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A FORTIORI LOGIC

© Avi Sion, 2013 All rights reserved.

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A FORTIORI LOGIC

CHAPTER 18 – Adin Steinsaltz

1. Qal vachomer and dayo

2. A recurrent fallacy

3. Lack of formalism

R. Adin Steinsaltz, aka Even Yisrael (Israel, b. 1937), translated the Talmud into Hebrew (and other languages) and published it with a new commentary, over many decades, starting in 1965. There is a website called The Aleph Society, where his biography[1] can be read, as well as his commentary on the Babylonian Talmud[2]. I searched there for his comments regarding (using the spelling there preferred) kal va-homer and the dayyo principle, and found[3] several posts which I will presently analyze.

I must say I am sorry my analyses turn out to be so critical, because I actually greatly admire R. Steinsaltz’s œuvre. Nevertheless, emotions cannot be allowed to deter us from honest logical assessments. I take Biblical statements like the following as guidelines in such contexts: “Ye shall not… deal falsely, nor lie to one another” (Lev. 19:11), “Thou shalt not respect the person of the poor, nor favour the person of the mighty” (Lev. 19:15), “Ye shall do no unrighteousness in judgment, in meteyard, in weight, or in measure” (Lev. 19:36).

1. Qal vachomer and dayo

In this section, we shall look into R. Steinsaltz’s descriptions of qal vachomer reasoning and the dayo principle in relation to it.

On Baba Kamma 25a-b. Having analyzed Baba Kamma 25a-b in great detail in an earlier chapter (7), I will not here say much about its content. Rather, I will concentrate on R. Steinsaltz’s remarks on the subject and see where he personally stands. What is amazing throughout R. Steinsaltz’s treatment here is the way he blithely ignores all the difficulties involved. He presents the matter very briefly, and on a very superficial level where everything seems obvious and harmonious, and he either does not realize or conceals the inherent difficulties.

As regards the Mishna, R. Steinsaltz only mentions and comments on R. Tarfon’s first argument, without mention of the second. The differences between the two arguments, and between the Sages’ dayo objections to them, are thus completely lost to him or at least skipped over. He interprets R. Tarfon’s first argument as a ‘proportional’ a fortiori (unaware that it could also be read as a mere pro rata argument), and the Sages’ dayo objection to it as a “ruling.” He does not notice that R. Tarfon’s second argument has the distinction that, whether interpreted as pro rata, a crescendo (proportional a fortiori) or pure a fortiori, it has the same conclusion, so that it is immune to the Sage’s previous dayo objection, so that the Sages’ renewed dayo objection must be understood differently. His definition of the dayo principle is therefore very simple:

“Limiting the conclusions that can be reached by means of a kal va-homer in this manner is called dayyo ‘enough.’ It is enough to learn a parallel halakhah from a kal va-homer, but not more than the original law itself.”

With regard to the Gemara, he has only this to say:

The Gemara explains that the concept of kal va-homer and dayyo stem from the story of Miriam who spoke inappropriately about her brother Moshe (see Bamidbar 12). As punishment, she was struck with tzara’at (biblical leprosy), and was forced to leave the encampment for seven days. The Torah explains that had her father banished her, surely she would have been embarrassed for seven days – now that she was banished by God, she will have to be removed for that length of time. Although logically banishment because of God’s anger should have lasted twice as long, dayyo limits the punishment to the same amount of time that she would have been embarrassed by her father.”

He does not notice that the Gemara takes for granted, on the basis of the baraita[4] it is quoting, which refers to the story of Miriam as its model, that qal vachomer is “logically” a crescendo in form, i.e. goes ‘proportionately’ from seven days banishment to fourteen days in the case under consideration, and that the dayo principle “limits” this penalty to seven days. This is of course, as we have shown, not true – qal vachomer may equally well be purely a fortiori argument, in which case there is no call for a dayo objection to it. This means that, in the Miriam example, the conclusion may well be immediately as the Torah has it seven days, rather than fourteen days reduced to seven as the Gemara naïvely claims.

Interestingly, although the Gemara does not explicitly say so, R. Steinsaltz claims that it says that “the concept of kal va-homer” – and not only that of dayo – “stems from the story of Miriam.” This is inaccurate, since qal vachomer is found earlier in the Torah than in Num. 12:14-15; and as just explained, even the reading of the dayo principle into this passage of the Torah by the Gemara (or the baraita it quotes) is open to debate on logical grounds (though not impossible). These inaccuracies show that R. Steinsaltz has not studied the mechanics of a fortiori argument, and instead simply taken erroneous traditional views for granted.

To these criticisms we should add that R. Steinsaltz fails to mention and analyze all the subsequent issues arising in the Gemara. First, the troubling fact that the Gemara does not notice or take into consideration R. Tarfon’s second argument in its explanation of qal vachomer and dayo; had it done so, it would have had to admit that a fortiori argument may be non-‘proportional’ and therefore that the dayo principle of the Mishna Sages is of two types. Second, in its headlong pursuit of proof that the dayo principle is “of Biblical origin,” so that R. Tarfon must know it and essentially agree with the Sages, the Gemara makes up an intricate scenario about their different viewpoints, which upon detailed logical scrutiny turns out to be specious. Unfortunately, none of this is hinted at in R. Steinsaltz’s treatment.

On Baba Kamma 63a-b. R. Steinsaltz’s commentary on Baba Kamma 63a-b is essentially the same, repeating verbatim the above quoted paragraph about Miriam. What is added here is his definition of qal vachomer, and accessorily (though he does not here name it) the dayo principle, as follows:

“The method of kal va-homer usually translated as an A fortiori argument allows us to learn one law from another by arguing that if the less stringent law included a stringency, we can conclude that the stricter law includes that stringency, as well. Although the method of kal va-homer is considered to be a powerful one, it is limited in cases where there is an attempt to derive more than the original law included.”

This is of course a traditional rabbinic definition; we have previously seen other very similar statements. R. Steinsaltz is of course not claiming it as original, though he does not mention its historical author (perhaps because he is unknown[5]). We can better analyze his statement by presenting it more formally, as follows:

Law P is more stringent (R) than law Q,

and, law Q is stringent (R) enough to imply stringency S;

therefore, law P is stringent (R) enough to imply that stringency S.



Notice firstly that this argument is purely a fortiori: the conclusion has the same (“that”) stringency as the minor premise. R. Steinsaltz does not remark on the difference between such argument and the a crescendo form assumed by the Gemara in Baba Qama 25a, where the conclusion would be ‘proportional’, i.e. contain a greater stringency. Yet, R. Steinsaltz goes on in the same breath telling us that “the method of kal va-homer… is limited in cases where there is an attempt to derive more than the original law included.” This is, as already pointed out, an allusion to the dayo principle. But then we have a contradiction, or at least a mix-up of genres! If the argument is as he depicts it here purely a fortiori (i.e. non-proportional), the dayo principle is irrelevant to it and should not be mentioned. If on the other hand the dayo principle is to be mentioned, then the argument must be presented as a crescendo (i.e. as proportional). He can’t have it both ways.

Secondly, said in passing, the above definition of qal vachomer lacks the usual reverse statement:

Law P is more lenient (R) than law Q,

and, law Q is lenient (R) enough to imply leniency S;

therefore, law P is lenient (R) enough to imply that leniency S.



This statement is implied, for examples, in R. Chavel’s definition: “A form of reasoning by which a certain stricture applying to a minor matter is established as applying all the more to a major matter. Conversely, if a certain leniency applies to a major matter, it must apply all the more to the minor matter;” and again in R. Feigenbaum’s: “Any stringent ruling with regard to the lenient issue must be true of the stringent issue as well; [and] any lenient ruling regarding the stringent issue must be true with regard to the lenient matter as well.”[6]

There is of course no doubt that R. Steinsaltz knows this; but he does not say it here. We do find a broader definition of qal vachomer in R. Steinsaltz’s Reference Guide to the Talmud. There he says[7] that this hermeneutic rule sets up a parallel between two laws, one of which has some stricter aspects than the other. If the stricter law has a certain leniency, then the more indulgent law must have it too; and “vice versa,” if the more indulgent law has a certain severity, then the stricter law must have it too.

Thirdly, note the change in the relative positions of P and Q, in the above two arguments. In the first, P is more stringent than Q; in the second, P is more lenient than Q. But as regards their form, both arguments are positive subjectal (or more precisely antecedental, since the subsidiary item S is implied). Therefore, both proceed (despite appearances) “from minor to major.” It follows that R. Steinsaltz’s definition of qal vachomer here, even if expanded as we have proposed, is too narrow, because it ignores the corresponding negative moods as well as all predicatal (or consequential) a fortiori reasoning. His definition is also too narrow because it is focused on legal matters, whereas in fact (even in the Bible and the Talmud) a fortiori argument can be used with regard to non-legal matters. But we can assume that R. Steinsaltz is well aware of the possibility of such wider use, since he quotes in his Reference Guide the example of Jer. 12:5: “If thou hast run with the footmen and they have wearied thee, then how canst thou contend with horses.”

Moreover, R. Steinsaltz makes no effort at validation of a fortiori reasoning. He does not explain why it is indeed logical to reason in this manner. He takes it for granted without further ado, which attitude is quite curious for a man who was trained in the ways of modern science and mathematics. In his Reference Guide, he informs us that this is the exegetical rule most often encountered; but he does not go any deeper into the subject than that.

As regards the above definition the dayo principle, it looks commendably broad because it is sadly vague. Disappointingly, R. Steinsaltz does not delve into the nature, source and justification of this principle, nor analyze when it is applicable in any detail. However, in his Reference Guide, he lists various traditional specifications concerning qal vachomer, such as the possibility of applying it to new situations without the sanction of tradition, and (of significance to analysis of dayo) the impossibility to infer by a fortiori (as unanimously admitted) a prohibition or (according to some opinions) a punishment.

On Zevahim 69a-b. We will skip the legal minutiae dealt with in Zevahim 69a-b, which R. Steinsaltz does not develop in detail anyway, and rather focus on his general comments. He repeats here, as does the Gemara, previous comments regarding the dayo principle, and then adds:

“One question raised by the rishonim is why logic would lead us to conclude that Miriam should have been banished for 14 days. Why not 8 days? Or forever?

Rabbenu Tam is quoted as connecting this with the idea that there are three partners in the creation of a person – his mother, his father and God. Thus God is the equivalent of both mother and father and offense against Him deserves double banishment.

Rabbenu Hayyim ha-Cohen suggests that Miriam deserved just a little extra banishment, but the minimum time that someone suffering from tzara’at is banished is a week, so any additional banishment must be for a full extra week.

The Ramban argues that no explanation is necessary, since this is merely the way the midrash halakhah speaks; that since she deserves more the expression is that she needs twice as much.”

This text is further confirmation that R. Steinsaltz – like many great rabbis before him – firmly believes that “logic would lead us to conclude” that an a fortiori argument yields a ‘proportional’ conclusion. He takes this for granted and merely like his predecessors questions why the Gemara specifies specifically 14 days as the logical inference from 7 days, and not more or less. Neither he nor they realize that (as I have explained in detail in an earlier chapter (8.2)) the issue of the quantity of punishment has nothing to do with qal vachomer as such, but relates to the separate operation of a principle of justice or of our sense of justice. Thus, though the question asked: “Why not 8 days? Or forever?” is pertinent, it is far less important than the unasked question: why not 7 days?

On Pesachim 81a-b. We need not here either be concerned with the legal details treated in Pessachim 81a-b; suffices for us to look at R. Steinsaltz’s following remarks relating to the range of applicability of qal vachomer reasoning:

“Although the Gemara on our daf (=page) tries to find a source in the Torah for this halakha, its conclusion is that there is no clear reference in the Torah for it, rather it is a halakha le-Moshe mi-Sinai, a law that was transmitted orally to Moses on Mount Sinai that was not recorded in the Torah.

… Although Rabbah tries to apply the rule of kal va-homer (a fortiori) to this case…, the Gemara rejects this, arguing that we cannot learn a kal va-homer from a halakha le-Moshe mi-Sinai.

Although we usually perceive the rule of kal va-homer as being a straightforward logical one, it cannot be used in the case of halakha le-Moshe mi-Sinai because of the unique quality of such halakhot. In general, a law that appears in the Torah can be used not only for itself, but also as a source for other laws that can be compared to it. A halakhah le-Moshe mi-Sinai, even as its strength and severity are equal to those of a law written in the Torah, is not seen as being grounded in the same set of rules as the written halakhot, so we cannot extrapolate other laws from it.”

This commentary contains three items of information: (a) a definition of the term halakha le-Moshe mi-Sinai; (b) the ruling of Rabbah that new laws cannot be deduced through qal vachomer argument from a premise characterized as halakha le-Moshe mi-Sinai; and (c) a highlighting of this logical phenomenon as exceptional. As commentaries go this strikes me as a bit thin, so I will now try to add my own reflections.

Regarding (a), what can be said (perhaps rather cynically) is that a law designated as halakha le-Moshe mi-Sinai is so labeled precisely because there is no written evidence that it was given to Moses at Sinai ! If anything, what we have here is an early example of the power of advertising, where the jingle counts for more than the product. What many modern commentators say (more moderately) is that such laws were so called simply because they were considered very ancient and already well-established in Jewish jurisprudence.

Regarding (b), the question logicians must ask here is: If X formally implies Y, does it logically follow that ‘X is imperative’ formally implies ‘Y is imperative’? That is, if we can deduce Y from X, can we deduce the legal necessity of Y from the legal necessity of X? Answer: suppose Z is our ultimate standard of judgment (in the present context, say Obedience to Divine Will). Then our question is: if Z is impossible without X, does it follow that Z is impossible without Y? The answer is, clearly, yes: given X implies Y, and not-X implies not-Z, it follows that Z implies X, then Z implies Y, then not-Y implies not-Z[8]. Thus, as regards formal logic, we ought in principle to accept any strictly deductive inferences, including those made through properly formulated qal vachomer arguments. This cannot be disputed, as just demonstrated syllogistically.

Nevertheless, I do not deny that the conclusions of certain qal vachomer may be regarded as having less legal weight than their premises, in acknowledgment that the premises used usually have some inductive origins. In the context of Jewish law, laws that are evidently and incontestably Scriptural are treated as axioms (i.e. as purely deductive in origin), and therefore formal inferences drawn from them are likewise considered reliable; whereas laws transmitted orally are more inductive in nature and thus retain some measure of uncertainty[9], so that even if they are per se conventionally granted credence, laws derived from them per accidens may still credibly be refused equal weight. In other words, Rabbah’s ruling is reasonable, even if it could have been otherwise.

Regarding (c), which is R. Steinsaltz’s own commentary to the preceding, what I would like to remark on is its relative passivity and superficiality. He notes descriptively that although halakha le-Moshe mi-Sinai is considered as binding as written Torah law, what is logically derived from the former is not as binding as what is logically derived from the latter. But he does not make any effort to reconcile this surprising phenomenon with the universal implications of formal logic. Instead, he claims that each type of law is subject to a different “set of rules” – suggesting, without any formal demonstration, that such relativism is logically conceivable. My contention here is that today’s more religious commentators must learn to overcome such intellectual restraint, and dare to ask difficult questions. They will find that the possible answers are usually not as frightening as they imagined. Credibility nowadays depends on readiness to question and if need be to honestly criticize.

2. A recurrent fallacy

In this section, we shall look into a couple of concrete applications, where the reasoning seems to be fallacious.

On Pesachim 23a-b. R. Steinsaltz presents the qal vachomer argument in Pessachim 23a-b as follows:

“The Gemara considers a number of cases of forbidden foods in an attempt to clarify whether an issur hana’ah – a prohibition against deriving benefit – is an inherent part of the issur akhila – the prohibition against eating something. One of the cases where we find a disagreement on this matter is gid ha-nashe (the sciatic nerve – see Bereshit 32:33), where Rabbi Shimon rules that we cannot derive benefit from it and Rabbi Yossi ha-Galili rules that we can.

The Gemara suggests that Rabbi Yossi ha-Galili learns this from a kal va-homer (an a fortiori argument) as follows: We know that the punishment for eating helev (forbidden fats) is very severe (karet), and that the punishment for eating gid ha-nashe is less severe (malkot). Since one is allowed to derive benefit from helev (this is clearly indicated in the Torah – see Vayikra 7:24), then certainly in the less severe case of gid ha-nashe one would be permitted to do the same.”

This presentation would seem to be an accurate rendition of the Talmudic argument. The problem is that R. Steinsaltz accepts its claims uncritically. Notably, the claim by R. Yossi ha-Gelili[10] that he has put forward a valid qal vachomer. Notice the former’s qualification of the conclusion as “certainly” following the premises.

However, on closer inspection, it is not obviously valid, because the terms used in the minor premise and conclusion (viz. deriving benefit from helev or gid ha-nashe) are not the same as those used in the major premise (viz. eating helev or gid ha-nashe). If there is indeed a valid qal vachomer, it must be less direct than it is made out to be; i.e. it must involve some tacit intermediate moves.

But further scrutiny shows that the putative conclusion cannot readily be derived from the given premises! Let us symbolize our terms as follows: P= helev, P1 = eating helev, P2 = deriving benefit from helev; Q= gid ha-nashe, Q1 = eating gid ha-nashe, Q2 = deriving benefit from gid ha-nashe; R = degree of punishment, so that R= 0 means ‘allowed’ and R > 0 means ‘forbidden’. R. Yossi’s argument can then be written as follows:

P1 is more R than Q1;

P2 is R not enough to be forbidden;

therefore, Q2 is R not enough to be forbidden.



This is a negative subjectal a fortiori argument; it has to be so, since the terms P, Q are subjects throughout it and the movement of thought is from major (P) to minor (Q)[11]. But this is not a valid argument, as already stated, because the major premise concerns P1 and Q1, whereas the minor premise and conclusion concern respectively P2 and Q2. We can, still, try to make it valid by proving somehow that “P2 is more R than Q2.”

(a) Knowing that “P1 is more R than P2,” we could through a generalization assume that “Q1 is more R than Q2;” but this does not permit us to infer that “P2 is more R than Q2.” More specifically, we are given that “P1 is more R than P2,” since eating helev is punishable (R), i.e. forbidden, whereas deriving benefit from helev (P2) is allowed, i.e. not punishable. From this, we could by generalization say: “for anything, deriving benefit is less punishable an act than eating.” It follows by application of this generality that “Q1 is more R than Q2,” i.e. that “eating gid ha-nashe (Q1) is more punishable than deriving benefit from it (Q2).” We are also given that “P1 is more R than Q1,” i.e. that the punishment of malkot (lashes) is less severe than that of karet (excision). From this we can deduce that: “P1 is more R than Q2.” But we still cannot deduce that “P2 is more R than Q2” – and without this proposition the a fortiori argument remains invalid. So this approach is not successful!

(b) Alternatively, we could try generalizing immediately from the given major premise “P1 is more R than Q1” to “P is more R than Q,” i.e. to “anything to do with P is more R than the same thing to do with Q,” and thence by application infer the needed major premise that “P2 is more R than Q2.” Although such more direct extrapolation is more far-fetched than the one tried previously, since it involves two distinct subjects in tandem, it at least yields the desired result!

Another way to approach this extrapolation would be to write the major premise as a hypothetical: “When (1) eaten, Helev (P) is more severely punished (R) than gid ha-nashe (Q);” then from this generalize to: “Under all conditions, Helev (P) is more severely punished (R) than gid ha-nashe (Q);” then apply the latter to: “When (2) deriving benefit, Helev (P) is more severely punished (R) than gid ha-nashe (Q).” We can now argue, regarding “deriving benefit”: “if gid ha-nashe (Q) is not punished severely enough (R) to be forbidden, then helev (P) is not punished severely enough (R) to be forbidden.” The problem with this approach is of course its credibility: it looks too much like deliberate manipulation to obtain the desired conclusion.

In sum: if deriving benefit from helev (P2) is not punishable (i.e. is allowed), it does not necessarily follow that deriving benefit from gid ha-nashe (Q2) is not punishable (i.e. is allowed). The latter conclusion is not logically impossible, and may even (as just shown) be produced by inductive means, but as far as deductive logic is concerned it is a non sequitur. There may be another proposition stated elsewhere or tacitly assumed in the Gemara, which makes possible the deductive generation of the required major premise “P2 is more R than Q2,” but I have not found any such intermediary; therefore, as far as I am concerned, the argument has to be judged as formally invalid.

Which means that R. Yossi was arguing in a fallacious manner. R. Steinsaltz, however, like the Talmud before him, takes R. Yossi’s a fortiori argument as essentially valid, though open to rebuttal (“ikka lemifrakh? literally, ‘you can break the argument’”). But note that this rebuttal is not an attack like mine above on the a fortiori process as such, but merely on one of its premises. He writes:

“The Gemara records the response of Rabbi Shimon, who forbids deriving benefit from gid ha-nashe, as arguing that we cannot see helev as being more severe, since there are certain rules where gid ha-nashe is more stringent. For example, gid ha-nashe applies to all animals, whereas helev is limited to domesticated animals (behemot) and does not apply to wild animals (hayyot).”

The thrust of this counterargument by R. Shimon seems to be the rejection of “Eating helev (P1) is forbidden, whereas deriving benefit from helev (P2) is allowed.” We are told that the interdiction concerning eating helev applies to domesticated animals, but not to wild ones. For the latter kind of animals, then, eating and deriving benefit are both allowed. Whereas the similar proposition on gid ha-nashe (Q) would have to apply to all animals. In short, the generality of “P1 is more R than P2” is not accepted by R. Shimon.

But anyway, as we have just shown, even if this generality were accepted, R. Yossi’s argument would still not be valid, since we cannot deduce through it that “P2 is more R than Q2.” Both R. Shimon and R. Steinsaltz do not seem to realize this more formal issue. This is a rather disappointing performance on the part of all three of these rabbis, and many others, which goes to show the importance of having formal models to go by.

On Baba Batra 111a-b. R. Steinsaltz describes the qal vachomer argument in Baba Batra 111a-b, after explaining how the premises were arrived at, as follows:

“The Gemara suggests a kal va-homer… If a daughter, who has less rights of inheritance from her father’s estate, nevertheless inherits her mother, certainly a son, who has stronger rights in inheriting his father’s estate, will inherit from his mother.”

If we try to present this reasoning in more formal terms, we get the following:

A son of a man (P1) has more (or stronger) rights of inheritance (R) than a daughter of a man (Q1),

and, a daughter of a woman (Q2) has rights of inheritance (R) enough to inherit from her (the mother) (S);

therefore, a son of a woman (P2) has rights of inheritance (R) enough to inherit from her (the mother) (S).



This argument looks at a glance like an a fortiori, but is not really one, since the major and minor terms are different in the major premise (P1, Q1) and in the minor premise (Q2) and conclusion (P2), although the middle term (R) and the subsidiary term (S) are uniform throughout. We can turn this argument into a genuine a fortiori, if we manage to deductively or inductively infer the required major premise: “A son of a woman (P2) has more rights of inheritance (R) than a daughter of a woman (Q2).” For a deductive solution, we need appropriate intermediate premises. For an inductive solution, we must accept the generalization of the given major premise, so that the needed major premise can be derived from it.

Alternatively, we could formulate the Gemara’s a fortiori argument with uniform major, minor and subsidiary terms, as follows:

A son (P) has more (or stronger) rights of inheritance (from father) (R) than a daughter (Q),

and, a daughter (Q) has rights of inheritance (R) enough to inherit from her mother (S);

therefore, a son (P) has rights of inheritance (R) enough to inherit from his mother (S).



This would be a valid a fortiori argument if we could ignore the specification “from father” (which I have put in brackets) in the major premise. Otherwise, the middle term R (“rights of inheritance”) would not be the same in the major premise (where “from father” is specified, as given) and in the minor premise and conclusion (where it is irrelevant, and therefore cannot be specified). In order to remove the specification “from father” in the major premise, we need to generalize the given proposition from “A son (P) has more rights of inheritance from father (R) than a daughter (Q)” to “A son (P) has more rights of inheritance from anyone (R) than a daughter (Q)” – i.e. to move from a relative proposition to an absolute one. In concrete terms, we must presume a son to be generally more privileged than a daughter in matters of inheritance.

Such generalizations are legitimate provided they are performed overtly and explicitly acknowledged to be inductive acts. From a deductive point of view they are of course akin to circular argument or tailoring a premise to obtain the desired conclusion (see similar comments of mine relative to Pessachim 23a-b above, though the present case is a bit simpler). Thus, the author(s) of the Gemara containing this argument may be reproved, either for failing to realize and admit the inductive underpinning of the argument or for unconsciously engaging in fallacious deduction. In other words, there would have been no logical inconsistency if the Torah had prescribed that sons inherit from fathers more readily than daughters do, and daughters inherit from mothers more readily than sons do.

R. Steinsaltz next presents us with an attempted application of the dayo principle:

“Following this argument, the Gemara continues and concludes that since both sons and daughters inherit their mothers, the sons have priority in this case just as they do in cases when their father passes away. This position is rejected by Rabbi Zekharia ben ha-Katzav who believes that sons and daughters should share equally in the mother’s estate, because of the concept of dayo….

The Gemara relates that several amora’im wanted to accept Rabbi Zekharia ben ha-Katzav’s ruling, and the Talmud Yerushalmi reports that the Babylonian sages had a tradition that followed his teaching. Nevertheless, the halakhah follows the other opinion, and boys receive preference in inheritance laws also in the case of a mother’s estate.”

Apparently, the rabbis read the previously mentioned purely a fortiori argument as a crescendo, i.e. an argument involving a quantitative comparison, in this case a comparison of ‘priority’. For instance, the second version of it may be supposed to contain an additional premise about ‘proportionality’ as follows:

A son (P) has more (or stronger) rights of inheritance (from father) (R) than a daughter (Q),

and, a daughter (Q) has rights of inheritance (R) enough to inherit from her mother with some ‘priority’ (S1);

the ‘priority’ of inheritance (S) is proportional to the ‘rights’ of inheritance (R);

therefore, a son (P) has rights of inheritance (R) enough to inherit from his mother with greater ‘priority’ (S2).



Thus, the subsidiary term (S) is different in the minor premise (S1) and conclusion (S2), with S2 > S1. According to R. Zekharia, this inference is to be interdicted by means of the dayo principle; whereas others accept it as is. Is this indeed, as the former claims, an argument subject to dayo application? We could say so, with reference to the daughter’s position, since the conclusion diminishes what might be supposed to be her rights (to equal or even prior inheritance). On the other hand, from the son’s viewpoint, since the conclusion improves his position, giving him first priority, dayo is not called for. So, there is room for debate[12].

Perhaps a few more words on this sugya would clarify matters a bit more. The Torah laws of inheritance (specifically, Num. 27:8) give the daughters of a man second priority compared to his sons: “If a man die, and have no son, then ye shall cause his inheritance to pass unto his daughter.” The term “priority” used in this halakhic context refers to a “winner takes all” order of precedence.

The sons inherit (almost) all of their father’s wealth, his wife and daughters being effectively excluded (except for certain provisions that need not concern us here). If a son predeceases his father, his own sons are next in line for his share, then his daughters[13] if sons are not available. If no sons, or male or female offspring of theirs, are alive, then and only then do the daughters (of the father), and their offspring, get the (whole) inheritance. Thus, the “right of inheritance” of daughters is potential, not actual. It is contingent: it is conditioned on there being no male child, or grandchild through a male child, available to receive the (whole) inheritance.

This concerns inheritance from a father. What of inheritance from a mother? The Torah does not explicitly answer this question; so, the rabbis try to answer it, in Baba Bathra, 111a, as follows[14]:

“[It is written.] And every daughter that possesseth an inheritance in the tribes of the children of Israel; how can a daughter inherit [from] two tribes? — [Obviously] only when her father is from one tribe and her mother from another tribe, and both died, and she inherited [from] them. [From this] one may only [derive the law in respect of] a daughter.”

Thus, as already mentioned, the rabbis first establish the rights of daughters to inheritance from their mothers. This serves as the minor premise of the a fortiori argument they use to derive the rights of sons to inheritance from their mothers:

“Whence [may the law respecting] a son [he derived]? — One may derive it by an inference from minor to major: If a daughter, whose claims upon her father’s property are impaired, has strong legal claims upon the property of her mother, should a son, whose claims upon the property of his father are strong, not justly have strong legal claims upon the property of his mother?”

This purely a fortiori argument is thereafter, as already shown, turned into an a crescendo (i.e. a ‘proportional’ a fortiori) argument, and the question of dayo arises at this stage:

“And by the same argument: As there, a son takes precedence over a daughter, so here, a son takes precedence over a daughter. R. Jose son of R. Judah and R. Eleazar son of R. Jose said in the name of R. Zechariah h. Hakkazzab: Both a son and a daughter [have] equal [rights] in [the inheritance of] a mother’s estate. What is the reason? — It is sufficient [etc.]” … “And does not the first Tanna expound. ‘It is sufficient [etc.]’? Surely, [the exposition of] Dayyo is Pentateuchal!” … “Elsewhere he does expound Dayyo, but here it is different, because Scripture says, in the tribes, thus comparing the mother’s tribe to the father’s tribe: as [in the case of] the father’s tribe a son takes precedence over a daughter, so [in the case of] the mother’s tribe a son takes precedence over a daughter.”

In this way, the law for inheritance from a mother is made to mirror that for inheritance from a father. Note that the sons are given precedence over daughters with regard to inheritance from the mother, even though the sons’ rights are inferred from the daughters’ rights; but this is not logically problematic, since the daughters’ rights referred to are only potential and therefore not altogether displaced by the inferred sons’ rights.

Here again we may express disappointment at the rabbis in general, and R. Steinsaltz in particular, for not analyzing the various logical issues dealt with in this section with appropriate rigor; and here again, their lack of formal understanding of qal vachomer and the dayo principle is to blame. I am of course not contesting the law[15], but only wish to point out that it is arrived at by means of logic which is not purely deductive (though it is made to seem so, somewhat) but which depends on some inductive leaps that we have above duly exposed. There is nothing wrong with induction, provided it is frankly recognized as such and not presented as deduction.

Various forms. To recapitulate: we examined the above two Talmudic examples of a fortiori argument to test R. Steinsaltz’s understanding of such reasoning. Both arguments, though unrelated, displayed the same sort of fallacious thinking (judged by strict deductive logic standards) – so it looks like we stumbled upon a recurring fallacy in Talmudic logic, and no doubt in people’s thinking in general. It is interesting to note that R. Steinsaltz did not notice these errors. To be fair, this was an easy mistake to make, even if a little careful reflection would have quickly gotten alarm bells ringing.

This fallacy has many possible forms. The forms we encountered above are the following. The most typical is an argument that resembles positive subjectal a fortiori, except that the major and minor terms (P and Q) are not uniform throughout – being greater in the major premise than in the minor premise and conclusion.

P1 is more R than Q1,

and P1 is greater than P2 and Q1 is greater than Q2.

“Therefore,” if Q2 is R enough to be S,

then, all the more, P2 is R enough to be S.



This argument is fallacious because the givens before the “therefore” do not always imply the formally required major premise “P2 is more R than Q2,” even if the latter is sometimes true. This formal requirement cannot be ignored or vaguely assumed; it has to be proved by some means.

We can show this argument is fallacious by simple mathematics. If we symbolize ‘the value of R for some variable x’ by R{x}, we can put it as follows: given that R{P1} > R{Q1}, and that R{P1} > R{P2} and R{Q1} > R{Q2}, it does not follow that R{P2} > R{Q2}. This can be seen in the diagram below. All we can deduce is that R{P1} > R{Q2}; regarding R{P2} and R{Q2}, the first may be greater (or equal) or lesser than the second.

Clearly, if we can somehow show that R{P2} > R{Q1}, then it would prove that R{P2} > R{Q2}. If on the contrary R{P2} < R{Q1}, the relation between R{P2} and R{Q2} remains undetermined; in such case, to prove the desired relation, we would need to refer to some other intermediary, say R{y}, such that R{P2} implies it and it in turn implies R{Q2}.[16]

Diagram 18.1

Variants of this are: where the major and minor terms are uniform, but the middle term (R) is explicitly or implicitly relative (to some item X, say) in the major premise and the same relativity is not mentioned or intended in the minor premise and conclusion, or where the major premise is explicitly or implicitly conditional (again on some item X, say) while the same condition is not mentioned or intended in the minor premise and conclusion. That is to say:

P is more R (relative to X) than Q,

“Therefore,” if Q is R (relative to something else) enough to be S,

then, all the more, P is R enough to be S.



(On condition X,) P is more R than Q,

“Therefore,” (on some other condition,) if Q is R enough to be S,

then, all the more, P is R enough to be S.



In these two forms, the fallacy lies in ignoring that the tacitly intended ‘relativity to X’ or ‘conditioning upon X’ in the major premise may well not be also applicable in the minor premise and conclusion, whereas the minor premise would still be true if it applied to non-X but did not apply to X. The three arguments are, then, from a strictly deductive point of view, fallacious – i.e. the putative conclusion does not necessarily follow from the premises. This does not mean that there are not special cases where the required premises can be produced inductively or even deductively – it just means that they are not universally present.

We can from these positive subjectal forms predict three negative subjectal ones, where the minor premise is negative with the major term (P) as subject and the conclusion is negative with the minor term as subject (Q). We can also conceive of implicational and hybrid equivalents of these various forms. A similar set of fallacies can be expected to occur with predicatal a fortiori arguments. For instance, the following one would be typical:

More R is required to be P1 than to be Q1,

and P1 is greater than P2 and Q1 is greater than Q2

“Therefore,” if S is R enough to be P2;

then, all the more, S is R enough to be Q2.



3. Lack of formalism

Although it is pretty obvious that the rabbis involved in the above described fallacious arguments were intending valid deductive arguments, we can ‘save face’ for them by suggesting that (though they did not say so out loud) they did not really mean their arguments as strictly a fortiori, but were consciously engaged in arguments that are ‘roughly a fortiori’. They are mere analogies that resemble a fortiori but are not truly so, being more inductive than deductive. They were ‘pseudo-fortiori’, or ‘quasi-fortiori’, but not really fortiori.

In any event, as I have shown in the present and other chapters (7-9), the main cause of problems in rabbinic use of a fortiori argument (and this is also true of other forms of argument, of course) is their eschewing formal logical studies. This lack of formalism is confirmed by R. Steinsaltz in his work The Essential Talmud (chapter 30)[17], where referring to the specificity of the “talmudic way of thinking” he writes:

“A basic factor is the attitude towards abstraction. In the Talmud, as in most areas of original Jewish thought, there is deliberate evasion of abstract thinking based on abstract concepts. (…) The Talmud employs models in place of abstract concepts. (…) Kal va-homer, for example, is a method applied to a certain model in order to adapt it to another model. Thus there is a high degree of mechanical thought, and no attempt is made to clarify practical or logical problems per se; (…) it is not always possible to understand the convoluted methods of the operation itself” (p. 263).

R. Steinsaltz, of course, is not intending any radical criticism thereby. But I would say that this is the core problem, and if we ever hope to modernize and improve upon past Jewish legal thought, and credibly further develop it, we have to learn and adopt more formal methods of inquiry. There is in my view just no excuse for hanging on to ways of thinking that (occasionally, if not often) lead to error. Logic is not something arbitrary, which can be ignored or shunted aside when we dislike its results. Logic is a way to test if a theory is true, together with empirical data. If a theory goes against logic, and/or against empirical data, it must be rejected or at least reformulated. There is no credible escape from this methodological requirement. It is an absolute, applicable to religion as well as to everything else.



[2] The essays there posted are described as “based upon the insights and chidushim [i.e. novelties] of Rabbi Steinsaltz, as published in the Hebrew version of the Steinsaltz Edition of the Talmud.”

[4] A baraita is a statement of Tannaic origin, i.e. antedating the Gemara.

[5] I suspect that such statements were derived from the Mishna Beitzah 5:2; but I do not know who did that first.

[6] Chavel: Encyclopedia of Torah Thoughts, p. 27, n. 106. Feigenbaum: Understanding the Talmud, p. 88.

[7] I do not quote him verbatim, because I have before me the French version of his text. I presume the English edition says pretty much the same thing.

[8] I tacitly assume that the implications mentioned here are all normal; i.e. that not-X does not imply Y, X does not imply not-Z, and Y does not imply not-Z. The first two of these are tacit premises, and the third is a conclusion (demonstrable ad absurdum: if Y did imply not-Z, then since X implies Y it would follow that X implies not-Z – which is given as untrue).

[9] Or, if you prefer, such oral laws are more subject to faith, because their emergence in the present cannot be exactly traced all the way back to Sinai. Of course, this is objectively true even of the written law, but a difference of degree can still be claimed.

[10] Who is presumably no other than R. Jose, the father of R. Eliezer to whom the list of 32 hermeneutic rules is attributed.

[11] R. Yossi’s actual formulation, according to the Soncino Talmud, is: “If heleb, for which there is a penalty of kareth [if eaten], is permitted for use, how much the more the sinew [is permitted for use], for which there is no penalty of kareth [if eaten]” (brackets mine). Note in passing that the Socino Talmud wrongly refers to this as argument “a minori” (ad majus) – whereas it is clearly in fact a majori ad minus (from kareth to no kareth). Note also in passing that there is no issue of dayo in this context, and none is raised; yet no one marvels at the fact and its implication that the dayo principle is not always relevant.

[12] I would say, however, that dayo is not relevant here, since what is at issue is not strictly speaking a penalty to be applied by the court.

[13] This logically follows from Num. 27:8 by reiteration, i.e. applying the law first to the father and then to his sons.

[14] I quote from the Soncino edition, because I do not have at hand R. Steinsaltz own English translation (if he has one). The “it is written” reference here is to Num. 36:8.

[15] Although, in all honesty, I personally find the Jewish law of inheritance unfair to women, denying them independence as human beings. It was, no doubt, at the time it was promulgated a reflection of the existing patriarchal society, and probably an improvement in the status of women. But times have changed, and in today’s society women must be acknowledged as equals under the law.

[16] Note in passing that, given R{P1} > R{Q1}, we would likewise be unable to establish that R{P2} > R{Q2}, if R{P1} < R{P2} and R{Q1} < R{Q2}. On the other hand, if R{P1} < R{P2} and R{Q1} > R{Q2}, it would follow that R{P2} > R{Q2}.

2016-07-04T06:58:44+00:00