A FORTIORI LOGIC
CHAPTER 21 – Gabriel Abitbol
Gabriel Abitbol’s 1993 book, Logique du droit talmudique is a detailed study of Talmudic logic (in French, obviously), including a long chapter devoted to qal vachomer argument (pp. 126-169). I found a copy of this book (which is, by the way, beautifully bound) by chance in a Paris bookshop in 1995, when my book Judaic Logic was almost finished. I did not read it all then, but only scanned through it enough to make sure Abitbol had not anticipated my findings. I will here look at his work more closely, especially as regards a fortiori argument.
Abitbol begins his study in a promising manner, remarking that due to qal vachomer reasoning not being defined in Talmudic literature there are ambiguities regarding its application, and boldly stating his intention to discover the exact nature and logical structure of qal vachomer reasoning and, after examination of some examples of its use in Talmudic jurisprudence, to define it and determine the modalities of its application.
Terminology. He proceeds by first clarifying the expression qal vachomer, which may be identified with a fortiori argument. The word qal, which in Hebrew means “light” (i.e. of little weight), designates “a prescription that is less grave or easier to observe than another, or anything permitted, pure, without fault;” while the word chomer, which means “matter, heavy thing,” has the opposite legal connotations. Since there are two inferences characterized by the conjunction qal vachomer, viz. a maiori ad minus and a minori ad maius, which serve respectively to attenuate or aggravate, it is clear that the expression does not indicate “the process through which we arrive at a conclusion,” but simply recalls “the elements constituting the two premises of the reasoning.”
Here, I should say a few words. Abitbol is essentially right in this analysis of the qal vachomer expression, but wrong in characterizing the two parts of it as “premises.” Qal and chomer in fact refer to terms found in the premises; they are not themselves premises. Both terms are found in the major premise, and only one of them appears in the minor premise while the other appears in the conclusion. In argument a maiori ad minus (i.e. from major to minor), the chomer is in the minor premise; while in argument a minori ad maius (i.e. from minor to major), the qal is in the minor premise.
Evidently, Abitbol thinks of the word qal as referring to the proposition containing the qal term, and the word chomer as referring to the proposition containing the chomer term; but these propositions cannot both be premises – one is a premise and the other a conclusion – so this is a bit thoughtless. As regards the characterization of these two directions of argument as “attenuation” and “aggravation,” he cites R. Samson de Chinon in Sefer Keritut as the source. Personally, I think these characterizations are inappropriate in the context of a fortiori argument, since they signify a change of stringency and no change occurs in it – we just draw out implicit information.
Next, Abitbol informs us that the two “premises” (as he calls the propositions, again) are called in the Talmud melamed and lamed, i.e. the one that teaches and the one which is clarified by a teaching. He insists that these “must imperatively be constituted by prescriptions of the Written Law,” and that any qal vachomer not based on Torah legislation (but, e.g., based on some oral tradition, even if that is attributed to Moses at Sinai) is not a qal vachomer in the sense intended in R. Ishmael’s list of hermeneutic principles (but merely an “argumentative” a fortiori). Here, of course, I agree with him, and say he is right to point it out, as some commentators do ignore or forget this truth. However, to be precise, it is only one of these two propositions, viz. the minor premise, which is inscribed in the Torah; the other proposition, being the conclusion of the argument, is merely inferred. They are both Torah based, in a broad sense; but one is directly so, while the other is indirectly so.
Mechanism. After these preliminaries, Abitbol begins his analysis of qal vachomer in earnest, with reference to God’s reaction to Miriam’s gossip concerning Moses (Numbers 12:14), i.e. to the statement: “If her father had but spit in her face, should she not hide in shame seven days? Let her be shut up without the camp seven days, and after that she shall be brought in again.” According to Abitbol, the qal vachomer involved here is, as a baraita (given in Sifra, a legal commentary on Leviticus dating from Mishnaic times) asserts, that Miriam who offended God (and not merely her father) should have been shut up for fourteen days. Here, note well, the given predicate is “seven days” incarceration, whereas the inferred predicate is “fourteen days” of it.
In other words, he agrees with those commentators (including the Gemara, in Baba Qama 25a which refers to the same baraita) that a fortiori argument is inherently ‘proportional’; i.e. that it is a crescendo argument. This does not mean that it is always ‘proportional’, since Abitbol also gives as example the qal vachomer proposed by Hillel to the Bene Bathyra in Pesachim 66a, which goes (in the Soncino Talmud): “if the tamid, [the omission of] which is not punished by kareth, overrides the Sabbath, then the Passover [neglect of] which is punished by kareth, is it not logical that it overrides the Sabbath!” Here, the given and inferred predicates are clearly one and the same (viz. “overrides the Sabbath”). However, note well, Abitbol does not remark upon the difference between a ‘proportional’ conclusion (which the Miriam argument has, in his eyes) and a ‘non-proportional’ one (which the Hillel has, though he does not mention the fact).
Abitbol explains the “mechanism of the reasoning” concerning Miriam as follows: “comparing two situations,” one where someone offends a neighbor (i.e. a daughter offends a father, in this case), for which the penalty is known, and the second where someone offends God, for which the penalty is not known, it is “deduced” from the fact that God is hierarchically above Man that the sanction is to be greater in the case of offense to God. As regards Hillel’s argument, the reason why it does not similarly conclude with a greater penalty is due to the fact that the hierarchy between God and Man in the first example is obvious, whereas that between the tamid (i.e. perpetual) sacrifice and the Passover (i.e. pesach) sacrifice has to be sought out.
Both these explanations are unconvincing; they are non sequiturs. As can be shown formally, the fact that two subjects are in a hierarchy does not in itself prove that they are subject to different predicates (or for that matter the same predicate); there has to be a separate justification for any claim of proportionality (otherwise, the minimum implied by non-proportionality must be assumed). Moreover, it does not matter how complex the process of justification is, i.e. whether the hierarchy is obvious or difficult to establish; what matters is only the end product of the process, i.e. the fact of the claim concerned being credible. Incidentally, the distinction made here by Abitbol between obvious and sought-out hierarchies calls to mind the distinction made by Louis Jacobs between simple and complex a fortiori.
Clearly, Abitbol (like many commentators before and after him) has not grasped the fact that a fortiori argument is essentially pure (i.e. non-proportional), and only occasionally (when an additional premise of proportionality is duly supplied) a crescendo. This means that he has not succeeded in his initially announced quest to discover the exact nature and logical structure of qal vachomer reasoning. As we shall see, Abitbol conflates a fortiori argument with analogical argument throughout his analysis.
Abitbol presents a fortiori argument as a means of inference, from a textually given legal disposition, of an unknown (i.e. not textually given) legal disposition, through analogy to another pair of comparable legal dispositions. He proposes to graphically “represent” it in tabular form, essentially as follows (Table 21.1):
legal norm A
legal norm C
legal norm B
legal norm D
Actually, Abitbol labels subjects 1 and 2 as P1 and P2, and situations 1 and 2 as S1 and S2, respectively. I avoid doing so, so as not to cause readers confusion with my standard symbols P and S. Moreover, he does not have the symbols A, B, C, D, but instead has a + or – sign in each of the four “legal norm” cells. Positive signifies “being permitted, easy, pure or without fault,” and so on; and negative means “forbidden, grave, liable to, impure,” and so on. Thus, the “legal norms” are value judgments. However, since these signs vary from one example to the next, I have opted for the said variables. How is this table to be read – vertically or horizontally? It is evident that Abitbol is unsure on this issue, or at least unaware of the question and therefore inconsistent.
Judging by his stated intention (“the reading of this table is then as follows,” p. 132), the reading should be vertical, i.e. as follows:
Since in situation #1, subject #1 is A and subject #2 is B (melamed), it follows that in situation #2, where subject #1 is C, subject #2 will a fortiori be D (lamed). When A > B, then C > (or =) D (this is from major to minor); and when A < B, then C < (or =) D (this is from minor to major).
However, judging by most (though not all) of the examples he gives, he actually usually reads the table horizontally, i.e. as follows:
Since subject #1 is A in situation #1 and is C in situation #2 (melamed), it follows that subject #2, which is B in situation #1, will a fortiori be D in situation #2 (lamed). When A > C, then B > (or =) D (this is from major to minor); and when A < C, then B < (or =) D (this is from minor to major).
Apparently, Abitbol does not fully realize the logical difference between these two processes. This is perhaps due to the fact that, in both the vertical and horizontal readings, if A, B and C are respectively +, – and –, then D will be –; and if A, B and C are respectively –, + and +, then D will be +. But these formulae are superficial, because they ignore the generalizations involved behind the scenes (more will be said on this further on). Note that the bracketed “or =” is my own addition, based on some of his examples, in which there is a legal maximum, so that “more” effectively means “as much”.
An example Abitbol uses for illustration is the a fortiori of Hillel to the Bene Bathyra (Pes. 66a), which we have already transcribed above. The way this example is originally worded (p. 130), with tamid sacrifice, kareth penalty and Shabbat mentioned first, followed by pesach sacrifice, kareth and Shabbat, suggests the following reading: since as regards the tamid sacrifice, the penalty of kareth for omission is not applicable, whereas overriding the Sabbath is applicable, it follows as regards the pesach sacrifice, for which the penalty of kareth for omission is applicable, that overriding the Sabbath must be applicable. This is reasoning from the minor to the major: the given qal and chomer being no-kareth and overriding of the Sabbath, and the inferred qal and chomer being kareth and overriding of the Sabbath. Abitbol represents this (p. 133) as a table with tamid and pesach as the “subjects” and kareth and Shabbat as the “situations;” therefore, his reading of the table here must be horizontal.
Another example he uses for illustration is from Beitzah 20b. Its original wording is (roughly, p. 133): if on the Sabbath food preparation is forbidden while the tamid sacrifice is permitted, all the more on a Festival day, when such tasks are permitted, the tamid should be permitted. Here, the reasoning is from major to minor, with the given chomer and qal being Sabbath food preparation and tamid sacrifice, and the inferred chomer and qal being Festival food preparation and tamid sacrifice. Abitbol represents this (p. 134) as a table with food preparations and sacrifices as the “subjects” and Shabbat and Festivals as the “situations;” therefore, his reading of the table here must be vertical. This change of direction of reading goes to show that Abitbol tabulates the information in rather haphazard fashion, without attention to consistency. As we shall see, he sows additional confusion in this matter further on, when dealing with the two arguments of R. Tarfon. Although this issue is not crucial, it is indicative of a certain lack of professionalism.
Deficiencies. Abitbol’s tabular representation is obviously incomplete, since it leaves much important information unspecified. When the inferred norm (D) is said to be is greater or lesser than the norm from which it is inferred (C or B, depending on the reading direction), we are not told by how much it is so; presumably, the change of magnitude would be proportional, but this is too vague a characterization for clear judgment. Very often, too, the inferred norm is in fact equal to the given norm (as the two examples just given illustrate). Moreover, the major premise is not explicitly given in Abitbol’s table, even if it is implicit in the + and – signs placed in cells A and B, or respectively A and C (obviously, if A is + and B is –, then A < B, and if A is – and B is +, then A > B; likewise, if A is + and C is –, then A < C, and if A is – and C is +, then A > C).
It cannot be said that Abitbol is unaware of the major premise, since he repeatedly refers to a hierarchy between the subjects. Indeed, further on (on p. 165), he explicitly refers to a judgment that subject #1 “is less (or more) grave than” subject #2, in which the two subjects are placed in a “scale of values” (échelle évaluative) in accord with their legal value (as permitted, forbidden, etc.). Moreover, he even there shows awareness of the middle term (which was thus far missing, or merely implicit) when he adds that the same subjects may be “ordered in another scale” (i.e. they may be differently “grave” in some other respect, in relation to some other middle term) and thereby turn out to have opposite relative values (i.e. the major term in one scale may be the minor term in another, relative to one and the same other term). But the problem is that his tabular scheme does not allow him to represent these subtleties.
But the main criticism I would level against Abitbol’s treatment of a fortiori argument is his failure to realize the role of the middle term in the minor premise and conclusion. As just stated, he does acknowledge the middle term implicit in the major premise, in the context of the possibilities of “refutation” of a qal vachomer (as we shall see further on); but this does not imply that he is aware of its presence and crucial function in the minor premise. Nowhere does Abitbol explain the a fortiori inference of a conclusion as made possible by the passing of a threshold value of the middle term implied in the minor premise. Without this idea, he cannot be said to have grasped the full essence of a fortiori argument and what he refers to by that name is nothing more than mere analogy.
It is because of his failure to notice the need for a subject to have a sufficient amount of the middle term to gain access to a certain predicate (which condition is given in the minor premise, and makes possible the conclusion) that he remains stuck in the realm of mere analogical argument. At one point (p. 136) Abitbol seems about to hit upon this ignored element, but he surprisingly misses it. He says, much too vaguely, that the conclusion can be established only if two situations are shown directly or indirectly to be “tied by a certain relation” which allows the establishment in the second situation (i.e. between C and D) of the hierarchy observed in the first (i.e. between A and B). But he does not tell us what that “certain relation” might be, other than to refer to “reasoning by analogy.” Thus, Abitbol cannot claim to have truly understood and correctly defined qal vachomer reasoning.
Standardization. These criticisms become clearer when we try to put Abitbol’s effective formula for a fortiori argument in standard form for him. The following positive subjectal mood would correspond to his minor-to-major argument in the vertical reading:
Since subject #2 (= P) is more “grave” (in some respect, R) than subject #1 (= Q)
(as we know by generalization from situation #1, by comparing A and B),
and, in situation #2, Q is (R enough to be) C (= S);
(and S is constant, or varies in proportion to R;)
it follows that P is (R enough to be) D (= S, or more than S).
As regards his major-to-minor argument, it would needs be put in negative subjectal form to be valid, even though he effectively presents it as equally positive reasoning. Note the reversal of order of the subjects in the major premise:
Since subject #1 (= P) is more “grave” (in some respect, R) than subject #2 (= Q)
(as we know by generalization from situation #1, by comparing A and B),
and, in situation #2, P is (not R enough to be) C (= S);
(and S is constant, or varies in proportion to R;)
it follows that Q is (not R enough to be) D (= S, or less than S).
Evidently, Abitbol’s two “subjects” are indeed logical subjects (the major and minor terms, P and Q), while his “legal norms” A, B, C and D are their predicates (C and D being the subsidiary term S). However, it is clear that the two “situations” stand outside the a fortiori arguments as such: they are actually linked by a generalization from the first situation to all situations and thereafter a subsumption from all situations to the second situation. Abitbol evidently does not realize this; i.e. he does not realize that the first situation, involving the predicates A and B, is relevant to the a fortiori argument only insofar as it helps us to establish, by generalization, its comparative major premise; but it is not in fact part of the a fortiori argument. As for the second situation, it is the framework within which the a fortiori argument per se occurs, but it plays no active role within the argument.
We could, of course, similarly standardize the horizontal reading. The minor-to-major mood would here read:
Since situation #2 (= P) is more “grave” (in some respect, R) than situation #1 (= Q)
(as we know by generalization from subject #1, by comparing A and C),
and, in subject #2, Q is (R enough to be) B (= S);
(and S is constant, or varies in proportion to R;)
it follows that P is (R enough to be) D (= S, or more than S).
And the major-to-minor mood would read:
Since situation #1 (= P) is more “grave” (in some respect, R) than situation #2 (= Q)
(as we know by generalization from subject #1, by comparing A and C),
and, in subject #2, P is (not R enough to be) B (= S);
(and S is constant, or varies in proportion to R;)
it follows that Q is (not R enough to be) D (= S, or less than S).
Here, note well, Abitbol’s two “situations” are the logical subjects, while his two “subjects” are the outside conditions. Also, the generalization occurs with reference to subject #1, and is then applied to subject #2. Therefore, it seems to me, if we are going to use the terminology of “subjects” and “situations,” we would be better off sticking to the vertical reading, where these terms are literally most appropriate. In other words, Abitbol’s initial theoretical intent to go by the vertical reading was correct, whereas his subsequent frequent use in practice of the horizontal reading was not correct. To repeat, this is not a crucial matter – but it implies thoughtlessness and sows confusion.
More deficiencies. Anyway, from these more accurate formalizations, we can also see more clearly the missing or merely implied elements of Abitbol’s representation. He lacks an explicit major premise and middle term; he lacks an additional premise relating to proportionality, where needed; and most important of all, he totally lacks the crucial clause “R enough to be” (or “not R enough to be,” as appropriate). For these reasons, he cannot in fact logically validate the argument, even though he thinks he has done so by merely (roughly) representing it in a table and giving a few explanations. Moreover, Abitbol evidently does not realize that the above listed major-to-minor argument (i.e., in the vertical reading, when C < D, since A < B) is essentially negative in form. His failure to realize this is no doubt in part due to his opting for a graphic representation. It is difficult if not impossible to represent the distinction between positive and negative argument in a table; one might resort to ‘crossing off’ cell contents to signify their negation, but this is rather messy.
Furthermore, Abitbol does not show awareness of the distinction between predicatal argument and subjectal argument. I guess he would just tabulate predicatal arguments he came across in the same way, without highlighting the differences since he has no means to do so. That is to say, he would reword a predicatal argument he came across in superficially subjectal form, and thus manage to fit it into his tabular representation. However, since none of the examples of a fortiori argument Abitbol gives happen to be predicatal, it is not possible to say for sure how he would react to such eventuality. Maybe it would make him change his whole outlook.
Non-universality. Moreover, as we shall now demonstrate, contrary to the impression that the above selected examples give, it is not always the case that we have two “situations” to deal with. We may well be given the general major premise at the outset, without having to derive it from a specific instance in this way. In other words, we do not always have to engage in preparatory work to obtain the needed major premise: it may be given in a text and taken for granted, or it may be intuitively obvious or generally believed to be so. Indeed, Abitbol himself earlier (pp. 130-1) effectively admits this when he makes a distinction between an obvious hierarchy (as that between God and Man implied in the Miriam example) and a hierarchy that has to be sought out (as that between the perpetual and Passover sacrifices involved in the a fortiori spoken by Hillel to the Bene Bathyra).
Therefore, Abitbol’s table is not universally applicable, note well. Consider, for example, the following a fortiori argument presented by Abitbol (based on Makot 5b). It originally reads, roughly, as follows: “if false witnesses are liable to capital punishment if their lying is discovered by the court while the person they falsely accused is still alive, then qal vachomer they will be liable to capital punishment if their lying is brought to light after their victim has been executed.”
I would put this argument in standard form (minor premise and conclusion) as follows: ‘if false witnesses discovered while the victim of conspiracy is yet alive (Q) are criminal (R) enough to be liable to capital punishment (S), then false witnesses discovered while the victim of conspiracy is already dead (P) are criminal (R) enough to be liable to capital punishment (S)’. This is clearly positive subjectal, minor-to-major reasoning, made possible by the tacit but intuitively obvious major premise that ‘false witnesses discovered while the victim of conspiracy is already dead (P) are more criminal (R) than false witnesses discovered while the victim of conspiracy is yet alive (Q)’ (since a successful criminal conspiracy is obviously worse than an unsuccessful one). Abitbol, on the other hand, tries to represent the argument in tabular form as follows:
Status of the accused
Status of the witnesses
Refutation before the execution (subject 1)
Not death sentence
Refutation after the execution (subject 2)
Not death sentence
This is manifestly a fabrication. He is forcing the given material a fortiori argument into a preconceived form, which is clearly not appropriate to it. For, where did he get the two “situations” from, here? Certainly the first situation is nowhere mentioned or needed in the original argument. What does “status of the accused” (the heading of the first column) mean exactly? We might say that if the witnesses are shown up to be liars before the accused is executed, then indeed he would not be sentenced to death; but we cannot logically say that if the witnesses are shown up to be liars after the accused is executed, then he would not be sentenced to death, since that is said and done (the poor man might have his name cleared, but his life is gone). The tacit major premise is intuitively obvious; it is not a generalization from one situation to another.
Note incidentally that Abitbol here again must be reading the table horizontally, for only in this way can the argument be said to be from minor to major (since A < C and B < D). If he read the argument vertically, he would be comparing A and B (which are equal) and applying the same relation (equality) to C and D; this would not be minor-to-major reasoning like the original argument, but a pari. This is one more inconsistency between his practice and his theory.
Abitbol tries to stuff the given argument into his tabular format by changing its terms somewhat. But his table doesn’t make sense. If read horizontally, as he seems to read it, it says, in plain English: since in the event of the witnesses being discredited before execution of the accused, the latter is not liable to capital punishment whereas the former are so, it follows that in the event of discredit of the witnesses after execution of the accused, the latter is not liable to capital punishment whereas the former are so. Clearly, this new argument is not the same as the given one! In fact, it is rather meaningless and unconvincing. The proposed re-presentation is thus very artificial and inappropriate. This is not due to some problem in the original argument, but simply means that Abitbol’s tabular scheme is not fit to represent all conceivable a fortiori arguments. There are cases, like this one, that the scheme did not foresee, because they do not involve two “situations.” Instead of realizing this and reviewing his scheme accordingly, Abitbol tried to fake compliance with it. Evidently, not knowing quite how to handle this argument, he forced things a bit, hoping to get away with it.
Abitbol merits commendation for his taking into consideration, in his discussion of the dayo principle (pp. 140-8), both of R. Tarfon’s a fortiori arguments in the Mishna Baba Qama 2:5 – some commentators (including the Gemara commentary in Baba Qama 25a-b) merely focus on the first. Both these arguments are necessary for a full understanding of the dayo (“it is sufficient”) objections to R. Tarfon’s conclusions by his colleagues, known collectively as the Sages, on which the dayo principle is based. However, as we shall see, he does not represent these two arguments in his tabular scheme quite correctly, and moreover does not grasp exactly where in each of them the dayo principle actually operates.
R. Tarfon’s two arguments. R. Tarfon’s two arguments are should by now be familiar to readers of the present volume, so we can here reproduce them very briefly, as follows. They concern damages caused by a bull, either by ‘tooth & foot’ (munching or trampling) or by ‘horn’ (goring), either on public grounds or on private property; and their purpose is to determine, from information given in the Torah regarding the appropriate compensation in some cases, what the appropriate compensation is to be in cases not mentioned in the Torah – more specifically, in the case of damage by horn in the private domain. R. Tarfon argues in two ways that the owner of the bull should pay full compensation for damages in that context. He argues as follows (I paraphrase):
· Argument #1 – since in the case of tooth & foot the owner needs pay nothing for damages in the public domain and must pay in full for damages in the private domain, it follows in the case of horn, where the owner must pay half for damages in the public domain, that he has to pay in full for damages in in the private domain. The Sages contend that the owner need only pay half damages by horn on private property, just as he does on public grounds.
· Argument #2 – since in the case of the public domain the owner needs pay nothing for damages by tooth and foot and must pay half for damages by horn, it follows in the case of the private domain, where the owner must pay full for damages by tooth & foot, that he has to pay in full for damages by horn. The Sages contend that the owner need only pay half damages by horn on private property, just as he does on public grounds.
Now, before we look into how Abitbol represents these two arguments in his tabular scheme, let us see how we would represent them, given that ideally they are supposed to have “subjects” as row headings and “situations” as column headings and should be read vertically (i.e. first down column 1 and then down column two).
R. Tarfon’s 1st argument
Tooth & foot damage
In the public domain
In the private domain
R. Tarfon’s 2nd argument
In the public domain
In the private domain
Tooth & foot damage
Clearly, in the first argument, the “situations” are tooth & foot and horn and the “subjects” are the public and private domains; whereas in the second argument, the “situations” are the public and private domains and the “subjects” are tooth & foot and horn. In each argument, A, B and C are given (in that order) and D is derived (as the conclusion). We generalize the relation between A and B, and apply that generality to the relation between C and D.
Abitbol’s tables, on the other hand, are turned the other way; i.e. he uses the first of the above tables to represent the second argument, and the second table for the first argument. This is not a big deal, but just means that his tables are to be read horizontally instead of vertically, for it is clear that he does understand the two arguments. As we have seen, he does usually (though not always) set up his tables for horizontal reading, even though he initially (rightly) defines the reading direction as vertical. Even so, this turning of the tables should be pointed out here as further indication that Abitbol experiences a certain amount of confusion in this issue. All the more so since in his first table the subjects are symbolized as P1 and P2, and the situations as S1 and S2 (as per his initial definitions of those symbols), while in his second table the subjects are symbolized as S1 and S2, and the situations as P1 and P2 (he obviously does this intentionally, so as to stress that the two pairs of terms have changed position, but this is not formally correct procedure).
The Sages’ two dayo objections. These few criticisms are not of great import, to repeat. But the fact that Abitbol does not interpret the relation between R. Tarfon’s two arguments fully correctly and that he fails to see precisely what parts of them the Sages’ two dayo objections are aimed at – that is of considerable import.
While Abitbol evidently (as we have just seen) is aware of the difference of direction in R. Tarfon’s two arguments, he does not ask why the second argument constitutes a valid retort to the Sages’ first dayo objection. The answer to that question is that whereas R. Tarfon’s first argument infers full damages (for horn in the private domain) from half damages (for horn in the public domain), his second argument infers full damages (for horn in the private domain) from full damages (for tooth & foot in the private domain). From this reformulation of his argument, using the same information in a new way, it is clear that R. Tarfon interpreted the Sages’ dayo objection as an objection to the inference of full compensation from half compensation, and thus sought to avoid this weakness by proposing a renewed argument that infers full compensation from full compensation. In the latter case, the Sages’ objection would seem to be successfully taken into consideration, for the conclusion refers to the same amount of compensation as the minor premise does.
Yet the Sages respond to this ploy by repeating the same objection, word for word (in what looks like a redactor’s or editor’s copy-and-paste job)! Abitbol here again fails to ask the obvious question: in what way does this repeated dayo objection interdict R. Tarfon’s second argument? Obviously, the second dayo objection cannot really be identical to the first, since R. Tarfon’s second argument was precisely designed to logically avoid what he thought was the Sages’ first objection. Therefore, the Sages’ second objection must be aimed at some other aspect of R. Tarfon’s reasoning. Indeed, it is aimed at the formation of the major premise of his second argument. This premise relies on generalization from the given information (derived from Ex. 21:35) that compensation for damages caused by horn on public grounds is half.
Thus, the Sages’ second objection must mean that, no matter how the given information is used in an a fortiori argument, whether as a minor premise (as in the first argument) or as the basis of a major premise (as in the second argument), the concluding amount (of compensation, in this case) cannot surpass the given amount. That is to say, the dayo principle is not merely that such information cannot be used as a minor premise in an a crescendo argument, but more broadly that such information also cannot be used as a major premise in a purely a fortiori argument. No conclusion can exceed in magnitude the given information from the Torah (concerning a penalty), no matter how (i.e. by what process of reasoning) such conclusion is drawn. That is the full meaning of the dayo principle, and that is why both of R. Tarfon’s arguments are rejected by the Sages.
Abitbol has evidently not realized all this, because he makes no mention of it even though it is essential to true understanding of the Mishnaic discussion. It should be said that Abitbol is not alone in this failure of realization. The Gemara commentary relative to this Mishna is also totally unaware of these issues, and instead of addressing them gets bogged down in tangential issues of very doubtful relevance. It is only much later – I suspect thanks to some genial Tosafist (though I have not yet found who exactly, and when and where) – that Jewish tradition acknowledged a difference between the said two possibilities of dayo application. It was then stated that the Sages’ dayo to R. Tarfon’s first argument was aimed “at the end of the law” (al sof hadin), whereas their dayo to his first argument was aimed “at the beginning of the law” (al techelet hadin).
This traditional distinction does not fully clarify the issues, but it does take us a long way in that direction. It is surprising that Abitbol does not mention the distinction, since he refers in his bibliography (on p. 350) to the Encyclopedia Talmudit article on the dayo principle (in vol. 7, pp. 282-290) where this distinction is clearly mentioned. The explanation, I suggest, is that Abitbol is not aware that generalization is involved in the formation of the major premises of both of R. Tarfon’s arguments. Due to the constraints of his tabular representation of the reasoning processes, he does not constantly have in mind the major premise (even though, as we saw, he somewhat realizes its presence in the background), nor see how precisely it is derived from the Torah-given data. His tables, which are end products of reasoning processes, appear like raw givens.
To be fair, Abitbol does clearly say (p. 146-7) that the dayo principle is to be applied “whatever the hierarchy of reference defined at the start,” adding by way of explanation: “what permits the interpreter to reason by qal vachomer is precisely the fact that the Torah has furnished a value that allows the definition of a hierarchy… as well in the first qal vachomer as in the second.” By the latter explanation he means that since the major premise (“the definition of a hierarchy”) is based on a “value” given in the Torah (i.e. the half payment for horn damage on public grounds), it would be wrong to exceed it. Furthermore, he says that the reason why the Sages “limit in one case the conclusion of the qal vachomer” and in another case “annul it” – presumably referring to respectively the first and second arguments of R. Tarfon – is their belief that a penalty should not be imposed on the basis of an interpretation (more on this topic further on).
He then rightly says that “all this cannot be understood if one does not turn towards the structure of the argument in question.” However, his proposed structural explanation is incorrect. In his view, qal vachomer argument, which consists in reasoning from an admitted hierarchy to a contested one, “cannot in all cases lead to a complete certainty due to the analogy it involves;” such reasoning can only therefore result in a “probable” conclusion, which is neither “absolutely true” nor “absolutely false.” The conclusion, according to him, “does not exclude the possibility of another decision, even one contrary to it.” He cites in support of this view various medieval commentators.
But they are only pointing out in a general way the possibility of human error of reasoning and the danger of thereby wrongly punishing an innocent person, whereas he is claiming that qal vachomer is structurally (i.e. inherently) incapable of guaranteeing the conclusion. By thus qualifying qal vachomer as “reasoning with an uncertain result,” Abitbol shows that he has not grasped the essentially deductive, as against inductive, nature of a fortiori argument. The reason he has not done so is of course, as we have seen, that he has failed to formalize and validate a fortiori argument. He thinks of it as mere analogy, and indeed his tabular representation of it is applicable to mere analogy. Thus, I am justified in saying, as above, that Abitbol has not fully understood the two arguments of R. Tarfon and the two dayo objections to them by the Sages.
The dayo principle. Let us now look at Abitbol’s description and interpretation of the dayo principle as such (pp. 137-157). The latter principle, as we have seen, is based on the Sages’ reply to both of R. Tarfon’s arguments in the above described Mishnaic dialogue; it reads (in the Soncino Talmud): “it is quite sufficient that the law in respect of the thing inferred should be equivalent to that from which it is derived.” Abitbol interprets it as: “it suffices for the conclusion of the reasoning to conserve the quantity of the premise.” Although the original statement does not explicitly mention quantity (it says ka, meaning: as, like), this interpretation shows his awareness of the quantitative aspect of a fortiori argument.
Interestingly, Abitbol cites a second rabbinic principle as relevant in this context, viz. ain onshin min hadine, meaning literally, no punishment may be based on inference, or as he puts it: “the abrogation of a penal sanction based on a reasoning.” An example of this that he gives is the above mentioned argument concerning false witnesses (based on Makot 5b). According to him, the conclusion of this argument, viz. that the false witnesses should be put to death if the accused has already been executed, is not abided by, in accord with the said principle of ain onshin, which he later tells us applies to all conclusions enjoining capital punishment or flagellation (whipping).
Thus, judging by Abitbol’s presentation, it appears that two principles are involved in preventing penalties based on reasoning. The ain onshin principle would be the main principle, preventing the penalties of death or flagellation (which are irreversible), but not lesser penalties, based on any sort of inference; while the dayo principle would be called upon to prevent lesser penalties (such as imprisonment or financial compensation for damage), possibly only those proceeding from qal vachomer inference. This is new to me, but could well be true. Note however that Abitbol identifies the dayo principle with the “axiom well known in logic [that] the conclusion must not surpass the premises;” whereas of the ain onshin principle, he says that it can go against logic. If these viewpoints are truly rabbinic, they are open to criticism.
Identifying the dayo principle with the principle of deduction (viz. that the conclusion cannot contain more information than explicitly or implicitly given in the premises), as Abitbol does, and as many other commentators do too, is incorrect. That is because, though in some cases (e.g. in the Sages’ first objection) the principle is indeed aimed at the a fortiori argument as such, in other cases (e.g. in the Sages’ second objection) it is aimed instead at the generalization preceding the a fortiori argument (which means that is not part of the a fortiori argument as such). This criticism may seem like a bit of hair-splitting (pilpul), if we think broad-mindedly of all pertinent information used in the construction and execution of a qal vachomer as “premises.” But if we reflect that preparatory steps prior to the qal vachomer are not technically part of the a fortiori argument since it could well occur without them, and they without it, then we must either say that the dayo principle is not exclusively about qal vachomer as such, or we must reserve it for the Sages’ first objection and assign the Sages’ second objection another label, if not altogether deny it.
Furthermore, we must ask how far the dayo and ain onshin principles overlap. If we assign a general form to the dayo principle, and view it as interdicting all inference of penalties, including therefore the death penalty and flagellation, then we apparently have no need of the ain onshin principle, at least with regard to qal vachomer inferences. But of course this is not correct – the dayo principle does not interdict all inferred punishment, but more precisely the inference of proportionately increased punishment, whether due to qal vachomer as such or due to the generalization preceding it; whereas the ain onshin principle, if I understand correctly, interdicts all inference by whatever means of a death penalty or flagellation, even if there is no proportional increase. Thus, their scopes differ quite a bit, and at the same time there must be some overlap between them.
The dayo principle only forbids inference of penalties greater than those textually given, whereas the ain onshin principle only forbids inference of the more serious penalties of death or flagellation. Thus, the ain onshin is needed where the dayo principle is technically inoperative because the inferred penalty is not greater than the textually given one, and yet the penalty (being death or flagellation) is too serious to be applied by virtue of a mere inference. Thus, ain onshin is operative beyond the scope of the dayo principle; and conversely, the dayo principle is applicable to cases (involving lesser penalties) where the ain onshin principle is inoperative.
Abitbol, as we have seen, equates qal vachomer with analogy, and as a result does not see precisely when and where in the overall thought process the principle actually operates. He can only vaguely say that the dayo principle “limits” or “annuls” the proportional conclusion the argument seems to warrant. Nevertheless, he must be commended for being aware of the quantitative aspect a fortiori argument and of the dayo principle’s focus on “penal sanctions.” His understanding is superior to that of many commentators before and after him, in that he is (as evident throughout his presentation) aware that the principle concerns specifically the inference by rabbis (legislators or judges) of a greater penalty for a tort or crime not accounted for in the Torah from the lesser penalty for a similar tort or crime accounted for in the Torah. As he points out, though the conclusion of a qal vachomer has “probable character,” it does not “determine the degree of sanction to be applied.”
One reason, according to Abitbol, why a penalty must not be based on an inference is that all sanctions must be preceded by warnings. For example, the Torah sanction against someone who works on the Sabbath (Ex. 31:15) is preceded by the warning not to work on the Sabbath (Ex. 20:9). He gives many such examples. In other words, justice demands that the laws individuals are subject to be explicit (regarding both prohibitions and penalties for their breach). Abitbol points out that this principle of Mosaic law is also affirmed in the Latin adage nulla poena sine lege (no punishment without law), and more clearly and forcefully as of the 18th century, being stated in the 1789 Declaration of Human Rights (art. 8) as: “no one may be punished except by virtue of a law established and promulgated before the offence was committed and legally applied.”
This explanation looks good, but it is logically wobbly. For, if the rabbis inferred prohibitions and penalties, and duly decreed and publicized them, then the reasonable demand for explicit law would be satisfied. Clearly, then, there must be another reason for not inferring laws and sanctions from the Torah. The main purpose seems rather to be, as Abitbol also points out, to limit the power of human legislators and judges (i.e. of the rabbis), so as to maintain the primacy and sanctity of the Torah; for people, no matter how wise and devoted, can never be at once omniscient, perfectly just and merciful, and infallible like God, the giver of the Torah. Of course, historically this rule has not always been strictly upheld: the rabbis have in fact at times made innovations in laws and sanctions. Moreover, nowadays many Jews are not so sure that the Torah is entirely, if at all, of Divine origin, or for that matter that the rabbis are especially wise – so, the moral authority of both the Torah and the rabbis is doubted and we resort more readily to secular jurisprudence (which is far from always wise or just, I hasten to add).
Further evidence of Abitbol’s incomplete understanding of a fortiori argument can be found in his treatment of the doctrine of “refutations” of qal vachomer (pp. 157-66). There are, he correctly informs us, two sorts of refutation, as follows. Note that I use Abitbol’s symbols here, i.e. P for subjects and S for situations (because I do not need to refer to my standard symbols P, Q, R, S).
The first process of refutation consists in demonstrating that the relation between the two subjects (P1 and P2) does not always hold. Given that in some situation S1 this relation is P1 > P2, it is inferred by qal vachomer that in second situation S2 the relation is similar (i.e. again P1 > P2); however, the latter conclusion can be refuted by showing that there is a third situation S3 where the relation between the subjects is known to be the reverse, i.e. where P1 < P2; in that event, the previously proposed conclusion concerning S2 is put in doubt. This process is in the Talmud called pirkha a-iqara dedina, meaning refutation directed at the starting point of the reasoning. Abitbol proposes the following tabular representation of this process:
Abitbol refers to an actual example, drawn from Chullin 114a. As can be seen (reading vertically), we cannot be sure that in S2, given that P1 is forbidden it follows that P2 is forbidden, as the state of affairs in S1 implies (since there P1 is permitted and P2 is forbidden), because there is a conflicting tendency implied by S3 (wherein P1 is forbidden and P2 is permitted). In other words, if we had started with situation S3 instead of S1 as our model, our conclusion concerning situation S2 would have been the opposite (from chomer to qal instead of from qal to chomer). This understanding cannot be gainsaid; but it cannot be said to be accurately represented. Drawing a table with six cells does not explain why the initial argument ceases to function; it would have been more instructive to draw two tables of four cells for the same subjects P1 and P2, one with columns S1 and S2 and the other with columns S3 and S2 (in that order).
What has to be stressed is that the new column S3 serves to deny generality to the first column S1, so that the content of S2 (i.e. the value of P2 to be inferred from that of P1 in this situation) becomes uncertain. This cannot be expressed within this tabular format, but only through full disclosure of the form of a fortiori argument. We have to recognize that the relation of P2 to P1 in S1 serves, after generalization, as the argument’s major premise, from which together with the minor premise concerning P1 in S2, the putative conclusion concerning P2 in S2 is drawn. The refutation with reference to the opposite relation of P2 to P1 in S3 serves to deny validity to the generalization, i.e. to particularize the major premise – in which case, the putative conclusion can no longer be drawn. Furthermore, notice that there is no mention anywhere here of the operative middle term.
The second process of refutation is called in the Talmud yokhiah mimaqom acher, meaning refutation from another place, and is represented by Abitbol by means of the following table (please note that he, no doubt accidentally, here uses the + and – signs with the opposite senses, but I have for the sake of consistency stuck to the previous senses):
This table was constructed with reference to another Talmudic example, drawn from Qiddushin 7b. Here (reading the table vertically), the initial argument is that since in situation S1 subject P1 is forbidden while subject P2 is permitted, it would follow qal vachomer that in situation S2, where subject P1 is permitted, that subject P2 should be permitted; however, this cannot be, since in situation S1 subject P3 is permitted and in situation S2 subject P3 is forbidden, which means that if we similarly argued that since, under S1, P1 is forbidden whereas P3 is permitted, it would follow that, under S2, where P1 is permitted, P3 should be permitted, contrary to the given fact that it is forbidden. Clearly, here again, it would have been more accurate to illustrate the problem through two tables, instead of a single larger table.
Now, what is the exact nature of this second refutation? This can be better understood if we express the arguments involved in standard format, using some middle term (say, R). Clearly, just as it is true that given that P1 is more R than P2 and P1 is not R enough to be forbidden, it logically follows that P2 is not R enough to be forbidden, it would be equally sound to say that given that P1 is more R than P3 and P1 is not R enough to be forbidden, it logically follows that P3 is not R enough to be forbidden. Yet P3 is known to be forbidden! Therefore, there must be something wrong in one or both preceding a fortiori arguments (which are, notice, negative subjectal in form, going from major to minor). Granting that the middle term (R) is constant throughout, either the major premise or the minor premise or both must be wrong in at least one of the arguments.
We cannot offhand tell precisely what is wrong, but the refutation serves its purpose just by putting the original line of thinking in doubt. That is why in other accounts of the doctrine of refutation we find the first sort of refutation labeled “refutation of a premise” and the second sort labeled “refutation of a conclusion.” The difference between them is that the first sort constitutes a direct attack on the major premise, whereas the second sort constitutes an indirect attack (since it puts one or both of the premises in doubt simply by denying their joint conclusion). It should be said that Abitbol, although he does refer to the first sort as “the most direct process” of refutation, does not explain precisely how the second sort functions. His tabular representation is somewhat illustrative, but it is not explicative.
Finally, Abitbol describes – again by means of examples and tables – how such refutations might occasionally be neutralized by counter-objections. Judging by one example he gives, this is sometimes achieved by pointing out that something claimed to be permitted in a certain context is in fact not permitted in it, or that something claimed to be forbidden in a certain context is in fact not forbidden in it. In other words, the material truth of part or all of one or both of the premises may be challenged. There may be a counter-objection, and the initial premise may be reinstated, perhaps with slightly modified content. But all this, note well, is essentially not of formal significance – it has to do with the content. The rabbis are trying, through debate, to establish what premises to collectively adopt before drawing a final conclusion; the argument per se comes after.
It is at this point that Abitbol shows his clear understanding that the argument depends on a “hierarchy” between two subjects, and that the same subjects may have opposite positions within different hierarchies. He uses this understanding to explain the process of rabbinic give and take, rightly identifying the positions of disputants at successive stages as “hypothetical.” Each rabbi’s position is relative to some information that he has taken into consideration, but another rabbi is free to bring to bear additional information that changes the initial judgment. For this reason, he believes, qal vachomer arguments are never final, and always potentially capable of refutation. However, he is amiss in not making a clear distinction between the form and the content of argumentation, which is essential for truly logical analysis.
Clearly, Abitbol is not a formal logician. His study of Talmudic logic in general, and the qal vachomer in particular, contains interesting information and reflections, but it is not based on deep study of universal logic.
As we have seen, his tabular approach to a fortiori argument is of limited value, because it does not and cannot distinguish a fortiori argument from analogical argument. He fails to highlight all the factors involved in qal vachomer reasoning, notably the middle term and the idea of a threshold value of it as determining the possibility of drawing a conclusion, not to mention the prior formation of the major premise. Furthermore, his effectively lumping together of material and formal issues makes him wrongly view the argument as “only probable and subject to contestation,” even if he goes on to declare that despite its “inconveniences” it has “a certain logical security that guarantees the validity of its conclusion.”
In his opinion, what distinguishes the qal vachomer argument from other rabbinic hermeneutic techniques is that it calls on any interpreter resorting to it to be extra careful in its elaboration so as to preempt all possible challenges such as those above described. This requires broad knowledge of the Torah, subtlety, attention to detail and nuances, rigor, a critical spirit. It is these controlling factors in the background of rabbinic discourse that give the final conclusion of such argument its logical credibility and legal value. There is, of course, much truth in that reflection, but it is not the whole story.
I will not here examine the rest of Abitbol’s book, even though it looks interesting. What I can at a glance see, however, is that his treatment of the other rabbinic hermeneutic principles is even less formal than his treatment of qal vachomer. He works essentially through traditional examples and rabbinic descriptions and rules, even if he goes into considerable detail in an effort to clarify the principles. Of course, the principles cannot all be formalized, or at least not to the same degree. But some can in fact be expressed in formal terms, i.e. using symbols like X and Y in place of terms or theses, and thereby shown to be logically valid or invalid as the case may be. I have definitely shown that to be possible in my book Judaic Logic, which however appeared a couple of years after the book by Abitbol here discussed. Nevertheless, to repeat, I think this book merits further attention.
 Paris: Edition des Sciences Hébraïques, 1993.
 Translations from French into English are my own, throughout the present chapter.
 The word “premise” etymologically refers that which comes before, while the word “conclusion” refers to that which comes after. The same terminological error is found in other accounts, notably that of Louis Jacobs in his article on Hermeneutics in the Encyclopaedia Judaica (1971-2). Note also that Abitbol in this context makes an error of inattention: while rightly describing the major-to-minor process as “inference from a grave prescription to one that is less so,” he wrongly describes the minor-to-major process as “passage from an important prescription to one that is less so.” I mention this in passing, but it is unimportant since as is clear from his other statements he knows the correct reading.
 New York, 1971 (publisher not specified), p. 3.
 He gives a couple of examples of those, drawn from the set of ten or so found in the Torah – namely, Gen. 44:8, Ex. 6:12 and Dt. 31:27.
 “Kareth” refers to a Divine punishment, known in English as “cutting off.”
 Here Abitbol cites R. Aaron Ibn Chaim’s Midot Aharon (Amsterdam, 1742) as asserting that comparison and hierarchization are the foundations of all qal vachomer.
 Note that Abitbol is not the first (or last) to have opted for a tabular representation. As we have seen in the previous chapter, he was preceded in this by Michael Avraham (chapter 20), at least. We find similar tables used later, in the joint work of Abraham, Gabbay and Schild (chapter 25) and in that of Andrew Schumann (chapter 27).
 Actually, Abitbol does not consistently stick to these definitions. Thus, in a later example (p. 138) he has + designating death sentence and – the opposite, i.e. + as the more severe sentence and – as the less so. Also, in the second argument of R. Tarfon, the symbols P1 and P2 change places with the symbols S1 and S2 (p. 145).
 I could have added the special case of a pari (egalitarian) a fortiori argument, in which if A = B, then C = D – or if A = C, then B = D. Abitbol does not here mention this.
 I have not found these exact words on that page, but I presume this is an interpretation of the sentence: “If when thy hearth is closed, the hearth of the Master is open, how much the more must the hearth of thy Master be open when thy hearth is open.”
 I have called this process traduction. See chapter 3.5 above. As I show there, this is merely verbal change – the argument’s inherent form as predicatal is in fact unaffected.
 There is no point my trying to do the job for him, since I do not think his tabular representation method is of much use, anyway.
 See pp. 137-9, and further details on pp. 148-51. Note that the Sages eventually reject this argument, judging that if the accused is already dead the false witnesses need not be executed, based on the Torah (Deut. 19:18-19) saying “ye do unto him [any false witness], as he had purposed to do unto his brother” instead of “as he had done.” I have changed Abitbol’s wording a bit in the course of translation, only so as to briefly clarify the context. This argument is only alluded to in the Gemara (“…is there not an argument a fortiori?”), but the Soncino Talmud has a footnote making it explicit as follows: “If zomemim are put to death when their plot failed, it is surely all the more necessary that they should be where their plot had succeeded!” The word zomemim refers to false witnesses.
 Note that Abitbol has a + sign to denote “death sentence” and a – sign for “the contrary” (i.e. not death sentence). This is, incidentally, inconsistent with his earlier convention, according to which the qal (i.e. not death) should be a +, while the chomer (i.e. death) should be a –. This is not, of course, a big deal – except that it again shows some disorderly thinking on his part.
 If read vertically, it runs as follows: since as regards the status of the accused, if the witnesses are discredited before his execution he is not liable to capital punishment, whereas if they are discredited after his execution he is not liable to capital punishment, it follows as regards the status of the witnesses, where witnesses discredited before their victim’s execution are liable to capital punishment, that witnesses discredited after his execution are likewise liable to capital punishment.
 Note in passing that Abitbol wrongly interprets the Sages’ dayo here as “reducing the payment by half” (p. 144). This is not strictly correct. Although in this particular case the payment advocated by the Sages is half that advocated by R. Tarfon, this proportion is mere happenstance. What the Sages are really saying is that the payment should remain unchanged. Clearly, if the original payment was, say, a third of full payment, R. Tarfon would have still concluded with full payment and the Sages would have concluded with a third (not a half). Abitbol is obviously influenced here by the Gemara’s approach to a fortiori argument.
 As Abitbol points out (p. 146), the conclusion could have been “more than the full payment,” and thence “has to be at least full.” He surprisingly does not go into the question why R. Tarfon concludes with full payment and not proportionately “more than full.” The reason is, of course, that no punitive indemnity is intended, only restitution of financial losses.
 Since he explicitly says (p. 145) that the “analogy” can be “either between the hierarchies defined by the situations, or between those defined by the subjects.” However, when he adds “which makes possible the formulation of the same qal vachomer in two ways,” it is clear that he considers both arguments as one and the same qal vachomer, whose terms are merely reshuffled. The latter perspective is not correct – the second argument differs more significantly from the first than Abitbol implies, due in fact to the different generalization preceding the major premise in each case.
 The symbols should stick to the formal functions, not to the contents.
 All he says about it is that “the conclusion of this qal vachomer was not upheld by the Sages due to the restrictive principle of dayo.”
 He does, further on, allude to this issue in his account of the Talmudic doctrine of refutations of qal vachomer; but he does not allude to it in the present context, in relation to the two arguments of R. Tarfon.
 This goes to show, incidentally, how exclusive or excessive reliance on gadgets (like tables, graphics or symbols) can blind and mislead the researcher instead of enlightening and guiding him.
 Namely the authors of Halikhot Olam, Beit Ha-otsar and Magid Taalumot. The first says: “since qal vachomer is pure reasoning, it may happen sometimes that a man unintentionally makes an error by formulating a faulty qal vachomer.” The second says: “one may fear an error of reasoning which would allow for its refutation.” The third says: “the interpreter can make a mistake in the logical reasoning and as a result his judgment will be open to refutation without his being aware of it.” None of these statements affirms that qal vachomer is inherently refutable, but only that people might well err in formulating such argument (as indeed any argument).
 This is forcefully brought home on p. 156, where Abitbol refers to the two qal vachomer arguments of R. Tarfon, and then points out that the conclusion of “more than half” of the damages could mean “the totality or more than the totality.” From an analogical perspective, his comment is reasonable for both arguments. But from an a fortiori perspective, while his comment is reasonable for the first argument, which infers a crescendo from half to full compensation, it cannot apply to the second argument, which infers purely a fortiori from full to full compensation.
 “This conclusion however logical it be is not retained by the jurisprudence due to application to the qal vachomer of the principle ain onshin…” (p. 139).
 Although, as we saw earlier, he distinguishes between “limit[ing]” and “annull[ing]” the conclusion of a qal vachomer, still he identifies the dayo principle with the axiom that “the conclusion must not surpass the premises,” which concerns limiting rather than annulling.
 In the case of capital punishment, one cannot kill a person twice, but one could additionally torture him first or disrespect his body afterwards or harm his family or seize his property. In the case of flagellation, one could conceivably increase the number of blows or the way they are delivered.
 I wonder whether it is true throughout the Torah that every sanction is preceded by a warning. This seems to go against the rabbinic principle that the Torah is not necessarily chronologically ordered.
 Abitbol says of the starting point that it is “at P1,” but surely he means ‘at S1’, i.e. at the relation between subjects P1 and P2 in situation S1 (a reference to the major premise). Unless he means ‘at P1 for S2’ (a reference to the minor premise), which would be inaccurate (in most if not all cases).
 Which I will not repeat here, for brevity’s sake. Note that I have placed the terms in the table as he has. But looking at the original argument, it seems to me that the table should have been rotated (i.e. the terms called P1, P2 should have been S1, S2, and conversely; as for S3, it should have been called P3). And indeed, Abitbol reads his table horizontally (see top of p. 159). But there is no need for us to get into this issue here, as it has already been dealt with earlier.
 Which I will not repeat here, for brevity’s sake. Note that I have placed the terms in the table as he has. But looking at the original argument, it seems to me that the table should have been rotated (i.e. the terms called S1, S2 should have been P1, P2, and conversely; as for P3, it should have been called S3). Abitbol seems to read his table horizontally (see p. 159-60). But there is no need for us to get into this issue here, as it has already been dealt with earlier.
 See the discussion of this topic in the chapter on Mielziner (13.4).
 Which we have drawn attention to previously (21.2).
 In particular, note that Abitbol does not attempt to formalize the binyan av principle (3rd in R. Ishmael’s list of 13), or the harmonization principles (numbered 8-11, 13), which are the most susceptible to formal treatment besides qal vachomer (the 1st principle).