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A FORTIORI LOGIC

© Avi Sion, 2013 All rights reserved.

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A FORTIORI LOGIC

CHAPTER 23 – Alexander Samely

1. General definition

2. Descriptive formula

3. Three alleged techniques

4. Bava Kamma 25a-b

5. Samely’s online database

6. My critical researches

Alexander Samely published in 2002 a book called Rabbinic Interpretation of Scripture in the Mishnah, in which one chapter is devoted to a fortiori argument[1]. According to the Google Books blurb, “This volume offers a systematic and detailed description of early rabbinic hermeneutics as it can be reconstructed from the Mishnah (third century CE)”[2]. This is a very ambitious work, of wide scope and great detail, with many interesting innovations. I do not however propose to thoroughly review it, but will concentrate all my efforts on its treatment of a fortiori argument. I will also frequently refer to the author’s associated website[3].

1. General definition

The following sentence may be taken as Samely’s general definition of a fortiori argument:

“The mechanism of the inference involves an assignment of (mostly halakhic) categories to the two subjects; a ranking of these categories in a dimension of comparison; and the transfer of what is known about one of them to the other based on its higher rank in the comparison of categories” (p. 174).

The “two subjects” in question are those generally called the major and the minor term, although Samely does not so call them; the “dimension of comparison” refers to what I have called the middle term; and “what is known about” one of them that is “transferred” to the other refers to what I have called the subsidiary term. Note that Samely’s understanding of a fortiori argument is entirely focused on the positive subjectal mood, which goes from minor to major. His definition is pretty good for that mood, since it refers to four terms and correctly places them in relation to each other. But it misses out on the many other moods.

However, Samely is wrong when he then claims that it is the “differential of ranks” between the major and minor terms “which leads to the claim that for the second subject [i.e. the major term] the validity, certainty, or reasonableness of the inferred proposition is even greater than for the subject from which it is inferred [i.e. the minor term]” (p. 174, square brackets mine). He is here misled by colloquial expressions, like “all the more,” which are commonly used to signal a fortiori argument, into believing that (given the major premise, of course) the conclusion is more reliable than the minor premise from which it is derived. This is untrue: logic simply cannot perform such a feat; in any kind of argument, the conclusion is always and of necessity no more reliable than the premise(s) it is derived from. This shows that Samely does not fully understand the nature of a fortiori inference, being misled by the wording we commonly use in it.

He is of course correct in pointing out that the function of “a fortiori inference, or qal wa-homer… in the Mishnaic discourse is often explorative rather than apodictic” (p. 174). And again further on: “the Mishnah uses the a fortiori more for probing the consistency of a normative position, or for exploring its consequences, than for categorically determining it” (p. 176); also: “the halakhic a fortiori is used heuristically, not apodictically” (p. 177). But that is not what is at issue as regards the validity of the process. Samely has evidently not studied the question of validity to sufficient depth.

The closest he comes to such general considerations is when he writes: “Although not syllogistic in nature, the a fortiori argument has the potential for being presented in an apodictic and formal manner” (p. 190). However, his presentation of this “apodictic and formal manner” is neither appropriately abstract nor valid. He merely proposes an illustration, which may be briefly paraphrased as: ‘If single-sector consumer groups can influence things, then multiple-sector consumer groups can influence things even more’. The problem with it is the ‘proportionality’ of its conclusion – the suggestion that consumer groups can, in proportion to their numbers, jointly have more influence. That may be true empirically, but it is not a purely a fortiori deduction. No contradiction would have ensued if it was found that the influence is not so cumulative.

We can readily put the argument in standard form: the major premise would be: multiple-sector consumer groups (P) have more clout (R) than single-sector consumer groups (Q); whence: if single-sector consumer groups (Q) have clout (R) enough to influence things (S), then multiple-sector consumer groups (P) have clout (R) enough to influence things (S). But according to formal logic, there is no “even more” about it: the subsidiary term (S) cannot be greater in the conclusion than in the minor premise. By purely a fortiori argument, we can only infer that numerous consumer groups together will have at least as much influence as any of them singly (assuming no mutual interference). To draw an a crescendo conclusion as he does, we would have to append a pro rata argument. Samely has not clearly grasped this, and it colors all his analysis.

Samely does display considerable understanding of formal logic in general, when he rightly points out the flaws in Louis Jacobs’ analysis of a fortiori argument (p. 177, footnote). Regarding Jacobs’ “simple type,” viz. “If A has x, then B certainly has x,” Samely remarks that though this reproduces “the grammar” of the argument, it cannot do the same for its logical form, “since for the logical form some comparison between A and B is always necessary.” As regards Jacobs’ “complex type,” viz. “If A, which lacks y, has x, then B, which has y, certainly has x,” Samely comments: “the comparison at the centre of the argument is in any case not accurately represented by the idea of ‘lacking’ or ‘having’ y, but rather by ‘having y in various degrees.’” However, though perspicacious, this comment is not entirely correct, in that the ideas of lacking and having can be placed in the wider context of the idea of degrees. This means that Jacobs’ “complex type” can also be looked upon as a grammatical form rather than a logical one.[4]

However, Samely seems to basically distrust formal logic, when he writes: “And one should certainly avoid ‘translating’ a rabbinic argument so that it fits into a predetermined, ‘known’ logical format, in particular if that leads to the ‘discovery’ of Greek logic in the rabbinic procedures. In most cases a translation of this type does indeed succeed in reducing the ambiguities of the original argument. But the price that has to be paid for this is high, because the new language, ‘logic’, is so much poorer. A century of intense investigation of the relationship between logic and ordinary language has resulted in an attitude of extreme caution towards such translations even within a contemporary setting, let alone for arguments from culturally or historically distant contexts” (p. 181). While there is considerable truth in what he says here, it is not entirely accurate.

It is evident that he here has in mind Adolf Schwarz in particular, who is generally regarded as having had a preconceived and misconceived idea that a fortiori argument can be equated to syllogism, and thus that rabbinic procedures can be forced to fit into Greek logic. Samely is probably, subconsciously, also thinking more broadly of modern logic, which (in my view) is indeed so superficial and simplistic that it cannot reflect all the nuances and complexities of ordinary language. He suggests, on the authority of Wittgenstein, that “language is tied to the situations of its use (i.e. its illogicality, in a manner of speaking);” but in fact, this view however fashionable is not based on a correct understanding of the relation of formal logic to everyday thought and language.

Briefly put, though our theories of formal logic are based initially on intuitive use of logic by people in everyday discourse, the science of logic eventually transcends the practice of logic. That is to say, it passes from description to prescription. How people commonly verbally express their logical thoughts (which are not always in accord with formal logic, not only linguistically but also logically) then becomes a special object of study, which is relevant to the science of logic in a broad sense but not to be confused with the core field of study known as formal logic. Similarly, though the history of logic is part of the science of logic in a broad sense, it can never be a substitute for formal logic.

Certainly, there is no justification for taboos: we must refer to formal logic to evaluate given material arguments, however venerable they are. In other words, Samely’s relative avoidance of formal logic (I say ‘relative’, because he does in fact appeal to it to some extent) is not based on a wide knowledge and understanding of the subject. Moreover, as we shall see, this prejudice constitutes an obstacle to insight in his work.

2. Descriptive formula

Samely conceives of a fortiori argument more specifically as having a “five-point structure, which takes the argument to have four premises and one conclusion”: (1) There are two “norms” n and m, which (2) belong respectively to “norm types” N and M; and (3) when these are compared in a “dimension” X, the former are graded higher than the latter; given also that (4) to the lower grade (n, N) “applies feature A,” it follows that (5) to the higher grade (m, M) applies feature A (p. 179).

If now I try to recast this set of propositions in standard a fortiori format (in terms of P, Q, R S), I obtain the following valid positive subjectal argument:

Norm m of type M (P) is graded higher in dimension X (R) than norm n of type N (Q) is [major premise, from “points” 1, 2, 3],

and norm n of type N (Q) is graded in dimension X high (R) enough for feature A to be applied to it (S) [minor premise, from point 4];

therefore, norm m of type M (P) is graded in dimension X high (R) enough for feature A to be applied to it (S) [conclusion, as in point 5].



There are a number of problems with this descriptive formula, which I will now elucidate:

Asymmetry. Actually, Samely’s concluding point is worded as: “therefore: to norm m feature A should apply even more.” But I do not see why he fails to mention there the norm type M, while he mentions the norm type N in the preceding point. Similarly, I do not see why he there says “feature A should apply even more” instead of simply saying “applies feature A” as before. The expression “applies” is vague enough as it is – there is no profit in adding ambiguity by changing the wording from “applies feature A” in the minor premise to “feature A applies” in the conclusion, giving a confusing impression (though I doubt it to be intentional) that A is a predicate in the former and a subject in the latter.

A redundancy. Another question one can ask is: why is it necessary to mention both “norms” (n, m) and “norm types” (N, M), as Samely proposes? If we want to classify the norms in their types, the major term (P) may be expressed as ‘norm m of type M’ and the minor term (Q) as ‘norm n of type N’ – and these may be quantitatively related via the middle term (R), viz. “dimension X” in the major premise. There is no need for three propositions when one suffices. Splitting the major premise up into three components is not only redundant[5], but causes confusion because it conceals the operative totality.

Alternatively, use the “norms” as the operative subjects and put the “norm types” in parentheses next to them. Actually, Samely’s exact wording of the terms of the third (i.e. the major) premise are: “type M (norm m)” and “type N (norm n)” – which suggests he is not too certain as to whether the types or norms should be included here, so he fudges the issue. Possibly, what Samely intended here is that the major premise should be exclusively in terms of the “norm types,” i.e. very broadly stated, and then the minor premise and conclusion should be exclusively in terms of the more narrow “norms.”

In that case, there would have to be, at least implicitly, a substitutive syllogism that applies the given major premise ‘M is more X than N’ to the corresponding narrower terms (m, n), saying: since ‘m is M and n is N, it follows that m is more X than n’. However, we must be careful here to engage in valid deduction. The proposition ‘M is more X than N’ should mean that ‘every case of M is more X than any case of N’, or at least that ‘every M is more X than the N corresponding to it’ – otherwise, even if m and n are known to be subcategories of M and N respectively, we cannot directly deduce that ‘m is more X than n’ from it.

In other words, it is conceivable that, in some cases, the vague generality ‘M is more X than N’ is true and yet the more specific ‘this m is more X than that n’ is not true – so we must be careful. In any case, when such deduction is indeed possible, it is clear that the proposition ‘m is more X than n’ would be the operative major premise of the a fortiori argument. In such cases, the original broad premise ‘M is more X than N’ would strictly speaking not be part of the a fortiori argument as such, but only serve a preparatory act of reasoning – so it should not be presented as part of the a fortiori structure per se. Samely has apparently not realized all that.[6]

In practice (I refer here to the Talmud in general and related documents, and not solely to the Mishna), the major premise is expressed (if at all) sometimes in terms of broader categories (like M, N) than those in the minor premise and conclusion, and sometimes in terms of the same categories (like m, n) throughout the argument. If only for this reason, the formal formulation of a fortiori argument must be in terms of the operative categories, i.e. those appearing or intended in the minor premise and conclusion, and not in terms of broader categories that may sometimes appear or be intended in the major premise. This is obvious, since the form we adopt as standard must cover all possible cases.

A deficiency. Another issue worth raising regarding Samely’s formula is the absence of the notion of sufficiency in minor premise and conclusion, i.e. of the minor and major terms being X (i.e. the middle term, R) ‘enough’ for the application to them of feature A (the subsidiary term, S). Although it is true that this notion is very often absent in everyday and in Talmudic a fortiori discourse, this does not mean that it is at all expendable. It is in truth an essential part of the argument, without which the argument cannot be validated. Though it is not always explicit, it is certainly always at least implicit. What makes it possible for the norms n or m to have the feature A is that they have reached the level of dimension X required for A. Reaching or surpassing the value of X corresponding to the minor term is what makes possible and justifies the predication of the subsidiary term to the major term.

This value constitutes a benchmark or turning point. The argument tacitly intends that if the condition is not satisfied, the putative inference is not possible. Therefore, if the benchmark value of X (i.e. of the middle term, R) is not clearly made part of the minor premise (and thence, the conclusion), the involvement of X in the major premise remains unexplained. Samely has apparently not understood this crucial point, since he does not explicitly specify “X enough” in his formula. Perhaps we should consider his use of the vague relational expression “applies” as unconsciously intending this subtext. But such vagueness is unacceptable in formalization efforts.

Limitations. Another fault in Samely’s structure, if it is intended as a general statement, i.e. one applicable to all possible cases of a fortiori argument, is that it is too limited, being restricted formally to the positive subjectal mood, without mention of the negative subjectal mood, or the positive or negative predicatal moods. While negative moods may be regarded as implicit in positive ones by reductio ad absurdum, predicatal moods are significantly different from subjectal ones with regard to the form of both premises and conclusion and therefore cannot be glossed over. Many commentators through logic history have like Samely ignored predicatal a fortiori; but it is alluded to in some rabbinic accounts.

Also not mentioned by Samely are the four implicational moods corresponding to these four copulative moods. Did he perhaps intend the vague word “applies” to suggest either predication or implication? It is possible. As we shall see in the next section, on hermeneutic techniques, Samely seems to be aware of implicational a fortiori argument, though he uses the Greek terms protasis and apodosis instead of the more common Latin terms antecedent and consequent. This is presuming I interpret his intention correctly. But in any case, he does not explicitly make and discuss the distinction. Furthermore, just as his copulative form is limited to the positive subjectal mood, so his apparently implicational form is limited to the positive antecedental mood.

Also missing in Samely’s account is a distinction between superior, inferior and egalitarian a fortiori argument. That is, he does not notice that the major premise may be stated either as ‘this term is more X than that one’, or as ‘this term is less X than that one’, or as ‘this term is as X as that one’. The superior and lesser versions may be converted to each other, but the egalitarian (i.e. a pari) version has both features (i.e. works as well in both directions). Samely does not apparently realize that this third version is commonly used (even in the Mishna) and formally significant.

I would also object to the apparent limitation by Samely of the subjects to “norms” – though this might be largely explained by his concentration on Mishnaic qal vachomer. But let us ask: are these two terms truly normative (in an ethical or legal sense)? I would say: usually not (though in some cases they happen to be). The usual pattern, when there is a normative intent (which is not always the case) is roughly as follows:

Action P is better/worse (more R) than, or as good/bad (as much R) as, action Q (is R):

and action Q is good/bad (R) enough to be (as the case may be) permitted, forbidden, exempted or prescribed (S);

therefore, all the more (or equally), action P is good/bad (R) enough to be (as the case may be) permitted, forbidden, exempted or prescribed (S).



As can be seen in this general formulation, the major and minor terms, P and Q, e.g. two voluntary actions, are non-normative. But they are compared in a normative scale, viz. the middle term R; i.e. they are classified as in some sense good or bad. The conclusion may also be normative in the sense that the subsidiary term (S) permits, forbids, exempts or prescribes such behavior. In some cases, the conclusion is non-normative, with S simply denoting some physical or mental result. Thus, for Samely to call P and Q “norms” is misleading; it misplaces the possible normative elements, which are rather concentrated in R and S. His thinking here was too vague.

No validation. Most important, I see no effort by Samely here to formally validate the argument – i.e. to explain, other than merely ‘intuitively’ why the conclusion follows – indeed, must follow – from the premises. This is a major issue, which has to be addressed. We cannot just assert something, without proof. But maybe it is not of great interest to Samely, because he comes on more as a non-judgmental reporter of Mishnaic discourse than as a logician intent on evaluating it. He is perhaps more interested in the ‘how’ (the rhetoric) than the ‘why’ (the logic), i.e. in what convinces some (possibly even many) people than in what ought to convince everyone (as it does all people skilled in logic).

Samely’s use of “should apply” in the conclusion instead of “applies” suggests, however, that he does not consider the a fortiori argument to formally yield a deductive conclusion, but merely at best an inductive one. He is certainly marking some hesitation, thinking that in at least some cases the conclusion is open to doubt. This is indicative of a failure of understanding on his part – it means he is unaware that validity can be strictly demonstrated. It could of course be that he is merely trying to reflect rabbinic uncertainties in this issue. But then he should have clearly stated that such uncertainties are due to an erroneous conflation in their minds between doubts in a premise (usually the major, but sometimes the minor) and doubts in the process of inference (i.e. in the implication of the conclusion by the premises jointly).

As regards Samely’s use of “even more” in the conclusion, this may merely be intended to reflect the expression “all the more,” which is traditionally used to signal a fortiori reasoning. Or he may have observed that a fortiori argument in some cases seems intuitively to merit an a crescendo conclusion, without realizing that such cases are made possible by an unstated additional pro rata argument. Thus, his use of “even more” may also (and more likely does) signify that he is unsure as to whether the ‘dayo’ principle (or more precisely put, the principle of deduction with which it is generally confused) is universally applicable, or maybe sometimes ‘proportionality’ is legitimate[7]. Such uncertainty is inevitable if no efforts at validation or invalidation are invested. Or maybe, again, he is just trying to reflect rabbinic vacillation in this issue. But then, again, he should have explicitly made his dissent clear, explaining on logical grounds why the conclusion (qua conclusion) must exactly mirror the minor premise.

Precedence. Albeit these various faults, Samely’s descriptive formula is very good. It is so, because it correctly contains four explicit terms (n, m, X and A – corresponding to my P, Q, R and S), laid out in ordered premises and conclusion, including an explicit major premise with an explicit middle term (X) and a minor premise and conclusion with a clearly identified subsidiary term (A). These formal insights are so crucial and novel (in comparison to the proposals of the past authors he mentions) that I wondered where he got them from! The resemblance to the positive subjectal form presented in my book Judaic Logic is considerable. Yet Samely nowhere mentions my work on the a fortiori argument, though that book was published seven years before his. He does mention many other authors, so he must have made considerable efforts of research. It is surprising that he apparently never stumbled upon my work in the course of such research[8].

I put the question to Samely by e-mail recently, and he assured me that he had never before come across my work. I do not doubt his word, albeit the evident similarities, in view of the many differences, mentioned above and below, between his treatment and mine. In any case, for the record, I claim precedence as well greater accuracy[9].

It is a great pity that he did not discover and study my work before he engaged in his own research. It would have saved him from a lot of mistakes and made possible a much richer and more useful treatise. This remark applies primarily to a fortiori argument, but (knowing the originality of my findings) I expect it to equally apply to other Talmudic hermeneutic principles (notably those aimed at harmonization). As things stand, I would recommend to him to seriously study my Judaic Logic and the present work, A Fortiori Logic, and then review his whole treatise and do what is necessary to bring it up to date. I do not wish to sound condescending, but give this as friendly good advice.

3. Three alleged techniques

Samely has developed his own list of “hermeneutic resources” used in the Mishna, given in Appendix 1 of his book and also posted in his website[10]. This list, if I understand correctly, is not (or not exclusively) based on the traditional Jewish lists (namely, Hillel’s, R. Ishmael’s, R. Eliezer’s and the Malbim’s), but is an attempt to independently, systematically and exhaustively identify, in a modern perspective (i.e. scientifically, rather than dogmatically), the actual hermeneutic techniques of the participants in and compilers of the Mishna. This is, in principle, a very commendable enterprise. I presume, offhand, it has contributed a lot of insight and information to modern Talmudic studies. In truth, I have not read the whole work and so cannot judge it for good or bad, but at a glance it looks impressive and valuable.

All that interests me for the present are the three techniques of a fortiori inference that Samely lists in chapter 7 of his book. The following defines the technique he labels “Analogy4.2” (or “A4.2”), which obviously corresponds to the formula above reproduced and analyzed by me:

“Inference by analogy that norm m possesses predicate A, in the following manner: If norm n which belongs to the category N, which category is lower on scale X, has predicate A; then norm m which belongs to the category M, which category is higher on scale X, logically also has predicate A (or: logically must have more of the quality A).”

The technique labeled “Analogy5” (or “A5”) is similarly worded, with the following exception: instead of “norm” n or m, it speaks of “subject” n or m. These differences in wording suggest that Samely found that some Mishna qal vachomer discourse is not normative. The technique labeled “Analogy4.1” (or “A4.1”) is also similarly worded, with the following exceptions: instead of “norm” n or m and of “predicate” A, it speaks of “the protasis” n or m and of “the apodosis” A. These differences in wording suggest that Samely found that some Mishna qal vachomer discourse is not copulative, but implicational.

Thus, Samely claims to have identified three distinct techniques of a fortiori argument in the Mishna. Even so, it is evident from this list and must be stressed that Samely has not identified all moods and types of a fortiori argument. Samely’s listed arguments are apparently, to repeat, all of the positive subjectal (A4.2 and A5) or positive antecedental (A4.1) kind. Perhaps the reason is that he has found only these three varieties in the Mishna; but this needs empirical verification, with reference to the broader list of a fortiori moods given in my work. (As I show further on, such a claim is empirically inaccurate.)

Another difference in wording is that (my italics) A5 says “predicate A applies to subject m” and A4.1 says “the protasis of norm m has the apodosis A,” whereas A4.2 says “norm m possesses predicate A.” But all these are clearly intended as the same, as the relation “has [predicate or apodosis] A” is thereafter used throughout the definitions. They are mere stylistic variations in wording.

A detail to note in the three techniques is in the conclusion the bracketed clause: “or: logically must have more of the quality A” (in A4.2 and A5) or “or: logically must have an intensification of the apodosis A” (in A4.1). Notice the qualification “logically.” This seems to confirm that Samely believes ‘proportional’ a fortiori arguments to be valid, or at least that he has found such apparent arguments in the Mishna. Of course, as I have shown through formal logic, ‘proportional’ a fortiori argument is in fact invalid. To draw a ‘proportional’ (a crescendo) conclusion, one needs additional information, besides the a fortiori premises, enabling us to formulate an appropriate pro rata argument.

My own reaction to the said three techniques is that making a distinction between them was unnecessary hair-splitting. True, there are noticeable differences between them; but (granting momentarily for the sake of argument that only these three moods exist) do they justify distinct categorizations? It is a bit like dividing the class ‘flowers’ into ‘red roses’, ‘pink roses’ and ‘red tulips’! The differentia used are inessential; and the proposed division is obviously not exhaustive, since it lacks symmetry[11].

And why are these arguments not candidly called “qal wa-homer,” and instead referred to as numbered “analogies”? One gets the impression that there was a deliberate intent to appear different and innovative. Samely’s explanation for his use of artificial names (“Analogy4 and Analogy5”) for the “Mishnaic a fortiori argument” is that “the hermeneutic argument known in rabbinic literature and in modern scholarship as qal wa-homer… is not clearly defined.” Moreover, though this descriptive label is often used in the Mishna, it also frequently employs the vaguer expression din, meaning judgment or inference, with reference to the same structure of argument. The word din, he adds, may also be used for other types of inference. “More importantly,” he argues, “although the Mishnah does not seem to use the terms qal and homer in conjunction with any [other] hermeneutic resource… the possibility of that happening somewhere in rabbinic literature cannot be ruled out” (p. 176).

These explanations are unconvincing. Although the concept of a fortiori argument is indeed insufficiently defined by the rabbis, and the terminology for it varies and may even not be exclusive to it, I see no reason for changing its accepted name. It would suffice to just point out these problems and specify one’s more precise position, conventionally choosing one name (or more) from among the existing names. The real and only reason Samely changes the name is that he wishes to fit the argument into his wider system of classification and labeling. That being a methodologically acceptable reason, his stated lame excuses for doing it were unnecessary.

4. Bava Kamma 25a-b

To assess Samely’s understanding of a fortiori argument more precisely, it is necessary to look at his interpretations of actual examples of Talmudic reasoning. Let us first look at Samely’s analysis, given in his website[12], of Mishna Bava Kamma 2:5[13] and the related Gemara on pp. 25a-b of the Talmud tractate. He there says a few things I agree with, and many I disagree with. Regarding my own detailed analysis of this passage, the reader is referred to the earlier chapter of the present work, ‘In the Talmud’ (7). The reader must study that chapter in order to understand where I am coming from in the present critique. This saves me from having to repeat relevant findings here.

Samely’s reading of this important sugya (i.e. Talmudic discussion on a specific topic) is original in some respects; but it is also seriously inadequate in certain respects. It is interesting to note that he realizes that R. Tarfon’s two arguments need not be taken as a fortiori in form – as (to my recollection) all past commentators have assumed – but may, one or both of them, be viewed as mere analogy. He writes:

“I take the two presentations of Tarfon’s argument to be open to two different constructions: while the first sounds more like an a fortiori argument, the second sounds more like an ‘analogy of relationships’ argument. However, I am assuming that the argument in both formulations is the same, so that one of the two constructions is the meaning intended by both formulations.”

Actually, I would say that the most likely scenario is that R. Tarfon’s first argument is merely analogical (pro rata) whereas his second is (purely) a fortiori. It is however also conceivable that both R. Tarfon’s arguments are analogical, or alternatively that his first is a crescendo (i.e. a fortiori cum pro rata) and his second purely a fortiori. But in any case, we cannot logically suppose that both R. Tarfon’s arguments are purely a fortiori, unless we also suppose that R. Tarfon’s first argument was fallacious (since it does not obey the principle of deduction, which in this instance corresponds to the dayo principle). Still less can we assume R. Tarfon’s two arguments to be respectively a fortiori (the first) and analogical (the second), as Samely initially imagines.

Thus, Samely’s just quoted interpretations are mostly incorrect, showing he does not clearly understand the differences between a fortiori argument and mere argument by analogy. The way he distinguishes the two types of reasoning is by saying:

“If it is an analogy of relationships argument (A0.5), its terms happen to provide for the inference of an increase in punishment, but are otherwise quite independent of any ‘all the more so’ inference typical for the a fortiori.”

What does this distinctive “all the more so” phrase do, exactly? He does not know or say. He seems to think that the distinction lies in preliminary generalization. It is to his credit that he realizes that the proposed a fortiori arguments are preceded by generalizations, which establish the needed major premise. However, he does not realize that identical generalizations precede the proposed mere analogies, as I have shown. As he puts it:

“If it is an a fortiori argument then it incorporates within itself … a tacit generalisation from a specific cited case to a whole class (the case of tooth/foot to the case of all damage where the “public domain versus private domain of the person injured” distinction applies); in either case, there is no involvement of Scriptural text.”

Note that Samely views R. Tarfon’s two arguments as essentially one and the same. He says so explicitly earlier (“the argument in both formulations is the same”) and further confirms it here by using of the singular (“if it is…” etc.). However, in truth, not only are the structures of R. Tarfon’s arguments potentially distinct (as already mentioned), but though both are derived from the same Scriptural givens their major premises are built on different directions of generalization from these Scriptural givens. In the first case, the greater stringency of the private over the public domain is generalized from ‘tooth/foot’ to ‘all cases’ – whereas in the second case, the greater stringency of horn over tooth/foot is generalized from ‘the public domain’ to ‘all cases’. These preliminary inductive acts make the two arguments very different, whether they are conceived as analogical or a fortiori.

However (to repeat), in Samely’s view R. Tarfon’s two arguments are one and the same, and the only issue is whether the process is interpreted as analogical or a fortiori. Looking at his attempted explanations of the difference between these two processes of inference, however, I do not discern any formal difference! The conclusion is either way the same (as R. Tarfon indeed claims) – viz. that liability for “horn damage in private” property “should be” / “ought to be” “more severe” than in the public domain, i.e. “full damages.” Evidently, he accepts ‘proportionality’ without reflection in both cases. Note also that his use of the wording “should be” or “ought to be” is indicative of some uncertainty in his mind as to the logical necessity such conclusion, either way. All this is evidence that Samely’s understanding of the forms of reasoning he seeks to inventory is limited.

Moreover, it is not at all clear what Samely means when he writes “in either case, there is no involvement of Scriptural text.” The generalizations are evidently generalizations from Scriptural givens. The whole discussion is an attempt to derive new laws from Scriptural givens (namely, Exodus 21:35 and 22:4).

As regards the Sages’ replies to R. Tarfon, viz.:

“They answered: It is enough if the inferred is as strict as that from which it is inferred: if in the public domain half-damages, so also in the private domain of him that was injured, half-damages.”

(אמרו לו דיו לבא מן הדין להיות כנדון מה ברהר חצי נזק אף ברשות הניזק חצי נזק)

Samely deserves credit for noticing that they are identical in both cases, for this is often glossed over. This reiteration is surprising, since – as Samely also notices – R. Tarfon’s two arguments proceed along different routes, as already seen. However, Samely’s explanation of this is simply to say:

“The Sages … do not accept his explanation. In their final reply they still depict him as inferring from the half-damages of the horn in public (the half-damages come only from the horn; had they taken on board what he said, they should have spoken about the full-damages of the tooth/foot).”

Evidently, Samely has not understood the difference between the Sages’ first and second dayo objections. Though the language of the two objections is identical, their meanings differ significantly in relation to R. Tarfon’s two formulations. Given that R. Tarfon’s first argument goes from half damages for horn in the public domain (minor premise) to full damages for horn in the private domain (conclusion), the Sages’ dayo objection can here be construed as a statement that the conclusion must mirror the minor premise (i.e. remain ‘half’). But given that R. Tarfon’s second argument goes from full damages for tooth & foot in the private domain to full damages for horn in the private domain, the Sages’ dayo objection here obviously cannot be construed as a statement that the conclusion must mirror the minor premise, since it already does!

It follows that in the latter case, the Sages must – albeit using the very same language – be interdicting another part of the reasoning process, viz. the inductive formation of the major premise, i.e. more specifically, the generalization from “in the public domain, horn damage implies more legal liability than tooth & foot damage” to “in all domains (including the private), horn damage implies more legal liability than tooth & foot damage.” If this generalization were not rejected or at least devalued by the Sages, R. Tarfon’s conclusion of “full damages” would be logically inevitable, whether by analogy or a fortiori. By interdicting this generalization, the Sages thus ensured that the only possible conclusion was “half damages” – as before.

Samely, to repeat, did not grasp this reasoning, because he lacked precise analysis of analogical and a fortiori argument forms. His attempted explanation in fact does not explain anything. He should at least have read the explanation given by a Tosafot commentator, which clarifies things considerably. With regard to the Gemara relative to this Mishna, Samely remarks only: “The Gemara at bBQ 25a treats the whole as an a fortiori, naming it a qal wa-homer but the Gemara focusses on the acceptance or rejection of the idea of dayyo.” This is true – but of course says very little. Samely does not subject to any critical analysis and evaluation the Gemara’s idea that qal vachomer is naturally ‘proportional’ and that the dayo principle is a Torah-based limitation on this logic. He seems to just take it all for granted.

Following this, Samely draws attention to the first of the three related a fortiori arguments proposed further on in the Gemara, saying: “A reversal of this argument is found in bBQ 25b bottom” (actually, this continues on p. 26a). However, though he reads the argument correctly[14], and he derives from it the first argument of R. Tarfon[15], Samely makes no attempt to clarify the logical intricacies it involves. Moreover, he inexplicably makes no mention of the other two additional arguments in the Gemara.

All this leads me to conclude that Samely is not entirely clear with regard to a fortiori logic, whether theoretically or in practice. This conclusion naturally casts a shadow on his understanding of the Talmud’s hermeneutic principles in general (and not only the first, “qal vachomer”). Sorry to say. On this basis, I would re-examine very carefully all the results of his research before granting any of it credence. It looks interesting enough offhand, but one cannot expect accurate results if one does not really understand the forms of discourse and reasoning involved. I would again recommend to that author to study my past and present work and then conscientiously review all his own past work. I cannot take the time to do all this verification and correction work for him; it is his responsibility.

5. Samely’s online database

One very interesting aspect of Samely’s research is that he has tried to list and analyze every hermeneutic act in the Mishna, categorizing them all in accord with his system and making the results publicly available through a search engine in his website[16]. I cannot, to repeat, say whether the work done was all appropriately and accurately done, but do wish to express approval of the task he set himself and the effort made anyway. Such empirical and thorough research is truly what is needed and deserves applause. I called for such meticulous and wide ranging research already in my book Judaic Logic. What I can say in the way of general criticism, off the cuff, is that he does not seem to give sufficient attention to the hermeneutic principles conceived by the rabbis themselves. If so, this is a serious lacuna, since we cannot possibly hope to understand their way of thinking if we do not pay close attention to how they themselves conceived it.

As regards qal vachomer specifically, Samely has (to date[17]) allegedly identified and analyzed 44 occurrences in the Mishna (including 25 of A4.1, 13 of A4.2 and 6 of A5). The following is an alphabetically ordered listing of the references.[18]

Alexander Samely’s list of 44 qal vachomer cases in the Mishna

Arakin 8:4

Bekoroth 1:1 II

Bekoroth 9:1 I

Berakoth 9:5 V

Demai 2:2

Eduyyoth 6:2 I

Eduyyoth 6:3 I

Eduyyoth 6:3 II

Eduyyoth 6:3 III-IV

Eduyyoth 6:3 V

Hullin 2:7

Hullin 10:1 II

Hullin 10:1 III

Hullin 12:5

Makkoth 1:7 VI

Makkoth 3:15 II

Makkoth 3:15 IV

Menahoth 8:5

Nazir 7:4

Nedarim 3:11 VII

Nedarim 10:7 I

Negaim 10:2 I

Negaim 10:2 II

Negaim 12:5 V

Pesahim 6:2 I

Pesahim 6:2 II

Pesahim 6:2 III

Pesahim 6:2 IV

Sanhedrin 6:5 II

Shebuoth 3:6

Sotah 6:3 I

Sotah 6:3 III

Temurah 1:1 III

Terumoth 5:4 I

Terumoth 5:4 II

Yadayim 4:7

Yadayim 4:8 I

Yebamoth 8:3 II

Yoma 8:9 III

Yom Tov 5:2

Zebahim 7:4

Zebahim 7:6

Zebahim 8:12

Zebahim 12:3

My initial purpose here is merely to pass on the information, so as to make it more widely known. Looking at such a list, we need to answer the following questions. Are the 44 instances Samely found all indeed fortiori, or are some are not a fortiori at all? Are they all positive subjectal (or antecedental) as he claims – or do some of them have other forms? Are some of them negative and/or predicatal (or consequental)? Are they all superior (or inferior) in form, or are some egalitarian (a pari)? Were some a fortiori arguments missed by him, and if so what forms do they have?

Moreover, underlying these questions is another, more basic one: are all these attempted a fortiori arguments truly a fortiori, or are some of them merely pseudo a fortiori? Are there arguments that superficially may seem like a fortiori or be alleged to be so, and yet in reality are not so? Whoever tries to enumerate the occurrences of an argument form must be able to indubitably recognize such form, and tell real and valid cases of it from illusory and invalid cases of it. It does not seem that Samely has attained this level of awareness, and so his enumerations must be verified independently by more competent logicians before being accepted.

I have finally actually looked at all of his findings and his reflections on them, and can at the outset testify that not all of these are indeed instances of a fortiori argument (whether valid or invalid). Also, not all arguments have the forms he thinks they have. Moreover, his list is not exhaustive. Funnily enough, this inventory does not include Bava Qamma 2:5, the crucial examples and discussion of a fortiori argument in the Mishna (which we examined in the previous section); this may have been a mere error of inattention, because Samely does discuss these arguments[19]. But also excluded are four cases that I found independently: two in tractate Avot (1:5 and 6:3), one in Kelaim (8:1) and one in Pesahim (6:5)[20]; so, there may well be other oversights.

6. My critical researches

Having ascertained that Samely’s theoretical and practical understanding of a fortiori argument is imperfect (to put it nicely), I decided reluctantly to closely examine the relevant material in the database posted in his website. I say ‘reluctantly’, because I could see that would be a daunting task, and I am as lazy as the next man. As it turned out, I did well to do so, because I discovered many more imperfections in his treatment. The job took me a few months to complete.

Samely has not helpfully provided visitors to his website with a downloadable one-document listing of all the data in his database. The material is buried in scattered places, to be recalled one bit at a time. This may be intended to make data theft more difficult, but it also constitutes an obstacle to ready verification of his results by his peers. Nevertheless, I went through the forty-five web pages where the data is posted, and collected it all into one document about twenty pages long (which I am holding on to as documentary evidence). This included the given “text” and his “analysis” of it.

As regards the original Mishna text, I take it for granted that he accurately transcribed it. I do have some doubts as to that, in view of the many errors I found in his typing, as well as many repetitive spelling mistakes (e.g.: aught i/o ought, purety i/o purity), all of which are indicative that he did not double-check his own work. Moreover, there are many explanatory interpolations in square brackets, and other kinds of brackets, and it is not clear whether they are his own additions, or commentaries by past authorities that were already attached to the text. Sometimes he seems to be quoting verbatim; sometimes to be paraphrasing.

Strictly, I ought to have gone back to his sources and verified all the Mishna passages in question to see whether he reported them correctly. However, I decided not to take on this big additional job, considering that my purpose here was primarily to verify Samely’s interpretations, and not so much to definitively study the source document in question. In truth, I did some spot-checking (again with reference to the Soncino edition), and found the half-dozen texts examined on the whole passable. I thus, to repeat, took the texts proposed by Samely for granted and proceeded from there. I then wrote down, more or less independently, my own analysis of each text; and then, after that, my analysis of Samely’s analysis. I also tabulated for myself various features of these findings to obtain an overview.

These analyses of mine added over twenty-five more pages to the document I had constituted. I thought to attach this detailed document to the present book, knowing from experience that when one criticizes material displayed in a website (as against in a published book), the erroneous information is usually wiped out very soon after, without any admission of error or trace of correction. But I decided against such inclusion in view of its considerable bulk. Instead, what I have done is offer the reader of the present book various summaries of my main findings. I hope Samely will have the good grace to acknowledge his errors and thank me for finding them.

The main results of my research on Mishnaic a fortiori argument are exposed in detail and summarized in tables (A2.1 and A2.2) in Appendix 2. This exposition does not include my above mentioned critical notes on Samely’s prior work, which are too rough for publication; but it is based on them. The comments I will now make are intended as a brief summary of my critique of his work, including some general remarks (which all interested readers should look at), backed by further details in smaller print (which some readers may prefer to skip or skim over).

What I can say at the outset about this portion of Samely’s work is that it is far from the scientific research project it is trumpeted to be. There is considerable lack of rigor in its methodology and slackness in its execution. All too often, the interpretations proposed seem quite subjective – just one man’s reading, without convincing proof. Although he has manifestly invested much effort in the project, much more effort is needed on his part to make it fully credible. We shall presently go over the main faults in Samely’s treatment, under various headings. But first, let me overview my findings statistically.

Summary of findings. Of the 45 Samely references that I examined, 2 had no relation to a fortiori argument (here henceforth abbreviated to ‘afa’) that I could discern; 6 did not contain any afa, but only one or more rebuttals of such; the remaining 37 did contain a total of 42 afa, of which 38 were valid and 4 were invalid. The 4 invalid afa included 1 positive subjectal and 3 negative subjectals. The 38 valid afa included 23 positive subjectals, of which 4 were a pari, plus 9 negative subjectals, of which 1 was a pari; 1 positive predicatal and 1 negative predicatal; 3 positive antecedentals and 1 negative antecedental. Most references (31, to be exact) are to a single valid afa, while two references are to 2 of them and one is to 3. Note additionally, as already mentioned, Samely misses out on 4 more valid positive subjectal a fortiori arguments[21].

Let us now examine more closely the main faults in Samely’s treatment of Mishnaic a fortiori argument. We have already mentioned his inclusion of some non-afa in his listing of cases and his missing some genuine cases of afa. The following areas also deserve critical attention: the absence of accurate statistics; no effort at validation; peculiar categorizations; difficulties with the middle and subsidiary terms; difficulties with the major and minor terms; ignoring the sufficiency of the middle term; choosing the reverse of the effective middle term; failure to see and acknowledge unusual features, notably, negative arguments, ‘major to minor’ arguments, predicatal arguments and implicational arguments; failure to stick to the original wording and structure; confusion regarding ‘proportionality’; and other problems.

Absence of accurate statistics. Our first impression, looking at the list of references relating to a fortiori argument in Samely’s database – i.e. 25 cases of A4.1 (26 cases, if we add the missing reference to mBQ 2:5), 13 cases of A4.2 and 6 cases of A5, is that this is an enumeration of individual occurrences of the argument. However, when we look more closely at his data, we find that some of his references in fact contain no a fortiori argument, but either a mere objection to such an argument or another sort of discourse; others contain an invalid a fortiori argument; and others contain not just one, but two or even three valid arguments. Evidently, Samely needs to improve his mode of presentation, and give clear statistics on each type and subtype of argument.

As already mentioned, Samely’s listing of afa (generated by his database search engine) does not include the two arguments in Bava Kamma 2:5 (though these arguments are present and treated in his database), as well as four arguments in the tractates Avot, Kilaim and Pesahim. Concerning the Samely cases I have above excluded from the modified total of 42 afa, the following explanations are in order:

· The two Samely references unrelated to afa were: Nedarim 3:11 and Yoma 8:9. The first case reads: “Great is the circumcision which for the sake of Moses the righteous was not suspended as much as an hour.” Samely tries to make something of it, saying: “there may be an implied dimension of a fortiori here (if even for Moses it was not suspended even for one hour, all the less so for ordinary Israelites…).” Although we could construct a valid positive predicatal argument[22] from the given proposition, I do not see that this is the intent of the Mishna. The second case reads: “As the Miqweh cleanses the unclean so does the Holy One, Blessed be He, cleanse Israel.” Presumably, Samely reads into this the a fortiori: “if the ritual bath can clean the unclean, then all the more so can God do so.” But here again, this does not seem to be the intent of the Mishna, which is just making a comparative statement: “just as this, so that.”

· The 6 Samely references not containing any afa, but only one or more rebuttals of such, were: Eduyyoth 6:3 II & V, Hullin 10:1 III, Negaim 10:2 II, Pesahim 6:2 II & III. The problem here is simply that Samely sometimes treats the objections to an afa as equivalent to an afa, perhaps because they have ‘to do with’ afa. On the other hand, he does not follow this policy consistently. There are many cases where he does not treat the objections apart from the afa, such as: Eduyyoth 6:3 III-IV, Menahoth 8:5, Shebuoth 3:6, Sotah 6:3 I, Yadayim 4:7[23], Zebahim 12:3, to name some. It would be best for Samely to always group rebuttals together with actual afa.

Although Samely does somewhat distinguish arguments and objections, by usually labeling them oppositely (for example, if the argument is A4.1, then the objection to it is “non-A4.1”), it is evidently not always clear in his mind what the difference is between an argument of a fortiori form and the rejection of such an argument. He sometimes seems to regard the rejection as itself a sort of a fortiori argument. This is due to the fact that he has not formally analyzed the different ways a fortiori arguments may be rejected. Sometimes, we have to choose between rival a fortiori arguments; sometimes, the proposed a fortiori argument is rejected as an illicit process; but mostly, the opponent merely puts in doubt one or both of its premises, without denying validity to the process.

· In some cases, one senses a sort of perplexity in Samely, in that he is unable to mentally separate the a fortiori argument from its subsequent rejection. A case in point is Sotah 6:3 III, where he describes as: “non-A4.2 an a fortiori whose rejection is anticipated.” That is to say, he is not entirely clear as to whether, when an a fortiori argument is proposed hypothetically, with the intention to immediately reject it, it is to be considered as an effective a fortiori argument or a nonexistent argument. The status of a counterfactual argument is uncertain for him. In fact, if one understands the formalities of a fortiori argument, one clearly sees the difference between rejecting an argument as badly-formed and rejecting a well-formed argument by denying one or both of its premises.

No effort of validation. Samely is basically forced to take all the rabbis’ arguments at face value, because he has no technical means by which to judge their validity or invalidity. He almost never pronounces clearly “this argument is right/wrong,” for the simple reason that he has found no theoretical way to tell the difference, or even sought for a way. He is not a formal logician, but only at best a reporter of rabbinic discourse. To his credit, he has effectively identified one form of a fortiori reasoning – namely, the positive subjectal (or antecedental) mood – and this helps him to standardize rabbinic discourse somewhat. This is no mean achievement, to be sure. But he is unable to deal with issues of validity: he lacks the tools for it.

We can see this for instance in his treatment of Nazir 7:4, for which Samely’s only comment is “refuted by tradition” – implying that there is no need to analyze the argument independently since it is allegedly dealt with by tradition[24]. But it is especially evident with regard to the 4 references containing invalid afa, namely: Makkoth 3:15 II[25], Temurah 1:1 III, Terumoth 5:4 I & II.

· The Makkot 3:15 passage, reads: “If the one who commits one transgression has his life taken away, [then] all the more will the one who performs one commandment be given his life!” Although roughly positive subjectal in form, it is distinctive in that its subjects (‘one who commits one transgression’ and ‘one who performs one commandment’) are contrary and its predicates (‘his life shall be taken away’ and ‘his life shall be given to him’) are contrary; this means that the argument is a contrario, and therefore invalid[26]. Samely sees it as somewhat reasonable, saying: “the punishment for transgression is, measure for measure, smaller than is the reward for fulfillment,” although he finally taxes it as “merely ‘rhetorical’” on the ground that “Life or death are on/off qualities, so the idea of ‘increase’ is impossible (no person can have be given several concurrent lives).” Thus, although Samely does judge the argument to be invalid, he does not do so for the right reason.

The Temurah 1:1 passage contains two rival afa, of which one (by R. Yohanan ben Nuri) is invalid, because it is negative subjectal, yet minor to major. It goes like this: ‘If the sin/guilt offering, which becomes priestly property only after slaughter (Q), is fully owned (R) not enough to be substitutable (S); then, all the more, the firstling (P) is fully owned (R) not enough to be substitutable (S)’. Samely does perceive and try to explicate the invalidity of this argument. He denies that a putative conclusion follows from the given premises, saying: “from the prohibition of sin and guilt offerings for substitutions follows thus nothing for the substitutability of the firstling” (my emphasis). However, he is not here engaging in independent criticism, but merely reflecting what the original speaker, R. Yohanan ben Nuri, is saying.

In fact, Samely has not fully understood the discussion. He fails to notice that before this non-sequitur, R. Akiva proposes a valid a pari negative subjectal that can be paraphrased as: ‘The sin/guilt offering (P) is as much a gift to the priest (R) as the firstling (Q) is; whence, if the sin/guilt offering (P) is a gift (R) not enough to be substitutable (S), then the firstling (Q) likewise’. R. Yohanan counters this with an intentionally invalid argument, above described. This fake argument is put forward only to formally ridicule the previous; but in fact it does not do the job intended, because the previous argument is egalitarian, whereas this one is minor to major. This is why R. Akiva replies, if I understand correctly, with a claim that both offerings are equally holy and that holiness takes effect as soon as it comes into the owner’s home, so that substitution can occur at once. Samely makes no mention of R. Akiva in this context.

As regards the two invalid afa in the Terumoth 5:4 passage: one, by Beit Hillel, can be stated as: ‘If the clean heave offerings, which are forbidden to non-priests [but not to priests], (Q) are restricted (R) not enough to prevent them being outweighed by clean common food (S), then, all the more, the unclean heave offerings, which are forbidden [even] to priests, (P) are restricted (R) not enough to prevent them being outweighed by clean common food (S)’; the other, by Beit Shammai, can be stated as: ‘If the “light” common produce, which are permitted to non-priests, (Q) are restricted (R) not enough to prevent them from being neutralized by clean common food (S), then, all the more, the “weighty” heave offerings, which are forbidden to non-priests, (P) are restricted (R) not enough prevent from being neutralized by clean common food (S)’. These arguments are both invalid, because they are negative subjectal, yet minor to major. The difference is that the first is apparently put forward in earnest, whereas the second is apparently put forward with the intent to show up the invalidity of the first. I should add that there are two implicit afa following the above two, which argue from the possibility of outweighing or neutralizing unclean heave offering by clean common food to a like possibility by clean heave offering. I have not counted these afa, though they are valid, because they are not explicitly formulated, even though they are necessary to obtain the desired final conclusions.

As regards Samely’s treatment of this Terumoth passage, he does not declare the two arguments invalid. He tries to represent the argument of Beit Hillel as follows: His middle term for this inference is “the scale of restriction,” relative to which unclean heave offering would be the major term and common food the minor term, and presumably (though he does not explicitly say so) clean heave offering would be an in-between term. On this basis he argues: If “unclean heave offering can be neutralized (even) by clean common food,” then “unclean heave offering should all the more be capable of being neutralized by clean heave offering.” Note that his minor premise and conclusion both have the major term as subject, and respectively the minor term and the in-between term as predicates. This gives the impression that the argument is positive predicatal, with the major term as the subsidiary term and the minor and in-between terms as respectively the major and minor terms. From a formal viewpoint, this spells utter confusion! The confusion is, I would say, due to Samely’s attempt to telescope two arguments into one. As regards the argument of Beit Shammai, it leaves him perplex: “Is this meant as a counterargument, and reductio ad absurdum of the preceding A4.1? If so, it does not itself have the structure of an a fortiori?”

Samely’s peculiar categorizations. Samely’s categorization of a fortiori arguments, as A4.1, A4.2 or A5, is rather subjective and arbitrary. Based on his definitions of these labels, I identified them as possibly referring respectively to implicational, and normative and non-normative copulative, a fortiori argument, respectively, leaving aside gaps and overlaps for now. However, looking at the cases listed in his database, I must say that I do not see such assumed differences between them! Comparing his practice to his theory, it is evident that even he finds these categories confusing. His success rate here is a paltry 4/24, regarding A4.1, and 12/42, regarding A4.2 and A5.

· Samely’s A4.1 category seems by its definition intended to refer to implicational a fortiori argument, yet of 24 cases he so labels, only 4 cases are indeed implicational (viz. Bava Qamma 2:5, 2 cases[27], Hullin 10:1 II and Zebahim 8:12), the 20 others being copulative. Samely’s A4.2 category seems by its definition intended to refer to normative a fortiori argument, and the 12 cases he so labels are indeed normative – but then so in fact are all 30 other cases he lists! Samely’s A5 category seems by its definition intended to refer to non-normative a fortiori argument, but the 6 cases he so labels are all normative. I take as normative any concept involving a value judgment, be it intuitive or based on Scripture. To my mind, “trustworthiness” (Demai 2:2), “transgression, fulfillment of commandment, reward, punishment” (Makkoth 3:15 II), “the dimension of value” (Negaim 12:5 V, 3 cases), “just, unjust, punishment” (Sanhedrin 6:5 II[28]) are all value judgments, which are all classed as A5, are just as normative as “exempt from obligation” (Bekoroth 1:1 II), “commandment, reward” (Hullin 12:5), “transgression, fulfillment of commandment, reward, punishment” (Makkoth 1:7 VI), “forbid, reward” (Makkoth 3:15 IV), “requires, [having] to be used” (Menahot 8:5), “prohibition, authority” (Nedarim 10:7 I), “forbidden, sets aside prohibition” (Pesahim 6:2 I & IV, 3 cases), “need for ritual, bars, guilt” (Sotah 6:3 I & III, 2 cases), “respect due, can [i.e. is allowed to]” (Yadayim 4:8 I), which are all classed as A4.2. Indeed, notice, one case of A5 (Makkoth 3:15 II) involves exactly the same concepts as a case of A4.2 (Makkoth 1:7 VI).

Thus, though his definitions seem boisterously confident, their proper applications in practice are far from evident. He nowhere justifies himself, explaining why this case falls under one heading and that case falls under another heading. He does not demonstrate, as he should, the relevance and consistency of his labeling; indeed, he cannot. This is an inexcusably careless way to work. All this leads me to regard, regrettably, his categorizations as pseudo-intellectual window dressing, rather than seriously scientific observations.

Difficulties with the middle and subsidiary terms. Although in his theory Samely has clearly marked the difference between the middle term R (which Samely calls “the scale or dimension of comparison, X” between two items), and the subsidiary term S (which he refers to the “predicate or apodosis, A” which is “transferred” from one item to the other), his behavior in practice shows that he nevertheless often (though not always) confuses or conflates them. Such ambiguity means that his subsidiary term is usually anticipated in his major premise, and his middle term is usually left out in his minor premise and conclusion, so that his arguments are often in fact not true to a fortiori form. Usually, the difficulty he has with the argument is one of interpretation, for the original argument is itself clear enough, but he does not make enough of an effort to discern and depict its real intent. This concerns at least 8 cases.

· Thus, in Demai 2:2, he uses the compound term “scrupulous=trustworthy” for both terms, although the middle term is in fact “scrupulous” and “trustworthy” is the subsidiary; his using them both, in this equated form, in the major premise (given in his “points 1-3”), minor premise (“point 4”) and conclusion (“point 5”) shows that he was unable to decide which is which. In Yadayim 4:8, Samely’s middle term in the major premise is “respect due to the names,” and in the minor premise and conclusion his subsidiary term is “without this meaning irreverence/disrespect” and no middle term is mentioned, so that the argument appears invalid. In Eduyyoth 6:2 I, he does not show awareness that his middle term “uncleanness” refers to potential conveying of uncleanness, whereas his subsidiary term “conveys uncleanness” refers to actual conveying of uncleanness. In Eduyyoth 6:3 II, similarly, he should have clarified the difference between his middle term “corpse-uncleanness in persons alive and dead,” and his subsidiary term “corpse-unclean;” the former apparently referring to ‘potential or actual’, and the latter to ‘only actual’, uncleanness[29]. We must assume the same ambiguity for Eduyyoth 6:3 III-V, which he claims without going into details to be “structurally exactly analogous” to the preceding.

In Hullin 2:7, the division of labor between his middle term “susceptibility to invalidation” and his subsidiary “invalidation” is clear enough, but he again does not remark upon the implied difference between potentiality and actuality. Similarly, in Negaim 10:2 I, his middle term is “power to render unclean” and his subsidiary is “renders unclean;” in Sanhedrin 6:5 II, his middle term is “grievous… in the dimension of God’s compassion” and his subsidiary is “grieves God;” in Zebahim 7:4, his middle term is “susceptibility to sacrilege” and his subsidiary is “subject to sacrilege” – these three cases again suggesting a potential-actual relation between the two terms. In Menahoth 8:5, Samely wrongly declares the middle term to be the generality “quality of material (purity)” and the subsidiary to be the more specific “pure beaten quality” (whereas the middle is in fact quite clearly: having or not-having “to do with eating”). This, together with the preceding examples, suggests that Samely imagines there has to be a relation of inclusion of some sort between the middle term and the subsidiary – which is untrue: the two concepts may be conceptually quite unrelated.

Difficulties with the major and minor terms. Again, although Samely’s theoretical treatment suggests a relatively clear understanding of the positions and functions of the major and minor terms, in practice he sometimes indulges in some astounding mental gymnastics with them, as in the following 4 cases.

· Thus, in Arakin 8:4, his major premise suggests that his major and minor terms are the subjects “divine demands” and “human demands,” whereas in his minor premise and conclusion the subjects are respectively “God” and “man” while “divine demands” and “human demands” appear as modifications of the predicate “taking care to preserve man’s property,” so that we are faced with a mongrel argument, half subjectal and half predicatal, with five terms instead of four. In Makkoth 1:7 VI, Samely admits two versions of each term: the minor term is “[person who commits] transgression” in the major premise, but “person joined to a transgression” in the minor premise; the major term is “[person who commits] fulfillment of a commandment” in the major premise, but “person joined to the fulfillment of a commandment” in the conclusion; moreover, the middle term is “God metes out punishment or reward” in the major premise, but “punished” in the minor premise and “rewarded” in the conclusion; and likewise, the subsidiary term is “actual transgressor” in the minor premise and “actual performer of the commandment” in the conclusion. This makes a total of eight terms instead of the standard four! Of course, a valid reading is possible[30], but Samely did not manage to sort it out.

In Makkoth 3:15 II, he has a similar major premise with two middle terms, and again a change (though slight) of minor term in the minor premise (to “one-off transgressor”) and ditto (implied though not stated) for the major term in the conclusion, plus the subsidiary term is contrary in the conclusion (“gain or get back his life”) to what it is in the minor premise (“lose his life as punishment”). Again, all this goes against a fortiori logic, i.e. it is formally invalid; and it demonstrates that Samely does not realize the importance of proper form for purposes of validation[31]. Admittedly, the confusion here is largely due to the original formulation by R. Hananiah, which is exceptionally a contrario; but in the next example this excuse does not apply.

I am referring to Terumoth 5:4 I, where the subjectal major premise has “common food” as the minor term and “unclean heave offering” as the major term; while the latter term is the subject of both the minor premise and conclusion, whose predicates are respectively “can be neutralized by clean common food” and “can be neutralized by clean heave offering.” This means that the minor premise relates the major term and the minor term, while the conclusion relates the major term (again) and an additional term (presumably lying in between the major and minor, but not mentioned in the major premise)! The middle term is given in the major premise as “the scale of restrictions,” but does not reappear in the minor premise and conclusion. The subsidiary term, common to the minor premise and conclusion, appears as “can be neutralized by;” but in fact, without the further clause saying by what neutralization occurs, such a term is incomplete, and as already seen it is differently completed in the minor premise and conclusion, so that we have two subsidiary terms in fact.

A veritable mixed salad! Samely’s major premise has in fact the wrong minor term (it should have been “clean heave offerings”) and his conclusion has the wrong subsidiary term (it should have been “can be neutralized by clean common food” as in the minor premise). Moreover, as I show in Appendix 2, the argument by Beit Hillel that concerns us here is even when properly formulated actually invalid, since it is negative subjectal, yet minor to major. Invalid also is the rival argument, in Terumoth 5:4 II, put forward by Beit Shammai for the purpose of showing up the invalidity of the preceding one. Samely, having apparently not understood this[32], tries to present the first argument in positive subjectal form (and thus as valid)[33]. But the truth is that his minor premise and conclusion, formulated with the phrase “are neutralized by,” seem positive because they lack the middle term. If we insert the middle term (R), we see immediately that they are really negative propositions, with the phrase “are restricted (R) not enough to be prevented from being neutralized by.”

Ignoring the sufficiency of the middle term. One important failure of Samely’s attempted formalization of a fortiori argument is his missing out on the idea of the subject being “R enough” or “R not enough” for the predicate to be predicated of it. This idea is essential to understanding the minor premise and conclusion, and the logical relation between them and with the major premise. It is only by satisfying the condition of a certain threshold value of the middle term (R enough) that a subject and predicate become joined; and if that condition is not satisfied (R not enough), then they are not joined. Because he has not grasped that idea theoretically, Samely in practice usually makes no mention of the middle term in the minor premise and conclusion (though perhaps this is sometimes in part due, as we have seen, to his confusing or conflating the middle term with the subsidiary). He can get away with this in many cases, but in some 4 cases this results in his misunderstanding the argument.

· We have just seen a case in point, in our discussion of Terumoth 5:4 I, where the phrase “are restricted not enough to be prevented from being neutralized by” had to be used in lieu of Samely’s “are neutralized by” in order to see the negativity and thence invalidity of the argument. Other cases worth mentioning include Yebamoth 8:3 II[34], where he should have had “are restricted not enough to be prevented from being allowed forthwith” instead of merely “are allowed forthwith;” Zebahim 7:6, where he should have had “… is able as carrion to convey of uncleanness etc. not enough to prevent its slaughtering from rendering clean the uncleanness of its terefah” instead of merely “the terefah of … is rendered clean by slaughter;” and Zebahim 8:12, where he should have had “is halakhically significant not enough to invalidate,” instead of merely “does not invalidate.” In these three cases, the extra words are necessary in order to make the ‘major to minor’ argument valid; in the first two, a double negative is introduced (“not enough to prevent”), while in the third, the negation is shown as applying to the sufficiency of the middle term (“not R enough to S”) instead of directly to the subsidiary term (“not S”).

As regards mere failure to mention R, without significant effect, see for instances: Shebuoth 3:6, Yadayim 4:7, Yom Tov 5:2 and Zebahim 12:3. In Berakoth 9:5 V, Samely mentions neither the middle term nor the subsidiary term, saying only “from shortcut to spitting” (meaning from minor to major, though no major premise is given). He does sometimes mention the middle term, perhaps as a qualification of the subsidiary term. An example of this is Hullin 2:7, where Samely argues: since “sacrificial slaughter cannot be invalidated by wrong intention of the owner,” then “non-sacrificial slaughter should also not be invalidated by wrong intention of the owner.” Here, objectively, the subsidiary term is “invalidated” and the middle term is “by wrong intention of the owner;” but, noting that the middle term is defined in the major premise as “susceptibility to invalidation by intention,” we can suspect that the two terms are here again somewhat conflated in Samely’s mind.

Choosing the reverse of the effective middle term. Another difficulty Samely has in practice with the middle term is that he does not realize that the direction of the relative term chosen as middle in the major premise affects the validity of the conclusion, since a major term in one direction is a minor term in the other direction, and vice versa. Such subtleties may look like linguistic manipulations to the untrained eye, but they are crucial to formulating valid a fortiori arguments.

· Thus, in Bekoroth 1:1 II, Samely’s major premise states that (my italics) “the Israelites’ first-born are more distant than the Levites’ own first-born” (implying that the former term is the major and the latter term is the minor), whereas his minor premise and conclusion concern the first-born of respectively the Israelites and the Levites (implying that the former term is the minor and the latter term is the major). To form a valid argument, the major premise should have been “the Levites’ first-born are closer than the Israelites’ own first-born.”

In Pesahim 6:2 I, it makes a big difference whether the major premise is stated as “Sabbath work rules have a more stringent status than Sabbath rest rules” or as “Sabbath rest rules have a more lenient status than Sabbath work rules.” In the first formulation (which Samely chose), the works rules are the major term and the rest rules are the minor term, and the proposed argument being positive subjectal and from major to minor is invalid, whereas in the second formulation (which he should have preferred) the works rules are the minor term and the rest rules are the major term, and the proposed argument being positive subjectal and from minor to major is valid. Samely could have more easily spotted and avoided this error if he had put the middle term in the minor premise and conclusion. He would then have felt the important difference between “is stringent enough to set aside” and “is lenient enough to set aside,” and realized the need to correct his major premise.

Similar comments can be made in relation to the two rival arguments in Pessahim 6:2 IV. Using the middle term “stringent” again, Samely has the first argument (by R. Eliezer) as positive subjectal yet major to minor, and the second argument (by R. Akiva) as negative subjectal yet minor to major. But in this guise, both arguments are formally invalid! Yet if he had reversed the polarity of his middle term, using “lenient” instead, as just explained in the previous example, he could have made both arguments valid. Alternatively, he could have fixed the two arguments by changing the polarity of the sufficiency of the middle term in the minor premises and conclusions, as earlier explained. That is, for the first argument, he could have used the negative “is not stringent enough to prohibit the setting aside of” instead of the positive “sets aside;” and for the second argument, he could have used the positive “is stringent enough to prohibit the setting aside of” instead of the negative “does not set aside.”

Another case of this sort is Sotah 6:3 III, where Samely has as his major premise: “Compared for the duration of her being barred from her husband, the Sotah-initiating testimony is less weighty than the testimony establishing her guilt.” This means that “the Sotah-initiating testimony” is the minor and “the testimony establishing her guilt” is the major, in the scale of “duration” – i.e. lengthiness of separation. Now, this major premise happens to work well in the previous case he considers (Sotah 6:3 I), where the argument from “the Sotah-initiating testimony” to “the testimony establishing her guilt” is a valid positive subjectal, from minor to major. But Samely foolishly sticks to the same major premise when formulating the Sotah 6:3 III argument, thereby producing an invalid positive subjectal, yet major to minor! Here again, he should either have reversed the polarity of the middle term (from lengthiness of separation to briefness of separation) or alternatively changed the polarity of his minor premise and conclusion (from “is lengthily separating enough to require only one witness” to “is lengthily separating not enough to require more than one witness”).

Failure to see and acknowledge unusual features. Samely’s theoretical model for a fortiori argument is essentially limited to the positive subjectal form. Consequently, he fails to see and acknowledge less typical forms; namely: negative arguments, major to minor arguments, predicatal arguments and implicational arguments. These unusual features are, of course, often interconnected – but it is well for us here to look on each of them separately.

· Negative arguments. Although Samely formulates 6 arguments negatively (namely, Bekoroth 9:1 I, Demai 2:2, Hullin 2:7, the 2nd afa of Pesahim 6:2 IV, the 2nd afa of Temurah 1:1 III, Zebahim 8:12), he does not stop and reflect on such negativity so as to integrate it into his theoretical modeling[35]; as a result, he fails to notice the negativity of 5 other arguments (namely, Arakin 8:4, the 1st afa of Temurah 1:1 III, Terumoth 5:4 I & II, Yebamoth 8:3 II, Zebahim 7:6[36]). And of course, not having reflected on the issue, he does not realize that the negativity of an argument affects its conditions of validity. He does exceptionally, with regard to the 2nd afa of Temurah 1:1 III, deny validity to the inference (saying “follows thus nothing”); but the explanation he gives, though it refers to the major and minor terms, does not raise the issue of polarity. Consequently, since he lacks formal understanding, he does not criticize the 2nd afa of Pesahim 6:2 IV in the same manner, which he should have done.

· ‘Major to minor’ arguments. Although Samely formulates 13 arguments, rightly or wrongly, in major to minor form (namely, Arakin 8:4, Bekoroth 9:1 I, Demai 2:2, Hullin 2:7, Nedarim 3:11 VII, Nedarim 10:7 I, Pesahim 6:2 I, the 1st afa of Pesahim 6:2 IV, Sotah 6:3 III, Yadayim 4:8 I, Yebamoth 8:3 II, Zebahim 7:6, Zebahim 8:12), he usually does not notice himself doing so. In the 2 cases when he does notice (namely, Pesahim 6:2 IV and Sotah 6:3 III), it is without reflection and moreover he commits errors of logic[37]. Given such failure to notice and reflect, it is not surprising that he does not integrate such occurrences in his theoretical modeling; and given such failure of theoretical treatment, it is not surprising that he makes errors. It should also be noted that some arguments that he formulates as minor to major should have been formulated as major to minor (namely, the 2nd afa of Pesahim 6:2 IV) or as a pari (namely, Bekoroth 1:1 II, Eduyyoth 6:3 I, Makkoth 1:7 VI[38]).

· Predicatal arguments. Samely unconsciously formulates 2 predicatal arguments, as is evident from his use of the words “required” and “due” in them. In Nedarim 10:7 I, his major premise defines the middle term as “the authority required to cancel [vows]” and his minor premise and conclusion have “he can cancel vows which…” (“he” being the subsidiary term, and “can” here meaning “has the authority required”); this is a well-formed positive predicatal argument, though Samely does not realize he is not engaged in the usual positive subjectal form of argument. In Yadayim 4:8, his major premise defines the middle term as “respect due to the names,” but his minor premise and conclusion get a bit confused, making the argument look positive subjectal (and therefore invalid, since major to minor), whereas the intent is obviously positive predicatal (whence valid)[39]. A third predicatal argument, namely Demai 2:2 (which is negative, minor to major), is not similarly verbally marked by him (he presents it as effectively negative subjectal, major to minor).

· Implicational arguments. I would characterize 4 arguments in Samely’s list as having implicational form. Namely: the two afa of Bava Qamma 2:5, Hullin 10:1 II, and Zebahim 8:12 – the first three being positive antecedental (minor to major) and the last being negative antecedental (major to minor); there are no consequental arguments, note in passing. The reason these arguments cannot be taken as subjectal rather than antecedental is simply that the subsidiary item in each case is not a predicate but a consequent of the minor and major items[40]. While Samely formulates these arguments in a way that is objectively implicational, he does not verbally mark their difference as such. As already seen, we cannot reasonably accept his categorization of these four cases as A4.1 as signifying implicational argument, since there are twenty other cases that are so labeled and yet are clearly not implicational. Thus, here again, we must consider that he is unconscious of formats other than the positive subjectal.

The distinction between “from minor to major” (Latin: a minori ad majus, Hebrew: miqal lechomer), “from major to minor” (Latin: a majori ad minus, Hebrew: michomer leqal) a fortiori argument is nothing new, but known with some clarity since antiquity in both the Greco-Roman and Jewish worlds. The ancients were also aware of egalitarian (a pari) argument and, though somewhat less clearly, of the possibility of negative argument. It is very surprising, therefore, to find Samely setting off on a systematic analysis of Mishnaic thought while apparently totally unaware of these different possibilities. The concepts of predicatal and implicational a fortiori argument being less widely known, Samely can be excused concerning them; but for the rest he needs to study the matter much more and revise his past work.

What is evident is that ignorance of the impact that polarity and orientation have on validity have caused Samely serious errors of logic. I would say that one practical reason he failed to notice such deviations from the forms he assumed to be general was that he did not actually label the terms (or theses) in his analyses. That is to say, at least in his database (as distinct from his book), he never inserts his own abstract labels N, M, n, m, X, and A, in the examples, but is content to just treat cases vaguely, in wholly concrete terms. Had he always inserted his labels, so as to demonstrate that the examples do indeed fit his forms, he would have quickly discovered that some of the examples in fact do not fit his forms[41]. This would have led him to improve his analyses and definitions. Thus, at the root of his theoretical and practical errors is a too casual approach, an absence of formalism.

Failure to stick to the original wording and structure. Another fault in Samely’s approach is failure to stick as closely as possible to the wording and structure of the original text. This is, to be sure, not always possible or wise, for the job of an interpreter is not merely to reflect the surface presentation, but to bring out the deeper intent. An argument may appear to be invalid if taken too literally, but be found quite valid if an effort is made to capture the underlying logical intuition that gave rise to it. In everyday speech, we allow ourselves considerable brevity and poetic license; when such speech is examined more strictly, with logical evaluations in mind, some corrections are often called for. For instance, very often we have to propose an appropriate middle term, because it is left unstated in the original text. Nevertheless, we should not too readily depart from the original text, thinking to improve on it, for it may have unexpected lessons to teach us.

· For example, in Arakin 8:4 R. Eleazar ben Azariah says: “And if Man does not even have authority to devote to the High One everything he owns, how much more is Man obliged to protect his possessions!” This is naturally read as a negative subjectal a fortiori argument: ‘Man’s use of his possessions for holy ends (P) is more religiously valuable (R) than man’s use of his possessions for profane ends [including the waste of his possessions without purpose] (Q); whence, if P is R not enough to be authorized without limit (S), then, Q is R not enough to be S’. Samely’s interpretation of it is very similar, except that it is made in positive terms: “Compared in the dimension of a claim on man’s property, divine demands are higher on the scale than human demands;” and “God has taken care to preserve man’s property by limiting dedications to the Temple;” therefore, “man should take even greater care to preserve his property with regard to human wishes or priorities.” In the original text, we find both the negative form “does not have authority to devote” and the positive form “is obliged to protect,” so either way might seem okay prima facie. But if we reflect on the issue of validity, it is clear that the argument has to be formulated negatively, and not positively, since it is subjectal and major to minor. It is not sufficient to use positive terms with negative connotation like “limit” and “preserve,” because this misplaces the negation, which formally must be applied to the “enough” of the middle term R.

Regarding Bekoroth 1:1 II, Samely’s interpretation is in a way closer to the rabbis’ than mine. The Mishna claims a minor to major inference, and Samely tries to comply with that (though in a gauche manner, as already seen), whereas I propose an egalitarian major premise, seeing no reason offhand to treat the two parties involved differently. That is, I assume that all first-born (including Israelites and descendants of Levi) were together released from their traditional duties; and thereafter a selection of Levites (namely, Aaron and his descendants) were inducted for future priestly duties; whence, it is easy to argue from Israelites to Levites as regards the exemption from the earlier régime. On the other hand, Samely’s formulation of the major premise introduces a notion of “distance” that does not reflect the original argument and would require complex additional explanations and justifications.

Samely formulates the Demai 2:2 argument as a negative subjectal, disregarding the negative predicatal format used by the rabbis when they say: “He is not trustworthy concerning himself, how can he be trustworthy concerning others?” Here, notice, “he” is the subsidiary term (S) and “trustworthy concerning himself” and “trustworthy concerning others” are respectively the minor and major terms (Q, P); since the latter are predicates, the argument ought to be formulated in predicatal form. In Eduyyoth 6:2 I, the rabbis argue: “For the living being, which is clean, a limb severed from it is unclean – so is it not to be inferred that for a corpse, which is unclean, a limb severed from it is unclean?” I read this simply to mean: ‘If the limb severed from a living body (Q) is unclean (R) enough to be declared unclean (S); then the limb severed from a corpse (P) is unclean enough to be declared unclean’. Samely, however, effectively takes the subsidiary term to be “capable of conveying uncleanness.” Unless he bases this on some Gemara or other traditional commentary (he does not mention any), I do not see the reason for this interpolation.

Similarly, Samely unnecessarily complicates the matter with regard to Eduyyoth 6:3 I. In his interpretation of Menahoth 8:5, Samely takes the middle term to be “quality of material (‘purity’),” whereas in the Mishna it is clearly “to do with eating.” With reference to Nedarim 3:11 VII, Samely proposes an afa (saying, “there may be an implied dimension of a fortiori here”) though there is no hint of such intention in the given text[42]. We have earlier come across other cases where the polarity of the argument is misconceived, or the middle or subsidiary term is badly chosen, or the middle and subsidiary terms are tied together; I shall not repeat my analysis of them here: suffice it to note that many of those cases are not only in error but fail to mirror the original text.

Confusion regarding ‘proportionality’. Samely is just about silent on the theoretical issues relating to ‘proportionality’, and his recognition in practice of the difference between purely a fortiori argument and a crescendo (i.e. a fortiori cum pro rata) argument is far from manifest and consistent. Sometimes he judges correctly (24 cases), sometimes incorrectly (12 cases), and sometimes he fudges (6 cases). This demonstrates he does not have a clear and certain understanding of the subject-matter, even on an intuitive level, let alone on a formal level.

· He rightly suggests proportionality in his conclusion in 7 cases, namely: the first argument of Bava Kamma 2:5 (language “ought to be more”), Makkoth 3:15 II (“at least”[43]), Makkoth 3:15 IV (“should be even greater”), Negaim 12:5 V (3 cases, “even more”) and Sanhedrin 6:5 II (“even more”). But he wrongly suggests it in 6 cases, viz. Arakin 8:4 (“even greater”), Demai 2:2 (“can only be less”), Shebuout 3:6 (“should cause even more”), Terumoth 5:4 I (“should even more”), Zebahim 7:4 (“should be even more subject”) and Zebahim 12:3 (“should even more have”). In 1 case, Hullin 12:5 (“or even some greater”), Samely’s language is tentative though it should have been definitely proportional. In 6 cases, viz. the second argument of Bava Qamma 2:5 (“should be more”), Eduyyoth 6:2 I (“should also or even more so”), Pesahim 6:2 I & IV (3 cases, “should even more/less”) and Yadayim 4:7 (“should apply also/even more”), his language wrongly suggests proportionality as a possibility. In 3 cases, viz. Bekoroth 9:1 I (“should be even less substitutable, with even greater certainty not substitutable”), Makkoth 1:7 VI (“even more certainly”) and Yom Tov 5:2 (“must be even more certainly”), the intent is epistemic rather than ontical. In 17 other cases, Samely rightly makes no suggestion of proportionality. And lastly, in 2 cases, namely Berakoth 9:5 V and Terumoth 5:4 II, he offers no analysis, so we cannot determine what he thinks (neither of these cases are in fact a crescendo).

If I had any doubt, looking at Samely’s theoretical treatment, as to his posture regarding the validity of ‘proportional’ a fortiori argument, all such doubt disappeared when I saw his practical treatment. Samely has obviously not given any thought to this crucial issue. What is clear is that Samely is unsure what to make of it. Samely’s numerous wrong suggestions of ‘proportional’ a fortiori argument are inexcusable, since he has come across and devoted some attention to the rabbis’ dayo principle, given in mBQ 2:5. Not once[44] does he again raise the issue in all the other examples of a fortiori argument that he deals with. Apparently, this encounter did not even awaken in him (as it did, for instance, in Maccoby) an awareness of the principle of deduction, i.e. that the conclusion of a deduction cannot claim more information than its premises logically provide. But in any case, he should have at least asked the question on the basis of Talmudic hermeneutics, i.e. in the way of a religious rule. Evidently, he did not internalize the dayo principle.

Additional criticisms. Some more criticisms, some general, some specific, can be leveled against Samely’s treatment of a fortiori arguments in the Mishna. One worth mentioning is that Samely does not seem to be aware of the difference between the davka reading of a text (where what is not explicitly included is taken to be implicitly excluded) and lav davka readings (where things not explicitly included may yet be argued to be included). That is an important aspect of rabbinic hermeneutics which, judging offhand, he seems not to have grasped (see his comments on Hullin 10:1 III and Menahoth 8:5). But I think that enough has been said. We have found various sorts of faults: methodological, conceptual, interpretative, and most important—logical.

Samely has done a lot of work, and a lot of it deserves praise, but he clearly needs to overhaul his database – and before that, he needs to study a fortiori logic more carefully. Glancing very briefly at his “techniques of interpretation” other than those relating to a fortiori argument, I tend to suspect the rest of his work also needs to be thoroughly reviewed and revised. And for that purpose, Samely should further study works on the rabbinical hermeneutic principles (including my Judaic Logic), before trying to impose his own vision. One reason I engaged in such a detailed examination of Samely’s work was to show him, his Oxford UP publishers, and indeed all readers, that my criticism of it is fair, being based on sustained study and not merely on general impressions. Certainly his effort deserves such proportional attention, whatever its shortcomings.

Moreover, it occurred to me as I analyzed his work that his many errors have some utility – they are lessons learned. By showing them up for all to see, I was providing future researchers with a valuable gallery of possible errors. When future researchers try to interpret and evaluate a fortiori argument in the Gemara or any other document, they will be more careful to avoid repeating such errors. A researcher must especially be sure to master the logic theory, before venturing to examine documents.



[1] Oxford: Oxford UP, 2002. See chapter 7.

[4] Regarding Samely’s proposed distinction between grammatical form and logical form, I would say that a more accurate representation would be a distinction between surface form (the way an argument often appears in everyday discourse, which is often abridged and casually worded) and deep form (the way an argument academically ought to be presented, with all the elements relevant to its logical validity made manifest). Jacobs presented some (though not all) common surface forms of a fortiori argument, without succeeding or even trying to formulate deeper levels of discourse. Samely, for his part, as the present essay makes clear, tries to dig deeper but fails.

[5] It is evident that Samely found this gradual approach useful, to order things in his own mind; but we are here concerned with logic not psychology.

[6] I should also here draw the reader’s attention to the recurring fallacy identified in the chapter on Adin Steinsaltz (18.2), where six terms are also involved. It could be that some of the cases Samely had in mind belong under that fallacy. A fortiori argument as such must be thought of as a four-term argument form, to avoid all possible error. When more than four terms are involved, the required major premise must first be separately proved.

[7] In the context of Samely’s narrow view of a fortiori argument as positive subjectal or antecedental forms, ‘proportionality’ signifies that the conclusion’s predicate is quantitatively greater than the minor premise’s predicate. This is of course contrary to the findings of formal logic for purely a fortiori argument, but Samely apparently does not know it. For a ‘proportional’ conclusion to be justified, we must have additional information (i.e. a further premise).

[8] I self-published my book Judaic Logic in the summer of 1995, and distributed many copies to various persons and libraries round the world, including a copy to the Jewish and National University Library in Jerusalem in October 1995 and one to the Bodleian Library in Oxford in March 1996 (which are still listed in their online catalogs today). An edition intended for the general public appeared in the spring of 1997, published by Editions Slatkine, Geneva, and many copies were sold. Furthermore, an extract of my book appeared in Higayon: Studies in rabbinic logic, Vol. 4 (1997), a journal edited by Moshe Koppel and Ely Merzbach of Bar Ilan U., and published by the Center for Jewish Public Policy / Alumna, in Jerusalem under the title “Forms of A-Fortiori Argument,” which listed the main eight forms in bold type with appropriate explanations. Finally, I put the whole book online in my new website, www.TheLogician.net, as of May 2001, and it appeared very soon after in many search engines including Yahoo (which at the time was the foremost one). The title page of Judaic Logic had 1350 visitors in the first year of its exposure on my website. In other words, my work was not exactly unknown in 2002.

[9] In any case, the Ramchal (R. Moshe Chaim Luzatto) can be credited with the main aspects of a fortiori argument formalization, back in the first half of the 18th century. See my essay devoted to his findings in the present volume. As I there explain, I only recently discovered his work, and was certainly not influenced by it when developing my theory of a fortiori. (The English translation of the Ramchal’s The Book of Logic was published in Jerusalem the same year as my Judaic Logic appeared in Geneva. Had I known about it, I would certainly have looked into it and mentioned it, as I did for the Ramchal’s Ways of Reason. In any case, I formulated my theory well before I published it. The book took me four years or so to complete, and the chapters on a fortiori argument were among the first I dealt with. I can probably find the exact date my formulations of the eight main moods of a fortiori argument crystallized, through old backup diskettes.)

[11] Why is A5 not called A4.3? Surely, A5 is closer in form to A4.2 than A4.2 is to A4.1? Also, is A4.1 normative or non-normative or both? If only the one, why not the other? Or why is there not a form that is to A4.1 what A5 is to A4.2? These are issues of symmetry, which put the proposed list and labeling of just three types in doubt.

[13] I mainly use Samely’s transliterations in the present chapter, so as to facilitate references in his own work. (The transliteration for this tractate’s name that I use in other chapters is Baba Qama.)

[14] Except that his wording of the conclusion, viz. “should even more be,” seems to suggest a crescendo argument whereas the Gemara’s finale is clearly purely a fortiori.

[15] Effectively by reductio ad absurdum, since he speaks of “reversal” and after presenting the former he says “So:” and restates the latter. If we deny the putative conclusion that there is some liability for tooth and foot damage on public grounds, and combine this denial with the other premises, we obtain as conclusion that there is full liability for horn damage on private property. Actually, this is not quite accurate, because the minor premise of the “reversal” argument is that there is some liability for horn damage on public grounds, whereas that of the “derived” argument is that there is more precisely half liability. However, R. Tarfon’s first argument would work as well with the vaguer minor premise; so that’s ok.

[17] I say ‘to date’ because the number of cases identified by Samely has increased over time, as is evident from the fact that in his book (p. 414) he lists only 22 occurrences (including 14 of A4.1, 5 of A4.2 and 3 of A5). This is of course to his credit, signifying that his research is ongoing; but it also means there may be still more cases that he has not yet discovered. I examined the database in the Summer of 2011.

[18] The roman numerals seem to be subdivisions added by Samely, though I have found no explanation by him. He also adds Arabic numerals in brackets, which having no idea what they might mean I have omitted.

[19] That is to say, when one searches the database for all instances of “A4.1,” the results returned do not include mBQ 2:5. This could merely be a programming problem, because the database does have an entry with the arguments of R. Tarfon (labeled as “A4.1”). Even so, the absence of this reference is indicative of carelessness in Samely’s work.

[20] The two cases in Avot I found by chance, while reading in that book. The two other cases I found by means of a search for key phrases in a document downloaded from sacred-texts.com, called Eighteen Treatises from the Mishna, by D. A. Sola and M. J. Raphall (1843). That search yielded only 6 cases in all, namely 5 cases of ‘a fortiori’ (Chullin 10:1, Kilaim 8:1, Pesachim 6:2 and 6:5, and Beitzah 5:2) and 1 case of ‘how much more’ (Chullin 12:4-5). Still, it uncovered two new cases.

[21] He does mention the Avot 6:3 case in a footnote to p. 183 of his book, but does not say why he excluded it from his list. Note that this tractate is certainly regarded as part of the Mishna, even though there is no Gemara for it.

[22] That is: “Stricter standards (R) are required of righteous individuals like Moses (P) than of ordinary Israelites (Q). If the command of circumcision (S) was strictly enough applied (that it was not suspended even for one hour) for Moses (P), then the command of circumcision (S) is to be strictly enough applied (that it be not suspended even for one hour) for ordinary Israelites (Q).”

[23] This argument is mentioned in his book, p. 183.

[24] Another telling example of such naivety is Bekoroth 9:1 I, where Samely comments: “The way the subsequent hermeneutic operation rejects this a fortiori argument is interesting: none of the ‘premisses’ seem to be undermined: it is merely shown that the conclusion, point 5, is contradicted by Scripture.”

[25] This argument is mentioned in his book, p. 179.

[26] See my fuller analysis of this argument in Appendix 2.

[27] Which two cases Samely says might alternatively be construed as A0.5: “I take the two presentations of Tarfon’s argument to be open to two different constructions: while the first sounds more like an a fortiori argument, the second sounds more like an ‘analogy of relationships’ argument [A0.5]. However, I am assuming that the argument in both formulations is the same, so that one of the two constructions is the meaning intended by both formulations.”

[28] This argument is mentioned in his book, p. 186.

[29] In truth, Samely misconstrues both these terms. R. Eliezer’s argument is really: since a limb severed from a living being (P) is just as dead (R) as a whole corpse (Q), then the uncleanness (S) of the latter (Q) is also true of the former (P). Samely senses that his a fortiori construction does not make sense, and therefore suggests that the argument may alternatively be “a fairly straightforward analogy (of part-whole relationships).”

[30] See Appendix 2.

[31] See my full analysis of this argument in Appendix 2. Samely does characterize this argument as “rhetorical,” but he does so not for formal reasons but because “life or death are on/off qualities, so the idea of ‘increase’ is impossible.” Still, it is interesting that he appeals to the “measure for measure” principle to form his major premise: “the punishment for transgression is, measure for measure, smaller than is the reward for fulfillment.” Although, note, this is an erroneous formulation of the principle as such, because there is no a priori reason why “the punishment for transgression” should be smaller than “the reward for fulfillment.” One might declare the two as equal offhand – but one would have to give good reasons why one should be smaller than the other. The idea is in fact based on Scriptural passages like Exodus 20:5-6, where God states that he visits “the iniquity of the fathers upon the children unto the third and fourth generation” of them that hate Him, whereas he shows “mercy unto the thousandth generation” of them that love Him and keep His commandments. The latter is an additional principle, not identical with the principle of measure for measure.

[32] Samely remarks: “(non-A4.1/A4.1) Is this meant as a counterargument, and reductio ad absurdum of the preceding A4.1? If so, it does not itself have the structure of an a fortiori? It is denied that the greater leverage attaches to the common food.”

[33] Since both arguments go from minor to major, they have to be formulated as negative subjectal to be indeed invalid. So, Samely’s attempt to build a valid positive argument for Beit Hillel misses the whole point.

[34] This argument is mentioned in his book, p. 184.

[35] His comment concerning the afa in Demai 2:2, viz. “it has a negative form, but it is a positive analogy: he is unreliable concerning himself, how much more so concerning others,” is an attempt to to dismiss the phenomenon rather than acknowledge and try to explain it.

[36] In the first two of these six cases, Samely should have noticed that the original arguments in the Mishna are clearly negative; in the last four cases, Samely may be excused in that the rabbis themselves formulate the arguments positively, although it should have been obvious to him and them that they had to be formulated negatively (using the double negative “not enough R to prevent”) to be made valid. The latter arguments could conceivably be recast in positive form, but such traductions are usually rhetoric rather than logic.

[37] He does comment that the two arguments of Pesahim 6:2 IV are “going in opposite directions,” but he does not clarify just what this means and what it implies regarding validity. And in fact, he formulates both arguments in the wrong direction – the one that should be minor to major as major to minor, and the one that should be major to minor as minor to major! Similarly, his comment concerning Sotah 6:3 III, viz. “Mirror-image of the earlier a fortiori inference: compared with mSot 6:3 I (4), propositions four and five have switched places” shows awareness that the minor premise and conclusion (i.e. more precisely put, the minor and major terms) have switched places – but Samely offers no explanation or validation for this phenomenon. Moreover, his formulation of the Sotah 6:3 III argument is invalid!

[38] Another a pari argument, namely the 1st afa in Temurah 1:1 III, by R. Akiva, is not even noticed or mentioned by Samely.

[39] His argument goes: Given more respect is due to the name of God than to that of Moses: if “the name of God appears in Scripture underneath that of the ruler without this meaning irreverence,” then “the name of Moses can appear in the bill of divorce together with that of the ruler also/all the more without this meaning disrespect” – whereas it should have been: Given more respect is due to God than to Moses: if occurrence of name in a document (Scripture, in this case) side by side with a ruler’s name (Pharaoh’s) is respect enough for God, then occurrence of name in a document (a bill of divorce, in this case) side by side with a ruler’s name (current ruler) is respect enough for Moses. Samely has the negative subjectal format “if PRS, then QRS” – whereas the correct format for positive predicatal argument is “if SRP, then SRQ.”

[40] Thus, in the two Bava Kamma arguments, the damages brought about by an ox imply various legal liabilities on the owner of the ox (not the ox itself); in the Hullin case, offerings imply the payment certain dues to priests (the offerings do not pay, it is the one offering them who does); in the Zebahim case, wrong sprinkling of blood does not render the remaining blood unfit (the antecedent and consequent do not have the same subject, i.e. the same portion of blood).

[41] See for instances Makkoth 1:7 VI and 3:15 II in this regard.

[42] There is, to be sure, a rabbinic idea (see the 5th and 6th rules of R. Eliezer ben Yose haGelili) that a fortiori argument may be “meforash” (explicit) or “satum” (implicit); so Samely’s suggestion is not in principle impossible. But it is not his task here to propose innovations. And besides, countless comparisons could similarly be turned into a fortiori arguments!

[43] We can safely say that Samely here intends proportionality, even if the language he uses is indefinite, in view of the special difficulty of the case. As already stated, this argument is invalid, because it is a contrario. Nevertheless, it does involve ‘proportionality’, i.e. its predicates ‘having life taken away’ and ‘having life given’ can be viewed as two points along a common continuum. However, Samely’s addition of “at least” in his conclusion is gratuitous.

[44] Except in Zebahim 7:6, but only because the issue is there apparently raised by R. Yose. However, though the latter uses the language of dayo, saying “it is enough,” I doubt he is really invoking the principle as Samely suggests. I see no quantitative or even qualitative change in R. Meir’s proposed conclusion. R. Yose seems to be denying that the conclusion of the a fortiori argument follows from its minor premise (which he accepts), arguing: “[Even though] slaughtering the carrion of a beast renders clean, [still] nipping off [head of bird] does not [render clean].” This implies that he is doubting the major premise, somehow, rather than literally appealing to dayo.

2016-08-23T09:50:37+00:00