the problem of anchoring the oral law so-called, i.e. the Jewish legal tradition existing at a certain period of history, in the more authoritative written Torah. Sometimes that anchoring is possible by quite natural (i.e. purely logical) means; but sometimes some intellectual artifices are necessary to achieve the desired end. With this frank admission in mind, we can more clearly trace the history of commentaries on the hermeneutic principles and practices in general, and on the a fortiori argument in particular, from two points of view.

The first viewpoint is that of the uncritical traditionalists. Their writings or lectures on the a fortiori argument or on hermeneutics are simply designed to pass on as clearly as possible the information received from tradition. This teaching is presumed true and valid without question, and the only role of the teacher is to clarify it and give examples of it. The second viewpoint is that of the critical logicians, among which I count myself. Their written or oral reflections on the subject are aimed at scientific evaluation, and are therefore perforce more formal and not necessarily in agreement with tradition. Truth and validity are not automatically granted, simply because the argument in question is claimed to be, directly or indirectly, of Divine or prophetic origin, or to have the stamp of approval by rabbinical or whatever authorities. These two viewpoints are pretty well bound to be at odds in some cases, though not in all.

Many indices can be used as litmus tests for the classification of a commentator in one camp or the other. We must look and see where each commentator stands in relation to the debate between R. Tarfon and the Sages in the Mishna; how he perceives the argument(s) of the former and the objection(s) of the latter. We must also pay attention to his eventual reactions to the Gemara: to its general equation (on the basis of a baraita) of a crescendo argument with a fortiori; to its readings of the argument about the isolation of Miriam in Num. 12:14-15; to its claims about R. Tarfon’s ideas about when the dayo principle may or may not be applied to an a fortiori argument. In short, we must look out for the depth and breadth of a commentator’s awareness of the issues involved. Certain authors will judge such matters dogmatically: they are the traditionalists. Others will be more circumspect: to the extent they are so, they belong to the critical school.

That is our theoretical stance; but in practice, as we shall presently discover, there is rarely need to get that fancy, because most commentators on the a fortiori argument treat the issues relatively superficially.


2. Philo of Alexandria

Philo of Alexandria, aka Philo Judaeus or the Jew (20 BCE – 50 CE), has some importance in the history of philosophy, because he embodies an early confluence between Jewish religion (and philosophy) and Greek (religion and) philosophy, and also of course because he later had considerable influence on Christian religion (and philosophy). He interpreted the Jewish Bible (the Septuagint version, in Greek) allegorically, by adaptation of Greek and Hellenistic ideas (derived from works of Plato, Aristotle, and the Stoics) as well as reference to Jewish traditions.

I include him here because he lived, though outside of the Holy Land, in the Mishnaic period. According to Encyclopedia Judaica article on him, there is no mention of Philo is any ancient Jewish source other than Josephus (Antiquities 18:8). This could mean that mentions of him by rabbis of the Mishnaic and/or Talmudic periods have been lost; or that they did not know of his work; or that they knew of his work but disapproved of it. Philo has occasionally anticipated positions that rabbis in the Talmud later adopted, presumably independently, since they did not acknowledge his authorship. He was subsequently (much later) not considered by the rabbis as particularly kosher, despite his evidently intense faith in God and the Torah, and his services to the Jewish community of Alexandria[2], due no doubt partly to the Greek influences evident in his views and partly because Christians eventually adopted many of them.

Philo did not, to my knowledge, discuss a fortiori argument, but he did use it. I found one instance of a fortiori argument in a work by Philo, namely De Fuga et Inventione[3] (§84, p. 163), translated from the Greek original into French by Esther Starobinski-Safran: “No blasphemer has a right to forgiveness. For if those that cursed their mortal parents are chastised, what punishment ought to be meted to those who dare blaspheme the Father and Creator of the universe?” The minor premise of this argument comes from the Biblical passage: “May he who curses his father and his mother perish” (Exodus 21:17).[4]

The reference to blasphemy in the conclusion is derived from the earlier Biblical passage: “And if a man come presumptuously upon his neighbor, to slay him with guile; thou shalt take him from Mine altar, that he may die” (Exodus 21:14). According to Philo, an intentional murderer seeking refuge at the altar is effectively guilty of “blasphemy,” since by this gesture he blames God for his sins. The “blasphemy” perceived by Philo is, note well, not explicit, but implicit in the blame of God, which is in turn implicit in the act of seeking refuge with Him albeit guilty of intentional murder (as explained in §79-80)[5].

The proposed argument is thus: if someone who curses his parents deserves punishment, then someone who blasphemes God deserves punishment – possibly greater punishment[6]. Evidently, the argument is positive subjectal; and possibly a crescendo (note that Philo only asks “what punishment?” – he does not explicitly give an answer, whether equal or greater). Notice its similarity to the argument in the Torah concerning Miriam (Num. 12:14-15), which also proceeds from parents to God. I have not looked for use of a fortiori argument by Philo in other books.

I do not know why Philo finds it necessary to infer from the penalty for cursing parents an equal (or maybe greater) penalty for blaspheming God, when the Torah already states, in Leviticus 24:15-16: “Whoever curseth his God shall bear his sin. And he that blasphemeth the name of the Lord, he shall surely be put to death; all the congregation shall certainly stone him….” Perhaps he differentiated these two verses, since v. 15, which refers to cursing (Heb. yeqalel), does not mention a death penalty, whereas v. 16, which refers to blasphemy (Heb. noqev), does mention a death penalty[7]. But in that case, his purpose would be to prove (by a fortiori argument from parents) that the death penalty for cursing parents also applies to cursing God. But if so, why use the word “blaspheme” instead of “curse” in his stated argument? So there is some confusion or redundancy, here. In any case, note that he does not mention the death penalty at all in his argument, but only chastisement or punishment[8].

I should say that I noticed Philo’s a fortiori argument thanks to Starobinski-Safran, who in her introductory commentary (p. 38) reads the argument as: “If the outrage inflicted on parents is punished by [bodily] death (Ex. 21:15-16[9]), then the outrage to God, which is infinitely more grave, calls for death of the soul, an even heavier penalty” (again, my translation from the French). She does not specify ‘bodily’ death (my addition in square brackets), but we may assume this to be her intent from the explicit contrast to death “of the soul.” Note that the penalty for cursing parents is implicitly specified in Leviticus 20:9 as death by stoning (according to the rabbis, who point out that it there says: “his blood shall be upon him”), the same as the penalty explicitly specified in Lev. 21:16 for blasphemy. So as far as this-worldly punishment goes, the two cases are identical; but nothing is said (or denied) concerning other-worldly punishments.

The distinction here made by Starobinski-Safran between death (of the body) and death of the soul, and the proportionality of punishment she assumes, are not (as far as I can discern) explicitly given in Philo’s wording. The commentator is reading these details into the text. Perhaps she mentions “death of the soul” because no actual verbal blasphemy (meriting bodily death under Jewish law) has occurred in the case under consideration, but only blasphemy by implication (from the attempt to take refuge at the altar illicitly). She does not say what she precisely means by “death of the soul;” maybe she means the penalty of karet (usually translated as ‘excision’)? Certainly, even if Philo did have similar intentions in mind, the matter is open to debate, just as the intent of the argument concerning Miriam is debatable.

The trouble with allegory, as we see in this example, is that while it may provide a neat explanation and integration of the text at hand, there is a risk that it ends up being taken literally by people and thus have untoward legal consequences. The plain meaning of the Torah text referred to here is that an intentional killer who seeks refuge at the altar may be forcibly taken from there and executed anyway, because he committed murder and this crime is liable to the death penalty (Ex. 21:14). On the other hand, Philo allegorically explains the killer’s seeking refuge at the altar as effectively an act of blaming God for his sin, and thence accuses the killer of “blasphemy;” this gives the impression that when the killer is pulled away from the altar to be executed, he is being executed for “blasphemy;” but what he has done is not strictly-speaking blasphemy according to the law (Lev. 24:16).

An additional note. In today’s political context, when fundamentalist Moslems are trying to impose worldwide restrictions on the freedom of speech, by pressuring governments to outlaw what they call “blasphemy,” even by non-Moslems, and when many Western politicians, journalists, academics and sundry institutions, seem about ready to abjectly comply, it is important to add the following disclaimer. Whereas we frequently read in news bulletins about Moslems savagely lynching people they accuse of “blasphemy” (meaning, any doubt or criticism of Islamic beliefs), it must be stressed that Judaism – like Christianity – nowadays no longer applies past laws against blasphemy (meaning, speaking ill of God). Jews today believe in and are happy with the rights of thought, speech and conscience they have in the West, including Israel; and moreover they do not approve of capital punishment for blasphemy or other religious transgressions anywhere in the world. I am not aware of any segment of the Jewish population who would say otherwise.

The outlook of today’s Jews on Torah passages like those imposing the death penalty for blasphemy or for cursing parents is not literal, but symbolic. Their practical purpose is simply to make people aware of the great immorality of such acts. The more severe the penalty theoretically prescribed by the Torah for a certain act, the greater the implied immorality of that act. But the prescribed penalty is merely potential, not actual. That is, retribution for such immoral deeds is today de facto and de jure no longer a rabbinic responsibility; it is left to God[10]. Thus, the aim of such passages is no more than moral education – to raise awareness of the full weight of thoughts, words and deeds. Each person remains free to choose – and, of course, to pay the natural consequences for his or her choice (including Divine retribution, if God so chooses). And that is a good thing. We certainly do not want the development of a Taliban mentality in our community, and most of us would surely combat it as regressive nonsense if it ever arose.

In this context, it is worth citing the Mishna: “A Sanhedrin that effects an execution once in seven years, is branded a destructive tribunal. R. Eliezer b. Azariah says: once in seventy years. R. Tarfon and R. Akiba say: were we members of a Sanhedrin, no person would ever be put to death. [Thereupon] Rabban Simeon b. Gamaliel remarked, [yea] and they would also multiply shedders of blood in Israel!” (Makkoth 1:10)[11]. For most crimes deserving capital punishment, with the possible exception of murder, the penalty was never in fact applied past the Biblical period. Moreover, capital punishment was effectively abandoned altogether during the Mishnaic period, although of course it has remained a topic of theoretical discussions. Jewish law is clearly very different in spirit and practice from Islamic law.


3. Sifra

The Sifra is a halakhic midrash to Leviticus, which is occasionally called Torat Kohanim like the Torah book (Leviticus) that it is an exegesis of (JE[12]). According to Jacob Neusner (USA, b. 1932), in Rabbinic Literature: An Essential Guide[13], it is considered as dated ca. 300 CE (p. 3). I nevertheless include it in the present chapter – as an extra-Talmudic document, rather than as a post-Talmudic one, for lack of a better place. Indeed, though later than the Mishna, it is often referred to in the Talmud (JE). Neusner describes it as an effort to more thoroughly anchor the ‘oral Torah’ – meaning the Mishna (and the Tosefta) – in the ‘written Torah’ – i.e. essentially the Pentateuch (pp. 56-57). Neusner does not mention the work’s author, but JE discusses the matter[14].

My interest here is in certain features of Sifra’s logic that are mentioned by Neusner. I have not personally read Sifra, but take Neusner’s description of these features for granted. As Neusner puts it, Sifra’s purpose is to show that “the Mishnah is subordinated to Scripture and validated only through Scripture;” and it does so by means of a “critique of the Mishnah” which too often seems to rely on its own logic rather than explicitly refer to the Pentateuch (p. 58ff). This critique, as we shall see, focuses both on syllogistic and a fortiori logic. Sifra reportedly makes (or seems to make) assertions concerning these fields that simply, as I will definitively show in formal terms, cannot be upheld.

Syllogism. First, Sifra disputes “that we can classify things on our own by appeal to the traits or indicative characteristics, that is, utterly without reference to Scripture.” According to Sifra (or according to Neusner’s reading of it), “on our own, we cannot classify species into genera. Everything is different from everything else in some way. But Scripture tells us what thing are like what other things for what purposes, hence Scripture imposes on things the definitive classifications, not traits we discern in the things themselves.” And again:

“The thrust of Sifra’s authorship’s attack on taxonomic logic is easily discerned… things have so many and such diverse and contradictory indicative traits that, comparing one thing to something else, we can always distinguish one species from another. Even though we find something in common, we can also discern some other trait characteristic of one thing but not the other.”

If I understand such statements correctly, what Sifra is saying (or more probably, just implying through its many particular discursive acts, since rabbinic literature is rarely if ever so abstract in its approach) is that antithetical syllogisms can consistently be constructed. This would mean the following in formal terms:

All S1 are G;

No S2 are G;

and X is S1.

and X is S2.

Therefore, X is G.

Therefore, X is not G.


On the surface, such a situation might seem conceivable. The individual or class called X might be classified under species S1 in some respects and under species S2 in other respects; and S1 might fall under genus G, while S2 does not fall under genus G. The two major premises do not seem incompatible, since they concern different subjects, S1 and S2; and the two minor premises do not seem incompatible, since a term may well have different predicates, S1 and S2. Yet the two conclusions are clearly incompatible!

However, logic is quite able to show where in the said premises the contradiction lies, by constructing a 2nd figure syllogism using the two initial major premises:

No S2 are G

All S1 are G

Therefore, No S1 is S2.


Using the latter conclusion as our new major premise, it follows by syllogism that if X is S1, it cannot be S2, and vice versa. That is, despite surface appearances, the two species, S1 and S2, are in fact mutually exclusive, by virtue of being related in contrary ways to the genus G. Thus, in fact, the two minor premises ‘X is S1’ and ‘X is S2’ cannot both be true at once. Therefore, the contradiction between ‘X is G’ and ‘X is not G’ will in fact never arise.

That is to say, the apparent argument of Sifra that contradictions are possible if we rely only on logic, so that appeal to Scripture is necessary to help us choose one side or the other, is not credible. It only seems credible due to superficial appeal to syllogistic reasoning; but in fact such quandaries cannot occur in practice for someone who truly knows logic. It should be said that the supposition that such quandaries are conceivable is not peculiar to Sifra; Greek and Roman sophists have also often imagined them possible.

Of course, Sifra may not be saying what I have here assumed it to say. It may just be saying that without Scripture’s guidance we cannot know whether X is S1 or S2; or perhaps (more likely) we cannot know whether species S1 or S2 falls under genus G or not, where G is some law or legal ruling. Such arguments would be logically acceptable. But what is sure, anyway, is that no one can legitimately argue that the initially listed two 1st figure syllogisms are compatible. This is not open to discussion.

The same of course can be said with regard to rival hypothetical syllogisms:

If B1 then C;

If B2 then not C;

and if A then B1.

and if A then B2.

Therefore, if A then C.

Therefore, if A then not C.


The if–then premises of such arguments may offhand seem compatible, but their conflicting conclusions (assuming thesis A is not a paradoxical proposition) show them to be in fact incompatible. However, it should be obvious that this restriction is only applicable in cases of strict implication; if some of the implications involved are less firm, a situation of rivalry might conceivably occur. If A deductively implies B and B deductively implies C, then the conclusion is that A deductively implies C. But if A merely inductively implies B and/or B merely inductively implies C, then the conclusion is that A merely inductively implies C. Whereas ‘deductive implication’ signifies a 100% certainty, what I call ‘inductive implication’ refers to a looser relationship where the antecedent probably (with less than 100% certainty) implies the consequent. In such cases, the conclusions ‘if A, maybe then C’ and ‘if A, maybe then not C’ may both be justified, even though there is some degree of tension between them.

We can similarly admit that potential (though not actual) conflicts might occur in categorical syllogism. If for instance the rival syllogisms have as major premises that Most S1 are G and Most S2 are not G, and as minor premises that X is S1 and X is S2, then the conclusions will be respectively that X is probably G and X is probably not G. Though these two conclusions are in tension, they are not strictly speaking incompatible, and therefore they might conceivably occur together (especially if their probabilities are expressed so vaguely). It is probably the possibility of such tendencies to conflict that the author of Sifra had in mind. Another possibility is that there are unstated conditions to the premises of the categorical or hypothetical syllogisms, which make the rival arguments compatible although they superficially seem incompatible.

A fortiori argument. Second, concerning the argument a fortiori or qal vachomer, Neusner tells us in the name of Sifra that it “will not serve” – for “if on the basis of one set of traits that yield a given classification we place into hierarchical order two or more items on the basis of a different set of traits, we have either a different classification altogether or, much more commonly, simply a different hierarchy.” This is intended as a critique of “the Mishnah’s… logic of hierarchical classification.” To wit: “Things are not merely like or unlike, therefore following one rule or its opposite, Things are also weightier or less weighty.”

Here, the suggestion is that we can construct compatible a fortiori arguments, with reference to different middle terms (R1, R2), which yield contrary conclusions. This is a very similar suggestion to the previous one, but one specifically centered on a fortiori argument. It should again be stated that Sifra is not alone in this error (if it indeed makes it) – many people seem to think that such a situation is logically possible. Such people do not truly understand the logic involved, as I will now formally show. Consider the following two arguments:

P is more R1 than Q is;

P is more R2 than Q is;

and Q is R1 enough to be S.

and Q is R2 enough not to be S.

Therefore, P is R1 enough to be S.

Therefore, P is R2 enough not to be S.


On the surface, looking at the premises superficially, such a situation may seem possible. After all, such major premises certainly occur in practice. P and Q may be in a certain relation within the hierarchy R1 and in a very different (even opposite) relation in another hierarchy R2. But such differences could not give rise to contrary conclusions, one implying that ‘P is S’ and the other that ‘P is not S’ – for the simple reason that the minor premises are incompatible, one implying that ‘Q is S’ and the other that ‘Q is not S’. Thus, in fact, such a situation is logically inconceivable.

Thus, contrary to what Sifra seems to be (according to Neusner’s analysis) suggesting, we do not need to appeal to Scripture to choose between this hierarchy and that one so as to avoid contradiction. Two hierarchies that lead to contrary conclusions will never be true together. This is logically obvious and demonstrable. Of course, here again, we might defend Sifra by saying that it perhaps does not claim such antithetical a fortiori arguments, but merely says that Scripture is required to establish the major and/or minor premises. This would present no problem. But if the claim is indeed one to viable antitheses, it is untenable.

We could also defend Sifra by pointing out that many rival arguments that seem to adhere to the formal conflict presented above are in fact not intended so strictly. The major premises, which tell us that P is more R than Q, may be tacitly intended to mean that P is usually (though not always) more R than Q; and/or the minor premises may really have the form ‘Q is usually R enough to be S’; in which cases, the conclusions will also be probabilistic at best. Thus, there may be an appearance of conflict, when in fact there is only some logical tension. This, I believe, often occurs in practice, and may well be what the author of Sifra had in mind when he raised this issue. Or again, there may be unstated conditions to the premises of the rival a fortiori arguments, which make them compatible although they superficially seem incompatible.

To sum up and conclude. If, as Neusner seems to be implying, Sifra criticizes the Mishna on the ground that it relies on logic independently of Scripture, and that by doing so it opens itself to irresolvable contradictions, Sifra can and must be opposed on purely formal grounds. Logic does not lead to contradictions, but on the contrary deflects them, or uncovers and resolves them. If, however, Sifra is only saying that the Mishna has to refer to Scripture for its major and/or minor premises, i.e. for the content of its propositions – that is another matter entirely: it is then an issue not of logic, but of fact or even of moral and legal evaluation of fact.

But upon reflection, even in the latter cases we must distinguish between deductive and inductive logic. It is true that deductive logic cannot prescribe facts and even less so their evaluations (though it can be used to ensure that such prescriptions are kept internally consistent). But inductive logic certainly can strongly impinge on issues of fact or even of moral and legal evaluation of facts. Through experience and scientific method we can, for instances, contest that hare are to be classified with ruminants, or that there are no fish that have scales but lack fins. And moreover, from purely factual material, we can put in doubt the credibility of evaluations; for example, how we conceive sunrise and sunset to occur directly affects the times prescribed for beginning and ending the Sabbath.[15]


4. The Korach arguments

The Midrash called Bemidbar (Numbers) Rabbah, which is closely related to the Midrash called Tanhuma (named after a rabbi), is (or at least its earliest portions are) thought to date from the 5th century CE, apparently before the completion of the Babylonian Talmud[16]. What interests us in it here is its commentary regarding Numbers 16:1, which reads:

“‘Now Korach… took’. What is written in the preceding passage (Num. 15:38)? ‘Bid them that they make them… fringes (Heb. tzitzith)… and that they put with the fringe of each corner a thread of blue (Heb. techeleth)’. Korach jumped up and asked Moses: ‘If a cloak is entirely of blue, what is the law as regards its being exempted from the obligation of fringes?’ Moses answered him: ‘It is subject to the obligation of fringes’. Korach retorted: ‘A cloak that is entirely composed of blue cannot free itself from the obligation, yet the four blue threads do free it?!’

He [Korach] asked again: ‘If a house is full of Scriptural books, what is the law as regards its being exempt from the obligation of mezuzah [a small scroll with a selection of Torah verses, which is affixed to the doorposts of Jewish gates and homes]?’ He [Moses] answered him: ‘It is [still] under the obligation of having a mezuzah.’ He [Korach] argued: ‘The whole Torah, which contains two hundred and seventy-five sections, cannot exempt the house, yet the one section in the mezuzah exempts it?! These are things which you have not been commanded, but you are inventing them out of your own mind!’”

There is, note well, no evidence of this discourse in the Torah itself; it only appears much later in history, in the Midrash. These two arguments attributed to Korach are traditionally regarded as samples of qal vachomer, although (I presume) most commentators view them as qal vachomer of a fallacious sort. For that reason, they are especially interesting, in that they illustrate a possibility of erroneous reasoning in the a fortiori mode. We may paraphrase the two arguments briefly as follows:

a) If mere threads of blue wool (on each of the four corners) are sufficient to make a garment lawful to wear, then surely if the whole garment is made of such blue wool (even if without the corner threads) it is likewise lawful.

b) If a few passages of the Torah (in a mezuzah affixed to the doorposts) are sufficient to make a house lawful to live in, then surely if the whole Torah is stored in a house (even if without a mezuzah) it is likewise lawful.

These two arguments have the following form in common: If a small quantity of something (Q) is sufficiently in accord with the norm (R) to make so-and-so be declared lawful (S), then surely a large quantity of that thing (P) is sufficiently in accord with the norm (R) to make so-and-so be declared lawful (S). The preceding hypothetical proposition comprises the minor premise and conclusion of the a fortiori argument. Its tacit major premise is therefore: a large quantity of something (P) is more in accord with the norm (R) than a small quantity of same (Q). The argument is clearly positive subjectal, from minor to major.

What is wrong with this argument? The answer is obvious: its major premise does not have the logical necessity it is implied to have. While on the surface it might seem like a large quantity is preferable to a small one, this is not necessarily the case, because the two quantities may present significant qualitative differences. That is, the terms of the proposed major premise are incompletely specified, and therein lies the fallacy. The minor premise, regarding the sufficiency of a small quantity (Q) to satisfy the norm (R) for a certain result (S), may be true only provided that this quantity fulfills certain qualitative criteria (which may have additional quantitative aspects). If the larger quantity (P) does not fulfill these same qualitative criteria, it may well not be able to satisfy the norm (R) for a certain result (S). Therefore, the major premise should, to be truly universal, more precisely read: a large quantity of something precisely specified (P) is more in accord with the norm (R) than a small quantity of the exact same thing (Q).

Returning now to the two Korach arguments for the purpose of illustration, we can say the following. In both cases, the sophistry consisted in occulting the details given in brackets. In (a), what makes the garment kosher is not merely that it contains blue threads, but that it contains them on the four corners. In (b), what makes the house kosher is not merely that it contains Torah words, but that it contains them on the doorposts. The details do matter – they are not expendable. Therefore, in effect, Korach’s two arguments may be said to commit the fallacy of having more than four terms. The major and minor terms in the major premise are made to appear the same as the subjects in the minor premise and conclusion, but they are in fact different from them.

The two arguments might have been a bit more credible, had they respectively advocated an inference from a garment not made of blue wool yet having kosher tzitzit, to a garment entirely made of blue wool as well as having kosher tzitzit; or from a small mezuzah affixed to the doorposts, to a giant mezuzah affixed to doorposts. But even then, such inference would not be necessarily true, because there is no formal reason why the law might not interdict garments made entirely of blue wool (even with kosher tzitzit) or giant mezuzot (even affixed to doorposts). The major premise in use in any argument must be in fact true, for a true conclusion to be drawn from it. Very rarely is the major premise logically necessary; it is only so if its contradictory is self-contradictory. In most cases, the major premise has to be determined empirically – or, in such a religious context, be given in the proof text.

In my opinion, the two arguments attributed to Korach are not factual reports, but post facto fabrications with an educational purpose. Because: either Korach had some brains and could see for himself the fallacy of his reasoning, but he cynically proposed these arguments anyway, thinking no one else would notice; or he was not intelligent enough to realize his own errors of logic. But in either case, surely Moses had the intelligence needed to see through the fallacy, and would have publicly reproved Korach for his dishonesty or his intellectual deficiency, so as to stop the rebellion in its tracks by discrediting its leader. However, since according to the Torah account Divine intervention was used, we can infer that Moses did not use this logical means. I think the arguments were imagined by the author(s) of the Midrash for three reasons. One was to flesh out the story of Korach with some (most likely anachronistic) Talmudic-style legal debate, showing up the perversity and stupidity of the rebel. Another was perhaps to intimidate eventual readers, saying in effect: if you behave like Korach, expressing doubts in the law of Moses, you will be punished like Korach. The third was perhaps to teach people some a fortiori logic, to make sure they do not make similar errors of reasoning.

However, this is not how some later commentators have understood the purpose of this Midrash. They have taken it to mean, not that Korach was arguing fallaciously, but that Korach was being too logical, so that we ought to learn from this story to suspend our rational judgment now and then. For instance, R. Ephraim Buchwald, in an essay called “The Excesses of Rationality” (2007)[17], explains the matter as follows:

“According to Korach, human logic always prevails. Korach is certain that the rational processes are the ultimate determinant of right and wrong. Since the laws handed down from Moses and Sinai have no internal logic, they must be summarily rejected. It is for that very reason that parashat Chukat follows parashat Korach. The Torah, in Numbers 19:2, declares: ‘Zoat chukat HaTorah’: This is the statute of the Torah! There is no logic to the laws of the Red Heifer. Reason is of little value when it comes to this irrational ritual. The Red Heifer comes to confirm to Korach and all his fellow rationalists, that the ultimate authority is the law of Moses and Sinai, not mortal logic! … While Judaism in general is a most rational and logical faith, true believers must eventually conclude that there are certain aspects of the religion that one can not rationally fathom or master. It is that leap of faith that a believer must make, and this doubt that we all must overcome, and for which we are ultimately rewarded.”

This is obviously, in view of what our analysis above has demonstrated, an erroneous interpretation of the Midrash. The commentator evidently does not have great logical knowhow, since he seems to think that the two Korach arguments are valid. He is therefore not qualified to discuss the limits of human logic. Korach cannot be presented on the basis of the two arguments attributed to him as a “rationalist,” or proponent of reason, since they are in fact not in accord with logic. If he was not an idiot, he was a sophist who cynically faked logical argument. In the Midrashic story, Moses does not answer Korach by sullenly saying: “your arguments are sound, but I will stick dogmatically to my positions,” as our commentator implies. Rather, I’d say, Moses refutes Korach, as often done in Talmudic debate, by denying his conclusion, thereby tacitly implying that at least one of his premises is incorrect; and since the minor premise is in accord with the law of Moses, it must be the major premise that is mistaken. In other words, the correct interpretation is that Moses does not concede Korach’s reasoning powers, but rather challenges them.

R. Buchwald is, of course, relying on the traditional commentaries regarding the statute of the red heifer (Numbers 19). They find it odd that the ritually clean people involved in preparing the ashes of the red heifer should be made unclean (v. 7, 8, 10), while those ashes are used to ritually clean people who are unclean due to having come in contact with a dead person (v. 12). Rashi comments, citing Yoma 67b: “Because Satan and the nations of the world taunt Israel, saying, ‘What is this commandment, and what purpose does it have?’ Therefore, the Torah uses the term “statute.” I have decreed it; You have no right to challenge it.

But in truth, what has this to do with logic? It is not logically inconceivable that the same substance (the ashes of the red heifer) might have one effect (ritual uncleanliness) on one set of people (the people producing or handling it) and another, opposite effect (ritual cleaning) on another set of people (the people it is sprinkled on). Such complex relations can readily be found in nature – e.g. a chemical substance might be harmful to one kind of organism and beneficial to another. Or consider, to take an extreme example, the particle-wave duality in quantum mechanics, where the same phenomenon seems different viewed from different perspectives.

The red heifer ritual is no more ‘illogical’ than the ritual of sacrificing animals to purify people of their sins, or the rituals of tzitzit or mezuzah, or that of matza, or those of shofar, lulav and succah, or any other religious ritual. When dealing with the supernatural, everything is equally artificial, i.e. inexplicable by natural means. Rituals are not given in nature, or rationally inferred from it. Such truths (if they are indeed true) can only be known through revelation or similar (alleged) extraordinary means. Belief in them – at least in the case of people without prophetic powers of their own, and maybe even for prophets – depends on faith. Even prayer, the most natural expression of belief in God, depends on faith.[18]

Moreover, the inexplicability of alleged spiritual practices is not a reflection on human logic. Human logic does not promise omniscience. There are many things we do not, and perhaps can never, understand, even in the natural world; all the more so, in the (presumed) spiritual world. The fact that there are limits (whether short or long term) to the power of logic can never be used as an argument against the power of logic within its natural limits. There is no logical argument by which logic might be invalidated, because such argument would be claiming to have some logic, and thus be self-defeating. Even if logic admittedly cannot predict all truth, it can certainly eliminate quite a bit of falsehood. For this reason, we should not hasten to ditch it just because it does not deliver everything we wish for.

R. Buchwald’s attempt to compare the Korach argumentation to the red heifer statute is, anyway, ingenuous. He regards the Korach arguments as perplexing because though sound (in his view), they lead to conclusions that are contrary-to-fact (i.e. to Biblical fact); and he regards the red heifer ashes as perplexing, because (I presume, though he does not say so) they have contrary behavior patterns in relation to different subjects. But even supposing these two perplexities are justified, they are certainly logically very different and cannot be lumped together. If they are, as he supposes, both ‘illogical’, they are ‘illogical’ in significantly different ways.

In any case, there is one kind of illogic that no amount of faith can ignore or cure – and that is any breach of the laws of thought. Faith is acceptable where there is some gap or uncertainty in knowledge; but if a claim – however ‘authoritative’ – goes against these fundamental laws, we can be absolutely sure it is incorrect. This applies equally well to other-worldly claims as to this-worldly ones. Our reaction in such case should not be blind faith, but to demand a credible resolution of the paradox. This is the adult, mentally-healthy reaction to such conundrums. In this sense, logic has much to say even about spiritual claims. Logic is mankind’s main protection against falsehood of any kind.


5. Saadia Gaon

When I found out that Saadia Gaon, ben Yosef (Egypt, ca. 882 – Iraq, 942), had written a short book, entitled in Hebrew Perush Shelosh Esre Midot (Explanation of the Thirteen Hermeneutic Principles), and actually found a copy of it on the Internet[19], I was overjoyed, hoping to find in it some interesting original insights into qal vachomer. However, upon reading it (with the help of a friend), I was rather disappointed. Saadia Gaon there in fact says nothing theoretical about qal vachomer, other than to say that it may be used for non-legal as well as legal purposes.

He does not analyze the argument in any way, but is content to present five rabbinical examples of it – without, by the way, explaining why he chose those particular ones. If a man is obligated to take good care of his second wife, all the more so his first wife. Since, if one finds one’s enemy’s strayed animal, one is obligated to return it to him, it follows a fortiori that one must do that for a friend. And so forth. All these examples are in fact legal in content; he does not actually give any with non-legal content, but simply repeats (somewhat lamely, as if he could not think of any offhand) that non-legal content is possible. That’s it. Of course, examples have their importance; but they are certainly not enough.

According to the introduction (in French) to the Œuvres Complètes[20], Saadia does not always thus limit his commentary on the midot to examples, but in some cases gives explanations, even if his explanations are sometimes obscure (e.g. as to what distinguishes the 7th and 8th rules). In any case, he does not go into the details concerning the rules. Moreover, we are told, Saadia considers that anyone has a right to put forward new applications of the thirteen rules, which liberty is far from admitted by other commentators. Nevertheless, I should add, Saadia is known to have defended the rabbinic tradition that the thirteen midot were Divinely revealed to Moses at Mt. Sinai[21]. He no doubt did so in the context of his polemics with the Karaites, who of course rejected rabbinic interpretation[22].

So I was taught, anyhow; but I have not offhand found an explicit statement to that effect. Perhaps he merely implied it. We might, for example, so interpret his citation of Sanhedrin 88b, “With the increase in numbers of the disciples of Shammai and Hillel, who did not advance far enough in their studies, the controversies increased” (The Book of Doctrines and Beliefs, pp. 32-33), to explain the existence of disagreements between rabbis. The implication is that originally, when the Torah was first given, there were no doubts; these developed over time, when levels of learning diminished. This matter could be further pursued, but I will leave it at that for now and move on.

I would like, rather, to take this opportunity to quote Saadia Gaon on the value of empiricism and rationalism:

“Furthermore [authentic tradition] verifies for us the validity of the intuition of reason. It enjoins us, namely, to speak the truth and not to lie. Thus it says: For my mouth shall utter truth…. Besides that it confirms for us the validity of knowledge inferred by logical necessity, [that is to say] that whatever leads to the rejection of the perception of the senses or rational intuition is false…. Next [tradition] informs us that all sciences are [ultimately] based on what we grasp with our aforementioned senses, from which they are deduced and derived.” (The Book of Beliefs and Opinions, Pp. 18-19.)

It is also interesting to note here certain rules for inference set by Saadia Gaon:

“In endeavoring to establish the truth of inferential knowledge, we shall henceforth be on guard against these five possible forms of mistakes, namely: (1) that it does not conflict with knowledge established by sense-perception; (2) that is does not conflict with knowledge established by Reason; (3) that it should not conflict with some other truths; (4) that it should not be self-contradictory; still more, that it should not (5) involve a difficulty more serious than the one intended to avoid.” (The Book of Doctrines and Beliefs, p. 42.)[23]

Taking Saadia at his word, we can predict that were he placed squarely before new facts and shown the validity of certain logical inferences, he would have the intellectual and moral integrity to admit them, and would not dogmatically insist on contrary, more traditional ‘facts’ or ‘inferences’. Unfortunately, there are still today some people who think they do religion a service by refusing to face facts and logic. Just yesterday, I had the hilarious experience of watching an online video showing an Islamic apologist claiming in 2007 on Iraqi TV that the earth is flat and much larger than the sun, which is also flat![24] Fortunately, apologists for Judaism never go so far; but they also sometimes show considerable resistance to change.

I say this here because readers of the present volume must obviously be prepared to adapt to new discoveries and insights, and not cling at all costs to traditional views. I want to emphasize in passing that to be critical does not signify to be hostile and willfully negative. Though critical, I have personally no desire to contradict or denigrate our religious tradition. Not all critical commentators are so moderate in their views or intentions; some are very eager to find fault with the rabbis or the Torah. For my part, I would prefer to always justify the rabbis and the Torah, and confirm their wisdom, and it is only reluctantly that I criticize some of their claims. Nevertheless, I try to be scrupulously fair and honest – i.e. to be scientific – and to admit that there is a problem when there indeed appears to be one. This is the golden mean – neither dishonestly attacking nor dishonestly defending, but sincerely looking for the truth.


6. Rashi and Tosafot

Concerning the contribution of Rashi, i.e. R. Shlomo ben Yitzhak (France, 1040-1105), to the understanding of the hermeneutic principles, Mielziner tells us that he “occasionally explained, in his lucid way, the single rules where they are applied in the Talmudic discussions.” There is, he adds, “a separate treatise on the hermeneutic rules ascribed to this commentator and published under the title of Perush Rashi al Hamidot,” which however “seems to be spurious.” This is found “in Kobak’s Jeschurun, vi, Hebrew part, pp. 38-44, 201-204; the remaining commentaries on the thirteen rules are enumerated by [Adolf] Jellinek in Ḳonṭres ha-Kelalim, Nos. 163-175.”

I have no access to these various sources, so must make do with a more ad hoc treatment. The question that interests me here is: firstly, what does Rashi say about the qal vachomer in Numbers 12:14-15 (and eventually, the other cases found in the Torah, and maybe also those in the Nakh)? And secondly, what does he say about the discussion concerning the dayo principle in Baba Qama 25a-b? I shall also try and determine the viewpoints on these topics of Rashi successors, the Tosafot. The basic issue to my mind is: do these post-Talmudic commentators accept the idea seemingly advocated in the Gemara (based on a baraita) that qal vachomer is naturally ‘proportional’ and the dayo principle is designed to reign in such velleity in it? The answer to expect is, obviously: yes, they do.

First, let me mention in passing Rashi’s comments on other a fortiori arguments appearing in the Torah. Concerning Genesis 44:8, all Rashi says is: “This is one of the ten a fortiori inferences that are found in Scripture, which are all listed in Bereishit Rabbah (92:7).” For Exodus 6:12: he is likewise content to say: “This is one of the ten a fortiori inferences in the [Tanakh],” although he additionally explains Moses’ speech defect as an “obstruction of the lips.” He has no comment regarding Deuteronomy 31:27. Evidently, Rashi does not question the Midrashic statistic of just ten qal vachomer in the Tanakh.

As regards Numbers 12:14, Rashi’s comment is: “If her [Miriam’s] father were to display, to her, an angry face, would she not be humiliated for seven days? Certainly, then, in the case of the Divine Presence, [she should be humiliated] for fourteen days. However, it is sufficient that the derivative equal the source of its derivation. Therefore, even with My rebuke, let her be confined for seven days.” As can be seen, this is just a repetition of the thesis given in a baraita transmitted in the said Gemara. If we look for Rashi’s comment opposite that baraita in the Gemara, we find that he has none. That means he considers the matter sufficiently clear as it is and sees no point in adding anything to it. There you have it. Rashi does not ask or answer any theoretical questions concerning qal vachomer reasoning or the dayo principle, but takes them for granted.

Rashi comments somewhat more extensively on another qal vachomer and dayo principle application, namely in Tractate Zevachim 69b (Seder Kodashim)[25]. There, the Mishna explicitly refers to both the argument (by R. Meir) and the application of the principle (by R. Jose), and the Gemara expounds almost exactly in the same words as in Baba Qama 25a, saying: “Does not R. Meir accept the principle of dayo [it is sufficient]? Surely the principle of dayo is Biblical, for it was taught: How is a qal vachomer applied? And the Lord said unto Moses: If her father had but spit in her face, should she not hide in shame seven days? How much more should a divine reproof necessitate [shame for] fourteen days; but it is sufficient for that which is inferred by an argument to be like the premise!” But Rashi does not add much, other than to (rightly) point out that the qal vachomer in the Miriam story is implicit rather than explicit.

That Rashi uncritically accepts the common notion that a fortiori argument is ‘proportional’ is evident not only in his acceptance without comment of the “fourteen days” given in the Gemara of Baba Qama 25a, but also in his comment to Genesis 4:24 (which, it should be noted, is not included in the traditional list of ten a fortiori arguments in the Tanakh). There, based on Tanchuma Bereshit 11, Rashi elucidates Lamekh’s statement “If Cain shall be avenged sevenfold, truly Lamekh seventy and seven-fold” as a qal vachomer, as follows: “If Cain killed intentionally, [and yet] his punishment was delayed for seven generations, [then] I, who killed unintentionally, surely will have my punishment deferred for many periods of seven generations.”

Note, however, that though in the case of Miriam Rashi acknowledges the dayo principle, he does not mention its application in the case of Lamekh; nor does he tell us why he doesn’t. I suggest that the reason why it seems reasonable in one case and not the other is the following. In the example of Miriam, the conclusion (14 days penalty) is more stringent than the minor premise (7 days penalty), in accord with the principle of midah keneged midah (measure for measure), so the dayo principle is required to mitigate the punishment; whereas, in the example of Lamekh, the conclusion, though likewise quantitatively superior (77 instead of 7 generations), is more lenient as regards the sanction than the minor premise (i.e. signifies longer deferral of punishment), and so is not subject to the dayo principle (which if applied would speed up the punishment).

Successors of Rashi, known as Tosafot[26], comment on the Mishna and Gemara in more detail. Essentially, they subscribe to the scenario apparently advocated by the Gemara when interpreting the Mishna of Baba Qama 25a. That is to say, they accept uncritically that R. Tarfon’s two arguments are a crescendo. Nevertheless, to their credit, they consider that his first and second try are logically (and not merely rhetorically) different, due to reshuffling. They also agree with R. Tarfon that his second argument is able to avoid the dayo restriction as set by the Sages against his first argument, because while the first argues from half to full damages, the second argues from full to full damages. As a result of which, they intelligently explain the Sages’ continued insistence on dayo application with reference to premises antecedent to the qal vachomer itself.[27]

However, since Tosafot accept the Gemara reading of both arguments as a crescendo, they also accept the “fourteen days” notion proposed by the Gemara (following a baraita), i.e. the claim that qal vachomer naturally yields a ‘proportional’ conclusion. Without questioning this claim, they only focus on trying to explain this number (rather than any other large number[28]). An explanation they give is to refer to seven days as the minimum period of quarantine in the event of leprosy (Leviticus 13:4); a more severe confinement must be at least another seven day period.[29] Tosafot also consequently make efforts to defend the obscure notion, ascribed by the Gemara to R. Tarfon, that the dayo principle can on occasion be ignored, specifically where it would “defeat the purpose of” the qal vachomer.[30]

But such glosses are superficial in their concerns; they gloss over the more serious underlying issues. I have shown in my detailed analysis of this sugya in the two preceding chapters (7-8) that we cannot countenance some of the commonplace interpretations of this Mishna and Gemara without getting ensnared in a multitude of logical errors, which make at least some of the rabbis involved look very foolish. Once the logical errors are understood, it is seen that many of the explanations proposed in the Gemara, and later by others, including Tosafot, are vain attempts to uphold a very wobbly structure. If we want to redeem the rabbis involved, we must approach the whole matter much more lucidly, and consider a moral instead of logical explanation of the dayo principle.

I do not want to seem to be dismissing Tosafot in a debonair manner, being fully aware of their importance, but simply see no point in repeating here what I demonstrated earlier. So I invite the reader to go there.


7. Kol zeh assim

A thorough study of the logic in Tosafot, and even just of its a fortiori logic, would doubtless result in a thick and interesting book. Not having the necessary language skills, I cannot myself undertake such a study; but I would certainly recommend that someone duly qualified in both logic (especially as taught in the present volume) and the Talmud do the job. But we can here get an idea of the logical resourcefulness of Tosafot through one example, which has to do Baba Qama 25a. This is thanks to Yisrael Ury, who in his book Charting the Sea of Talmud[31] provides an English translation of a commentary by Tosafot and some useful clarifications as to its intents[32]. This passage of Tosafot is only incidentally concerned with Baba Qama 25a, using it to illustrate a certain form of argument; so we shall not here cite all of it, but only quote or paraphrase the parts of it relevant to our narrower purpose.

The Tosafot commentary, whose precise author is not named, proceeds in three stages, we might say. In a first stage, it refers to one of the arguments originally given in the Mishna Baba Qama 2:5, which it paraphrases as follows:

“Whereas tooth and foot, for which damages are not paid for damage done in the public domain, yet are liable for full damages for damages done in the domain of the damaged party, then horn, for which half damages are paid for damage done in the public domain, certainly should pay full damages for damage done in the damaged party’s domain.”

Looking at this argument, we easily recognize the first argument of R. Tarfon, since it proceeds by mentioning first tooth & foot damage in the public and private domains and then horn damage in the same domains. As I have shown previously, this argument can be put in standard a fortiori form as follows:

Private property damage (P) implies more legal liability (R) than public domain damage (Q) [as we know by extrapolation from the case of tooth & foot[33]].

Public domain damage (Q) implies legal liability (Rq) enough to necessitate half payment for damage by horn (Sq) [this is derived from the Torah[34]].

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

Therefore, private property damage (P) implies legal liability (Rp) enough to necessitate full payment for damage by horn (Sp = more than Sq).


We shall here label this argument as argument (1a). Notice that it is positive antecedental. The major premise is obtained by generalization from the givens regarding damage by tooth & foot. The major and minor terms are ‘damage on private property’ (P) and ‘damage on public domain’ (Q). The middle term is ‘legal liability’ (R); and the subsidiary term is ‘to make the payment for damage by horn have a certain magnitude’ (S). In fact, note well, the argument is not purely a fortiori but a crescendo, since the magnitude of S in the conclusion is greater than that in the minor premise. This means there is a tacit premise to take into consideration, about the proportionality of ‘payment due’ (S) to ‘legal liability’ (R).

Although not directly mentioned by Tosafot, the second argument of R. Tarfon is, as we shall see, also (if not more) relevant to the present discussion; so we shall restate it here, in standard form:

Horn damage (P) implies more legal liability (R) than tooth & foot damage (Q) [as we know by extrapolation from the case of public domain].

Tooth & foot damage (Q) implies legal liability (R) enough to necessitate full payment for damage on private property (S).

Therefore, horn damage (P) implies legal liability (R) enough to necessitate full payment for damage on private property (S).


We shall here label this argument as argument (1b)[35]. Notice that it is also positive antecedental. The major premise is, here, obtained by generalization from the givens regarding damage on public grounds. However, the major and minor terms are ‘damage by horn’ (P) and ‘damage by tooth & foot’ (Q). The middle term is again ‘legal liability’ (R); but the subsidiary term is ‘to make the payment for damage on private property full’ (S). Note that this argument is purely a fortiori, and not a crescendo. But it is clear that it could also be stated in a crescendo form, and that if it were would yield the same conclusion (viz. full payment for horn damage on private property), since no payment greater than full is admitted by the Torah or the rabbis. For this reason, it suffices to state it in pure form.

The second stage of our Tosafot commentary concerns an objection, and the reply to it, put forward in the past by a commentator called the Ri (presumably this refers to R. Isaac ben Samuel, a 12th century French Tosafist). The Ri’s objection is described as follows:

“But consider that the damages of tooth and foot are common!”

To which objection the Ri himself replies:

“Paying full damages in the damaged party’s domain is not a severity (chumra) to be used in an objection (pirka), for it does not at all cause tooth and foot to lead to the requirement of half damages for damage done in the public domain as does horn.”

I have to say that I only understood the Ri’s objection thanks to the clarifications given by Ury, which I presume are traditional. He explains it as follows: because damage caused by tooth & foot is “commonplace,” the ox’s owner is obligated to take extra care “that his animal not cause damage when it comes in proximity to the property of others;” so that if such damage does indeed occur, he is more open to blame. As regards damage by horn, since the goring of another animal by an ox is “a rare event,” it is unexpected by the ox’s owner and he is justified in not taking special precautions against it; so that if such damage does indeed occur, he is not as liable.

Thus, the Ri’s objection means that, whereas on public grounds tooth & foot damage implies less liability than horn damage (no liability against half liability), as the Mishna teaches (based on the Torah), it may well be that on private property tooth & foot damage implies more liability than horn damage (full liability against, say, only half). This reasoning thus constitutes an objection to the original Mishna argument – i.e. it is designed to show that the conclusion that seems inevitable in the latter (namely, full liability for damage by horn) is perhaps not so inevitable. Putting this reasoning in standard form, we obtain the following:

Tooth & foot damage (P) implies more legal liability (R) than horn damage (Q) [since the former is common and the latter is uncommon].

Tooth & foot damage (P) implies legal liability (R) enough to necessitate full payment for damage on private property (S).

From which it does not follow that horn damage (Q) implies legal liability (R) enough to necessitate full payment for damage on private property (S).


We shall label this as argument (2a). This argument should be compared to the second argument of R. Tarfon, which we labeled (1b). Notice that they are very similar, except that the major premise has been reversed so that the putative conclusion no longer follows. In (2a), tooth & foot damage is the major term, while the horn damage is the minor term. The middle term is unchanged. The subject of the minor premise is unchanged (still tooth & foot damage), but now this subject is the major term. The subject of the putative conclusion is unchanged (still horn damage), but now this subject is the minor term. Since the format of the attempted a fortiori argument is still positive antecedental, inference from major to minor is illicit. Thus, we can no longer draw the conclusion of (1b) that ‘horn damage on private property necessitates full payment’.

Such conclusion is now a non sequitur – it is not excluded by the new premises (it does not contradict them), but it is not justified by them, either. This argument is not itself an a fortiori argument, note well, but merely serves to put in doubt R. Tarfon’s second a fortiori argument. It obstructs his conclusion, without needing to actually contradict it. It rejects his argument by reversing its major premise[36]. If, as R. Tarfon takes it, the owner of an ox is more responsible for horn damage than for tooth & foot damage, then the inference from full liability in the latter to full liability in the former is perfectly logical. But if, as the Ri contends with reference to ‘frequencies of occurrence’, the owner of an ox is more responsible for tooth & foot damage than for horn damage, then the inference from full liability in the former to full liability in the latter is debatable.

Another way to look at the objection (2a) is to say that the major premise of R. Tarfon’s first a fortiori argument (1a) – which take note is the one that Tosafot mentions – is no longer granted. This premise, viz. “private property damage (P) implies more legal liability (R) than public domain damage (Q),” was obtained by generalization from the given that damage by tooth & foot implies no liability in the public domain and full liability on private property. However, now the objection makes us aware that this generalization is open to question, since the conditions for legal liability are not the same in the case of damage by horn, due to there being different frequencies of occurrence. Thus, analogy is blocked. What applies to tooth & foot does not necessarily apply to horn.

This, then, is the objection conceived of as possible by the Ri, stated in more formal terms. Let us now try to understand the way he himself neutralized the objection. Remember that we are given by the Mishna (based on certain Torah verses) that tooth & foot damage on public grounds does not necessitate any payment for damages, while horn damage on public grounds necessitates payment of half damages. On this basis, the Ri replies to the objection by saying: (i) that “paying full damages in the damaged party’s domain” ought to “cause tooth and foot to lead to the requirement of half damages for damage done in the public domain as does horn;” and (ii) that since this consequence does not in fact occur, “paying full damages in the damaged party’s domain is not a severity to be used in an objection.”

The first part of his remark (i) refers to an a fortiori argument with the same major premise as (2a) combined with the given information about horn damage on public grounds necessitating half payment; these premises would conclude that tooth & foot damage on public grounds necessitates half payment (at least – more than half, i.e. full, if proportionality is applied). We may label this argument (2b), and put it in standard a fortiori form (positive antecedental, from minor to major) as follows:

Tooth & foot damage (P) implies more legal liability (R) than horn damage (Q) [since the former is common and the latter is uncommon].

Horn damage (Q) implies legal liability (R) enough to necessitate half payment for damage on public grounds (S).

Therefore, tooth & foot damage (P) implies legal liability (R) enough to necessitate half payment for damage on public grounds (S).


But, the Ri tells us in the second part of his remark (ii), this conclusion cannot be true, since the Torah tells us that tooth and foot damage on public grounds is exempt from any payment! Therefore, he concludes, this last a fortiori argument must be rejected. This last argument, which is a reductio ad absurdum, can be labeled argument (2c). It says: since the minor premise of argument (2b) is Torah given, and the process is valid, the only way to reject it is by abandoning its major premise[37]. That is to say, tooth & foot damage cannot be taken to imply more legal liability than horn damage on the basis of the former being more common and the latter being less common, as the objection (2a) initially attempts. Thus, the Ri shows that the objection, although reasonable sounding in itself, leads to absurdity and must be dropped.

The third stage of the Tosafot commentary we are analyzing is introduced by the statement in Hebrew: “vekhol zeh assim bakal vachomer,” meaning in English: “and all this I will put into the a fortiori argument”[38]. The unnamed Tosafist then argues as follows[39]:

“Whereas tooth and foot, even though their damages are common, they are exempt from payment for damage done in the public domain, but necessitate a full payment for damage done on the property of the injured party – then horn, even though its damage is not common, and it necessitates payment of half damages for damage done in the public domain, does it not follow (lit. eino din) that it necessitates payment of full damages for damage done on the property of the injured party?”

The question posed is of course rhetorical – the author’s intention is clearly that the proposed conclusion does follow. Where the author says “even though” (lit. af al pi) – as in even though the damage is common or even though the damage is uncommon – he is obviously referring back to the objection of the Ri, which suggests an inverse proportionality between frequency of occurrence and legal liability, i.e. that the more common a certain kind of damage is, the less the liability for it, and conversely that the less common a certain kind of damage is, the more the liability for it.

The purpose of this Tosafot commentary is, as its introduction (“all this I will put into the a fortiori argument”) implies, to somehow merge together the original argument of the Mishna and the Ri’s objection and his retort to the objection. Obviously, “all this” refers to the two preceding stages. Our job now is to judge whether the argument here proposed by Tosafot does indeed perform what it is designed to do. We can, for a start, put the proposed argument in standard form, as follows:

Tooth & foot damage (P) is more common (R) than horn damage (Q) [since the former is common and the latter not so].

Yet, tooth & foot damage (P) is common (R) not enough to make the ox’s owner exempt from full payment for damages on private property (S)[since tooth & foot damage on private property necessitates full payment[40]].

Therefore, horn damage (Q) is common (R) not enough to make the ox’s owner exempt from full payment for damages on private property (S)[whence, horn damage on private property does necessitate full payment].


We shall label this argument as argument (3), or refer to it more familiarly and briefly as “kol zeh assim.” As can be seen, it is negative subjectal in form (it goes major to minor). Its major and minor terms are respectively ‘damage by tooth & foot’ (P) and ‘damage by horn’ (Q). Its middle term is ‘frequency of occurrence’ (R), and its subsidiary term is ‘to make the ox’s owner exempt from full payment for damages on private property’ (S). The major premise is known to us by generalization from the frequencies of occurrence observed in the public domain, where P and Q are characterized as common and uncommon, respectively. The minor premise is based on Torah information. The argument has to be put in negative subjectal form to be validated, note well, because it has as its subjects the two causes of damage and it goes from major to minor. Note that the subsidiary term is identical in minor premise and conclusion; this means that the argument is purely a fortiori. The negative conclusion can finally be restated in the more familiar positive form (this being a simple eduction).

Alternatively, we could formulate the argument in positive subjectal form (going from minor to major) as follows. Note the change of polarity in the middle and subsidiary terms, and the change in the order of the terms tooth & foot and horn. The net result is the same:

Horn damage (P) is more uncommon (R) than tooth & foot damage (Q).

Yet, tooth & foot damage (Q) is uncommon (R) enough to make the ox’s owner have to pay in full for damage on private property (S).

Therefore, horn damage (P) is uncommon (R) enough to make the ox’s owner have to pay in full for damage on private property (S).


The question we must ask here is: what does Tosafot mean by “all this”? In other words, what features of the preceding arguments (1a), (1b), (2a), (2b) and (2c), is argument (3) really referring to? “All this” is rather vague and needs to be specified more precisely. The two essential features that the Tosafot kol zeh assim argument shares with the discourse preceding it are the following: first, it has the same final conclusion as R. Tarfon’s two arguments, viz. that damage by horn on private property entails full payment; second, it takes into consideration the Ri’s objection, in that it is built around the observed fact of tooth & foot damage being more common than horn damage, and at the same time, it takes into consideration the Ri’s reply to the objection, in that the kol zeh assim argument abstains from inferring greater liability for tooth & foot damage than for horn damage from their different frequencies of occurrence.

Thus, it can be said that the unnamed Tosafist’s kol zeh assim argument does indeed, in a certain sense, conflate all the preceding arguments. Nevertheless, it does not annul and replace the preceding discourse. Especially note that we cannot formally derive the kol zeh assim argument from either or both of R. Tarfon’s arguments, or derive them from it. However, unlike the Ri’s objection (2a), this argument (3) is compatible with R. Tarfon’s (1a) and (1b), since the major premise here has a different middle term. Thus, Tosafot’s argument is a new, additional argument – not a substitute for the others. Its major premise comes from the Ri’s commentary, taking both the objection (2a) and the retort to it (2b) and (2c) into consideration; its minor premise comes from the Mishna, and before that the Torah; and its conclusion agrees with that of R. Tarfon. Therefore, the kol zeh assim argument is a clever artifice, a way to allude to a number of issues in one shot.

Nevertheless, it should be stressed that the Tosafot argument is logically quite redundant, once we have become aware of the Ri’s objection and his own retort to it. For the Ri’s objection to R. Tarfon’s original argument is that ‘frequency of occurrence’ may have an impact on ‘legal liability’, while his own retort to the objection is that if this impact were admitted a contradiction to Torah law would ensue; whence it follows that such objection is inadmissible. Once this inadmissibility is realized, there is no utility whatsoever in at all mentioning ‘frequency of occurrence’ as this Tosafot commentary so glibly does, since all connotation of ‘legal liability’ has been permanently removed from it.

Moreover, note well, Tosafot’s argument does not constitute decisive proof of anything, just as R. Tarfon’s arguments do not. We have to admit that these arguments are not decisive, anyway, if we wish to leave room in the discussion for the Sages’ dayo principle. For the conclusion that damage by horn on private property entails full payment is eventually denied by the Sages (the colleagues of R. Tarfon in the Mishna), when they say and insist: “dayo—it is enough!” Their preferred conclusion is that damage by horn on private property entails only half payment. The Tosafot commentary (at least that part of it translated for us by Ury, which is all I have on hand) does not deal with this important issue here, its purpose being only to illustrate kol zeh assim argument (i.e. it refers to Baba Qama 25a only incidentally, here, so as to clarify another issue entirely).

Upon reflection. After writing the above, it occurred to me that Tosafot’s argument (3), unlike R. Tarfon’s arguments (1a) and (1b), is immune to both of the Sages’ dayo objections. R. Tarfon’s two arguments, you may recall, were neutralized by the Sages’ two dayo objections, because they both relied in some way on the information that damage by horn on public grounds obligates the ox’s owner to half compensation, in order to arrive at the conclusion that damage by horn on private grounds entails full compensation. The kol zeh assim argument differs radically from those in that it does not rely on the said information to arrive at the same conclusion. This means that Tosafot’s argument is not logically affected by the Sages’ dayo rebuttals, and conversely that they are logically unable to neutralize it.

As far as I know, Tosafot did not realize the collateral damage his kol zeh assim argument was capable of causing in this sugya. His argument, as we have seen, was only intended to save some of the insight of the Ri on ‘frequency of occurrence’ (the leftover, as it were, after the Ri’s objection was neutralized by his retort) and to reaffirm R. Tarfon’s conclusion. But actually, Tosafot’s argument does not merely buttress R. Tarfon’s – it definitely proves it, since it is not subject to reproof by dayo. Does this then mean that the Sages’ dayo objections, and therefore (at least in this particular case) the dayo principle, are null and void? Hopefully not – but then, under what conditions, exactly, could we still sustain them?

Since Tosafot’s argument (3) is formally clearly valid, we can only find fault with its content – i.e. by denying its major premise and/or minor premise. The major premise, “tooth & foot damage is more common than horn damage,” does not seem easily deniable assuming it is based on empirical observation; if it is not based on empirical observation, however, it could be denied as factually inaccurate. The minor premise, “tooth & foot damage is common not enough to make the ox’s owner exempt from full payment for damages on private property,” was, you may recall, based on the information given in the Torah (Exodus 22:4[41]) that tooth & foot damage on private property necessitates full payment.

It could be argued that this Torah passage does not actually specify full compensation, but rather refers to the quality of the feed or food restituted, leaving open the issue of quantity. But the rabbis, to my knowledge, do not accept this interpretation, and probably would not do so. We could still, however, deny Tosafot’s minor premise by denying that there is a threshold of the middle term, i.e. a frequency of occurrence as of which the ox’s owner is exempt from full payment for damages on private property and before which he is not. This is a more subtle yet technically possible approach, aimed at still more thoroughly detaching the concept of legal liability from that of commonness.

That is, if we say that no matter how common or uncommon tooth & foot damage is, this statistical feature has no effect whatever on the legal liability for the owner of an ox to pay (in full or whatever) for depredations on private property – then the minor premise of the kol zeh assim argument is dissolved, and the conclusion of that argument (concerning horn) no longer logically follows. Indeed, if we reflect on the meaning of the minor premise, we see that it does not make sense, anyway. It seems to suggest that if tooth & foot damage was more common than it is, it might at some point be common enough to exempt from full compensation on private property, whereas the Torah seems to unconditionally impose full payment.

Thus, it is possible in various ways to attack Tosafot’s argument, and the probably best way to do so is by totally disconnecting the issue of legal liability from that of frequency of occurrence. In that event, nothing of the Ri’s initial objection would be left over, and the “all this” of the Tosafot all this I will put claim ceases to be credible. This would be, I daresay, an acceptable price to pay if we wish to continue to uphold the Sages’ dayo principle – at least in the present context, and more likely in all contexts, since the present context is in fact the root context for that principle, on which all subsequent appeals to that principle in the Talmud historically depend.

One thing is sure, we cannot cling to both the Sages’ dayo principle and Tosafot’s present kol zeh assim a fortiori argument – they are logically incompatible. We have to choose between them. It is obvious that we ought to choose to hang on to the dayo principle, which is more ancient (about late 1st – early 2nd century CE) and seems more important in Talmudic discourse, rather than on to the kol zeh assim argument, which appears much later in Jewish history (about the 12th cent. CE, say) and whose loss has less impact on Jewish jurisprudence. Therefore, the kol zeh assim argument seems condemned – at least in the present context (i.e. with reference to Mishna Baba Qama 2:5), even if a similar form of argument might be attempted in some other context(s) without unpleasant consequences.


8. Maimonides

Rabbi Moshe ben Maimon (Spain, 1135 – Egypt, 1204), known in Jewish literature by the acronym “Rambam,” and more widely as Moses Maimonides[42], wrote at about the age of sixteen[43] a treatise on logic, called Maqala Fi-Sana’at Al-Mantiq (in Arabic), first translated into Hebrew by Moses Ibn-Tibbon (France, ca. 1240-1283)[44], and thence into other languages, including Latin (Basel, 1527), German (19th century), French and English (20th century). The edition I have in hand is a 1982 reprint of a 1935 critical edition with the text in Hebrew and in French (translation by Moise Ventura); its title is Milot haHigayon in Hebrew and Terminologie Logique in French (meaning, in English, Terms of Logic)[45]. It also contains the original Arabic-language version (extant parts) written in Hebrew letters. The scope of this work is considerable; it is not a mere lexicon, as its title suggests. It is an earnest teaching of formal logic and many of the more philosophical concepts surrounding it.

Briefly put, the contents of Maimonides’ study are as follows, chapter by chapter. 1) proposition: subject, predicate, affirmation, negation; 2) quantity: universal, particular, indeterminate, singular; 3) terms, copula, tenses, modalities; 4) oppositions, modalities; 5) immediate inferences, conversion, inversion; 6) syllogism, premise, conclusion, major, middle and minor terms; 7) figures and moods of the syllogism, conclusive-inconclusive, hypothetical and disjunctive arguments, direct reduction and reduction ad absurdum, induction, analogy, juridical reasoning; 8) sensory experience, axioms of reason and their derivatives, widespread opinions, traditional assertions, true propositions, demonstrative syllogism, dialectical syllogism, rhetorical syllogism, sophistical syllogism, poetic syllogism, enthymeme; 9) the four causes, material, formal, efficient and final, proximate and remote causes, the four elements, the material substratum; 10) species, genus, difference, attributes per se and per accidens, substance, definition, description, the ten categories[46]; 11) essential and accidental, potentiality, actuality, contraries with or without an intermediary, property, privation, relative, opposite; 12) anteriority in time, in nature, in rank, in merit, in cause; 13) names of various sorts, synonyms, homonyms, amphibologies, metaphors; 14) logos as rational faculty, thought and verbal discourse, logic as art and science, divisions of theoretical and practical philosophy, logic as instrument of all other sciences.

Effectively, Maimonides was importing into Jewish culture some very powerful tools developed by Aristotle and his successors[47]. But, while this book contains an interesting exposé of the main elements of Aristotelian logic, what is surprising is that it does not mention a fortiori argument[48]. One might understand such silence regarding most of the other of the hermeneutic rules, since they are principles of interpretation used specifically in Talmudic contexts. But the a fortiori argument is, as well as one of the means used in the Talmud for exegetic purposes, a universal method of reasoning. One would therefore have expected Maimonides to have included this form of argument in his treatise on logic[49]. However, two excuses can be put forward on his behalf. The first is that Aristotle himself, and subsequent logicians to the time of Maimonides, hardly mentioned this form of argument and never treated it in any significant detail. The second is that Maimonides wrote this book at a very young age, and perhaps was not then fully aware of the great significance of a fortiori argument in Talmudic discourse.

However, it does not appear that Maimonides subsequently dealt with a fortiori argument, or for that matter the other hermeneutic rules. Maimonides of course freely used a fortiori discourse in his works. In his Guide for the Perplexed, for instance, I found 23 instances. But, such use is practice, not theory (I have checked them all). As far as I know, he did not anywhere specifically stop and reflect on the reason why this argument works, even though he was exceptionally conscious of logical issues. The following are some examples of use of a fortiori argument by Maimonides, which I found in the said work:

  • “If the firmament, with that which is over it, be supposed to be above the heavens, it would a fortiori seem to be unreal and incomprehensible.” (Part 2, chapter 30.)
  • “We thus learn that his prophetic perception was different from that of the Patriarchs, and excelled it; a fortiori it must have excelled that of other prophets before Moses.” (Part 2, chapter 35.)
  • “The best test is the rejection, abstention, and contempt of bodily pleasures: for this is the first condition of men, and a fortiori of prophets.” (Part 2, chapter 40.)
  • “Man is superior to everything formed of earthy matter, but not to other beings; he is found exceedingly inferior when his existence is compared with that of the spheres, and a fortiori when compared with that of the Intelligences.” (Part 3, chapter 13.)
  • “But I agree with Aristotle as regards all other living beings, and a fortiori as regards plants and all the rest of earthly creatures.” (Part 3, chapter 16.)
  • “The law forbids us to imitate the heathen in any of these deeds, and a fortiori to adopt them entirely.” (Part 3, chapter 29.)

Although Maimonides does not mention or discuss a fortiori argument in his treatise on logic, Terms of Logic, he does (in chapter 7) mention inductive reasoning and argument by analogy, both of which are involved in the background of a fortiori thinking. He describes induction as follows: “it proceeds from some particular assertions, admitted as true due to experience, to arrive at a general proposition that can be made into a premise of syllogism.” Regarding analogy he says: “when one of two objects that resemble each other by a certain trait has some attribute that is not apparent in the other [object], we affirm of the latter [object] the same attribute.” Also noteworthy is the great credence Maimonides gives to sense data and to the axioms of reason and deductive inferences from them (in chapter 8), saying: “all that is perceived by a healthy organ is indubitably true. And the same can be said of rational data… and their derivatives.”[50]

These remarks and similar ones of his strike me as very ‘modern’, because in the past (and in very many cases still today) people thought of logic as essentially a deductive enterprise. Maimonides here devotes some space to the more inductive and analogical aspects of reasoning (which are also found in Aristotle, of course[51]). Yet his outlook is not quite modern, in that he does not mention the all-important proviso that a generalization is always tentative, i.e. subject to rejection or particularization if subsequent experience reveals instances to the contrary. That is to say, he makes the common error of focusing on the positive side and ignoring the negative side. The same is true, of course, of analogy – it is an inductive act, which may later be repudiated; i.e. its credibility remains dependent on further experience. Moreover, Maimonides does not mention that induction and analogy are closely related logical acts. First, as already said, in that analogy is inductive. And second, in that every generality is a statement that the individuals constituting it have some attribute in common, i.e. are analogous in some respect; this was also known to Aristotle[52].

Still, Maimonides’ outlook is considerably different from that of Talmudic scholars who preceded him[53]. This is true not only in his frank acknowledgment of experience, axiom, induction and analogy, as important elements of human judgment, as already stated, but even with regard to deduction. Having been influenced by Aristotelian logic and philosophy, Maimonides’ understanding of deductive reasoning was no doubt a lot more structured and rigorous. I would speculate that, even if he did not openly criticize any of the Talmudic inferential processes, he was personally aware of the tenuousness of some of the arguments used. This may perhaps explain, at least in part, his viewpoint concerning the hermeneutic rules in his Sefer Hamitzvot (Book of Commandments). In this important halakhic work, he adopts a more sweeping and severe position than the Talmud itself regarding the legislative effectiveness of the hermeneutic principles, including a fortiori argument (presumably, since he does not explicitly except it):

“And now I will begin to discuss the Principles (shorashim, roots), totaling fourteen, to be relied upon in enumerating the mitzvot… The Second Principle: Do not include laws which are derived from one of the 13 principles of Torah interpretation [of Rabbi Yishmael] or from a ribui [an extra word, letter, etc. in a Scriptural verse].”[54]

Mielziner comments on this ruling as follows: “Maimonides holds that laws derived from the Mosaic law by means of the hermeneutic rules are, in general, not to be regarded as biblical laws (min hatorah) except when expressly characterized as such in the Talmud. But this somewhat rational view is strongly criticized by Nachmanides (in his annotations to that book) who shows that from the Talmudical standpoint every law which the Rabbis derived by the authoritative interpretation from sacred Scripture, has the character and sanctity of a Mosaic Law”[55]. To quote Nachmanides: “all elucidated in the Talmud through one of the thirteen methods are words of Torah and they are the interpretation of the Torah which was told to Moshe”[56]. Be that as it may, the fact remains that for Maimonides, even if an inference a fortiori is highly deductive, it does not pass the Biblical status of its premises onto its conclusion.

Other, more recent authors concur with this assessment. Halbertal writes: “Maimonides also defines mitsvot of a Scriptural status in terms of their traditional pedigree and their non-controversial nature. As a result of this definition, all laws derived from the application of legal hermeneutical principles, such as a fortiori and analogy, are relegated to Rabbinic status, not that of Scripture. This definition is a direct result of Maimonides’ theory of mitsvot, according to which direct linkage with Sinaitic Revelation is incompatible – at least at the level of Scripture – with controversial laws. This definition brought Maimonides into conflict with Nahmanides, who strongly criticized the Maimonidean position on this issue. Nahmanides’ critique is based upon both the corpus of Talmudic law and considerations of an ideological nature.” Sinclair likewise: “According to Maimonides, the status of Scriptural law (de’oraita) is conferred by tradition alone upon laws which are free of controversy. Any controversial law is ipso facto Rabbinical in nature, including a law which is derived from the Scriptural text by means of hermeneutic principles such as a fortiori and analogy.”[57]

We can dig more deeply into Maimonides’ thinking on this issue in his commentary on the Mishna, found in his introduction to Seder Zeraim[58]. He does believe that “all the commandments were stated with their generalities, specifics and fine details at Sinai…. [Moshe was given] the 613 precepts with their explanations; the commandments in writing, and the explanations by oral transmission.” He makes a similar statement in the introduction to his Mishneh Torah, citing Exodus 24:12. This is the doctrine that the revelation consisted of two components, viz. a written Torah and an oral Torah, which is of course relevant to any discussion of the hermeneutic principles.

Moreover, the Rambam explains, for those, like Joshua and the Elders, who received the Torah entirely and directly from Moshe Rabbeinu, there were no doubts or disputes (machlokot); it is only those who came after them that had to resort to inferences (svara), i.e. to the thirteen midot. Regarding the latter, some of the inferences made convinced everyone; but in other cases, there were disagreements concerning the inferences to be made: in such cases the sages resorted to majority vote (rov). Thus, apparently, he believed the hermeneutic rules were given at Sinai, even while acknowledging that some disagreements arose over time concerning them too.

I have not anywhere found more specific comments by Maimonides on the individual hermeneutic principles, and in particular on qal vachomer and the dayo principle. It may, however, be that he has scattered significant remarks in his halakhic works: I do not know. So I will stop here.


9. More on medieval authors

In this section, we will examine bits and pieces of additional information drawn from various sources regarding a fortiori and other reasoning found in Jewish medieval literature.

Moise Ventura, in his very fine 1935 critical edition of Maimonides’ Terms of Logic, which was based on thorough comparative research in numerous past editions and manuscripts, as well as various commentaries, wrote somewhat wonderingly:

“When one browses through the Hebrew manuscripts in the great libraries, one is struck to see the considerable number of works written in the Middle Ages to abridge or comment on Aristotle’s Logic. Among these writings, some are due to Moslem authors, whose works were subsequently translated from Arabic to Hebrew, and the others to Jewish authors who wrote in Hebrew on this subject. Almost all these works have remained unpublished.…” (p. 18, my translation from French).

He goes on, asking why Maimonides’ work received such special attention, that it was so often translated, published and commented on. Was it his authority or the literary qualities of the work that earned it such exceptional popularity? His explanation is that Maimonides’ book (written while yet in his teens) was not intended to vulgarize Aristotle’s Organon, but to prepare the ground for his own philosophical system in the framework of Judaism, which came to maturity decades later (when he was fifty-five) in his Guide for the Perplexed[59]. It would, of course, be very interesting to examine all the above mentioned manuscripts and to see what is said in them, if anything, concerning a fortiori argument, and to evaluate the level of understanding of such argument exhibited in them.

In a recent but unfortunately too brief article by Aviram Ravitsky, entitled “Aristotelian Logic and Talmudic Methodology: The Commentaries On The 13 Hermeneutic Principles And Their Application Of Logic,” included in Schumann’s collection Judaic Logic[60], we are informed that there are “probably… dozens of treatises” on this subject:

“In 1917 Aaron Freimann published a bibliographic list[61] of commentaries on the thirteen principles, in which he counted over fifty different commentaries. Today, some sixty manuscripts are known to consist of commentaries on the principles (though some of them overlap)” (p. 120).

Ravitsky rightly distinguishes between “material” and “formal” commentaries. The former class, which most commentaries fall into, make use of examples drawn from the Talmud and related literature to illustrate and explain hermeneutic principles. The latter refer to Aristotelian logic and philosophy to elucidate them (the qualification of ‘Aristotelian’ being here broadly understood to include later developments). Ravitsky informs us, based on his careful examination of some thirty documents, that “a recognizable trend of [such more ‘formal’] commentaries… began in the 14th century” (p. 117)[62].

This article is of considerable interest to us here, since a few of these commentaries (hopefully their most significant elements) are actually quoted. This gives us a chance to discover and evaluate the thinking of their authors, especially regarding a fortiori argument. The first quoted is R. Avraham Elijah Cohen (late 14th – early 15th centuries); referring to the argument of qal vachomer, he writes:

“And I contend that this would be… explained by the art of logic. […] A bull is robust compared to a donkey, and nonetheless it is not robust compared to a man; a cat is not as robust as a donkey, all the more it is not robust compared to a man.” (P. 122.)

Ravitsky regards this as an “instance of formalistic commentaries” because it uses non-halakhic concepts (in this case, features of animals) instead of the legal or rabbinical content usually found in Talmudic examples. I would not however call this statement an example of formal analysis, even if it does refer abstractly to “the art of logic.” But I do agree that its use of a secular illustration is significant (although, to be precise, such illustrations also do occasionally occur in the Talmud and related literature[63]), since it is indicative of recognition that the argument can be used in any context. In any event, let us examine this example in formal terms:

Donkeys (D) are less robust (R) than bulls (B).

Even so, bulls (B) are less robust (R) than men (A).

Cats (C) are less robust (R) than donkeys (D).

Therefore, cats (C) are less robust (R) than men (A).


Although some sort of a fortiori argument is explicitly intended here, if we label the five terms involved as shown above we see that what we are actually given is a chain of three quantitative comparisons (relative to R) resulting in a fourth: “A > B and B > D and D > C; therefore, A > C.” But this cannot be considered as a fortiori argument, for the simple reason that there is no predication involved – i.e. A, which seems to play the role of subsidiary term, is not a predicate of B or C (or even D). In other words, the author of this example did not (at least, not in this instance) understand a fortiori argument! (Nor, incidentally, does Ravitsky show understanding, since he does not raise the issue!)

The next author quoted is R. Isaac Aboab of Castile (1433-1493), a disciple of R. Isaac Canpanton (whose school, Ravitsky tells us (p. 139), was distinguished in that its interest in Aristotelian logic was not merely philosophical, but had a potential impact on halakha). R. Isaac Aboab describes “the essence of the argument” as follows:

“A fortiori is a principle that teaches the scale of astringency from lenient to strict, and the scale of extenuation from strict to lenient.” (P. 123.)

He then gives the following illustration of this argument: Even though Reuven received a scholarship, he was not given a place; therefore, Shimon, who did not get a scholarship, would “all the more” not be given a place. This author demonstrates some understanding of a fortiori argument, both in his abstract description of it and in the example he proposes for it. We can show this sample argument valid by casting it in standard (negative subjectal) form:

Reuven (P) was given greater regard (R) than Shimon (Q) was given, since the former received a scholarship whereas the latter did not.

Yet, Reuven (P) was not given enough regard (R) to be given a place (S).

Therefore, all the more, Shimon (Q) will not be given enough regard (R) to be given a place (S).


I wonder whether R. Isaac Aboab was the first to express a fortiori argument in this terminology of “strict” and “lenient,” which has remained the rabbinical norm to this day? This is a historical question that is worth investigating. If the answer is yes, that would make him a significant figure in the development of a fortiori logic[64]. Be that as it may, we have to note that his above quoted description of the argument is a bit vague. What does he mean by “the scale of astringency from lenient to strict” and “the scale of extenuation from strict to lenient”? All it tells us is that stringency increases as we go from lenient to strict and decreases as we go from strict to lenient. We have to refer to his example to get a better grip on what he is trying to say.

As for his example, it only illustrates the negative subjectal mood of a fortiori argument. He does not (at least, not in the segment of his discourse that Ravitsky has quoted for us) give examples of the other three (or seven) valid moods. On the other hand, Ravitsky mentions that this author noticed “the discrepancy … between the single Hebrew term of qal vachomer and the two forms of the application of this principle” (p. 132), as the earlier quotation (“from lenient to strict” and “from strict to lenient”) makes clear. We can wonder whether Isaac Aboab might not be the first Jew to have noticed this difference of direction. Nevertheless, it is not clear whether he identified it as one between positive and negative subjectal moods, or as one between positive subjectal and positive predicatal moods; I do not suppose he did either.

Moreover, although Ravitsky classifies this effort as “formal” analysis, and there is indeed some formalism in it insofar as abstract terms like “strict” and “lenient” are used, it is strictly-speaking not very formal. Aristotle’s theory of syllogism may be characterized as formal in that he used abstract symbols like A, B, Γ (or labels like “the minor,” “the middle,” and “the major,” or ordinal numbers) instead of concrete terms, and because he systematically developed all possible figures and moods and determined with reference to the laws of thought which are valid or invalid. R. Isaac Aboab, on the other hand, is still apparently stuck in the realm of sample concrete labels like “Reuven” and “Shimon,” and makes no attempt at systematization or validation. This is, admittedly, closer to formal than the earlier Talmudic total absorption in concrete cases; but it is not yet quite formal.

Regarding the issue of validity, Ravitsky quotes the unknown author of Sharei Tsedek (apparently in Spain, ca. late 14th – early 15th centuries)[65]:

“The reason [R. Ishmael] began with this principle [i.e. qal vachomer] is that it features self-explanatory truth more than the other [hermeneutic] principles, alike the first figure of logical syllogism that is more self-evident than the rest of the figures.” (P. 124.)

What this author seems to be saying is that a fortiori argument is (at least, comparatively to the other hermeneutic principles) self-evident, just as first figure syllogism is (compared to the other figures) an irreducible primary. But the analogy in fact stopped there. He was certainly not claiming, as Ravitsky seems to suggest, that by placing it in first position in his list of thirteen principles R. Ishmael was implying that the other hermeneutic principles can be reduced to qal vachomer, just as the other figures of syllogism can be verified by means of the first. In truth, as I have shown in my book Judaic Logic, a fortiori argument is not an irreducible primary, but is reducible to simpler forms of argument, including hypothetical syllogisms and quantity comparisons; as for the other hermeneutic principles, see my comments there.

Ravitsky goes on to quote other medieval authors regarding other hermeneutic principles: Moses of Narbonne, who equated gezerah shavah to analogical inference; R. Avraham Elijah Cohen, who analyzed mah matsinu in terms of the distinctive properties of subjects; R. David Ibn Bilia, who analyzed klal uphrat using the terminology of genus and species. We need not in the present context discuss these issues. I only wish to remark in passing that I agree with Ravitsky that the influence of Aristotelian logic (and more broadly, philosophy) is evident in all these cases.

Two other authors are quoted by Ravitsky on the subject of a fortiori argument. One is R. Immanuel ben Isaac Aboab (ca. 1555-1628), a great-grandson of the earlier quoted Isaac Aboab of Castille. He explained as it follows: “The initial principle is qal vachomer. Meaning, the Torah is expounded by the element and manner that lead from lenient to strict, and is what the logicians refer to as: Argumentum a minori ad maius, vel a fortiori” (pp. 131-2). The other is the much latter Isaac Samuel Reggio (1784-1855), who says essentially the same thing, viz. that the rabbinical hermeneutic principles are mostly “based on the rules of the art of logic. E.g. the first principle, named qal vachomer is extremely fluent amongst the scholars of the art of logic under the title of ‘Argumentatio a minori ad majus’…” (p. 131). Note that both these definitions focus solely on minor to major reasoning, ignoring major to minor, and making no distinction between positive and negative, subjectal and predicatal, forms.

These two authors are quoted in support of the notion that the hermeneutic principles and the art of logic are, on the whole, in agreement. Another, much earlier author, R. Hillel ben Samuel of Verona (ca. 1220 – ca. 1295), went so far in this optimistic vein as to declare sweepingly: “the Sages of the Talmud established all of their scrutinies (sic) on the methods of syllogism and demonstration” (p. 127)[66]. Naturally, some rabbinical authorities expressed their disagreement with such naïve statements. For instance, Isaac ben Joseph Ibn Polgar (ca. 14th century), who argued: “when they begin to study logic, foolery and error enter their minds, for they think that the conditions of syllogism are necessary in legal-religious matters [… whereas] our sacred Torah is expounded by the thirteen principles alone” (p. 129).

Having studied the issues involved in great detail in my earlier work Judaic Logic, I would place myself somewhere in between these various opinions; I will not go into detail here, but merely repeat some conclusions. The rabbinical hermeneutic principles are variously logical: some are quite logical (notably qal vachomer), some are more or less so, some are not logical (non sequiturs), and some are antithetical to logic (antinomies). Thus, it is inaccurate to regard them as either all logical or all illogical. In my view, they all ought to have been logical; logic is not something one can discard at will.

The claim that the hermeneutic principles were originally a secret code applicable only to Torah interpretation may seem conceivable prima facie; but once one considers it seriously, it is seen to be difficult to uphold. Briefly put: for a start, since this code is not given in the written Torah, to claim it as given in the oral Torah as a tool for the justification of the oral Torah is a circular argument. Secondly, one can imagine a secret code as being necessary, assuming that God wanted only some people (namely the Jewish people, or perhaps more specifically the rabbis) to truly understand the Torah; but once this code is no longer secret, the past argument in its favor falls apart.

Mielziner, who I quoted on this topic in an earlier section (9.1), rightly identifies the hermeneutic principles as developed ad hoc by the rabbis over time, as means by which traditional laws existing and developed from pre-Mishnaic times to post-Talmudic times, could be anchored – by hook or by crook, if I may so put it – to the written Torah. I suggested much the same in my own study, Judaic Logic. Ravitsky, I think, shows the same awareness when he defines them as “basic and fundamental rules by which the oral tradition is related to the Scriptures” (note the guardedly vague term ‘related’ he uses, p. 117).

It is of course not possible with so limited a sample to describe and evaluate medieval attempts to relate the rabbinical hermeneutic principles, and in particular the first of these, viz. qal vachomer, to Aristotelian logic. We cannot even be sure that Ravitsky, on whose brief study we have heavily relied in the present section, selected and quoted the most significant authors and works. We have seen that this commentator did not notice certain weaknesses in logic in his selections, notably R. Avraham Elijah Cohen’s confusion between simple quantitative comparisons and a fortiori argument. Moreover, we saw that Ravitsky was too quick to acknowledge as ‘formal’ arguments that were still, strictly speaking, material. Furthermore, his uncritical acceptance of Saul Lieberman’s claims in “Rabbinic Interpretation of Scripture” makes me doubt his judgment. So we cannot take for granted that he acquitted his set task in a fully representative manner.[67]

Much more detailed studies would be needed to arrive at some solid historical conclusions. Nevertheless, based on the data we have at hand, we can tentatively propose the following conclusions. Medieval Jewish commentators wished to correlate (at least some of) the rabbinical hermeneutic principles with Aristotelian logic, out of a desire to reconcile the philosophy and science of their day with the worldview and claims of the Torah and subsequent Judaic tradition. Some of these commentators were themselves rabbis, some were lay philosophers. They were on the whole not critical, in the modern sense; rather, they had faith that the two fields of human endeavor could indeed be reconciled. They remained in the mainstream of Rabbinism, although presumably some passed over to Karaism.

Logic was regarded by many as a neutral discipline, without conceivable negative impact on religion. R. Jedaiah ben Abraham Bedersi Ha-Penini (ca. 1270 – ca. 1340), for instance, wrote “this art [i.e. logic] is comprised of knowledge or views that would result in neither harm nor benefit to faith” (p. 135)[68]. Some rabbis, on the contrary, realized the dangers posed by logic for the Judaic viewpoint. As Ravitsky points out, “they cast restrictions on the study of logic or even opposed it;” some of them could well see that logic is “a discipline that educates for rational criticism, or even animadversion, of the type that would make it difficult to accept religious truths” (p. 134). Nowadays, no one can contest that the study of logic has both a positive and a negative impact on religious belief; mostly, perhaps, the latter. Ravitsky clearly agrees when he concludes: “Attempts to reconcile [the two are] farfetched and artificial.”

From a logician’s perspective (as far as I can see so far), these various medieval commentators cannot be claimed to have entirely succeeded in their endeavor to correlate hermeneutics and logic, because: (a) their approach was not formal enough; (b) they were not sufficiently systematic; and (c) they did not make the required efforts of validation. Aristotle and his successors had given them examples of formalism, systematic treatment and validation, in relation to the syllogism and other forms of argument; but they had not done the same job in relation to a fortiori argument or the logic of causation. So the later commentators were not able to draw on such past work. Of course, many of the hermeneutic principles could be explicated somewhat in non-formal terms. For instance, rules like gezerah shavah or klal uphrat could be adequately discussed informally. But some, such as qal vachomer and binyan av, to name but two, could only be dealt with credibly by formal means. These means were, in fact, largely available in the epoch under study; but apparently none of the medieval commentators surveyed had the logical competence needed to apply them.


10. Moshe Chaim Luzzatto

Although the Ramchal deserves in many ways to be classed as a modern author, I have put him here so as to count him among the post-Talmudic Jewish logicians[69]. Surprisingly, this important author is not even mentioned in many standard studies of Talmudic logic, such as Mielziner’s; somehow, and quite unfairly, he has passed unnoticed. Ravitsky, likewise, does not mention him.

Formulation. R. Moshe Chaim Luzzatto (Italy-Netherlands-Israel, 1707-1746), also known in Jewish literature by his acronym “Ramchal,” wrote two books on logic, namely Sepher haHigayon (The Book of Logic, 1741) and Derech Tevunot[70] (The Way of Understanding, 1742); he also wrote a couple of books on grammar which may have some logical significance, though I have not read them. Concerning Derech Tevunot, which is more intended as a teaching of Talmudic reasoning than of logic in general, I wrote the following in my Judaic Logic review of it (or more precisely, of a 1989 translation of it, called The Ways of Reason[71]):

As well, he mentions a-fortiori argument, in the form: X1 is greater than X2, and X2 is Y, therefore X1 is Y; we may notice, however … that the middle term which explains and justifies the process, being the respect in which X1 and X2 are compared, is lacking, and also that he is not apparently aware of the formal varieties of the argument (but the form of his argument is correct, as a positive subjectal).[72]

But at the time I wrote that comment, I had not seen Sepher haHigayon (i.e. the English translation of it, called The Book of Logic)[73], for the simple reason that it was first published in 1995, the same year my said book was first published. About this work by Ramchal much needs be said, but what will be said here is only what it says about the a fortiori argument (in chapter 14). I have to admit that R. Luzzatto’s understanding of a fortiori argument is surprisingly original and advanced[74]. On second thoughts, we should perhaps not be surprised; the mid-18th century is after all not so long ago, and writers of that period are normally counted as ‘early modern’.

Quantified commensurates [are terms that] share a common quality, but not in the same degree. One exhibits a Greater degree and the other a Lesser degree of the same quality. Rules of Greater Degree: 1. To whichever subject the greater is predicated, the lesser will also be predicated… 2. What cannot be predicated to the greater term cannot be predicated be predicated to the lesser… Rules of Lesser Degree: 1. To whichever subject the lesser is not predicated, the greater will not be predicated either. 2. Whatever is affirmed about the lesser will surely be affirmed about the greater.” (Pp. 89-90.)[75]

Let us examine these four “rules,” and see to which of the standard models of a fortiori argument they respectively correspond. The middle term (R) of each argument is left tacit in these rules, but may be identified with the “common quality shared in different degrees” referred to in the definition. The major (greater), minor (lesser) and subsidiary terms (P, Q, S) are noted symbolically (as P, Q, and S, respectively) by me in each rule. I give the Hebrew original so everyone can verify the accuracy of the translation:

  • מי שיפל בו היתר, יפל בו הפחות. “To whichever subject (S) the greater (P) is predicated, the lesser (Q) will also be predicated.” This, being major to minor and positive, refers to positive predicatal argument; note that P and Q are predicates.
  • מה שלא יפל ביתר, לא יפל בפחות. “What (S) cannot be predicated to the greater term (P) cannot be predicated to the lesser (Q)[76].” This, being major to minor and negative, refers to negative subjectal argument; note that P and Q are subjects.
  • מי שלא יפל בו פחות, לא יפל בו יתר. “To whichever subject (S) the lesser (Q) is not predicated, the greater (P) will not be predicated either.” This, being minor to major and negative, refers to negative predicatal argument; note that P and Q are predicates.
  • מי שמחיב בפחות, כל שכן ביתר. “Whatever (S) is affirmed about the lesser (Q) will surely be affirmed about the greater (P).” This, being minor to major and positive, refers to positive subjectal argument; note that P and Q are subjects.

Amazing! This is the first time I see all four moods of (copulative) a fortiori argument listed by anyone before me. They are classed in the following order: first the two major-to-minor moods, the positive and the negative; then the two minor-to-major moods, the positive and the negative. For this reason, their order of presentation seems odd by my standards: positive predicatal, negative subjectal, negative predicatal, positive subjectal. But that, of course, is an unimportant issue – the fact remains all four forms are clearly there.

Since these definitions are explicitly built around a “common quality shared with varying degrees,” we can say that[77] they do include the middle term (R). However, what is manifestly lacking in them is the notion of a threshold of R that any subject must cross before it gets the predicate. Yet, this is an essential feature of a fortiori that anyone must acknowledge who claims to understand the argument. We can therefore say without any exaggeration that R. Luzzatto correctly formulated the four moods of a fortiori argument, in the sense of perceiving their two possible orientations (subjectal and predicatal) and two possible polarities (positive and negative) some 250 years before I did. As far as I know, he was the first to do this important work (in or before 1741, presumably while a resident of Amsterdam). However, although his formulation does mention the middle term, it is still incomplete since it does not mention the crucial issue of the sufficiency (or insufficiency) of that term, without which the argument cannot be validated. Therefore, while Ramchal should be regarded as an important contributor to a fortiori logic, he cannot fairly be said to have been the first to formalize the argument correctly.

Assessment. That these four moods are not listed by him using symbols instead of terms (like my P, Q, R, S) is not important. Nor is it important that he does not devise descriptive names for the arguments (positive/negative, subjectal/predicatal). What we have here is still a considerable measure of formalization in the strict sense of the term, since abstract concepts are used instead of concrete examples. Expressions like “what,” “whatever,” “whichever subject,” “predicated,” “affirmed,” “the greater (term),” “the lesser” are all equivalent to use of symbols – they serve the same function of theoretical generalities allowing for any specific values that may occur in practice.

Note, too, that his definitions are not made in narrowly legal terms, but in terms adaptable to any subject matter. We can also say that he was clearly aware of the middle term underlying the major and minor terms, which puts him ahead of many past and present logicians. So, R. Luzzatto may almost be said to have been the first to formalize a fortiori argument, or at least its four forms (the primary copulatives). I say ‘almost’ – because some criticism of his presentation is possible and necessary. There are a number of significant deficiencies in it, from a formal logician’s point of view.

Firstly, the preamble, “Quantified commensurates share a common quality, but not in the same degree” (הם שחלוקים בכמות איכותם), is actually the unstated major premise of all four “rules,” telling us that the major and minor terms P and Q share a common quality R to different degrees. This premise should not be detached from the four arguments, because it is an integral part of each of them, making the inference possible. It should be repeated every time it is relied on.

Secondly, the middle term R should also be explicitly mentioned in the minor premises and conclusions, whereas it is left tacit in them. It is not enough to state there that P and Q are greater or lesser, with implicit reference to the major premise. The term R mentioned in the major premise must be repeated in the minor premise and conclusion, to ensure that it is with respect to that exact same term that they are intended; otherwise, we risk committing the fallacy of two middle terms. If the middle term is not mentioned in all three propositions, it is not fulfilling its role of intermediary, which explains why the putative conclusion follows the given premises.

Thirdly, and most importantly, note again the absence of the explanatory concept of sufficiency (or insufficiency) in the minor premises and conclusions, i.e. the awareness that there is in each case a threshold value of R as of which S is applicable (or not). Without this subtle feature, we have no explanation for the link between the subject and predicate in the minor premise, and therefore no explanation for our claiming the same link in the conclusion. The middle term, and its being present enough or not enough, are essential details to succeed in validating the argument. The minor premise must specify these details, even if the conclusion is stated without them.

Even so, it would not be fair to say that Ramchal confused a fortiori argument with argument by analogy. If he had done so, he would have allowed for inference in positive subjectal form from major to minor and in positive predicatal form from minor to major. The fact that he did not count such reasoning (and its negative corollaries) as valid shows that he was referring to a fortiori reasoning rather than to qualitative[78] analogy. For whereas the analogical argument is non-directional, able to function indifferently in either direction, a fortiori argument is distinctively directional.

In sum, although we can rightly attribute the formulation and listing of all four moods of (primary copulative) a fortiori argument to R. Luzzatto – assuming no one preceded him in this feat that I do not know about – we cannot say that he succeeded in fully formalizing these arguments. A little bit more work was needed to thoroughly define each unit of reasoning he listed, in a way that made their validations possible. The deficient way he has formulated the arguments (i.e. without mention of sufficiency or insufficiency of the middle term in the minor premise) makes their putative conclusions invalid – i.e. the contradictories of these conclusions remain logically possible.

We should also note that he does not actually analyze the four moods he has listed, and distinguish between those in which P and Q are subjects and those in which they are predicates. We can suppose that he was aware of the differences between major-to-minor and minor-to-major moods, and between positive and negative moods, because of the way he has ordered the material. But he does not seem to have clearly noticed the also important structural difference between subjectal and predicatal. Still, what he did achieve should not be belittled. It deserves high praise.

This historical finding is a quite unexpected and somewhat humbling for me. Although I independently formulated these four principal moods back in 1995 (no doubt some time before), I must now admit they were already roughly known. But I can still claim as original, their more precise formulation and analysis, as above detailed. I can also claim the discovery of the corresponding implicational forms and various derivative secondary forms.

Moreover, fourthly, as far as I know, R. Luzzatto made no effort of validation, but accepted the reasoning involved in his four “rules” as self-evident. But of course, that won’t do – logicians have to justify all arguments they acknowledge in appropriately detailed and convincing ways[79]. So, I can claim this important achievement, having formally demonstrated that a fortiori arguments can be reduced to more familiar and proven arguments. And that was of course made possible by my more precise formulations and analyses.

Needless to say, the said four deficiencies in R. Luzzatto’s treatment of a fortiori argument are not a reflection on his intellectual capacities. It is evident that he could easily have further developed his study of the subject in the stated directions had he wished to. Obviously, he was a logician more concerned with teaching practical logic than in researching theoretical issues. Of course, deeper theoretical analysis does improve practice, but it is a fact that most people do not feel the need to go that far.

An insight by R. Luzzatto also worth noting is the following:

“In addition, it is necessary to distinguish between quantified commensurate terms which are greater or lesser on the one hand, and more or less likely, on the other. For when a certain quality is exhibited to a greater degree, it is not, therefore, more likely to occur; in fact, it is often less likely.” (P. 90.)

Here again, we see the lucidity of the man. Many people, from Aristotle’s time to the present day, have made the mistake, when discussing a fortiori argument, of confusing ontical differences in degree between the major and minor terms in relation to an underlying middle term with epistemic differences in degree, i.e. with degrees of likelihood. R. Luzzatto is evidently aware of the alternative possibilities involved, since he stresses that the major term may in fact (in some cases) be less likely than the minor term. Clearly, this is an author whose work on logic deserves careful reading or rereading.

Nevertheless, it must be pointed out that the Ramchal’s above listed four arguments are all purely a fortiori; he does not like many logicians before and after him attempt to draw ‘proportional’ conclusions. This is in one sense to his credit, in that purely a fortiori argument is the essence of a fortiori. But in another sense this is a deficiency, in that a crescendo argument is also valuable, provided we understand that it involves an additional premise about ‘proportionality’.

The Ramchal’s non-mention of a crescendo argument – and for that matter of the dayo principle – is surprising, considering the large role such argument plays in the Talmud, and in particular in Baba Qama 25a. I do not know whether he has anywhere written any comments regarding Talmudic a fortiori argument. If he did, it would be very interesting to know what he said, in view of his above-average clarity of insight and logical skill.


11. More research is needed

What we have found so far in the preceding pages concerning the views of early Jewish commentators on a fortiori argument may look a bit slim. The truth is that my linguistic skills are insufficient to do a much more thorough job than that on them. Someone with better Hebrew and Aramaic than mine will have to look into this matter more fully, carefully examining any relevant comments in the literature, and publish a new work on the subject.

This of course means, for a start, examination of the later commentaries placed all around the Mishna and Gemara in current editions of the Talmud; but all other possible sources must also be investigated. A thorough, chronologically-ordered bibliography on Talmudic logic, hermeneutics and methodology needs, perhaps, to be drawn up for this purpose. This can be done using material from various sources. Possible starting points: the Jewish Encyclopedia article on Talmudic hermeneutics[80]; the ‘Logic and methodology’ section of the Wikipedia article on Talmud[81]; M. Mielziner’s Introduction to the Talmud, pp. 83, 96, 128-9; and others. These various lists are doubtless far from exhaustive[82]. The works they include seem offhand to relate to logic and the rabbinical hermeneutic rules in general, and thence to the subject-matter of qal vachomer in particular. But some or even many of them might have no distinctive additional information or even no relevant information at all.

The reason I mention these lists, and the potentially relevant authors and works in them, is in order to stress that the present work is far from complete: it is probably at best sketchy. The work of history and evaluation that needs to be systematically done is yet to be done. We need to collect all the relevant information in those and similar works before we can hope to write a thorough history of the subject and to more precisely trace the evolution in understanding of qal vachomer and other arguments among Jewish logicians and commentators. I of course should have – and would dearly have loved to – study all the works listed there, but I have so far not found many of them in English (or French) translation: most are, of course, in Hebrew.

I could only here, for now at least, do some of the work – whatever was within my linguistic purview. Ideally, the books listed in the eventual bibliography should all be translated into English by someone so as to be available for scrutiny by international scholars like me who do not necessarily master Hebrew. At least the segments relevant to our study should be translated; that is, those to do with the hermeneutic principles and practices of the rabbis, and in particular those to do with the a fortiori argument. The ideal would be to create a freely accessible ‘.org’ website in which such works would be collected and their translations posted for all to see and study.

In truth, the probability is high that most of the books listed – especially those by medieval rabbis – simply repeat the same old platitudes about the qal vachomer argument, the dayo principle, and other hermeneutic principles. The reason for this is simple – the obligation of rabbis to conform to orthodox standards in order to be accepted by their peers. There are, to be sure, sometimes disagreements and even very passionate disputes among them. But these are all probably within certain bounds, for otherwise they would not be considered as kosher and perpetuated and read.

We must, however, remain open to the possibility of the unexpected: the Ramchal’s seemingly original discovery of the four moods of a fortiori argument is a felicitous case in point.

[1] Further on, Mielziner remarks that there were “some legal traditions… for which the Rabbis were unable to find a biblical support or even a mere hint” (and informs us that 55 such cases have been enumerated). These were suggestively labeled as halakhot leMoshe miSinai – laws handed down to Moshe from Sinai.

[2] He was part of an embassy sent in 40 CE by the Alexandrian Jewish community to Rome, to plead their cause before the emperor Gaius Caligula.

[3] Paris: Cerf, 1970. The title means “escape and discovery.”

[4] My translation from the French of both Philo’s words and the Biblical passage.

[5] It is interesting to note, here, that the words ‘blasphemy’ and ‘blame’ are etymologically very close. Starobinski-Safran informs us in a footnote that the Greek words Philo uses in this context have many possible meanings, including blaspheming, cursing, slandering, or insulting. She further remarks that “to seek refuge at the altar without right is offensive towards the Divinity; in effect, it is discharging onto [the Divinity] the faults one is responsible for” (my translation).

[6] Note that Philo does not prove, by means of his a fortiori argument, his initial claim that “No blasphemer has a right to forgiveness;” he only proves that they deserve punishment of some sort, possibly death.

[7] Blasphemy, according to Ibn Ezra, could signify reviling or, more likely, use of the Tetragrammaton. Rashi states that only if the Tetragrammaton is uttered is blasphemy liable to the death penalty. (The latter comments are drawn from The Soncino Chumash, pp. 759-60.)

[8] Of course, if the death penalty is specified for cursing parents, then (a fortiori) the death penalty (at least) must be applied for cursing God. If we say that the death penalty is the maximal sentence, then the argument is purely a fortiori; it cannot be a crescendo. But if a more severe penalty is conceivable, as Starobinski-Safran suggests, then the argument might well still be a crescendo.

[9] Starobinski-Safran writes verse 16, but that must be an error of inattention, for v. 16 concerns kidnapping; it is v. 17 that concerns cursing of parents. Verse 15 concerns the smiting of parents; but this seems to me irrelevant in the present context, since no one can physically smite God.

[10] We might here refer to the statement “vengeance is Mine” (Deut. 32:35).

[11] Soncino Talmud, Tractate Makkoth 7a. See also on this topic: Mishna Sanhedrin 7:4, 9:1, 11:1 and Tractate Sanhedrin 41a.

[12] The Jewish Encyclopedia (JE) article referred to here may be consulted online at: .

[13] Nashville, Tenn.: Abingdon, 2005.

[14] Maimonides (in the introduction to his Yad haḤazaḳah), among others, considers its author to be Rab, aka Abba Arika (Babylonia, 175–247 CE), in view of the book’s other title, Sifra debe Rab. Another theory, proposed by Malbim (in the introduction to his Sifra edition), is that R. Ḥiyya b. Abba (ca. 180-230 CE), a late Tanna or early Amora who lived in Eretz Israel, was the book’s redactor. The latter is also credited with compilation of the Tosefta. A lot more is said on this topic, which need not concern us here.

[15] For these and many other examples, I recommend the reader to the website: , although I must stress I do not agree with its sweeping radical conclusions, against belief in God, against the Jewish religion as a whole, and against our national right to Israel.

[18] To form objective judgments on such matters, one must take into consideration not only one’s own religious beliefs but also those of other people. There are many religions in the world, each with its own rituals and its own rationales for these. They cannot all be absolute truths. Most, if not all, must be human inventions. Of course, each of us conveniently believes it is the others’ belief systems that are imaginary. But try proving that! Therefore, all of us should have a measure of modesty and tolerance in his beliefs. This is not relativism, but honesty.

[19] Originally written in Arabic; translated into Hebrew by Nahum ha-Maarabi in the 13th century. Full Hebrew text is given in the Œuvres Complètes. According to the introduction to this volume, the authenticity of the text has been demonstrated. Some comments are included in footnotes. You can also read it online at the Internet Archive: . The Wikipedia article on Saadia Gaon informs us that, according to Azulai, Saadia has also written (again, in Arabic) a methodology of the Talmud entitled Kelale ha-Talmud.

[20] Œuvres Complètes de R. Saadia ben Iosef al-Fayyoumi (vol. 9). Ed. Joseph Derenbourg, Hartwig Derenbourg and Mayer Lambert. Paris : Leroux, 1897.

[21] It is stated (apparently in the Sifra) that Exodus 21:1, “Now these are the ordinances which thou shalt set before them,” was said by R. Ishmael to refer to the thirteen rules for interpretation of the Bible revealed to Moses on Sinai. This equation may be convenient, but it is not based on a literal reading.

[22] His book, Emunot veDeot (Constantinople, 1562; in Hebrew), can be downloaded free at: .

[23] Needless to say, the two books quoted here, Of Doctrines and Beliefs (Abridged ed. Trans. Alexander Altmann. Oxford: Phaidon, date not specified.) and Of Beliefs and Opinions (vol. I. Trans. Samuel Rosenblatt. New Haven: Yale, 1948.) are two translations of Saadia Gaon’s Emunot veDeot. Incidentally, I am amazed how different they are; so much so that I had to quote them both because I could not find the same material in both!

[25] This can be read in English in the Soncino Talmud, at: .

[26] The Tosafot were medieval rabbis, active between the 12th and mid-15th centuries mainly in France and Germany, who elucidated and explicated many passages of the Talmud. Their commentaries (tosafot means additions) were very important to subsequent development of Jewish law. Many grandsons of Rashi are counted among them, by the way. I here refer to them collectively, because I do not know precisely which one(s) commented on the issue here concerning us; perhaps his or their names is/are known to experts.

[27] For a summary in English of the comments of Tosafot and others, see the Art Scrolls’ Talmud Bavli.

[28] Even thousands considering God’s exaltedness.

[29] Another commentator has suggested: “Why particularly fourteen? The Rabbis (Nidah 31a) remark that each parent provides a child with five essential parts (the father with bones, sinews, etc., the mother with skin, flesh, etc.), whereas God provides him with ten (spirit, soul, etc.). Since God doubles the father’s portion, the humiliation for his rebuke is also double, fourteen days to the father’s seven.” From the Metsudah Chumash w/Rashi at: .

[30] As I have shown, this notion is based on rhetoric; it has no basis in logic. But note that since I regard the dayo principle in its broadest sense as a moral rather than logical principle, I do not deny that it might have exceptions, as R. Tarfon claims in the Gemara’s scenario.

[31] Jerusalem: Mosaica, 2012.

[32] Although Ury does not clearly state where this Tosafot is found, it seems from the context to be opposite Kiddushin 4b. The relevant pages in Ury’s book are 113-118.

[33] Based on Ex. 22:4, and its extreme inversion, as explained in an earlier chapter (8.6).

[34] Ex. 21:35 – “And if one man’s ox hurt another’s, so that it dieth; then they shall sell the live ox, and divide the price of it; and the dead also they shall divide.”

[35] Its premises are based on the same Biblical information as the first argument.

[36] Such reversal of course means that the two major premises are in conflict, and therefore that the two arguments cannot be both upheld. It is not surprising, then, that they yield conflicting results.

[37] What the Ri actually says is: “paying full damages in the damaged party’s domain is not a severity to be used in an objection,” which could be taken to mean that he advocates denial of the minor premise of the objection (2a); but obviously, he cannot be intending that, since he knows that the minor premise is given in the Torah; therefore, it must be the major premise of the objection, which institutes the greater severity for tooth & foot damage compared to horn damage, that he intends to abandon.

[38] Note that Ury has it as “kol zeh achnis,” which he (or whoever) translates as “all this I will fold.” But the Hebrew portion he quotes clearly has “assim,” so I have preferred that word. Maybe there are different versions of the same Tosafot text. It is not an important issue.

[39] I have referred to the translation given by Ury, but modified it considerably so as to make it both more literal and more readable.

[40] Although strictly speaking “tooth & foot damage on private property necessitates full payment” does not imply “tooth & foot damage is common (R) not enough to make the ox’s owner exempt from full payment for damages on private property,” we can inductively assume this implication granting that there is a threshold value of the middle term (R) that allows access to the predicate.

[41] “If a man cause a field or vineyard to be eaten, and shall let his beast loose, and it feed in another man’s field; of the best of his own field, and of the best of his own vineyard, shall he make restitution.”

[42] The suffix ‘-ides’ means ‘son of’.

[43] This is according to Ventura’s Introduction to his Terminologie Logique (p. 7). The Wikipedia article claims he was “in his twenties.”

[44] The Tibbonides were a famous family of translators in the 12th-13th century, in the south of France. According to Ventura (pp. 14-17), it is they who translated the Arabic word mantik, which means much the same as the Greek word logos, into the Hebrew word higayon. A better choice would have been dibbur, but they avoided it because of certain philosophical connotations (association with the Arab dialectics of Kalam). Until then, the word higayon did not have the precise sense of ‘logic’. Opponents of Maimonides used this word (which the translators chose, not him) against him, quoting the Bab. Talmud (Berachot 28a): “Prevent your children from using higayon.” But other commentators, namely R. Joseph b. Caspi and Jacob Anatolio (a relative of the Tibbon family), objected to this reading, the former arguing that by higayon the rabbis meant pseudo-logical babble, while the latter pointed out that this statement referred specifically to children and not to adults.

[45] Paris: Librairie Philosophique, 1935. Reprinted, 1982.

[46] Here Maimonides lists the Organon of Aristotle as including eight works. Today, only six are included therein, the Rhetoric and the Poetics being excluded. But such inclusion, as Ventura points out (p. 13), is reasonable from Maimonides’ point of view, since for him rhetorical syllogism is based on traditional assertions and poetical syllogism is based on fictions or imitations. Incidentally, the latter concept was introduced by Al-Farabi.

[47] Maimonides apparently learned logic at least partly through his readings of Muslim commentators, especially Abu Nasr Al-Farabi (Central Asia, 872-950); he certainly mentions the latter (see e.g. p. 106). The impact of such early study of logic on the rest of his work is evident; such studies explain his orderly mind, rationalism and conceptual powers. It cannot be said that the rabbis after Maimonides all studied this work and took it to heart. Unfortunately, many still today carefully avoid studying logic, even as they refer to Maimonides’ halakhic works.

[48] Ventura expresses the same surprise, I think, when he remarks, in his Appendix to Chapter VIII (pp. 76-77): “Maimonides did not in his treatise mention a fortiori reasoning.” Notwithstanding, he then suggests that Maimonides’ mention of enthymeme (p. 72) might be construed as a tacit reference to a fortiori argument, since M. Lalande, in his Vocabulaire technique et critique de la philosophie, defines a fortiori as “an enthymeme that assumes a premise like the following: ‘Who can do the more can do the less.’” But this argument of Ventura’s is clearly spurious, since ‘enthymeme’ may refer to abridged argument of any sort. He goes on, describing how a fortiori is understood in the Talmud. Here, he adheres to the idea that “Miriam should have been sequestered fourteen days instead of seven,” suggesting that a fortiori argument is naturally ‘proportional’ and requires the dayo principle to restrict its excesses. This belief of Ventura’s is indicative of a lack of reflection on his part on the logic of a fortiori argument, no doubt under the influence of the Gemara.

[49] Ventura (p. 23) lists some modern authors who have made an effort to relate Aristotelian logic and Talmudic hermeneutics, namely A. Schwarz and M. Mielziner, and M. Ostrovski. He also mentions (p. 18) a commentary by Moses Mendelssohn on Maimonides’s treatise on logic (I have not read it).

[50] Pp. 64 and 69; translations from the French my own. Compare a similar statement by Saadia Gaon quoted earlier.

[51] See for instance Topics 1:12: “Induction is a passage from individuals to universals.”

[52] For instance: “For there is no name common to all the objects that I mean, but, for all that, these things are all in the same class by analogy.” (Meteorology, 4:9)

[53] This does not mean that the Talmudic rabbis did not engage in induction and analogy. As Ventura points out (pp. 77-78), they engaged in analogy (e.g. mah matsinu, hequesh, gezerah shavah); and in induction (e.g. binyan av), more or less consciously. But in either case they do not seem to have reflected on the issue from a logician’s viewpoint. This is of relevance to a fortiori argument, since it depends on induction for its premises and involves a sophisticated form of analogy. See also Saadia Gaon.

[55] Nachmanides, also known as Rabbi Moses ben Nachman, acronym Ramban (Spain, 1194 – Israel, 1270). Note that the Ramban was born ten years before the Rambam passed away.

[56] Cited in: . However, the Ramban is also there quoted as saying: “know that though they [the Sages] said a man does not rule via analogies by himself, they did not mean to say that all analogies were explicated to them from Sinai and given to them from the mouth of Moshe [etc.]; this is not true, since we have found them always disagreeing in many places [etc.], and were this received tradition from Sinai [etc.], there would be no occasion for these questions and for the answers that were said in the Gemara [etc.]. But the intent of an analogy which is from Sinai is that they had received a tradition that a certain ruling is learned from an analogy, but from where exactly it is derived was not a part of the received tradition.” In other words, the Ramban’s opinion is not as sweeping as it first seems, at least as regards analogical reasoning.

[57] M. Halbertal, Maimonides’ Sefer Hamizvot and the Structure of the Halakhah. (Heb.) Tarbiz 59 (1990), 457-480. D.B. Sinclair, Legal Reasoning in Maimonidean Jurisprudence. L’Eylah 29 (1990), 32-35. Both quotations found on the Internet at: .

[59] Ventura’s idea that Maimonides’ Terms of Logic was effectively a propaedeutic to his Guide is perhaps mirrored in Joseph A. Buijs essay Maimonides’ Use of Logic in the Guide to the Perplexed (in Schumann’s Judaic Logic collection), where the earlier logical and epistemological work is said to “infuse the development of issues in his later philosophical work.” Buijs does not, however, mention Ventura’s commentary.

[60] Piscataway, N.J.: Gorgias, 2010.

[61] “Die Hebräischen Kommentare zu den 13 Middot des Rabbi Ismail” in Festschift Adolf Schwarz, ed. S. Krauss (Berlin and Vienna, 1917).

[62] See also, by the same author, “Talmudic Methodology and Aristotelian Logic: David ibn Bilia’s Commentary on the Thirteen Hermeneutic Principles” in the Jewish Quarterly Review – Volume 99, Number 2 (Spring 2009), pp. 184-199.

[63] E.g. Chullin, 60a. Not to mention the Bible, where most of the a fortiori discourse has a non-legal content. Note too that one of the first medieval commentators, Saadia Gaon (in his Commentary on the Thirteen Midot), explicitly teaches that qal vachomer may be legal or non-legal in content.

[64] Of course, this is assuming these English words reflect similar ones in Hebrew, and are not mere interpolations by the translator.

[65] Whose identity is uncertain according to Ravitsky, though some have identified him with Gersonides. He refers us to his essay “On the Date of Sha’are Sedek, attributed to Gersonides” (in Hebrew), Daat, 63 (2008), pp. 87-102.

[66] Similarly optimistic statements by R. Avraham Shalom (15th century) and R. Elijah Galipapa (18th century) are quoted by Ravitsky.

[67] Regarding Saul Lieberman, see my comments in a later chapter (15). Another index that makes me wary of Ravitsky’s reliability is his failure to take account of my work Judaic Logic when discussing ‘modern research.’ He mentions this book in passing twice, on issues of minor import; the first time, regarding the expression ‘qal vachomer,’ and the second, only to imply that, like Jacobs, I perceived in binyan av reasoning a type of induction close to J.S. Mill’s Method of Agreement. All the original work in Judaic Logic, such as the formalization of qal vachomer and of the midot used for harmonization, is not even mentioned, let alone taken into consideration. This suggests to me that he did not take the trouble to study this important book, but only mentioned it to ‘pad’ his references. No wonder he can say in his abstract: “To date, the application of logic to the realm of the 13 principles has not received proper attention in the research literature.” This is of course factually inaccurate and only stated so as to amplify the importance of his paper. Nevertheless, his paper is informative and thoughtful.

[68] Ravitsky also quotes R. Joseph Ibn Caspi and the Moslem philosopher Al-Ghazali to the same effect. He also mentions Maimonides, R. Abraham Ibn Izra, and others.

[69] In this regard, it is interesting to quote Louis Jacobs in his Religion and the Individual (Cambridge: UP, 1992), p. 101: “Although Luzatto lived in the eighteenth century, the historian Zunz rightly remarked that the Jewish middle ages lasted until the end of the eighteenth century.”

[70] An image in pdf of this book in Hebrew may be viewed/downloaded at: .

[71] Trans. Rabbis D. Sackton and Ch. Tscholkowski. Jerusalem: Feldheim, 1989.

[72] Note that all symbols introduced here [viz. X1, X2, and Y] are my own. N.B. I do not have my copy of the book on hand, and therefore cannot quote exactly what is said in it, as I would have preferred to today. I assume my past summary was accurate, although it is possible that today I would see things differently.

[73] Trans. Rabbis D. Sackton and Ch. Tscholkowski. Jerusalem: Feldheim, 1995.

[74] I wondered at first if the translators had, perhaps unwittingly, infused their own relatively modern ideas into the original text – because it seems so modern! But the original Hebrew is shown and it is evident that the translation is correct. With regard to their translation of Derech Tevunot, I had in my Judaic Logic expressed strong disappointment – not because I doubted that they rendered the Ramchal’s words accurately, but because I felt that the English wording they used for various items and processes was not in accord with more familiar works and therefore could be misleading.

[75] Bold fonts used by the translators omitted by me.

[76] בשמירת היחס “…provided that the relationship of greater and lesser is maintained in regard to that predicate.”

[77] Contrary to what I say in my Judaic Logic, with reference to his treatment in Derech Tevunot (above quoted).

[78] The analogy would be qualitative rather than quantitative (pro rata), since in each of Ramchal’s four moods the subsidiary term S remains constant, i.e. the same in the minor premise and conclusion. As pointed out further on, Ramchal does not show awareness of a crescendo argument, or even (to my knowledge) of quantitative (pro rata) analogy.

[79] In their Foreword (p. xxi), the translators state: “He goes beyond the logical investigation of validity and nonvalidity to find the means of evaluating what is true and what is false. In his treatment of syllogisms, the Ramchal passes over how given premises make a conclusion logically necessary.” But of course, there is no exemption from the obligation to demonstrate validity – the “beyond” they claim for Ramchal is a cop-out.

[82] My passing on of this bibliographic material should not be construed as an attempt on my part to appear more “learned” than I really am. The material I list here is, for the most part, material I am ignorant of. Had I consulted it, I would have treated in within my book. The reason I include it here is merely to give other researchers a bit of a starting point for further investigation. See a rough and incomplete list here.