Original writings by Avi Sion on the theory and practice of inductive and deductive LOGIC  

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© Avi Sion

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© Avi Sion, 1995. All rights reserved.

Chapter 10.        THE THIRTEEN  MIDOT  (I).

            In the present and next two chapters, we shall indulge in a closer scrutiny and frank criticism of Talmudic/Rabbinic hermeneutics. In this first part of our analysis of the Thirteen Midot of R. Ishmael, we shall deal with Rules 1-7 and 12.


1.   Exposition and Evaluation.

2.   Inference of Information.

3.   Scope of Terms.


1.         Exposition and Evaluation.


            Traditional presentations of the principles and practise of Rabbinic exegesis consist in listing the Thirteen Midot of R. Ishmael (at least, though other techniques may be mentioned, in contrast or additionally), describing roughly how they work, and illustrating them by means of examples found in the Talmud or other authoritative literature.[1]

            Such an approach is inadequate, first of all, because the theoretical definitions of the rules are usually too vague for practical utility, and for purposes of clear distinction between similar rules. A simple test of practicality and clarity would be the following: if well defined, the rules should provide any intelligent person with a foolproof procedure, so that given the same database as the Rabbis, he or she would obtain the same conclusions as they did. The second important inadequacy in the traditional approach is the near total absence of evaluation; there are no validation procedures, no reductions to accepted standards of reasoning. There is no denying the genius of R. Ishmael and others like him, in their ability to abstract rules of intellectual behaviour from the observation of their own and their colleagues' thought-processes in various situations. Nevertheless, as we shall see, their failure to treat information systematically and their lack of logical tools, yielded imperfect results.

            We shall here propose some original ways to expose and evaluate Rabbinic hermeneutics (mainly, the 13 Midot). The most important step in our method is formalization; this means, substituting variable-symbols (like 'X' and 'Y') for terms or theses of propositions[2]. Formalizing an argument, note, means: formalizing all explicit and tacit premises and conclusions. The value of this measure is that it helps us to clarify the situations concerned, the Rabbinical responses to them, and the issues these raise. By this means, we move from a level akin to arithmetic, to one more like algebra. When we deal in symbols, we reduce immensely the possibility of warped judgement, due to personal attachment to some solution; all problems can be treated objectively. It should be said that logical formalization is not always the most appropriate tool at our disposal; in some cases, epistemological and/or ontological analyses are more valuable.

·             We have two sets of data to thus formalize, or analyze in some manner: (a) the theoretical pronouncements of Rabbis (defining or explaining the rules, or guiding their utilization), and (b) the practical examples they give in support (illustrating or applying their statements). This work allows us to compare, and if need be contrast, Rabbinic theory and practise. As we shall see, they do not always match.

·             Another utility of formalization or similar processes, is the possibility it gives us for comparing Rabbinic conclusions to the conclusions obtained by syllogism or other such established logical techniques. This is the ultimate goal of our study, to determine without prejudice whether or to what extent Rabbinic hermeneutics comply with deductive and inductive logic. As we shall see, they do not always parallel the course taken or recommended by ordinary logic.

            In anticipation of such divergences, it is important to study the Rabbinic hermeneutic principles carefully, and distinguish between their natural factors and their artificial factors. The natural aspects are immediately credible to, and capable of formal validation by, ordinary human logic, and thus belong to secular epistemology. The artificial aspects, for which Rabbis claim traditional and ultimately Divine sanction, are controversial and require very close examination, for purposes of evaluation or at least explanation. Our task with regard to such additives is to consider whether the rationales for them offered by the Rabbis are logical and convincing, or whether these factors ought to be regarded as human inventions and errors.[3]

            We shall in the rest of this chapter, and in the next, deal with the 13 rules of R. Ishmael under three large headings. "Inferences of information" - including rules 1-3, and 12, i.e. qal vachomer (a-fortiori argument), gezerah shavah (inference by analogy), heqesh, semukhim, meinyano, misofo (contextual inferences), and binyan av (causal inference). Then "scope of terms" - including rules 4-7, referred to collectively as klalim uphratim (genera and species). Finally, "harmonization" - including rules 8-11, and 13, about which much will be said.

            It should be clear that we have no intention, here, of masking any difficulties, but propose to engage in a "warts and all" exposé. The technicalities may be found hard-going by many people, but both secular and religious scholars, who endure through the ordeal, will be richly rewarded. They will find, not only an independent audit of Rabbinic hermeneutics, but a methodological demonstration of universal value. By the latter remark, I mean that the same method of exposition (by formalization) and evaluation (with reference to formal logic) can be applied to other movements of thought in Judaism, or outside it, in other religions or other domains (philosophy, politics, or whatever).


2.         Inference of Information.


            We shall first consider the exegetic rules whose purpose is essentially to infer new information from passages of Scripture, rather than to elucidate or harmonize the text (the division is, admittedly, to some extent arbitrary). Included here are both deductive and inductive processes, of varying degrees of formality and certainty.


         We have treated qal vachomer (R. Ishmael's Rule No. 1) in considerable detail already, and need only here remind of certain details. This refers to a natural thought-process, a-fortiori inference, the most deductive form of Rabbinic argument. The Rabbis of the Talmud and those which followed them, although they had an exceptionally well-developed understanding of this form of argument, did not have a complete understanding of it, such as one might expect in the event of Divine revelation. Their knowledge of it was not formal; they did not clearly distinguish inductive and deductive stages of reasoning; they misconstrued certain applications of the dayo principle[4]; and they erroneously counted the number of a-fortiori examples in the Tanakh.


               It ought to be remarked that R. Ishmael's formulation, just 'qal vachomer', is very brief - at best a heading; he does not define the processes involved. The distinction between miqal lechomer and michomer leqal is not given in the list of Thirteen Midot; I do not know whether it is explicitly found in the Talmud or only in later literature. To what extent were the Talmudic and post-Talmudic Rabbis aware of the difference between positive and negative a-fortiori; did they ever note the distinction between copulative and implicational forms of the argument, did they use the secondary forms; at what point in history were the more complex Rabbinic formulations that we find in contemporary literature developed: these are all questions I ask myself, but have not researched the answer to. Historians of logic have still much work to do.


            With regard to the legitimacy of the use of a-fortiori argument. We validated four (or eight) primary moods, namely copulatives, subjectal or predicatal, positive or negative (and implicationals, antecedental or consequental, positive or negative) and a number of derivative secondary moods. Since the process has naturally valid moods, it follows that if these moods are used properly, no formal objection to their use in contexts not sanctioned by tradition is possible. Tradition can only restrict their use with reference to the inductive preliminaries (as we discussed under the heading of objections); but with reference to the purely deductive aspects, no Rabbinic legislation is possible[5]. It would be like trying to conveniently exempt oneself from the obligation of honesty or consistency!

            The same freedom of thought must be acknowledged for all other purely deductive processes (or stages of processes), such as opposition, eduction, categorical or conditional syllogism, production, apodosis, and so forth. Any Rabbinic restrictions in such areas would be tantamount to an advocacy of antinomy, and cannot be tolerated. Rabbinic interference, on the grounds of some special Divine dispensation delivered at the Sinai revelation and transmitted by oral tradition, can only conceivably be applied to inductive processes; that is, with regard to situations which allow for more than one possible answers to a question, it is conceivable that there be a Divine decree as to which answer to favour in some specified situation(s) or all situations. However, we must keep in mind that the conceivability of such powers does not constitute proof that they exist in fact; it only makes logically possible a claim but does not justify it; and furthermore, that any controversy surrounding such powers throw doubt on their legitimacy.


         The technique of gezerah shavah (Rule No. 2) is also based on a natural thought-process, though a more intuitive and trial-and-error one. It consists in inference by analogy. The expression means "distinctive sameness", and therefore refers to the fundamental epistemological processes of comparison and contrast, which are jointly the basic technique of all concept formation. Applying them to textual analysis, we would quite naturally (i.e. without need of special communication or dispensation of Divine origin) look for homonyms and synonyms, to understand the language used and its conceptual references. In all discourse, we may find labels used which are analogous (similar in root, if not identical words), and apparently have similar or various meaning(s) in different contexts; or we may find different labels used in different contexts, with apparently the same meaning intended.


               The scientific-minded approach to gezerah shavah would run somewhat as follows. The meaning of a label, i.e. a word (every letter identical) or group of words (phrase), or word-root (having certain common consonants, in the same sequence; though possibly with some different vowels and consonants which indicate, on a wider grammatical basis, varying inflexions) or group of word-roots, is suggested by the various contexts in which it appears in the text(s) concerned, as well as in other texts and current usage, and through comparative etymology.

1.               Homonymy: If a, b, c... are all the occurrences of a label, and their assumed meanings (based on the above suggested methods) coincide, and no other assumed meaning(s) would be as coherent, then it may be assumed that the proposed single meaning is the intended meaning. If in some isolated context(s) the meaning of a label is uncertain, and it is coherent everywhere else, the same meaning can in all probability be generalized to the uncertain instance(s). But if the label is ambiguous elsewhere, there being one assumed sense in some contexts and some other sense(s) in others, then if no clear differentiating conditions are apparent, the sense most frequent elsewhere (if any) is the most probable, though some doubt remains.

2.               Synonymy: If A, B, C...  are various labels and their assumed meanings (based on the above suggested methods) are unambiguous, and mutually identical or at least similar everywhere they occur, and not even conditionally dissimilar anywhere, then these labels may be considered to be equivocal and interchangeable; that is, they are different labels for the same thing. If in some isolated context(s) the meaning of a label is uncertain, and it is coherent everywhere else, the same meaning can in all probability be generalized to the uncertain instance(s).

               Once the general meaning of a label or the equivalence of various labels is established, statements with the label(s) concerned may all be assumed to refer to the same subject-matter. A detailed example of the kind of analysis and synthesis here referred to may be found in the present volume, in our study of a-fortiori in the Tanakh (ch. 5-6).

               A traditional example of gezerah shavah is given by Enc. Jud. (with reference to Pes. 66a). The expression bemoado, meaning 'in its appointed time', is used both in Num. 9:2, concerning the Pascal lamb, and in Num. 28:2, concerning the daily offering (which includes the Sabbath); it is thence inferred that the Pascal lamb may be offered on a Sabbath (coinciding with Pessach), even though this entails activities forbidden on other Sabbaths.


            It is obvious that such reasoning is highly intuitive and dependent on one's overall context of knowledge. It is built up from the perception of words and the conception of their possible relations. The initial insights into possible meanings derived from immediate and wider context are conceptual acts dependent on the faculty of imagination; and subsequent ordering of the data, though a relatively mechanical process, is a function of the amount of data available at the time and taken into consideration. Such judgements can in no wise, therefore, be considered to have deductive value, but are eminently inductive.

            With regard to Biblical text, we have little material to refer to, other than the document itself. This means that our conclusions are virtually pre-determined, since the data available are finite, even if they constitute a sufficiently large and varied sample of the Hebrew of the time concerned. Actually, sometimes a word or phrase is only used once in the whole document, and its meaning becomes a subject of conjecture; obviously the more often a label appears in the text, the more certain its meaning. With regard to Hebrew usage later in history, it is of course very significant[6], but it must be kept in mind that it has been and still is culturally influenced by the interpretations suggested by the Rabbis, and therefore it cannot necessarily be used to further justify those interpretations.

            The natural interpretative process is adductive: an idea is floated, then tested in every which way for consistency[7]. It is, for this reason, susceptible to abuse. One may too easily stress similarities and ignore significant differences; and thereby stretch the application of an idea beyond its rightful borders. Or again, one may ignore similarities and emphasize incidental differences, and thus artificially restrict an idea. This is true of all argument by analogy; and repeated consistency-checking in an ever wider context of information provides the natural protection against error, as in all induction.


               Now, such a relaxed and patient attitude can hardly be practical in a legal framework, where some decisive position may be required 'right now'. On the other hand, the necessity to decide does not logically imply an impossibility to reverse the decision taken, later, in the context of new knowledge or modified conditions. The Talmudic authorities had debated matters and come to various conclusions which seemed wise to them. However, post-Talmudic authorities, intent on preserving these very decisions, proposed additional clauses to the hermeneutic principles which were to ensure they always resulted in the same conclusions, no matter how the data-context changed.

               Thus, in relation to gezerah shavah, they claimed that the Sages were occasionally informed by tradition as to which topics were open to legal analogy, but left to find the verbal analogy which would justify it; or again, that the Sages were in some instances informed of words which could be used for such inference, but allowed to find appropriate circumstances for their use; or again, that the Sages were told in advance the number of valid gezerah shavah arguments there would be![8] Now, I find all that hard to believe. Not only because it is very surprising that such alleged 'information' is (apparently) not explicitly mentioned by the protagonists themselves, but only makes its appearance in writing centuries later; but because the transmission scenario itself is unreasonable.

               Is it plausible that serious teachers would pass on vital legal information to their students in the form of riddles? Why would they engage in such games, and not get to the point, if they had the information? One cannot imagine a functioning law system in which it is not the law and its justifications which are transmitted from generation to generation, but conundra. For then, one would have to consider that the laws in question (i.e. those to be inferred by such means) had been inoperative until their formulation in the Talmud. In which case, surely, the more basic thesis that the law has gone on unaffected by time since Sinai - the very thesis these artifices were designed to defend - would be put in doubt. It seems obvious, therefore, that the above mentioned additional clauses are ex post facto constructs[9], based on no actual oral or written tradition.


               The controversies surrounding yet other additional clauses to the gezerah shavah process, provide still more cause for suspicion that such additional clauses are not Sinai traditions, but later constructs (in this case, Talmudic)[10].

               Thus, it is taught that the applicability of the gezerah shavah method depends on the 'freedom' of its middle term in one or both of its manifestations. This refers to whether each manifestation of the middle term involved, through which a legal factor is to be passed over from one issue to another, has already been utilized to justify some other Halakhah. Such a concern presupposes a principle that each unit of information in the Torah can only serve for one inference[11]; generic logic has no such restriction (a premise can be used in any number of arguments), but let us grant it to be a tradition. On this basis, three possibilities are considered: that the middle term is (a) 'free on both sides', (b) 'free on one side only', whether the source side or the target side, or (c) 'free on neither side'. Authorities say and agree among themselves that a gezerah shavah inference of type (a) is irrefutable. With regard to type (b), some say it is always valid, while others regard it as conditionally valid. With regard to type (c), some regard it as conditionally valid, while others say it is always invalid.

               Similarly, there is a debate as to how much legal detail a gezerah shavah allows us to pass over from premise to conclusion. There is also a debate as to whether once legal data has been transferred in one direction, other data may be transferred in the opposite direction, so as to equalize both sides, or whether the process is more restricted. It is irrelevant to us, here, which opinions are correct in these various debates - what is significant is simply the fact that there are at all disputes on matters so crucial.


               Regarding the 'freedom' (mufneh) concept, an interesting remark may be added: it can be viewed as an attempt, albeit a rather primitive one, to express the sort of syllogistic reasoning which follows the drawing of analogies. The Rabbis ask: once a term A is seen as analogous to a term B (gezerah shavah), can the laws applicable to A be applied to B and/or vice-versa? Their answers by means of the 'freedom' concept may be understood as follows.

               If both terms are 'free', it means that they were never before used in syllogistic inferences, presumably because they are both sui generis; consequently, the Rabbis assume them to be mutual implicants, and allow syllogism hither and thither between them. If only one is 'free', the Rabbis presume it to be a genus or species (I am not sure which) of the other, and thus allow syllogistic inference of laws from the genus to the species, though not vice-versa. If neither is 'free', it means that they have already led separate logical lives, so the Rabbis presume that the terms are unconnected (or at least that neither implies the other), and so avoid syllogistic inference.

               This perspective explains the Rabbis' concept, but does not fully justify it. For the basis of their syllogistic reasoning is too imprecise; they do not have a clear picture (even though this theory arose long after Aristotle) of the conditions of syllogistic inference. Similarity between terms and the histories of the use of such terms in inferences do not indubitably determine the implicational relations between these terms. The Rabbis lacked a clear understanding of opposition theory, as we shall see also in the section dealing with harmonization.


         We may, in my opinion, place under one heading, namely inference from context, the exegetic methods known as meinyano and misofo (Rule No. 12) and those known as heqesh and semukhim (regarded as part of Rule No. 2). All these take into account the textual closeness of an expression or sentence to certain other(s), and on this basis assume that there exists a conceptual relation between the passages under scrutiny, which makes possible an inference of certain attributes from the context to the expression or sentence. There is, we might remark, a small element of inference by analogy in such processes[12], though it might be characterized as extrinsic rather than intrinsic. The differences between these four techniques are, however, less clear (to me, at least).


               An example of contextual inference: the Rabbis inferred (by the rule meinyano) that the commandment "thou shalt not steal" in the Decalogue (Exod. 20:13), refers to kidnapping, on the grounds that the two preceding commandments, against murder and adultery, are both capital offenses, and kidnapping is the only form of stealing subject to the same penalty (Enc. Jud., which refers to Mekh., Ba-Hodesh, 8,5).


            Meinyano seems to loosely appeal to the surrounding subject-matter without precise definition of its textual position relative to the passage at hand. Misofo refers to a later clause or passage for the information it infers; though as some commentators have pointed out, it could equally well refer to an earlier segment of text. In these two cases, the conceptual common ground of source and target text is to some extent evident. In the case of heqesh and semukhim, however, the inference is based almost purely on textual contiguity, the contiguous passages (within the same verse or in two adjacent verses, respectively) having little evident conceptual relation[13].

            The natural justification of logistical inferences would be what we today refer to as 'association of ideas'. When two ideas are placed next to each other in our thoughts, speeches or writings, it may be because of some logical relation between them, or entirely by accident, or again because one contains some incidental reminder of the other. This last possibility implies that in some cases, even when purely logical considerations are lacking, an inference might yet be drawn from the fact of proximity. However, the possibility of chance conjunction still remains: topic X may be entirely spent and the narrative moves on to topic Y, an entirely separate topic[14]. This alternative possibility means that inference based solely on position is tenuous. The Rabbis were apparently aware of this uncertainty, and would use such processes only as a last resort, when the verse being interpreted involved a doubt which they had no other way to resolve.


               R. Ishmael did not mention the exegetic methods of heqesh and semukhim, and attempts by later authorities to explain this silence have a hollow ring. Thus, Bergman (with reference to the Sefer Hakerisus) says of R. Ishmael that "he regarded the hekesh as the equivalent of an explicitly written teaching". If R. Ishmael did not even mention the subject, how can the later Rabbis know by tradition why he did not mention it. How can they have information on his thoughts on an unspoken issue? The very notion is self-contradictory: proving again that the authorities often confuse their personal assumption concerning some matter with a 'received tradition' (refusing to admit that R. Ishmael might not have known about these things, or that there might be no tradition concerning them, and that such issues must be resolved adductively).[15]

               Again, R. Ishmael, apparently (and as the name given to the process implies), did not regard or was not aware that misofo inference was equally feasible in the opposite direction ('mitechilato', if we may say so), from an earlier to a later statement or clause. Later commentators (Bergman refers to Middos Aharon), who considered such reverse inference possible, explain R. Ishmael's silence by claiming, effectively, that in cases where the solution precedes the problem, the inference is so obvious that listing it would have been a redundancy. That is another anachronistic argument, whether we agree with the validity of such inference in both directions or not. The commentators must admit the possibility that R. Ishmael did not hold the same opinion, or more likely still (since he himself does not mention it) that he just did not think of the issue at all!


            Inferences of the binyan av type (Rule No. 3) seem to be a Rabbinical attempt at causal inference - using the term 'causal' in its widest sense, including any mode of causality; i.e. not only natural-mode causation, of motion or change, but also extensional causality, of 'static' (i.e. class) differences, as well as logical causality, or rational explanation[16]. Causal inference has been much clarified in more recent times by John Stuart Mill[17], who identified the 'methods of agreement and difference'. It results from observation of two kinds of events or things, such that the presence of one is always accompanied by the presence of the other, and therefore that the absence of the latter is always accompanied by the absence of the former. In such circumstance, one may, from observation of the first event or thing, presume the second even when it is not observable. This is an inductive process, involving analogy and generalization. Symbolically, broadly-speaking, the essential relation between a cause C and an effect E may be expressed by a hypothetical proposition and its contraposite:


            If C, then E (and if not E, then not C).


            However, the Rabbinical attempts at formulation of this natural principle stressed more the side of 'agreement' than that of 'difference'. R. Ishmael refers to an inference 'from one verse' or 'from two verses'. There were subsequently disputes as to the meaning of these subdivisions (which disputes, incidentally again tend to show the lack of a clear oral tradition). Some Rabbis understood them, respectively, as follows: if two topics (X, Y) have a certain feature (A) in common, then another feature (B) which the one (X) has may be assumed to be had by the second (Y)[18]; or, if three topics (X, Y, Z) have a certain feature (A) in common, then another feature (B) which two of them (X, Y) have may be assumed to be had by the third (Z). Other Rabbis claimed to understand R. Ishmael's formula differently[19]. They sought for a common feature[20] (A, say) of topics under comparison (X, Y) which would explain their having in common some other property (B), in which case the reappearance of that same feature (A) elsewhere (in Z) could be taken as a sign of the same property (B) there (i.e. in Z). In fact, this formula is formally identical to the second of the above mentioned[21], merely adding the (valuable) comment that A is to be considered as the cause of B.

            The difficulty in these statements is their emphasis on the positive, their attempt to generalize from a limited sample (X, or X and Y) without readiness to conceive the possibility of deviation from the apparently set pattern of conjunction (of A and B) in other cases, including, in particular, the case at hand (Y, or Z, respectively)[22]. Apparently sensing this weakness, the Rabbis tried to put a bit more emphasis on the negative, by pointing out differences in features between the (two or three) topics under scrutiny, thereby hoping to demonstrate other possible causes have been considered and eliminated. Thus, they might say, in the two-verse form of binyan av: X has C and Y lacks C, so that Z having C does not prove it has B; or again, X lacks D and Y has D, so that Z having D does not prove it has B. However, it should be clear that such statements are irrelevant to the main argument: they at best prove only that C or D do not cause B, but do not prove that A does cause B[23].


               An example of binyan av, given in Enc. Jud. (referring to B.M. 87b). The Rabbis attempt, with reference Deut. 23:25f., to determine whether a hired farm hand may eat produce, while working in fields other than those with vines or standing corn. To do so, they try to understand why the Torah allows him to eat in vineyards and in cornfields. They argue: it cannot be in relation to the obligation to leave gleanings for the poor (Lev. 19:10), since this applies to vine but not corn; and it cannot be in relation to the obligation to give the priest a portion of the dough (Num. 15:17-21), since this applies to corn but not vine; 'therefore,' it must be simply due to their being both plants, and the permission may be generalized to other produce.[24]


            It appears from such redundancies that the Rabbis confused somewhat the trial and error mental process of looking for a cause (ratio cognoscendi), and the formal conditions of the objective causal relationship (ratio essendi). Had they known the latter clearly, they would rather have systematically first made sure they had a complete enumeration of the appearances of features A and B in the Torah, alone and/or separately, as well as their negations if any. Then, to make possible the inference from A to B, in situations where A is mentioned in the text but B is not mentioned, they would have to check, not only that A and B are sometimes both affirmed together (at least once, but the more the better), but also that A is never affirmed with an explicit denial of B (that is the missing negative element). Furthermore, the probability-rating of the inference would be proportional to the frequency of conjunction of A and B, compared to that of A mentioned alone without mention of B[25]. It is possible that in the cases where the Rabbis applied this principle, they (who knew the Torah by heart) automatically performed these consistency tests and probability judgements; but they did not always do so explicitly.


               It must be stated that aetiology does not insist that the cause be one event or thing, or that the effect be one event or thing; each of these (cause or effect) may itself be two or more parallel things or events, provided the stated rules of induction (agreement and difference) are adhered to for them all singly. Furthermore, if the rules of induction are not invariably adhered to (whether by a single event or thing or many of them), they might still be found to apply conditionally or compositely: that is, provided we manage to identify and distinguish the conditions under which our partial causes become complete causes or our occasional effects become constant effects[26]. Consequently, too, there may be circumstances in which one event or thing is the cause of a certain effect and other circumstances in which another is so; or again, a certain event or thing has some effect in one set of circumstances, and another effect in another set.

               These details of causal logic were apparently not entirely understood by the Rabbis, judging from certain limiting suggestions they made. One such Rabbinic limitation was that (with reference to the symbols introduced above)[27] if X and Y have in common yet another feature (E, say) which Z lacks, then Z cannot be assumed to have B. I say, viewing 'A+E' as a joint feature, this objection seems reasonable; but it still remains possible that A causes B. Another Rabbinic limitation was the rejection of the possibility that distinct features of X and Y, such as C and D (see above), may independently cause B in their respective subjects, so that Z, which has only the common feature A, but neither of the distinct features C, D, may not have B. I say, it is conceivable that the two compounds 'A+C+nonD' and 'A+nonC+D' might be parallel causes, while the compound 'A+nonC+nonD' is not a cause: the issue depends on the negative side, which they ignored in their initial definition[28]. Such attempts at exception show, to repeat, that the Rabbis were not certain as to the precise conditions of causality[29].


            There is a manifest failure of theoretical research in logic, independent of any Torah related doctrines, by the Rabbinic authorities. Consequently, as may be expected, there is a lot of controversy between them on methodological issues (which, of course, ultimately affect the law); and worse still, sometimes the controversy revolves around a totally artificial issue (which naturally enough emerges from some general belief to which the disputants are all attached). What amazes me is that the existence of such controversies does not cause any of the people involved to frankly question the 'orthodox' doctrine that the hermeneutic principles, in their entirety, are Sinaitic revelations.


               A case in point is the discussion concerning the 'two verses coming as one' principle, according to which if a law L is stated in relation to two subjects, S1 and S2, and logically L(S1) implies L(S2), and/or vice-versa, so that one of the statements is redundant, then L may only be applied to the two situations specified. This principle has no natural basis, as far as I can see; i.e. it does not formally follow that L(S1) and/or L(S2) cannot logically imply some other, unstated application, say L(S3). If such implication does logically occur, it cannot be inhibited by Divine or Rabbinical fiat (Gd does not contradict His own natural laws, nor allow them to be by-passed by humans) [30]; for this reason, the restriction must be classed as artificial. Analytically, it seems to be merely an outcome of R. Akiba's claim that there is no superfluous statement in the Torah; which idea is itself controversial, since R. Ishmael (theoretically) rejects it, claiming that the Torah speaks in the language of men. Whatever its source, there is controversy concerning the precise formulation of the proposed principle, from Mishnah onward. Some authorities (among them Tosafot) claim that at least two redundancies are required for such restriction. Some (among them Rashi) say that the restriction is in any case not general to all inference, but limited to attempts to extend the law concerned by means of binyan av inference. But I see no formal basis for these subjacent disputes, either.


            The deep intellectual cause of such deviations from natural logic is, in my opinion, initially a naive non-formalism, gradually developing into a systematic anti-formalism (which is also naive, in other respects). The historical cause is an unfortunate, at first emotional and later ideological, antipathy to what they called 'Greek knowledge', which blocked any attempt to learn from the discoveries of others. For these people, logic has to adapt to the requirements of pre-conceived contents (the Halakhah), rather than all contents yielding to the dictates of an objective, formal logic. This Rabbinic claim of total control is evident, for instance, in Bergman's statement in this context: "The rules regarding Scriptural texts... reveal a law only in relation to the place they are applied and not elsewhere".

            The doctrine of the Sinaitic origin of Rabbinic hermeneutics is not primary, but a derivative of the doctrine of the Sinaitic origin of Rabbinic law. The Rabbis thought they could manipulate logic however they saw fit, so long as they arrived at the required legal results. Controversies occurred only in relation to the necessity or efficacy of this or that manipulation, but not in relation to the underlying epistemological assumption.

            These reflections need not be taken radically. Our concern, here, is with Judaic logic as such, not with Jewish law. If we throw doubt on the former, it does not necessarily follow that all, or any, of the latter is wrong; for, as logic teaches, denial of the antecedent (in this case, some aspects of Rabbinic hermeneutics), does not imply denial of the consequent (in this case, Rabbinic law) - unless their relationship happens to be exclusive. A law may be correct (i.e. truly Divinely-willed), but improperly derived from the text (i.e. from the wrong place or in the wrong manner). A law may, of course, alternatively, be incorrect, as well as improperly derived. These are not matters which can be dealt with in general ways; but each case must be reviewed carefully, after which the consistency of the whole must also be verified. In any case, logic cannot itself be made an issue of faith, something optional.


            A statistical note. I have not so far found out just how many times each form of exegesis described here is actually used in the Talmud and other Rabbinic literature. For the moment, here is some information gleaned from the Index Volume of the Soncino edition (1952) of the Babylonian Talmud. As we saw in an earlier chapter, this index contains some 137 references to qal vachomer argument (under various headings). With regard to various forms of argument by analogy, there are some 161 references (analogy, deduction by, 58; comparisons, for purpose of deriving laws, 1; gezerah shawah, 81; hekkesh, 17; semukin, 1; texts, proximity of, 2; textual reading, 1). Whether this index is complete, and whether each reference concerns a distinct sample or there are repetitions, and whether some references relate merely to theoretical discussions, I cannot venture to say[31].

            Note also that we must distinguish between use of an argument: (i) within the Bible itself (e.g. we know of four or five cases of qal vachomer in the Torah, and two dozen more in the rest of the Bible); (ii) by the Rabbis, especially those of the Talmud, in their efforts of exegesis from the Bible, as a document, of Halakhic or Hagadic material; and (iii) by the Rabbis, especially post-Talmudic ones, in relation to the non-exegetic pronouncements of other Rabbis. Clearly, the statistical question constitutes a large and difficult research project in itself.

See also Addenda 6 and 7. 


3.      Scope of Terms.


            Obviously, in the reading of any text, understanding the terms used is of the essence. This has two aspects: a qualitative aspect, which by its very nature presupposes knowledge of the language involved, and a quantitative aspect, which relates to determinations of scope. Rabbinic tradition has, of course, had much to say about both these aspects. The first aspect, elucidation of the denotations and connotations of terms, is in part dealt with within the hermeneutic principles, by way of inferences by analogy and context; and in part, it depends on cultural and religious tradition and the insights of commentators. The second aspect, concerning subsumptive issues, is covered by a set of hermeneutic principles which we shall now consider.

         The methods of exegesis known as collectively klalim uphratim[32], are efforts to interpret the effective subsumption of logically overlapping terms found in the Torah (and thence applicability of the proposition(s) involving those terms). Miklal uphrat (Rule No. 4) is the interpretation of a genus + species combination, in that sequence, as having a limiting effect, signifying 'only the species mentioned' (the species is mentioned for purposes of excluding others of the same genus); whereas miprat ukhlal (Rule No. 5) is the interpretation of a species + genus combination, in that sequence, as having an enlarging effect, signifying 'the species mentioned and others like it' (the species is mentioned as a sample of those in the genus). An easy way to remember these two rules is to say that the result is equal to the second of the two given terms. Other such rules, as we shall see, have an overall limiting result. (See Diagram 2.)


               To put us into the picture let us note that, in everyday discourse, we would (depending on the precise wording) understand the conjugate scope of logically overlapping terms as follows, granting that G is an overclass and S1, S2... one or more of its subclasses, and that G and G' are two classes which partly intersect without either subsuming all of the other's instances and S, say, is the entire subclass referring to the G' part of G. A statement whose subject is "GS" or "SG" would be interpreted minimally as concerning the species "S"[33], though in some cases the genus "G" might be appropriate. "G such as S1, S2..." might be read as "S1, S2..." (if 'such as' is taken to mean 'similar to') or as "G" (if 'such as' is taken to mean 'for examples'). A listing of the form "S1, S2... indeed G" would likely be intended as "G", though without a qualifier like 'indeed', a doubt might subsist. Lastly, "GG'" or "G'G" is usually intended as "G and G'" (i.e. "S", their common ground), though occasionally might mean "G or G'" (including both their grounds, as well as "S" or even possibly to the exclusion of "S").

               There is evidently much vagueness in ordinary language, which logical science can easily overcome by instituting conventions. This field of inquiry is not class logic proper, but a linguistic preliminary to it. Note that a mental act of 'reconciliation of conflicts' is involved, insofar as the terms dealt with are in some tension, according as we understand reference to a genus as concerning the whole of it (davqa) or most or an unspecified portion of it (lav davqa), and reference to a species as concerning it at least (lav davqa) or it exclusively (davqa). When the terms are mentioned together as subjects of propositions, there is therefore doubt as to whether the result is a generality, a contingency or an indefinite particularity. The logical rule, in case of doubt, is to acknowledge, by dilemmatic argument, the indefinite particularity (at least some) as true; deductively, we remain open-minded as to whether the generality (all) or contingency (some, but not all) is true; inductively, we opt for the generality, because it introduces no new polarity, unless or until conflicting evidence is found.


            For the Rabbis, in the klal uphrat case, the genus is mentioned as a first approximation of the meaning intended, and the species is added to more precisely pin-point that meaning. For instance, in Lev. 1:2, "of the livestock (behemah), of the herd and of the flock," the general term is one of variable connotation (it could be taken to include other types of animal, such as asses perhaps) and is clarified by means of the mentioned species. To explain why the species were not simply mentioned alone, without the genus, we are told that extensions unintended by the writer might then have been proposed, or alternatively that certain details suggested by the genus might have been missed; opinions differ among the authorities on this point. We could accept an amalgam of both as reasonable: the genus is there, effectively, to say "but do not include with these species, other dissimilar species of the same genus".[34]

            In the prat ukhlal case, some species are first listed to indicate the kind of thing intended, and the genus is added in conclusion to indicate "and other things of the same kind" to be also intended. For example, in Exod. 22:9, "an ass or an ox or a sheep, or any beast (behemah)," the species exemplify the things intended and the genus serves to extend the application of the law concerned to other similar things, implying the initial list not to be exhaustive. To explain why the genus was not simply mentioned alone, without the species, we are told that exceptions unintended by the writer might then have been proposed, or alternatively that certain details suggested by the species might have been missed; opinions differ among the authorities on this point. We could again accept an amalgam of both as reasonable: the species are there, effectively, to say "and be sure not to exclude from this genus, other similar species of the same genus".


               These methods, and other variations (mentioned below), are R. Ishmael's; R. Akiba proposed others in the same contexts: he determined the scope of statements with reference to the principles of ribui umiut and miut uribui (amplification and limitation, and vice-versa). The technical difference between these approaches is essentially one of emphasis. Whereas in klal uphrat, the mention of species serves to more precisely define the initial genus; in ribui umiut, the explicit mention of species stresses the exclusion of certain dissimilar things, not explicitly mentioned, belonging to the initial genus. And whereas in prat ukhlal, the final genus serves to more broadly define the full extent of the list of species, adding to those explicitly mentioned more species not explicitly mentioned, while also incidentally somewhat limiting, as all definitions do, excessive extrapolations; in miut uribui, the mention of the genus stresses the limits of extrapolation more, excluding certain unmentioned species of it too extremely dissimilar to the mentioned species, while also incidentally suggesting certain unmentioned species to be included[35].


            These forms of interpretation seem to me natural enough in themselves. In many cases, the wording is clear and no discussion is possible, anyway. However, in some cases, the results do not seem formally inevitable: one might sometimes view genus+species as signifying 'genus, of which a sample species is...'[36], and species+genus as signifying 'species, of which the relevant genus is...'[37]; in such cases, note, the term mentioned in second place is effectively in brackets, suggesting a proposition which communicates, in passing, some incidental information (not necessarily of immediate legal relevance). Consequently, if we take the rules as ex cathedra pronouncements, and attempt to always tailor our interpretations to fit their given formats, we are not unlikely to be occasionally misled. Clearly, behind such regulations is the rigid mode of thought which denies stylistic license to a document of Divine origin [38]

         With regard to the other combinations and permutations of these inferences of scope (classed as Rule No. 6), notably klal uphrat ukhlal and (apparently a later addition[39]) prat ukhlal uphrat, we need only add the following comments. These involve successive operation of the preceding two principles, with the klal stages having a broadening effect, and the prat stages a narrowing effect, the overall result being relatively narrow[40]. There is a proportionately greater opportunity to force the text into preconceived formats, rather than interpreting it naturally. It seems to me that we should always try to grasp the simple reading (pshat), and avoid deviation from it without overwhelming justification.


               An example: Num. 6:3-4 forbids the Nazirite from drinking wine or strong drink or their vinegars or liquor of grapes, or eating fresh or dried grapes or anything made from the grapevine, from the kernels (chartsanim) to the husks (zag)[41]. According to Bergman, Nazir 34b reads this passage as prat ukhlal uphrat, but this seems to me unjustifiable, if 'strong drink' (shekhar, which can make one drunk) refers to alcoholic beverages other than wine and such. For in that case, 'wine and strong drink' cannot be wholly regarded as a prat in relation to 'products of the vine', which is a klal at best only in relation to wine and other grape-based drinks, fresh or dried grapes, and kernels and husks. The way all these items are listed is natural enough, in three classes grouping together alcoholic drinks (not all grape-based), normally eaten forms of grape, and parts normally wasted by the consumer, respectively, and additionally mentioning a wider class which most (but not all) of the items fall under. To insist on fitting them into the format prat ukhlal uphrat is artificial and inaccurate.[42]


         We should also mention here the principles miklal hatsarikh liphrat and miprat hatsarikh likhlal (general term requiring a particular [complement], and particular term requiring a general [complement]), which R. Ishmael's list groups together as one (Rule No. 7)[43]. We may classify these, as Bergman does (presumably following previous authorities), with the klalim uphratim. The distinction between them is, he suggests, effectively: whereas klal uphrat and prat ukhlal and their ilk concern the collective effect of separately clear terms, the hatsarikh rules relate to vague terms whose precise meaning is only clarified by their mutual impact on each other. This distinction is very fine indeed, and rather forced judging by the examples given in the literature[44].


               I would say, rather, that a case could be made for distinguishing between functionally independent terms (broadly speaking, classes of entities, which may however have a hierarchical relation; e.g., 'animals' and 'bulls') and dependent terms (more precisely, a relatively independent term, like 'bulls', and its complementary clauses; e.g. 'the horns of' or 'the goring of people by'). The relation between independents is at best simply subsumptive (bulls are animals), whereas the relation between dependents is a more complex one, like possession or action (bulls have horns and bulls gore people)[45]. The former supplement each other, the latter complement each other. I do not mean to say that the Rabbis did classify their inferences under this or that heading on the basis of the distinction I am proposing (though perhaps they were trying to), but rather that if they insisted on making some kind of subdivision of the phenomena at hand, they might relatively usefully have selected it instead of the above mentioned[46]. I say 'insisted', because my distinction too is not radical enough to justify the formulation of additional hermeneutic rules. For one can usually (and much formal logic is based on this operation), perform what is known to logicians as a 'permutation'[47], and change the complementary term into an independent one (bulls are 'horn-having things' and bulls are 'goring things').


            A note on statistics. The Soncino general index has 77 references to the topic of klalim uphratim (supposedly, but I did not check). These come under various headings: amplification, 5; amplification and limitation, 15; amplification following amplification, 2; extension in exegesis, 5; general principles and exceptions, 1; general rulings, 2; generalisation, 2; generalisation and specification, 28; limitation in exegesis, 6; rule, general and particular, 1; rule, extension and limitation, 1; ribbui, 1; ribbui umiut, 1; specification, 1; specification as exegetical rule, 2; specification and generalisation, 4. As before remarked (in the discussion of Talmudic a-fortiori), to what extent such a list is exhaustive and non-repetitive, is hard to say without further investigation. In any case, it does not tell us precisely how many times each rule is actually used.  

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[1]              The present analyses were made possible thanks mainly to Bergman's detailed presentation of the 13 Midot. Though I dislike that author's pompous tone and unquestioning fanaticism, and disagree with many of his specific positions, he is to be commended for his unusual efforts to clarify the hermeneutic principles. All too often, authors are simply content with listing examples with a minimum of reflection; he at least tries (if not always successfully) to sort out logical relations explicitly.

[2]              The formalization of relations is not technically valuable (apart from saving space), and tends to alienate and confuse readers; for these more abstract features of propositions, we shall stick to ordinary language.

[3]              We may regard the Rabbinic principle ain mikra yotse miyedei pheshuto (quoted by Enc. Jud., p. 371, with reference to Shab. 63a and Yev. 24a, and there translated as "a Scriptural verse never loses its plain meaning", with the added comment "i.e., regardless of any additional interpretation"), as an implicit recognition that interpretations using the hermeneutic principles were not always natural. It may be asked how they managed to mentally accept conflicts between a midah-generated reading and a simple reading (pshat), given such a principle!

[4]              I am referring here to the Rabbinical subdivision of the dayo principle discussed earlier, in ch. 4, footnote 10.

[5]                Notwithstanding, the Sages were, in my view, very wise to reject corporeal punishment for breach of prohibitions discovered through qal vachomer argument, or for breach of Torah prohibitions whose penalties were inferred through qal vachomer argument. In practise, we can rarely if ever be 100% sure of having freed our deductions of all possible material uncertainties; and therefore some injustice might be caused. What is true of qal vachomer, the most deductive of Talmudic arguments, should be all the more true of the other hermeneutic principles. (According to Enc. Jud. pp. 371-2, this canon concerning inferred prohibitions or penalties, is R. Ishmael's: ain oneshim min hadin; R. Akiba disagreed. We are referred to J.T. Yev. 11:1, 11d.)

[6]              We are not here dealing with a dead language, so the job is less difficult.

[7]              e.g. does 'take' (qach) always signify acquisition by monetary payment, or must other meanings of the term be supposed, and if so how are they to be distinguished? It is not enough that an analogy is applied in one instance (without problems ensuing from that one application); it has to be tested in all other cases where the term appears, throughout the text at hand.

[8]              The references given by Bergman for these three provisions are all post-Talmudic: Halichos Olam for the first two, and Rabbeinu Tam and Tosafot for the third.

[9]              Or, to be more explicit: pretensions, lies - made in support of a certain ideology, that of unchanging oral law. Note well the basis of my accusation: precisely in the attempt to buttress their concept of the fullness and continuity of tradition, these people are forced to acknowledge the occurrence, in the case under consideration, of incomplete transmission of information, and thus imply a loss of data in the interim and the unreliability of the transfer. The proposed formulae are therefore inconsistent with their own motive, and therefore must be inventions.

[10]            Enc. Jud. , p. 368, refers us to Shab. 64a and J.T. Yoma 8:3, 45a. Enc. Jud. explains the development of the idea of "free" (mufneh) terms, as a way to prevent abuses of gezerah shavah in the schools; it adds that this is a R. Ishmael requirement, which R. Akiba apparently disagreed with.

[11]            Not to be confused with the principle of R. Akiba that in the Torah no word is superfluous and no word-placement is accidental.

[12]            Note, however, that in some cases, traditionally classed as heqesh (see Abitbol, pp. 100-104), Scripture itself explicitly establishes the parity between the two areas of law. For instance, Deut. 22:26, which compares rape and murder, saying "for as when... even so..." (ki kaasher... ken hadavar hazeh.). In such cases, an inference is much more certain, though it may well have some limits, because it is analogical rather than contextual.

[13]            Heqesh and semukhim are classified by commentators with gezerah shavah; but I am not sure why (except for analogies given explicitly in the text) - they are not really subcategories of it, being based on neither homonymy nor synonymy, but on textual proximity. Perhaps it is because they often involve elements of gezerah shavah, that they were rather grouped with it. Also note, the distinction between them is ambiguous: in text originally devoid of punctuation, like the Torah, how to tell the difference between the two parts of the same verse or two touching verses? It only becomes meaningful once the subdivisions of the text are established.

                Looking further into the issue, I found some interesting remarks in J.E. Apparently, gezerah shavah initially referred to "analogies in either word or fact", but eventually was restricted to verbal analogies (homonymies), while heqesh became used for factual analogies (synonymies). If that is true, my presentation of these terms, based rather on Bergman's account, is inaccurate (terminologically, though not in essence). With regard to semukhim, it is attributed by J.E. to R. Akiba, quoting him as saying "every passage which stands close to another must be explained and interpreted with reference to its neighbor" (with reference to Sifre Num. 131); it adds that 'according to Ishmael, on the contrary, nothing may be inferred from the position of the individual sections'. In view of this, semukhim cannot strictly-speaking be counted as implicitly included in the 13 Midot of R. Ishmael.

[14]            A case in point seems to be the above given example (Exod. 20:13): supposedly the next two commandments (against bearing false witness and coveting) are not subject to death penalty; in that case, why should stealing (or kidnapping) be a capital offense? Of course, if a proposition is surrounded on both sides by a certain subject-matter, it becomes more likely that the common subject-matter of the two adjacent propositions somehow concerns the boxed-in proposition. Nevertheless, the possibility of accidens remains; there may well be significant underlying differences, which can be pointed to. An example is Exod. 21:15-17, where a law concerning kidnapping is found between two laws concerning striking and cursing parents, respectively. (Note that, as an acquaintance of mine, Dr. M. Izbicki, has pointed out, the laws on striking of parents and kidnapping both concern violent acts. Also, see Cohen, p. 474; the law assigns a penalty of strangulation for these two, but stoning for cursing parents).

[15]                Regarding heqesh and semukhim, Bergman adds additional details, which we will not comment on, here, to avoid repetitions. I want only to point out that the semukhim inference he gives as an example at the end of the section is very odd: from (Deut. 22:11) the proximity of the prohibition of 'shatnez' and the prescription of 'tzitzit', the conclusion is drawn that shatnez is allowed in the case of the tallit! This is far from normal inference, since the conclusion is an exception to one of the premises, although there was no inconsistency between the premises. Formally, the argument runs as follows: 'You mustn't do X. You must do Y. Therefore, when you do Y, you may do X'. Since the conclusion formally implies 'you may do X' it is contradictory to the major premise; such argument thus depends on an anti-literal particularization of the major premise.

[16]            Our identification of binyan av with causal arguments may be too narrow; some examples in the literature seem like mere extrapolations with nary an underlying cause and effect thought-process (though we might construct one, ex post facto). An example given by Scherman illustrates this: Just as one may neither marry one's sister from the same two parents, nor one's father's full sister; then, since one may not marry one's sister from the same mother but different father, 'it follows that' one may not marry one's father's maternal half-sister (ref. to Yevamos, 54b). Scherman says binyan av is also known as mah matsinu? ("what have we found?"), though Bergman informs us that these two were counted separately in Hillel's list. I suspect the Rabbis at first engaged in generalizations with little reflection, and then gradually found it necessary to clarify conditions.

[17]            English logician (1806-1873), author of A System of Logic (1847). Mill's formulation of these methods is more complicated than the one proposed here: 'agreement' is observed constant conjunction of two phenomena, 'difference' is the constant conjunction of their negations; thus, the formal relationship is mutual, i.e. the same in both directions: this is the strongest form of causality, in which the cause is not only sufficient for the effect, but necessary or a sine qua non for it. Note that, in all such definitions (as Mill was aware), cause and effect are difficult to distinguish: to do so, we must look at their temporal or conceptual sequence. Note also that Mill suggested other important methods, namely 'residues' (elimination of alternatives) and 'concomitant variations' (see Appendix 1, section 3).

[18]            This approach is known as Chada mechada (Aram., 'one from one'). If the one-verse variant is so called, then the two-verse variant may presumably be called 'one from two' (but I do not know what it was actually called).

[19]                According to Bergman, this second school claimed the one-from-one inferences obvious and not needing to be included in the 13 Midot. How they viewed the 'one from two' formula, he does not say.

[20]            Heb. tsad hashaveh.

[21]            Added proof of which is the lack of distinction, in the 'common feature' approach, between one-verse and two-verse inference in accordance with R. Ishmael's formula. That is, if in this approach 'one-verse' means 'one from two', then what might 'two-verses' mean?

[22]                Logicians refer to the underlying logical fallacy as post hoc ergo propter hoc (i.e. 'after this, therefore because of this').

[23]            Unless, of course, it is already proven that A, C, D are the only alternative possible causes of B; so that the inference consists in elimination of two out of three possibilities. But that is not everywhere the case.

[24]            The logical naivety of such argument should be obvious. Say a boss calls three of his employees to his office (intending, say, to congratulate the first for his new-born son, give a raise to the second, and fire the third). No one knows the boss's reasons, but the third employee tries to guess why he has been invited, by looking for a 'common factor' in the two others. He says: "it cannot have to do with hair colour for the first is blond and the second is a brunette; it cannot have to do with honesty, because the first is dishonest and the second is honest; it must therefore have to do with nose shape, because both, and indeed I too, have the same nose shape." His whole argument is utterly fallacious and beside the point!

[25]            For it always remains conceivable that non-mention of B in certain passages signifies negation of B, even though there are no known cases of presence of A with absence of B, which fact inductively allows us (indeed, obligates us) to generalize and say that A implies B.

[26]                Conditional/partial causes, considered together with their conditions or allied parts (i.e. collectively), constitute unconditional/complete causes. Occasional effects, in loosely defined surrounding circumstances, become constant effects, when the significant circumstances are more precisely pin-pointed. All such variations on the theme of causality are obvious to anyone who has studied hypothetical logic, considering multiples and compounds of antecedents and consequents, and nesting; it is no great secret (at least nowadays).

[27]            Where X, Y, Z have in common A, and X, Y have in common B; and they would conclude that Z has B, also.

[28]            It does not surprise me, therefore, that, according to Bergman, 'the Gemara does occasionally raise such a refutation'. He adds that 'the commentators (to Kesubos 32a) have formulated principles to explain these exceptions'; I have not seen these principles, but we must keep in mind that they are ex post facto rationalizations, since natural logic allows alternative causes.

[29]            Another issue they raise is whether the exceptions have to be 'significant' or not - claiming that refutation of a 'one from one' binyan av requires a legal exception ('stringency or leniency'), whereas against a 'one from two' binyan av, any exception (legal or non-legal) will do. I can only say that such a distinction is not made by natural logic.

[30]            I refer here, of course, to deductive logic; if the 'implication' is merely inductive, i.e. not necessary but only recommended as probable, it might conceivably be preempted with reference to wider considerations.

[31]            See also, possibly, the items: exegetical principles, 2 references; rules by which law is expounded, 2; texts, exposition of, 3; texts, implication of, 1.

[32]            This phrase is wrongly translated by Bergman as "generalizations and specifications"; we would prefer "generalities and particularities" or "genera and species". Strictly speaking, the term generalization, in English, refers to a mental process moving from a particular statement to a general one, and specification refers to a movement from a vague statement to a more precise one. Here, however, the issue is finding out how wide or narrow was the intent of the writer of the text.

[33]            Note, however, that in English an adjective usually precedes a noun, while in Hebrew if follows it.

[34]            A way to explain the mention of the overclass is to regard it as intended to allude to its instances or subclasses not included in (or alluded to by) the mentioned subclass; which are to be the subject of an unstated proposition of opposite polarity, which makes the stated particular a contingent.

[35]            Albeit the theoretical formal convergence between these approaches, they in practise often led to divergent material conclusions (with R. Akiba's often ending up as Halakhah). I would need more specific knowledge of the Talmudic debates concerned to be able to explain why. But, assuming I have correctly understood the formalities and basing myself on the few cases I have studied, I would say that such differences simply go to show the amount of subjectivity involved in the 'inferences', in practise.

[36]            e.g.  'logical arguments, syllogism, adductions, are means to knowledge' does not restrict the genus to the listed species. One might construct a sentence in this way, if one thought some auditors unable to grasp the generic subject's name without apposition of some more familiar species names.

[37]            e.g. 'Aristotle's syllogism, a deductive argument, yields categorical conclusions' does not intend the genus as a whole but only the species. The genus, here, serves only to place the specific subject in a larger context.

[38]            It is interesting to note that the rationalistic philosophy, trying to explain every word and word placement in a certain preconceived manner, which is assigned to R. Akiba, is here, and in other instances, really at the basis of R. Ishmael's inferences, as well as R. Akiba's.

[39]            It is noteworthy that R. Ishmael mentioned only the first of these as one of his rules, ignoring the second and also klal-klal-prat and prat-prat-klal. My explanation of such repeated gaps in R. Ishmael's list is simple: as often in the early stages of an investigation, thinkers do not initially work out an exhaustive analysis of the topic concerned with reference to symmetries, but rather concentrate on cases which happen to arise before them in the context of their less abstract concerns. It is only later that they, or their successors, look into other possible combinations and permutations suggested by reshuffling the terms at hand. R. Ishmael did not get around to that systematizing stage; his successors did. (All well and good; what is annoying is their attempts to justify their results by imputing them anachronistically to him.)

[40]            A technical distinction suggested by the Rabbis between the two stated principles is that, quoting Bergman, the first "dictates that every item which bears even one similarity to the specification is included," whereas in the second "the item to be included must resemble the specification in at least two aspects." He adds: "How significant the resemblance had to be was left for the Sages to determine. Occasionally, they considered several aspects as one." Here again, I must say, I see no natural basis for these added details.

[41]            These words are variously understood by the Rabbis.

[42]            Another kind of example is Num. 18:16, which according to Bergman is interpreted in Shevuot 4b as klal ukhlal uphrat. Here, the forcing consists in reading a subject, predicate and qualification of predicate as classes (broad, broader and narrower), ignoring entirely their relative positions (logical roles) in the sentence.

[43]            Why he should do so, when he separated klal uphrat and prat ukhlal, is a mystery to me; or alternatively, why were not the latter two and their multiples grouped together, similarly? I would suggest, here again, a failure to stand back from accumulated knowledge and re-order and systematize the results obtained.

[44]            For instance, the example from Lev. 1:2 given earlier would, according to Bergman's definitions, constitute a klal hatsarikh liphrat rather than a klal uphrat, as well as I can tell. Note, however, that Bergman hints that 'there are many varying opinions' concerning the definition of hatsarikh processes; so we might reserve judgement.

[45]            This distinction is reminiscent of that which logicians make, between categorematic and syncategorematic terms, though not exactly identical. Note that, unlike independent terms, dependent terms are not conjoined by a mere 'and', but by more intricate relations like 'of', 'by', 'for', 'through'.

[46]            A Torah example might be Num 18:16: "according to your valuation, five silver shekels" (mentioned in an earlier note), where 'five silver shekels' makes no sense by itself, apart from the concept of 'valuation' (in this case, for redemption purposes).

[47]            It should be noted, however, that certain terms, which are colloquially put in a similar format (notably, 'capabilities of' and 'metamorphosis of') cannot be freely permuted without causing eventual contradictions. But such issues cannot be addressed in so narrow a context; they are fields of logic in themselves.



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