A FORTIORI LOGIC
CHAPTER 31 – Various other commentaries
The following shorter assessments are ordered chronologically. They are comparatively brief, either because I lack the information needed to say more about the author’s work, or because not much need be said concerning it. It should be said that this listing is no doubt far, far from exhaustive. Just as I seemed close to finishing the present book, I happened to type “a fortiori” in the search box of Google Books, and to my great surprise discovered that there are about 2.2 million books with this phrase in them! Of course, in many books the expression is used and not discussed; but surely also, in many something is said about it.
H. S. Hirschfeld is the author of two books, called Halachische Exegese (1840) and Hagadische Exegese (1847), dealing as their names imply with halakhic and haggadic exegesis. I have not read these books, but Samely mentions some of the contents of the first, and gives us some idea of Hirschfeld’s views on the a fortiori argument. Thus, he informs us: “Hirschfeld … was perhaps the first explorer of the qal wa-homer to attempt a symbolic representation of its logic. The formula he devised, for one main version of the argument, is ‘A – a = A + x :: B + a = B + x.’”
I can only guess what this symbolic formula is about, not having seen the context. Perhaps more information about it is given in Hirschfeld’s book. But, quite offhand, it seems to be saying: “If A without (or with less) a, has x, then surely B with (or with more) a, has x”. Hirschfeld’s representation of a fortiori argument thus seems to be a forerunner of Louis Jacob’s ‘complex’ a fortiori argument “If A, which lacks y, has x, then surely B, which has y, has x.” Whether the latter based his formulation on the former’s, or came to it independently, I do not know. In any case, this means that my criticisms of Jacob’s idea can equally be applied to Hirschfeld’s. In the last analysis, what we are offered in both cases is a description of the surface appearance of some a fortiori discourse, and not an explanation or validation of such reasoning. We are not told what the presumed relation is between A and B; nor how ‘A without a’ can tell us something about ‘B with a’.
Consider an example: ‘If a man without much money can find a wife, then surely a man with much money can do so.’ The reason this argument sounds credible is because we consider that money is an advantage that most women would appreciate having. But an equally cogent argument of the same form could be formulated with opposite effect. ‘If a man without children can find a wife, then surely a man with children can do so.’ The reason this argument does not sound very credible is because we consider that children (from a previous relationship) are an added burden that most women would rather do without. Thus, the credibility of the argument depends on a tacit additional premise. It is not enough to state a minor premise and conclusion; we also have to specify the major premise. In these examples, the major premise would tell us which attribute of the man is more attractive to the potential wife. Having money is more attractive than lacking it – so the first argument works; it is minor to major. Having children is less attractive than lacking them – so the second argument is unconvincing; it is major to minor. Thus, though both arguments are positive subjectal in form, they impact differently on the rational faculty due to unspoken differences.
It follows that a formula like Hirschfeld’s – or like Jacobs’ – cannot suffice as a representation of a fortiori argument. Note moreover that the argument form they propose is just one (albeit the most common one) among many. They do not mention negative subjectal, or positive or negative predicatal, or the implicational moods. They mention purely a fortiori argument, but fail to mention a crescendo argument. It is a pity that neither of these researchers found out about the much broader and more accurate representations by Moshe Chaim Luzzatto (mid-18th century).
I was inspired to start writing logic in my late teens after reading Introduction to Logic by Horace William Brindley Joseph. His long, free-wheeling discussions of general issues and particular points made the subject very interesting to me. I had already by then read some of Aristotle’s work, but his dry, more technical approach did not have the same effect on me. Naturally, then, today, when I looked for past commentaries on the a fortiori argument, I looked into Joseph’s book. Here is all that he says:
“… there are certainly forms which have not been examined. For example, there is the a fortiori argument. ‘He that loveth not his brother whom he hath seen’, asks St. John, ‘how can he love God whom he hath not seen?’” (Pp. 369-370.)
This is not much, but it tells us a couple of things. First, that H. W. B. Joseph was immediately aware that the a fortiori argument is not some sort of syllogism, but a distinct type of argument that had not yet been adequately examined (in 1916). Secondly, he gives us just one example of it, drawn from the Christian Bible, uncharacteristically without any attempt at analysis. Obviously, his statement was intended as a final note, outside the scope of his book.
The sample argument is indeed a valid a fortiori, as my formalization of it below shows. Specifically, it is a negative predicatal mood, whose implicit middle (R) would be a term like ‘capacity for love (heart)’ or ‘spirituality’, say:
More spirituality (R) is required to love someone unseen (P) than to love someone seen (Q).
Yet, he (some man) is not spiritual (R) enough to love his brother, whom he hath seen (Q).
All the more (or equally), he is not spiritual (R) enough to love God, whom he hath not seen (P).
Moshe Ostrovsky wrote a study of the rabbinical hermeneutic principles, called in Hebrew Hamidot She-haTora Nidreshet Bahem (meaning in English: The Rules that the Torah Requires), which was published in Jerusalem in 1924/5. I have not read the book, but some of its content relating to qal vachomer (a fortiori argument) are reported and discussed by Wiseman. The latter translates Ostrovsky’s description of the argument as follows: “The name Qal VaChomer applies only to teach what is a special judgement between two things that bear a graded difference, by means of which we judge what applies from one to the other”. As Wiseman puts it, this is “a minimally adequate definition.”
However, Ostrovsky gets more specific, and even waxes symbolic, when he analyzes the Mishnaic controversy between R. Tarfon and the Sages (i.e. Baba Qama 2:5). It is interesting to note that Ostrovsky views this controversy as one “between reason and the majority view,” where R. Tarfon upheld “ordinary reasoning” which the Sages overruled with reference to “tradition”; I would partly but not wholly agree with this understanding. According to Wiseman, Ostrovsky’s approach is “based partly upon the prior work of Schwarz;” although, looking at the information Wiseman gives, I do not see exactly how Ostrovsky is rooted in Schwarz, if by the latter we take to mean a reference to syllogistic interpretation of a fortiori argument.
Ostrovsky’s symbolic representation of the a fortiori argument is as follows:
A – a + b
(item A has less of something in case ‘a’ and more of it in case ‘b’);
B + a
(item B has a certain amount of it in case ‘a’);
B + b
(item B has a greater amount of it in case ‘b’).
This formula (the explanations in brackets are mine) is used to express the two examples given in the above mentioned Mishna (i.e. the two arguments of R. Tarfon), namely (the wording is mine, and the explanations in brackets, in terms of P, Q, R, S, are mine):
· If tooth & foot damage (A) engenders a lesser fine (less R; 0%) on public grounds (‘a’=Q) and a greater fine (more R; 100%) on private grounds (‘b’=P), and horn damage (B) on public grounds (‘a’=Q) engenders a certain fine (less R; 50%), then horn damage (B) on private grounds (‘b’=P) engenders a greater fine (more R; 100%) than that.
· Again, if on public grounds (A) tooth & foot damage Q) engenders a lesser fine (less R; 0%) and horn damage (‘b’=P) engenders a greater fine (more R; 50%), and on private grounds (B) tooth & foot damage (‘a’=Q) engenders a certain fine (less R; 100%), then on private grounds (B) horn damage (‘b’=P) engenders a greater fine (more R; 100%) than that.
I should first point out that, strictly speaking, in Ostrovsky’s symbolic formula, the first premise seems to say that A has less ‘a’ and more ‘b’ (less/more, relatively to each other); the second premise seems to say that B has more ‘a’ rather than as I put it ‘a certain amount of’ (more here being apparently in comparison to the ‘a’ of A, since 50>0 or 100>0), and the conclusion seems to say that B has more ‘b’ (more here being in comparison to the ‘a’ of B, since 100>50 or 100“>”100; and maybe also in comparison to ‘b’ of A, since 100“>”100 or 100>50). From the vagueness and inaccuracy of these details alone, we can see that this symbolic formula contains uncertainties and confusions.
We may now, for purposes of clarification, compare Ostrovsky’s symbolic formula for the Mishnaic illustrations to our standard form for them. For a start, we have to reason that since item A has less of a certain thing (R) in case ‘a’ and more of it in case ‘b’, then the same may (by generalization) be said for all items, including B; whence:
For all items like A, case ‘b’ (P) implies more of something (R) than case ‘a’ (Q).
So, for item B, if case ‘a’ (Q) involves R enough for a certain result (S),
it follows that case ‘b’ (Q) involves R enough for the same result (S).
Comparing this standard form (positive subjectal, from minor to major) to Ostrovsky’s symbolic formula, we see clearly that two terms are missing in it, viz. the middle (R) and subsidiary (S) terms. In his verbal description of the argument, he does have the middle term as “fine,” but he does not give this term a symbol in his abstract formula – meaning that he does not realize its central role in the thought process. Furthermore, he lacks the idea of a threshold value of R, i.e. that enough of R gives access to a certain result (S) and without that amount of R no access is possible. Also, the result (S) is not mentioned in his symbolic representation – i.e. it is not stated that the resulting “fine” is the same in the minor premise and conclusion. On the contrary, the conclusion seems to be that the “fine” is bound to be greater in the conclusion than it was in the premise. This suggests that Ostrovsky, like R. Tarfon, conceived the a fortiori argument here as a crescendo. However, he offers no additional premise justifying such ‘proportionality’. Rather, he seems to have mentally conflated the middle and subsidiary terms, and reasoned directly from the given that ‘b’ is greater than ‘a’ (as regards item A) to the conclusion ‘b’ from the given ‘a’ (as regards item B).
We can thus surmise that he conceives the argument as essentially analogical. His formula contains four terms (A, B, ‘a’ and ‘b’), as appropriate for mere analogy (if for A, b>a, then similarly for B, b>a). These four terms are not those of a fortiori argument (only two of them, viz. ‘a’ and ‘b’, are used in the a fortiori proper – the other two, viz. A and B, being external conditions). He shows no awareness that a generalization (from item A to all items, including B) precedes the a fortiori argument. His first premise seems to play the role of major premise, even though in fact it is only a given instance from which the effective major premise is obtained by generalization. The second premise (B + a), which serves as minor premise, seems to say that B has or involves more ‘a’ – whereas it should say that B has or involves a certain amount of the middle term in case ‘a’; and the conclusion (B + b) seems to say that B has or involves more ‘b’ – whereas it should say that B has or involves a certain amount of the middle term in case ‘b’, which is at least equal to (rather than necessarily greater than, as he seems to imply) the amount in case ‘a’.
Moreover, his symbolic formula is structured in such a way as to give the wrong impression that A and B are subjects, while ‘a’ and ‘b’ are predicates (or possibly, A and B are antecedents and ‘a’ and ‘b’ are consequents). If this were so, i.e. if B was the subject of the minor premise and conclusion (rather than ‘a’ and ‘b’, respectively), then the argument would be positive predicatal (rather than subjectal) – but if this were true, then it could not validly go from minor to major (since positive predicatal a fortiori argument is only valid from major to minor). Clearly, the argument must be structured as we have above proposed in our standard form rendering, with A and B as external conditions, and ‘a’ and ‘b’ as subjects. And for this, two additional terms must be introduced (viz. R for the middle term and S for the subsidiary). So, it cannot be said that Ostrovsky understood a fortiori argument.
Clearly, his symbolic formula is not even adequate as representation of the ‘if–then–’ givens, let alone as an explanation as to why the conclusion follows from the premises. The peculiar form of a fortiori argument is nowhere made explicit. There is no proposition in the formula that clarifies why what it tells us about item A is at all relevant to item B, and may be used as a source of inference; i.e. the major premise is missing. Moreover, if the inference is considered as deductive rather than merely inductive (as indeed seems to be the case in the examples given from the Mishna), there ought to be a credible validation procedure – which there is not. Ostrovsky’s attempt at formalization is thus only superficially descriptive, lacking depth and justification.
Furthermore, although (as already mentioned) Ostrovsky rightly presents the difference between R. Tarfon and the Sages as one between rational argument and majority ruling, he does not explain how a convention (majority ruling) can conceivably overrule an apparent law of logic (rational argument). Moreover, his presentation does not bring out the significant difference between R. Tarfon’s two arguments. Both his arguments yield a proportional conclusion, but whereas his first conclusion can be interdicted by the Sages by means of a rival purely a fortiori argument, his second argument yields the same conclusion whether it is read as a crescendo or purely a fortiori and therefore cannot be so easily interdicted. Because the greater cunning of R. Tarfon’s second argument compared to his first is not brought out, the important difference between the Sages’ two dayo (“it is sufficient”) objections cannot be made manifest.
Consequently, Ostrovsky (judging from Wiseman’s presentation) has to resort to non-formal explanations of the controversy. The Sages’ dayo principle is then made reasonable as a reflection of the uncertainties that may arise in judgment. As Wiseman puts it: “Inasmuch as a judge may not have adequate evidence to determine the truth of the counter claims, he… operates more by a general rule of likelihood than a theoretical construct;” meaning that “the same level of fine as that given (the dayo) is the norm, barring any other testimony that can be trusted to alter that judgement. It is practical (legal) reasoning rather than a formal, theoretical logic”.
Note again that (as far as can be seen from Wiseman’s account) Ostrovsky’s seems to view a fortiori argument as exclusively a crescendo, since he does not anticipate purely a fortiori argument. Note also that, though he presumably intended his analysis as traditional in orientation, he did not address the more tortuous interpretation of the Mishna proposed by the corresponding Gemara (Baba Qama 25a-b). With regard to formalities, too, we should note the absence of distinction between implicational and copulative a fortiori argument, between subjectal and predicatal moods, and between positive and negative moods. Ostrovsky’s presentation is (judging from what Wiseman reports) solely positive subjectal.
In conclusion, Ostrovsky did not formally solve the problem of a fortiori argument.
An outstanding definition of a fortiori argument that I have come across is that proposed in 1926 by Pierre André Lalande, perhaps with other authors, in his Vocabulaire technique et critique de la philosophie, viz.:
“Inference from one quantity to another quantity of similar nature, larger or smaller, and such that the first cannot be reached or passed without the second being [reached or passed] also.”
This definition is very accurate, even if not perfect. The starting clause, “inference from one quantity to another quantity of similar nature, larger or smaller,” refers to then major premise, which relates two terms – a “larger” (the major, P) and a “smaller” (the minor, Q) – which have a “similar nature,” i.e. a comparable property in common (the middle term, R). The next clause, “such that the first cannot be reached or passed without the second being [reached or passed] also,” refers to the minor premise and conclusion, and clearly explains that the inference depends on a threshold that once “reached or passed” in the one case (the larger or the smaller quantity) is necessarily reached or passed in the other case.
Note well the ontical reference to quantities (rather than epistemic reference to convictions), and the clear reference to the passing of a threshold as the explanation and justification of the inference. The realization that the passing of the threshold by one quantity necessitates that by the other is truly exceptional in its clarity of vision. Few commentators have come close to this degree of understanding of the argument; Lalande must have really thought about this a lot. Nevertheless, his definition is not perfect, for many reasons.
Most important, Lalande’s definition does not clearly specify that there is an implied threshold that is to be reached or passed, which is a certain value of the common property (which we have called the middle term, R) of the major and minor terms (P, Q) – which is specified in the major premise as their “similar nature.” The way he has formulated it, the middle term is left tacit in the minor premise and conclusion, giving the impression that it is the quantities of the major and minor terms themselves that are reached or passed, rather than a certain quantity of their common factor. Thus, he has the notion of “sufficiency” in his definition (implied by the words “cannot be reached or passed without”), but he lacks a precise notion regarding sufficient of what. He has, effectively (in the subjectal mood): “if Q is enough, then P is enough” (or vice versa), instead of the more accurate: “if Q is R enough, then P is R enough” (or vice versa).
Furthermore, notably absent in Lalande’s formulation is mention of the subsidiary term (S). He effectively says: “if Q is enough, then P is enough” (or vice versa), but he does not say for what. There is no predication. This may be due to an attempt to cover in the same statement both subjectal and predicatal argument; but he does not make that clear. It appears that his definition intends both directions of reasoning, since he apparently conceives the argument as proceeding from the larger to the smaller quantity (from major to minor) or vice versa (from minor to major), and at the same time only mentions a positive minor premise and positive conclusion (reaching or passing). If so, his definition is deficient in not mentioning negative subjectal and negative predicatal arguments (i.e. arguments involving not reaching or passing).
In truth, if what he had in mind in his above definition was positive subjectal and positive predicatal arguments, he would have seen fit to mention and emphasize the structural difference between them, and not left such an important matter tacit. More likely, what he had in mind was only positive and negative subjectal argument, and he did not realize that only the movement from minor to major is positive while that from major to minor is negative. It is also apparent from his definition that he only had in mind copulative forms of a fortiori argument, and did not become aware of the more complex implicational forms. Moreover, since he does not mention, let alone discuss, the subsidiary term, he could not have reflected on the difference between pure a fortiori argument and a crescendo argument (i.e. proportional a fortiori argument).
Note that he also describes a fortiori argument as “an enthymeme that assumes a premise like the following: ‘Who can do the more can do the less’.” He is here referring to the Latin legal rule: “Non debet, cui plus licet, quod minus est non licere,” which is better translated as: “one who is [logically] permitted to do the greater, is all the more [logically] permitted to do the lesser.” This is, of course, a more specific definition of a fortiori argument than the one discussed earlier, since it is limited to legal or eventually ethical contexts. Here, we might also (by contraposition, or more precisely by reductio ad absurdum) argue that “one who is not permitted to do the lesser, is all the more not permitted do the greater”. Note that these formulae, the positive and negative, are predicatal in form; and that they go, respectively, from major to minor and from minor to major.
Clearly, Lalande should have realized that there are four (or even eight) moods of a fortiori argument, and not just two as his main definition seems to imply. Nevertheless, his definition is way ahead of those of many other commentators, and highly to be praised.
Wiseman occasionally refers David Daube, author of the celebrated paper “Rabbinic Methods of Interpretation and Hellenistic Rhetoric” (1947), to buttress his own positions, although he does not agree with him entirely. A Wikipedia post describes Daube very enthusiastically, as “the twentieth century’s preeminent scholar of ancient law,” who “combined a familiarity with many legal systems, particularly Roman law and biblical law, with an expertise in Greek, Roman, Jewish, and Christian literature.”
According to Wiseman, Daube “suggests that much (if not all) of Hillel’s seven interpretative rules might have arisen from Hellenistic rhetoric, not just logic, as well as from its Roman incorporation in jurisprudence.” He considers Daube as having “overrate[d] Greek and Roman rhetorical influences on Jewish rules and on the QC particularly,” reminding that a fortiori argument (which he abbreviates as QC) appears in the Jewish Bible long before; however, he conceives it as possible that the orderly arrangement of the hermeneutic rules in lists may have been influenced by Hellenistic models. I agree with Wiseman, here.
Regarding qal vachomer, Wiseman informs us that Daube stated that at least two clearly proportional cases display “the methodological elaboration of law and theology by means of the norm a minori ad maius.” The two cases referred to are Matthew 12:10 and Romans 5:8 Daube, reports Wiseman, considered these two examples to be “legitimate, in form similar to Roman, juridical reasoning.” Indeed, Daube considered that, whereas the Jewish Bible uses of qal vachomer are “popular,” the New Testament (NT) cases – i.e. the two just cited, presumably – are “technical,” meaning “academic, ‘Halakhic’ applications of Hillel’s first rule of exegesis.”
Wiseman tells us all that in order to pit the authority of Daube against that of Maccoby, since he concludes: “by Daube‘s reckoning, since these QC’s express proper usage, Maccoby‘s critical view of the NT writers’ misunderstanding of the QC is largely empty or rebutted.” As for the case given by Paul that Maccoby dismisses as sheer “rhetoric,” Daube calls it “[n]o less significant.”
When I read this information about Daube I was, to put it mildly, nonplussed. Having earlier in the present study closely examined the a fortiori arguments in the NT, including the argument by Jesus in Matthew 12:10, and the four arguments by Paul in Romans 5:10, 17 and 11:15, 24 that Maccoby mentions, I can say with certainty that to claim these NT arguments as technically faultless, and indeed more skillful than the samples found in the Jewish Bible, is exaggeration if not deliberate disinformation. As for Romans 5:8, I do not see any a fortiori intent in it at all (but perhaps 5:10 is meant).
Note Wiseman’s characterization of these cases as “clearly proportional;” after which he expresses disappointment that Daube “does not address the dayo limit directly.” It is indeed surprising that Daube does not mention the distinction between proportional and non-proportional a fortiori argument, because it is very present in various ways in Mishna and Talmud, even though apparently unknown in ancient non-Jewish sources. If Daube was as thoroughly acquainted with both literary cultures as he is reputed to have been, he would surely have noticed that.
As regards the issue of proportionality, I agree with Wiseman that the a fortiori argument in Matthew 12:10 is intended as proportional. The one in Romans 5:10 (if that is the argument referred to as 5:8 by Wiseman) is perhaps intended as proportional, but being formally rather mixed up this cannot be said with certainty (the underlying a fortiori argument may be non-proportional). In any event – so what? How would adducing these NT arguments disprove Maccoby’s claim that a fortiori argument cannot be proportional? Even if a thousand examples of proportional such argument were brought to bear it would prove nothing, since Maccoby is not contending that people do not so argue in practice but that logically people should not so argue.
It is therefore clear that Wiseman is attempting here to lean on Daube’s prestige; this is argument by authority. Daube may have been an expert in many things, but he obviously did not understand logic, or at least a fortiori logic. It looks to me as if Daube was here intent on devaluating Judaism, and uplifting Christianity, for whatever motive. His attempt to thoroughly subsume Judaic hermeneutics under Hellenistic rhetoric is likewise suspect. He seems not to have been an impartial scholar, but someone with a partisan agenda. Thus, Wiseman can hardly rely on him to defeat Maccoby’s thesis. The latter admittedly had his own prejudices, but invoking Daube’s views is not the way to neutralize him.
Additionally, Wiseman’s attempt to oppose Maccoby by appealing to the concept of rhetoric defended by Daube is confused. ‘Rhetoric’, properly understood, is a wide concept that includes all human discourse, whether logical (formally validated) or illogical (whether non-sequitur or antinomic), aimed at convincing others of something for whatever purpose. In a narrower sense, when opposed to logic, ‘rhetoric’ refers to illogical discourse (i.e. sophistry). Some people cannot tell the difference. Some people, like Daube when he (according to Wiseman) “calls all the hermeneutical rules of the Rabbis rhetorical,” seem to try to blur the difference. Maccoby, for his part, takes the hard line that non-proportional a fortiori argument (which he and Wiseman both equate with the dayo principle) is logical, whereas proportional argument is rhetorical in the sense of illogical.
Wiseman attempts to discredit Maccoby’s stance by means of Daube’s fuzzy concept. He reproaches Maccoby for not admitting that proportional a fortiori argument has at least rhetorical value. Here, he relies on the vague generic connotation, which is partly positive. However, Maccoby insistence that proportional a fortiori argument is illogical does not deny it to be rhetoric in the negative sense, but only in the positive sense. So Maccoby does not deny it to be rhetoric in the all-inclusive sense, and Wiseman is here indulging in fallacious reasoning. His additional reproach that Maccoby by dismissing proportional a fortiori argument offhand is himself engaging in rhetoric in the negative sense (i.e. illogic) is fair enough; but it does not cancel out Wiseman’s own resort here to illicit rhetoric.
Anyhow, rhetoric is not the issue; the issue is logic. The questions to ask are obvious and essential. Can non-proportional and proportional a fortiori arguments respectively be formally validated? Or under what conditions can they be validated? And if so, exactly how? As Wiseman rightly points out, Maccoby does not formally validate his claims. As for Wiseman, he cannot formally validate his counter-claims because, as we have seen, his definitions of these arguments are too vague. For this reason, he is forced to resort to authorities and fuzzy concepts.
I often referred to R. Meir Zvi Bergman’s 1985 work Gateway to the Talmud when I wrote my Judaic Logic, having found him both scrupulously traditional and relatively clear in his expositions of the rabbinic hermeneutic principles. I did not at the time know of the work of Mielziner or other authors, so it would be fair to say that without Bergman I would not have progressed as far as I did at that time. In Judaic Logic, although I react somewhat critically to quite a few rabbinic doctrines relayed and defended by Bergman, my focus is not principally on critique. In the present brief exposé, I will examine his approach more critically, especially with regard to a fortiori argument.
Bergman devotes a few pages (pp. 121-129) to kal vachomer, the first of the thirteen “rules of Biblical exegesis.” He defines it as follows: “In this interpretation, inference of a stringency is drawn from a lenient case (kal) to a strict one (chomer), and inference of a leniency is drawn from a strict case to a lenient one.” This is of course the traditional rabbinic formula for such argument, which does not in fact cover all sorts of a fortiori arguments although it covers the main ones of interest to rabbis. Bergman has no theory of a fortiori argument to speak of other than that statement. By way of validation all he does is say: “the logic is compelling;” and he effectively repeats the same definition with the words “all the more… should” and “certainly… should” thrown in for emphasis. This is, needless to say, not formal proof, but mere declaration of intimate conviction.
Bergman then says: “Ten examples of this rule are explicitly mentioned in the Torah,” and proceeds to list the ten examples cited by R. Ishmael in the Midrash Bereshit Rabbah (without mentioning the two more cited separately in that volume, namely Gen. 4:24 and 17:20-21). Note that he says “in the Torah,” whereas he should have said “in the Tanakh” (i.e. the whole Jewish Bible), since only four of the ten examples he lists are from the Torah proper (i.e. the Pentateuch). His mention of only ten examples suggests that he is unaware that there are many more instances scattered throughout Jewish Scripture. To be precise, according to my most recent listing there are 46 instances. In any event, listing examples is not a viable substitute for formal treatment. The examples may strike us as reasonable, but we still need formalization and validation to be sure.
Also, the mere fact that an argument is used in the Torah is not proof of its validity. Its validity must still be established by formal logic. It can be said, within the framework of Jewish faith, that the Torah is the authoritative source of information from which our premises are to be formulated; but we cannot refer to the Torah to evaluate a process of reasoning per se. The Torah may well provide the content of Jewish discourse; but the form of valid discourse is something universal and independent of any religious faith. The process of moving from premises to conclusion has nothing to do with religious faith, but depends on the science of logic which is based on the five laws of thought – the three laws of identity, non-contradiction, exclusion of the middle, and the principles of induction and deduction. The Torah could not credibly deny these fundamental laws.
Bergman does ask the question: what determines whether a case is stringent or lenient. But his answer that it may be either “logic” or “the laws of the Torah that pertain to the case” is much too vague. He gives an example for each of these possibilities, but these examples do not clarify his meaning. In truth, the distinction between the major and minor items (terms or theses) depends in each particular argument on a tacit or explicit major premise, which specifies which item is which in relation to a specific middle term. This major premise might be known through rational insight and/or empirical observation, or it might be given in the Torah and/or be decided by the rabbis – but it must in any case be acknowledged as underlying the argumentative process. Bergman, not having to begin with analyzed the components of a fortiori argument, makes no mention of this major premise. All that his said initial formula includes are the minor premise and conclusion, and those without explicit reference to a middle term.
Bergman of course details the traditional doctrine of “refuting a kal vachomer” (pircha), but here again he is handicapped by his lack of formal tools, so he does not distinguish between challenging the content of an argument and challenging its form. He distinguishes, as traditionally done, between “challenging the origin” and “challenging the conclusion;” but he does not realize that these “refutations” are concerned with the content of a given argument rather than with its form. The kal vachomer argument as such is never “undermined” – all that is put in doubt is the information it contains. This may occur directly, by pointing out that the proposed major premise (“the origin”) is overly general, since there are cases where it evidently does not hold; or it may occur indirectly, by pointing out that the conclusion drawn is not in accord with certain known cases (which is a sort of reductio ad impossibile which calls for revision of a premise, usually the major though in some cases the minor); but in any event, this concerns the matter at hand not the logical process used.
As regards the dayo principle, Bergman says that “the full force of kal vachomer argument is qualified by” it. This is an inaccurate view, again due to absence of preliminary formal analysis. A fortiori argument, as a deductive process, is in fact unaffected by the rabbinical dayo principle; all the latter does is prevent the material formation of its major premise or of its additional premise about proportionality. Bergman also errs in following the Gemara’s claim, developed in Baba Qama 25a, that the dayo principle is “derived from Scripture (Num. 12:14).” According to this view, a fortiori argument is necessarily ‘proportional’; yet many arguments in the Tanakh, the Mishna and indeed in other parts of the Gemara and in later rabbinic literature are not ‘proportional’, so this is an absurd notion.
Bergman does acknowledge that the dayo principle may be “applied” in two ways – either, as he puts it, “on the origin” or “on the conclusion” of the kal vachomer. However, his explanations of these applications are not very clearly worded. He says that the first application refers to “the ‘lenient’ case,” whereas the second refers to “the ‘stricter’ case.” He does correctly say, regarding the Mishna Baba Qama 2:5, that the dayo objection by the Sages against R. Tarfon’s first argument relates to “the conclusion,” whereas the dayo objection by the Sages against R. Tarfon’s second argument relates to “the origin” – but his explanations are not technically correct. He does not show awareness of the ‘proportionality’ involved in the first argument and the generalization involved in the second, which truly explain the difference between the two ways of dayo objection. Bergman does not mention that this distinction is not found in the Gemara, but apparently emerged much later, probably in the medieval period thanks to some Tosafist.
Note lastly that Bergman uncritically accepts the Gemara’s invented debate between the Sages and R. Tarfon regarding the conditions of application of the dayo principle. This is of course to be expected from such a thoroughly traditional commentator. But in fact this Gemara can be subjected to scathing logical criticism.
Bergman’s approach throughout his study is decidedly uncritical. He recites traditional positions as if they are indubitable. This means that whatever he says must be received with extreme caution. If a commentator would not, even if he found some serious difficulty in the texts he analyzes, ever dare to question the correctness of a rabbinical pronouncement or an implied idea of theirs, his overall reliability is surely in doubt. His attitude may prove his great piety and love of the Sages, but it is not reassuring as regards objectivity and scientific accuracy. Whatever the difficulty, he will consciously or unconsciously do his best to ignore it or gloss over it, without respect for facts. There is a big difference between apologetics and critical study.
This section concerns the book Introduction to the Talmud and Midrash by H. L. Strack and G. Stemberger. Hermann L. Strack (Germany, 1848-1922) was a Protestant theologian and an authority on Talmudic and rabbinic literature. He authored, among many other works, a book called Einleitung in den Talmud & Midrasch, which was (it seems) first published in Leipzig in 1887, and then went through many editions. The fifth edition was translated into English and published by the Jewish Publication Society in Philadelphia in 1931. I gather that the nominal ‘co-author’, Guenter Stemberger (Austrian, still living), arrived on the scene long after and considerably ‘updated’ Strack’s work in successive editions. The first edition with his signature (in German) seems to have been in 1982. The first English translation of that seems to have been in 1991. According to Google Books, Gunter Stemberger later, together with Markus Bockmuehl, produced a revised edition of the latter in 1996. The edition I refer to here is a French translation and adaptation dated 1986.
Chapter 3, on rabbinic hermeneutics, presents the seven rules of Hillel, the thirteen of R. Ishmael, and the thirty-two of R. Eliezer ben Yose haGelili, in a standard manner, with some history and some examples. Qal vachomer is briefly described as “a reasoning that goes from the simple to the complex and vice versa” (my translation). There is no attempt at a more precise or formal definition, or to describe its varieties or workings, let alone at validation. It is not clear whether by “simple” and “complex” the author refers to the major and minor terms, or to two values of the subsidiary term. In other words, the issue of ‘proportionality’, i.e. the distinction between purely a fortiori argument and a crescendo argument, is not treated here. However, since he adds “and vice versa,” it is implied that he has at least vaguely become aware of the distinction between positive and negative subjectal argument, and/or between subjectal and predicatal argument.
With regard to history, it is suggested that Hillel, R. Ishmael and R. Eliezer probably did not author or compile their respective lists, but these were later attributed to them. Stemberger (obviously him, since this is very recent stuff) refers uncritically to the theories of Daube, Lieberman and Porton that we have examined (above and below) and found somewhat doubtful. One interesting (for me) item of information gleaned here is that the thirteen midot of R. Ishmael were believed by many doctors of Jewish law to be of Sinaitic origin as of the Middle Ages. They mention the Midrash haGadol, an anonymous work thought to date from the 14th century, as introducing the subject (in a comment on Exodus 21:1) as follows: “Rabbi Ishmael said: Here are the thirteen midot used for exegesis of the Torah, which were transmitted to Moses on Mount Sinai” (my translation and italics). However, there is a doubt as to whether this remark is as old as the work, because it was not included in fragments found in the Cairo ‘geniza’ (for this reason it has been excluded from some more recent editions).
Meir Brachfeld is the author of a paper entitled “A Formal Analysis of Kal VaChomer and Tsad Hashaveh,” published in 1992 in the Higayon journal (vol. 2, pp. 47-55). I have a copy of this article, but it is in Hebrew, and my Hebrew is not fluent enough, so I shall here again refer to Wiseman, who offers in A Contemporary Examination of the A Fortiori Argument Involving Jewish Traditions (Appendix E.3, pp. 242-5) a pretty thorough English translation of it, as well as his own discussion of it (on pp. 87-9). The article’s English abstract in Higayon is as follows:
“A section of ‘Halichot Olam’ [a rabbinical commentary] dealing with the mechanics of the principles of ‘Kal VaChomer’ and ‘Tsad Hashaveh’ is translated into the language of set theory. In this language, the principles can be concisely and precisely defined. Most interestingly, it emerges that a central concept underlying these principles is that of ‘irrelevance’ and that this concept is naturally defined using recursion.”
What is immediately evident, looking at the article, is that it takes a fortiori argument for granted without analyzing its form or validating it. Thus, there is no specific theory of a fortiori argument as such in it.
Brachfeld’s opening formula for the argument merely tells us that, under appropriate conditions, information (a din, a law or legal decision) is inferred from one context (the melamed, that which teaches) to another, similar context (the lamad, that which is taught), without specifying exactly how the process occurs. His formula is vague enough to be applicable to any sort of argument. He does not try and find the precise mechanics of a fortiori inference. The purpose of his article is not to analyze the argument per se, but rather to describe the ways it is used and challenged in halakhic (Jewish law) discourse. That is, he only proposes a sort of flow-chart, in the language of set theory, of typical rabbinic arguments and counterarguments in relation to it. This is of course interesting stuff, but not our main topic of interest in the present work.
I have, anyway, already analyzed this topic in some detail in the chapter devoted to Mielziner (13.4). Briefly put, the rabbis conceive of two ways that a qal vachomer may be “refuted.” It may be attacked by way of a premise or by way of the conclusion. However, their understanding of what is technically going on in such refutation is not entirely accurate. Both attacks in fact challenge a premise – the former rather directly, the latter rather indirectly. This means that the premise under attack is shown to be deficient in some way; usually, it is found not as general in scope as it was initially made out to be. When the conclusion is under attack, it is not the process of a fortiori inference that is being challenged, but a premise. Very rarely is the process of a fortiori inference itself challenged; and when it is, this simply shows that a particular argument was improperly formulated in some way, not that a fortiori argument in general is open to doubt.
The significance of these findings is that the belief by some rabbinical commentators that qal vachomer is not a deductive, always reliable, argument is based on incorrect analysis of the data they base their judgment on. Unfortunately, Brachfeld seems to take such erroneous assumptions for granted to a large extent. He is not, however, totally uncritical, since he points out that the traditional view creates certain logical problems. As Wiseman puts it, the author of ‘Halichot Olam’, is “also aware” of these problems, but “offers a mere excuse that just seems to brush aside the real issue, by delivering the comment of Rashi… that the rules [of interpretation, including qal vachomer] are handed down to us from Sinai.”
According to rabbinic tradition, Hillel I the Elder (Babylon, 110 BCE – Israel, 10 CE) drew up a list of seven midot (techniques of Biblical interpretation); later, R. Ishmael ben Elisha (Israel, ca. 50-135 CE) expanded this list to thirteen. It is generally considered that Hillel had learned these exegetic rules or principles, or at least the practices they represent, from his teachers Shemaya and Abtalion. Further developments in this field occurred through Nahum of Gimzo and R. Akiva, on the one hand, and by R. Ishmael, on the other hand. The latter’s list was identical to Hillel’s in some respects (namely, qal vachomer and gezerah shavah), but differed by merging certain rules, splitting others up, modifying still others, and adding at least one (namely the last).
In an essay called “Rabbinic Midrash” (1995), Gary Porton urges us not to (like Daube and Lieberman did) “take the existence, use, and attribution of the exegetical rules to specific sages as fact,” and not to “accept the rabbinic texts’ data as historically accurate and valid.” He informs us that in an earlier paper of his, “Rabbi Ishmael and His Thirteen Middot”, he raised “serious questions about Ishmael’s relationship to the thirteen exegetical rules attributed to him at the opening of Sifra and to the seven principles assigned to Hillel in several rabbinic documents.” More specifically, he informs us that:
· There are three divergent versions of Hillel’s so-called ‘list of 7 rules’: two having 8 rules and one having only 6 rules; and only 5 rules are found in common to them. (Porton does not here specify the core rules and the additional rules.)
· R. Ishmael’s so-called ‘list of 13 rules’ contains 16 rules, written “in four distinct literary styles.” (Porton seems to suggest here that the list had four different authors or compilers.)
· “We have no record of Hillel’s ever using any of the exegetical techniques attributed to him.” (Porton does not here say whether Hillel used other techniques or never publicly engaged in exegesis.)
· R. Ishmael uses only 8 of the 16 principles in the ‘list of 13’ attributed to him, and additionally 17 techniques “which do not appear in the list.” (Porton does not here specify which 8 principles were used and what the 17 unlisted techniques consisted of.)
· R. “Ishmael’s exegetical activity as recorded in the rabbinic corpus” is limited “with two exceptions… [to] exegetical principles which the two lists have in common.” (Porton does not here tell us which principles he considers as in common to both lists, and which not.)
I have quoted and paraphrased him at length because I find this historical thesis very interesting; not only because of its novel negative conclusions, but especially because of the methodology it presumably was based them on – namely, systematic empirical research throughout the Talmud.
The conclusions as such are not too worrisome (to my mind, at least): they can probably be explained away. Perhaps Hillel did list seven rules, but their transmission was faulty. Some of R. Ishmael’s thirteen rules might well have been lumped together from a longer list. Similarly, the lack of evidence of actual use by Hillel of any of the rules listed by him, or by R. Ishmael of half of the rules listed by him, can be explained by saying that their intent was not to describe their own personal practices, but the practices of their colleagues in general. Moreover, that R. Ishmael may have (with two exceptions) used only rules listed by Hillel is not too problematic, since after all his list is traditionally presented as a derivative of Hillel’s. As for the use of techniques by R. Ishmael that are not included in these two lists – well, that comes as no surprise: we have always known that these lists were incomplete, since for instance R. Akiva often used techniques not listed in them, and it is readily evident when studying the Talmud that reasoning processes are used in it which are not explicitly acknowledged, or which though named are not listed.
It should be added that speculation concerning the composition of the two lists is nothing new. I have also reported and made some suggestions of this sort in my Judaic Logic, with reference to R. Ishmael’s list. For a start, the various moods of qal vachomer could be counted as separate rules (this would make four or eight or more moods, according to how we go about it). Gezerah shavah may be based on verbal or material analogy (homonymy or synonymy). Hekesh and semukhim are early practices apparently not initially listed but later classified under that heading (i.e. as divisions of rule #2). Meinyano (Hillel’s rule #5) and misofo (new in R. Ishmael’s list) are two divisions of rule #12 (which in my opinion ought to be placed with or close to rule #2). Binyan av likewise has two varieties, mi katuv echad and mi shnei ketuvim (which were two rules #3 and #4 in Hillel’s list).
Rules #4-9, 11 may all be viewed as developments of Hillel’s #5 (which was itself two rules klal uphrat and prat ukhlal bunched together). To R. Ishmael’s rule #6, klal uphrat ukhlal, were apparently later added three other variants, viz. prat ukhlal uphrat, klal ukhlal uphrat, prat uphrat ukhlal. Rule #7 has two divisions, miklal hatsarikh liphrat and miprat hatsarikh likhlal. Rule #8, lelamed hadavar leshar haklal, is recognized as having two varieties (which I have named): the original lelamed oto hadavar, and an apparently later development, lelamed hefekh hadavar. Thus, R. Ishmael’s rules, though conventionally counted as 13 in number, are well known by tradition to include more than 13 exegetic devices (we could, based on the above enumeration, put the number as, say, 26), and Porton’s claim to have newly distinguished just 16 rules in them must be taxed as ingenuous and rather arbitrary!
Moreover, I am curious about the 17 techniques not on his list reportedly used by R. Ishmael. Are these techniques newly identified by Porton, or are they perhaps included in the list of 32 by R. Eliezer ben Yose haGelili or the list of 613 by the Malbim? It should be stressed that it is often very difficult to classify a given argument as falling under this or that heading. We have earlier seen, for example, how the arguments of R. Tarfon in Baba Qama 24b-25a could be construed as plain arguments by analogy rather than as a fortiori arguments. Porton’s claim that R. Ishmael’s practices can be classified in precisely 25 categories (8 listed by him and 17 not listed by him) sounds suspiciously confident, even if we assume he found just one case for each category.
Nevertheless, Porton’s thesis is intriguing, because he seems to be claiming that he has systematically researched the whole Talmud. We would expect him to at least scan through the Mishna, since both Hillel and R. Ishmael are Tannaic; but in truth, he would also have to look through the Gemara, and indeed other literature of the Talmudic period, since they might be mentioned there even if not mentioned in the Mishna. He has apparently searched for exegetic activity by Hillel and found none, or at least none included in Hillel’s list. And he has apparently examined all of R. Ishmael’s exegetic practices, and classified each instance under this or that rule, or under no listed rule, and then drawn the various conclusions mentioned above. If that is indeed the case, i.e. if Porton or anyone actually did this detailed research, he is to be highly commended for his empiricism and scientific method! This is just the sort of thorough effort needed.
I wonder, however, if this systematic effort actually did take place. I have not personally read his paper “Rabbi Ishmael and His Thirteen Middot” – which is presumably his research report – but only read his summary of findings in his later essay “Rabbinic Midrash.” However, I note that the earlier paper is only 15 pages long. That seems a bit slim, unless we assume that he did not publish all his findings (or maybe the data was scarce). For surely, to be scientific in his exposé, he would have to actually list all the arguments in the Talmud; or at least all those by R. Ishmael, since he claims there are none by Hillel, or at least none corresponding to any techniques on his list. It would not suffice for Porton to just give us a list of references; he would have to actually quote the passages referred to, so as to show us that Ishmael is indeed mentioned in them and so as to convince us that each passage was correctly identified as falling under this or that rule or under none of the rules.
Just how many arguments of each type in the Talmudic sources did Porton find and analyze and how many did he attribute to R. Ishmael? E.g. as regards qal vachomer specifically, how many occur in the Talmudic sources and how many of them was R. Ishmael the author of? He does not here give us even such basic statistics. Is the exhaustive research data perhaps made available somewhere for peer review? I do not know; I have found nothing posted to this effect on the Internet. Be that as it may, it is interesting to see someone at least thinking in terms of systematic empirical research. This is the sort of research I advocate. Traditional lists are interesting, but not enough.
My doubts as to the actuality or reliability of Porton’s alleged empirical research increased exponentially when, by coincidence, I came across a mention by Mielziner of Hillel applying, “among other arguments,” the “constructional” variant of the rule of gezerah shavah in Pessachim 66a. I looked up this Talmudic page, and sure enough Hillel is presented in the Gemara as having applied his second hermeneutic rule (and also his first, qal vachomer) to the issue at hand:
“They were told ‘There is a certain man who has come up from Babylonia, Hillel the Babylonian by name, who served the two greatest men of the time [Shemaiah and Abtalyon], and he knows whether the Passover overrides the Sabbath or not. [Thereupon] they summoned him [and] said to him, ‘Do you know whether the Passover overrides the Sabbath or not?’ … He answered them, ‘In its appointed time’ is stated in connection with the Passover, and ‘In its appointed time’ is stated in connection with the tamid [sacrifice]; just as ‘Its appointed time’ which is said in connection with the tamid overrides the Sabbath, so ‘Its appointed time’ which is said in connection with the Passover overrides the Sabbath. Moreover, it follows a minori [ad majus], if the tamid, [the omission of] which is not punished by kareth [cutting-off], overrides the Sabbath, then the Passover, [neglect of] which is punished by kareth, is it not logical that it overrides the Sabbath!”
We can infer from this error by Porton that his statistical claims must be received with the utmost caution. Maybe he looked throughout the Mishna only and did not scan the Gemara (let alone other literature of the period), forgetting that there are many statements in the latter too made in the name of the Tannaim. Many people, it seems, are tempted to make far out declarations so as to be noticed in the academic world; this motive may have played a role here.
R. Mordechai Torczyner is the creator since 1995 of www.webshas.org, a website “designed as a topical index to the Talmud.” This interesting resource includes a webpage devoted to “Talmudic methods of analyzing the Torah’s text,” including traditional hermeneutics, including kal vachomer argument. What he tries to do here is simply to list references for different aspects of kal vachomer use in Talmudic contexts.
Torczyner’s definition of kal vachomer as “learning more obvious lessons from less obvious lessons” expresses his pragmatic orientation. But of course, it offers no answer to the theoretical question as to precisely how “less obvious” lessons can teach us something about “more obvious” ones. It is just a vague statement, without logical explanation.
As regards the dayo principle, Torczyner defines it as follows: “A Kal VaChomer can’t teach a greater stringency from the lenient side to the stringent side, than the literal rule applicable to the stringent case (‘Dayo’): Bava Metzia 41b; Zevachim 43b-44a.” Surprisingly, he does not mention the Mishna Bava Kama 2:5 and/or the Gemara Bava Kama 25a-b! Yet these passages are crucial to rabbinical understanding of kal vachomer argument and the dayo principle.
Nevertheless, this is an interesting contribution, because of the references it gives. The following are some examples of the way information is presented:
“Deducing a Kal VaChomer on one’s own, without a received tradition: Niddah 19b.
A pasuk trumps a kal vachomer: Zevachim 3b….
The Torah writing out something which could have been learned from a Kal VeChomer: Pesachim 16b; Kiddushin 4a, 4b….
Punishment for a crime may not be deduced based on a Kal vaChomer (Ein Onshin min haDin): Makkot 5b, 14a, 17b; Temurah 9a; Keritot 2b-3a….”
Ron Villanova, in his work Legal Methods: A Guide For Paralegals And Law Students, defines a fortiori argument as follows: “This ‘with stronger reason’ argument implies a comparison of values;” it is “grounded on the common sense (and logical) convention that within the same category the greater includes the lesser (or, if you will, the stronger includes the weaker).” However, he warns us not to be misled by the word “includes” – by this term he intends “comparison,” as between the height of a taller man and that of a shorter one.
This definition is not very adequate, but it is improved somewhat by Villanova’s admonitions, such as that “the comparison should be one of factually like things and be factually meaningful,” and by the many legal examples he brings to bear. The comparative major premise of a fortiori argument is present in it; and it is understood that one cannot argue in any direction one pleases (e.g. negatively from a lesser predicate to a greater one). Also present are the idea of a middle term in it (implied by the words “within the same category”) and the idea that this same term must remain operative throughout the argument (as evident through examples).
However, the idea of a threshold of the middle term, as of which predication occurs, and before which it cannot, seems to be lacking here. This is evident in the author’s attempt to express a fortiori argument in the form of “conditional syllogism” (apodosis). He does this through examples, but we can do it for him more generally as follows:
If so and so is true of X, then (a fortiori) it is true of Y.
So and so is true of X.
(Therefore): it is true of Y.
Clearly, such presentation of a fortiori argument is simplistic. Just writing “a fortiori” in the consequent does not magically turn this argument into an a fortiori one! It does not tell us what a fortiori reasoning constitutes; it does not tell us just why and under what conditions the consequent follows logically from the antecedent.
No wonder then that Villanova sums up by claiming that “while a fortiori is a valid form argument, it is a form that is prone to weakness and, therefore, challenge.” He gives examples of possible error with reference to conditional syllogism; but these concern conditional syllogism generally, and are not specific to a fortiori argument. Without first clearly identifying the form of a fortiori argument – indeed its various forms – it is impossible to clearly identify the errors specific to such argument.
Giovanni Sartor, in Legal Reasoning: A Cognitive Approach to the Law, draws attention to some interesting forms of a fortiori reasoning. This is a large and interesting-looking treatise, but I will only here briefly look at its theory of a fortiori argument. I have here changed some of the terminology and symbols used by Sartor so as to simplify my analysis of his ideas as much as possible. Sartor considers a fortiori reasoning in general as heuristic and analogical, for reasons that shall become apparent.
“Factor-based” a fortiori reasoning compares the pros and cons for a certain decision. Sartor is thinking in terms of a “decision” because his concern is with legal reasoning, i.e. with judges deciding what judgment to apply in a “new case” given some “legislation, doctrine or precedents.” Suppose F1, F2 are reasons in favor of a certain decision D, and G1, G2 are reasons against it; then: given that the pro reason F1 outweighs the con reasons G1 and G2 taken together, enough to decide D, it follows that:
a) the pro reasons F1 and F2 taken together would outweigh the con reasons G1 and G2 taken together, enough to decide D (additive a fortiori), since F1 and F2 taken together are “more inclusive” than F1 alone;
b) the pro reason F1 would outweigh the con reason G1 taken alone, enough to decide D (subtractive a fortiori), since G1 is “less inclusive” than G1 and G2 taken together;
c) the pro reasons F1 and F2 would outweigh the con reason G1 taken alone, enough to decide D (bidirectional a fortiori).
We can put these three arguments in standard form for him, as follows (notice that the values of P, Q and S vary from one argument to the next, as appropriate):
a) Since F1 + F2 together (P) are more weighty (R) than F1 alone (Q), it follows that if F1 alone (Q) is weighty (R) enough against G1 + G2 to produce decision D (S), then F1 + F2 together (P) are weighty (R) enough against G1 + G2 to produce decision D (S) (positive subjectal, minor to major).
b) Since G1 + G2 together (P) are more weighty (R) than G1 alone (Q), it follows that if G1 + G2 together (P) are not weighty (R) enough against F1 to prevent decision D (S), then G1 alone (Q) is not weighty enough (R) against F1 to prevent decision D (S) (negative subjectal, major to minor).
c) Since F1 + F2 together (P) are more weighty (R) than F1 alone (Q), it follows that if F1 alone (Q) is weighty (R) enough against G1 to produce decision D (S) (as implied by conclusion (b)), then F1 + F2 together (P) are weighty (R) enough against G1 to produce decision D (S) (positive subjectal, minor to major).
d) Since G1 + G2 together (P) are more weighty (R) than G1 alone (Q), it follows that if G1 + G2 together (P) are not weighty (R) enough against F1 + F2 to prevent decision D (S) (as implied by conclusion (a)), then G1 alone (Q) is not weighty enough (R) against F1 + F2 to prevent decision D (S) (negative subjectal, major to minor).
Note that, though Sartor thinks of the “bidirectional” argument as one (c), it in fact has two forms ((c) and (d)), whose conclusions are of course essentially the same, though expressed in different directions (and therefore with opposite polarity); furthermore these two arguments depend on the preceding two ((a) and (b)). In all cases, notice, the operative middle term (R) is the “weightiness” of the factors concerned in producing or preventing a decision D in a given set of circumstances. The latter complex term is of course the effective subsidiary term (S). It is interesting that Sartor explicitly refers to the factors as “sufficient” or not for the decision; so he has grasped that crucial aspect of a fortiori argument.
Thus, Sartor’s “factor-based” a fortiori reasoning is valid a fortiori argument (of positive or negative subjectal forms). It is, however, just a special case of a fortiori argument, dealing with compounding or separation of terms (“factors”), assuming of course that such compounding or separation is in fact possible in a given case. By standardizing the reasoning we show its validity and also its ordinariness. Nevertheless, Sartor deserves credit for drawing attention to this particular sort of a fortiori reasoning, which is obviously valuable.
It should be pointed out, however, that Sartor considers the a fortiori conclusion to be “defeasible” because “there may be interference between the factors.” This may well happen; but if it were true in a particular case, we could not affirm the major premise in the first place. From this we see that Sartor is not fully conscious of the major premise, and of the fact that given that premise and the appropriate minor premise (and thus having the building blocks needed for a fortiori argument) the conclusion is logically inevitable, i.e. not “defeasible.” Of course, if we lack the needed premises, we do not have the means for engaging in a fortiori argument; but that would not constitute a weakness in a fortiori argument!
Moreover, Sartor conceives the above arguments as due to the relative inclusiveness or non-inclusiveness of the terms, i.e. (to use his exact words) as due to the fact that a compound is “at least as inclusive” as its components, and a component is “no more inclusive” than a compound with it. This is true specifically in “factor-based” reasoning; but of course it is not true in all a fortiori argument – so issues of inclusion should not be viewed as part of the essence of a fortiori argument.
But further on Sartor does enlarge his theory of a fortiori argument, stepping from “binary factors” to “dimensions (scalable factors).” He explains: “Categorising a situation as exemplifying or not certain binary factors is a superficial way of understanding how the features of that situation favour a certain outcome. In many cases, a binary categorisation results from transforming a deeper dimensional structure into a binary alternative.” He adds that dimensions may be “continuous” or “discrete.”
He then proposes a “reasoning schema” for “dimensional” a fortiori reasoning similar to that suggested above for (bidirectional) “factor-based” a fortiori reasoning. Briefly stated, he substitutes the notion of “dimensional strength” for what was previously regarded as a difference of “inclusiveness.” That is to say, instead of saying that one term is more or less “inclusive” than the other, he says that it is more or less “dimensionally strong.” This does the trick, but I have to remark (without going into the details here) that he unduly complicates matters in trying to convey this change of perspective.
Even so, the net result of his effort is pretty good compared to many other people’s attempts. His theory does acknowledge, albeit in a rather scattered way, the main formal components of a fortiori argument. There is no attempt at formal validation, but the argument is made comprehensible (albeit, to repeat, in a needlessly complicated manner). He does not seem to have come across my earlier work on the subject.
I have no doubt that there are many more studies relating to a fortiori argument besides those dealt with above. I can at this time list three more, two of which I came across surfing the Internet and the third I found mentioned in a publication.
The first is a book in German by Thomas K. Grabenhorst, Das argumentum a fortiori(1990). It is described by the publisher as “a pilot study using the practice of decision justification”. There is also a synopsis in German, which I translated very roughly using the Google facility. From it I gather that the author is claiming to have found a computer-assisted way to analyze a fortiori arguments used in legal contexts… or something of the sort. I cannot of course evaluate this claim without seeing an English translation of the book; but offhand I am skeptical. Why? Firstly, because I saw no mention in the abstract of preliminary formal work to determine the nature and varieties of a fortiori argument; and without such formal study, how could one program a computer to make correct assessments? Secondly, if even I, a human who has plenty of experience in formalizing material a fortiori arguments, must in each new case encountered make an intellectual effort to find the fitting form, how much more difficult would it be for a machine!
More recently, I came across a brief article by Tomasz Zurek, “Modelling of a fortiori reasoning” (2011). The abstract reads: “The paper presents the model of two variants of a fortiori reasoning applicable in the case of statutory law as well the example of the genuine law case, which has been modeled with use of established methodology. The model of reasoning assumes the existence of ‘less-more’ relation between the analyzed actions, which has been expressed by means of strict partial order and some additional assumptions. The paper also contains the implementation of the analyzed example.” I have not read this paper either, but mention it anyway because it cites the chapter on the formalities of a fortiori logic in my Judaic Logic, which is a good sign in its favor!
Thirdly, there is in a recent issue of the journal B. D. D. – Bekhol Derakhekha Daehu (No. 24, published in 2011) – an article in Hebrew by Avraham Lifshitz, called “Kal VaChomer and its Disprovings” (pp. 73-88). The English abstract is as follows:
“This article seeks to study the nature of the Kal VaChomer by examining those issues for which the G’mara disproves the Kal VaChomer in different ways.
The argument raised in the first part of the article is not only that the Kal Vachomer is not a necessary condition, but that it also not a sufficient condition. This is the conclusion drawn from the distinction between two disprovings: an inner disproving which undermines the relations of the Kal VaChomer, and an external disproving which demonstrates from an external data that the Kal VaChomer does not work. The invalidation of the Kal VaChomer as a sufficient condition stems from the ability to disprove it in an external fashion. Thus we have solved a severe Kushiya [difficulty] of the MaHaRil Diskin about the Tashbitu commandment.
In the second part of the article we raise the new insight that every internal disproving is in fact also an external disproving to the contradictory manner of the Kal Vachomer. By doing so we propose an answer to the Kushiya why the G’mara does not save Kal Vachomer from being disproved by means of its reversal.”
I have not read the article, but it seems clear from this description that its focus is specifically on Talmudic argumentation, and from its verbal expression that its approach is very traditional. When the author says that qal vachomer is neither a necessary nor a sufficient condition, he presumably means that such proof can be dispensed with and is in any case not decisive; I would be curious to see how he justifies those general claims. His distinction between inner and external “disproving” seems to refer to the commonly known distinction between invalidation of the process of inference and refutation (or at least putting-in-doubt) of a premise or two; I would have to see the article to judge his stated claims in that regard. The essay needs to be translated into English.
 In a footnote on p. 177, referring to Halachische Exegese (Berlin: Athenaeum, 1840), p. 227. Samely adds: “But the reduction of complex conceptual relationships with the qal wa-homer to the meaning of the arithmetic minus and plus offers no adequate analysis of the argument’s structure. Modern symbolic logic is certainly in a better position to capture the dynamics of the a fortiori argument, and Koch and Rüssmann, Juristische Begründungslehre, 259f, and Klug, Juristische Logik, 132-7, attempt to provide formulas.” Further on, in a foortnote on p. 179, Samely tells us that Hirschfeld “identifies the qal wa-homer as an inference from opposition (Gegensatz’, Halachische Exegese, 217f and 224).”
 Most probably the plus and minus signs signify presence and absence; but they might also signify increase and decrease (as stated in brackets). However, in the latter case, the plus sign has both meanings, since it is used with one meaning relative to ‘a’ and with the other relative to ‘x’.
 Needless to say I am not here trying to recommend to women any specific attitudes; the arguments here proposed are just intended as illustrations, as possible opinions.
 2nd ed. rev. Oxford: Clarendon, 1916. A full copy of this work may be downloaded at: ia700303.us.archive.org/8/items/introductiontolo00jose/introductiontolo00jose.pdf.
 England, 1867-1943.
 By K. I. Milman.
 In A Contemporary Examination of the A Fortiori Argument Involving Jewish Traditions (Waterloo, Ont.: University of Waterloo, 2010).
 Ostrovsky, p. 41; Wiseman, p. 30.
 Ostrovsky, p. 36; Wiseman, p. 73.
 Ostrovsky, p. 68; Wiseman, p. 73-4.
 Notice that this reading is essentially implicational, with antecedents ‘engendering’ consequents. We could also propose a copulative reading, by saying: ‘if the fine for tooth and & foot damage is lesser on public grounds and greater on private grounds; etc.” The difference lies in where we place the recurring word ‘fine’. However, I would say the implicational version is more accurate, because the terms ‘greater’ and ‘lesser’ are relative to each other and to the term ‘fine’.
 Wiseman, p. 74.
 France, 1867–1963.
 Paris: Alcan, 1926. Paris: PUF, 1972.
 My translation. I already called attention to this definition in my Judaic Logic.
 P. 32.
 Wiseman, in A Contemporary Examination of the A Fortiori Argument Involving Jewish Traditions, pp. 165-7. Daube’s paper in Collected Works of David Daube (Cambridge: Cambridge UP, 1947), pp. 333-355. See also my brief comment on this paper in the chapter on Saul Lieberman (15.2).
 Germany, 1909 – USA, 1999.
 Matthew 12:10 – “And a man was there with a withered hand. And they asked him, “Is it lawful to heal on the Sabbath?”—so that they might accuse him.” Romans 5:8 – “but God shows his love for us in that while we were still sinners, [Jesus] died for us.” (English Standard Version, 2001.)
 Daube was apparently of Jewish descent, but it is not stated whether he was in fact Jewish.
 New York: Mesorah, 1985.
 No doubt he skips these two cases so as to avoid the embarrassing question as to why they were not included in the list of ten. Louis Jacobs has suggested that this discrepancy is evidence that the Midrash has been augmented over time (see 16.4).
 And it nowhere explicitly denies them, although it does at times seem to de facto contravene one or the other of them, as when there is an apparent contradiction between two passages of the Torah, or between the Torah and certain empirical findings by the scientific method. Even the rabbis understand that inconsistencies are a serious problem, and they always try to explain them away.
 Bergman at one point says that a “refutation based on logic… cannot rebut a kal vachomer” (p. 124-5). But he is in fact discussing the kol ze assim argument proposed by a Tosafist, which is not really a formal issue but still a material one. I have dealt with that in chapter 9.7 and need not repeat my analysis here. Note that Bergman does not mention that this argument is from the Tosafot – because, I suspect, to him all rabbinic commentaries, from the Mishna to present day discourses, are timelessly Torah-given.
 See chapter 7.5 and compare it to Bergman’s presentation of this issue on pp. 128-9. I stress that there is no evidence of or call for any such debate in the Mishna; it is a gratuitous projection of the Gemara.
 Tr. and adapt. by Maurice-Ruben Hayoun (Paris: Cerf, 1986.)
 Ed. Moshe Koppel and Ely Merzbach. Jerusalem: Aluma, 1992.
 We can put the process in standard form for him, of course. In positive subjectal (or negative predicatal) a fortiori arguments, the melamed and the lamad are respectively the minor and major terms (Q and P), the similarity between them is the middle term (R), and the din is the subsidiary term (S). In negative subjectal (or positive predicatal) arguments, the melamed and the lamad are respectively the major and minor terms (P and Q). But Brachfeld himself shows no awareness of the actual form(s) of a fortiori argument.
 Nevertheless, we can question Brachfeld’s implicit assumption that he can fully sort out the technical possibilities of counterargument without first understanding the nature of the argument itself. See the next paragraph.
 Obviously, if a speaker uses an invalid mood, it does not follow that all moods of a fortiori argument are invalid. This is true of any form of argument, not just a fortiori.
 The Jewish Encyclopedia gives as references: for Hillel’s rules, Tosef., Sanh. vii; the introduction to the Sifra, ed. Weiss, p. 3a, end; Ab. R. N. xxxvii; and for R. Ishmael’s rules, the Baraita of R. Ishmael (intro. to Sifra). Some say Hillel composed or at least compiled the list; others that he merely expounded it (implying that it existed before him).
 See Mielziner, pp. 123-128. Nahum of Gimzo was opposed by R. Nehunia ben Hakana, “who insisted upon retaining only the rules of Hillel.” See also Neusner: Rabbinic Literature: An Essential Guide, pp. 56-63.
 In: Judaism in Late Antiquity. Ed. Jacob Neusner. Leiden, Netherlands: Brill, 1995. Pp. 217-238. The relevant part of this essay (p. 226) may be read online at: books.google.com/books?id=azrc9jU4YaUC&pg=PA270&dq=Saul+Lieberman+“Rabbinic+Interpretation+of+Scripture”+in+Hellenism+in+Jewish+Palestine&hl=en&ei=5vxgTd6SM9H44Aba7rCkCg&sa=X&oi=book_result&ct=result&resnum=5&ved=0CDsQ6AEwBDgU#v=onepage&q=Ishmael%27s and Hillel%27s Exegetical Techniques&f=false.
 In: New Perspectives on Ancient Judaism Volume One: Religion, Literature, and Society in Ancient Israel Formative Christianity and Judaism. Ed. J. Neusner, P. Borgen, E. S. Frerichs, and R. Horsley. Lanham: University Press of America, 1987. Pp. 3-18.
 I have for instance a text before me, the introduction by Joël Muller (1897) to the Oeuvres Complètes de R. Saadia ben Iosef al-Fayyoumi (vol. 9), which recounts various ideas on this topic. For example, one Aharon ben Chayim suggested that the three last rules of R. Ishmael (based on verbal differences) may be later additions, and that the rules numbered 3, 4 and 6 should each be counted as two. Also mentioned are differences in the order of listing of the rules; for instance, Saadia Gaon places rule 9 in 5th place.
 Mielziner moreover distinguishes exegetical, constructional and exorbitant varieties of gezerah shavah (pp. 143-150). Had I studied Mielziner’s work before writing my Judaic Logic, I would surely have integrated his insights and classifications into it.
 Mielziner lists semukhim (juxtaposition) as an “additional rule,” which “has some similarity to Heckesh,” and which “was probably introduced by R. Akiba;” he distinguishes two kinds of semukhim (pp. 177-179). In my Judaic Logic, I advocated lumping together all arguments by analogy under the loose heading of gezerah shavah. This was just a convenience measure, as R. Ishmael’s list of thirteen midot is considered post-facto as symbolically representative of all rabbinic hermeneutics, even though R. Akiva’s rules were originally competitive.
 Mah matsinu is counted by some commentators as a third variety of binyan av. Mielziner considers that incorrect, explaining that this refers to analogy from one case to a single similar case, whereas binyan av refers to generalization from one or more special provisions to all similar cases (p. 159); and thus he seems to think of mah matsinu as closer to gezerah shavah (p. 142). In any case, mah matsinu has to be fitted in somewhere in the list of thirteen midot, if it is to be regarded as exhaustive.
 See Mielziner, p. 127. R. Akiva’s alternative set of rules ribui umiut might also be listed and counted in this context, even though they were originally a competing viewpoint, if we regard R, Ishmael’s list as having become the symbolic representation of all rabbinic hermeneutics.See Mielziner, pp. 182-5.
 Mielziner only mentions only two variants: prat ukhlal ukhlal and klal ukhlal uphrat (p. 168).
 See Bergman (I do not have his book at hand right now, but I do quote him on this in my Judaic Logic, chapter 11). Bergman refers to these two variants as the particular teaching “about itself as well as the general law,” or (not about itself but) “only about the general law”. Examples are there given. Mielziner makes no mention of this variation.
 Including the Tosefta, Mechilta, Sifra, Sifre, Baraitot, etc. I should add that, in my opinion, it is imperative to always specify the exact documentary source of each instance of hermeneutic practice found; I cannot imagine that the same credence be given to a statement attributed to a Tanna found in the Mishna, and one found in the Gemara (which is comparatively hearsay evidence), or in some other document of the Talmudic period or later. Obviously, there are degrees of credibility in proportion to the distances in time and place.
 Most people would not go to the source text and check firsthand every case mentioned in a list of cases.
 From the Soncino Talmud (with minor adaptations).
 See www.webshas.org/torah/alpeh/midos.htm. While other topics dealt with there deserve further attention, we shall here only consider the treatment of “kal vachomer” (I use his spelling of the term). Torczyner is apparently American.
 Coral Springs: Llumina, 1999. Large parts of this volume can be consulted online at Google Books: books.google.ch/books?id=_mhnDRnoaOcC&printsec=frontcover#v=onepage&q&f=false.
 This is volume 5 of A Treatise of Legal Philosophy and General Jurisprudence. Chief ed. Enrico Pattaro (Dordrecht, Netherlands: Springer, 2005). Large parts of this volume (but not all of it) can be consulted online at Google Books: books.google.ch/books?id=udmrNHt4wB8C&printsec=frontcover#v=onepage&q&f=false.
 See chapter 8, pp. 221-240. I cannot fully analyze this work, because I do not have the whole text, and because I discovered it rather late, when my book was almost finished.
 Frankfurt am Main: Peter Lang, 1990.
 Proceedings of the 13th International Conference on Artificial Intelligence and Law (New York: ACM, 2011), pp. 96-100.