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A Formal Analysis of Biblical, Talmudic and Rabbinic Logic

The books displayed here can all be purchased online in print hardcover or paperback or e-book epub form at:. Future Logic is an original, and wide-ranging treatise of formal logic. It deals with deduction and induction, of categorical and conditional propositions, involving the natural, temporal, extensional, and logical modalities. This is the first work ever to strictly formalize the inductive processes of generalization and particularization, through the novel methods of factorial analysis, factor selection and formula revision.

This is the first work ever to develop a formal logic of the natural, temporal and extensional types of conditioning as distinct from logical conditioning , including their production from modal categorical premises. First published ; rev. Judaic logic is an original inquiry into the forms of thought determining Jewish law and belief, from the impartial perspective of a logician. It attempts to honestly estimate the extent to which the logic employed within Judaism fits into the general norms, and whether it has any contributions to make to them.

The author ranges far and wide in Jewish lore, finding clear evidence of both inductive and deductive reasoning in the Torah and other books of the Bible, and analyzing the methodology of the Talmud and other Rabbinic literature by means of formal tools which make possible its objective evaluation with reference to scientific logic.

Accordingly, the subsidiary term is the predicate of the minor premise and conclusion in subjectal a-fortiori, and their subject in predicatal a-fortiori. Because of the functional difference of the extremes, the arguments have opposite orientations. In subjectal argument, the positive mood goes from minor to major, and the negative mood goes from major to minor. In predicatal argument, the positive mood goes from major to minor, and the negative mood goes from minor to major. The symmetry of the whole theory suggests that it is exhaustive.

With regard to the above mentioned invalid moods, namely major-to-minor positive subjectals or negative predicatals, and minor-to-major negative subjectals or positive predicatals, it should be noted that the premises and conclusion are not in conflict. The invalidity involved is that of a non-sequitur, and not that of an antinomy. It follows that such arguments, though deductively valueless, can, eventually, play a small inductive role just as invalid apodoses are used in adduction.

We need not repeat everything we said about copulative arguments for implicational ones. We need only stress that moods not above listed, which go from major to minor or minor to major in the wrong circumstances, are invalid. The essentials of structure and the terminology are identical, mutatis mutandis ; they are two very closely related sets of paradigms. The copulative forms are merely more restrictive with regard to which term may be a subject or predicate of which other term; the implicational forms are more open in this respect. In fact, we could view copulative arguments as special cases of the corresponding implicational ones [47].

These are mere eductions the propositions concerned are equivalent, they imply each other and likewise their contradictories imply each other , without fundamental significance; but it is well to acknowledge them, as they often happen in practice and one could be misled. The important thing is always is to know which of the terms is the major more R and which is the minor less R. The arguments would work equally well P and Q being equivalent in them.

Nevertheless, we must regard such arguments as still, in the limit, a-fortiori in structure. It follows that each of the moods listed above stands for three valid moods: the superior listed , and corresponding inferior and egalitarian moods unlisted. In the latter case, R cannot serve as middle term, and the argument would not constitute an a-fortiori. Formal ambiguities of this sort can lead to fallacious a-fortiori reasoning [48]. A-fortiori logic can be extended by detailed consideration of the rules of quantity. These are bound to fall along the lines established by syllogistic theory. A subject may be plural refer to all, some, most, few, many, a few, etc.

The extensions the scope of applicability of any class concept which appears in two of the propositions the two premises, or a premise and the conclusion must overlap, at least partly if not fully. If there is no guarantee of overlap, the argument is invalid because it effectively has more than four terms.

In any case, the conclusion cannot cover more than the premises provide for. In that case, if the minor premise is general, so will the conclusion be; and if the minor premise is particular, so will the conclusion be indefinitely, note. This issue does not concern the middle and subsidiary terms R, S , since they are predicates. In predicatal argument, whether positive or negative, the issue is much simpler.

Since the minor premise and conclusion share one and the same subject the subsidiary term, S , we can quantify them at will; and say that whatever the quantity of the former, so will the quantity of the latter be. With regard to the remaining terms P, Q, R , they are all predicates, and therefore not quantifiable at will.

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The major premise must, of course, in any case be general. All the above is said with reference to copulative argument; similar guidelines are possible for implicational argument. In such situations, a separate inductive evaluation has to be made, before we can grant the a-fortiori inference. Another direction of growth for a-fortiori logic is consideration of modality. In the case of copulative argument, premises of different types and categories of modality would need to be examined; in the case of implicational argument, additionally, the different modes of implication would have to be looked into.

Here again, the issues involved are not peculiar to a-fortiori argument, and we may with relative ease adapt to it findings from the fields concerned with categorical and conditional propositions and their arguments.

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To avoid losing the reader in minutiae, we will not say any more about such details in the present volume. We can, additionally, easily convince ourselves of their logical correctness, through a visual image as in Cartesian geometry. Represent R by a line, and place points P and Q along it, P being further along the line than Q — all the arguments follow by simple mathematics.

However, the formal validation of valid moods, and invalidation of invalids, are essential and will now be undertaken. Consequently, a-fortiori arguments may be systematically explicated and validated by such reductions. We shall call the colloquial forms bulk forms, and the simpler forms to which they may be reduced their pieces. Again, this concerns the superior form.

Note that given the first two pieces, the superior, egalitarian and inferior bulk forms are exhaustive alternatives, since the available third pieces are so; that is, if any two are false, the third must be true. But in the broadest perspective, Rp and Rq may each be an exact magnitude, or a single interval, ranging from an upper bound to a lower bound including the limits , or a disjunction of several intervals; this can complicate things considerably. To keep things simple, and manageable by ordinary language, we will assume Rp and Rq to be, or behave as, single points on the R continuum; when P or Q are classes rather than individuals, we will just take it for granted that the propositions concerned intend that the stated relation through R is generally true of all individual members referred to, one by one.

We need also emphasize, though we will avoid dealing with negative commensuratives in the present work so as not to complicate matters unduly, that the strict contradictory of each bulk form is an inclusive disjunction of its three pieces. We may continue to use the same labels superior, egalitarian and inferior for negative propositions, even though in fact the meaning is reversed by negation, in order that the intent of the original positive forms be kept in mind. Thus viewed in pieces, the negations of major premises are clear enough; but we must forewarn that the negative versions of the bulk forms are easily misinterpreted.

Other interpretations might be put forward. For logicians as against grammarians the precise interpretation of variant forms is not so important; what matters is what conventions we need to establish, as close as possible to ordinary language, to assure full formal treatment. We can do this without affecting the versatility of language, because it is still possible to express alternative interpretations by means of the language already accepted as formal.

In subjectal argument the minor premise and conclusion have P or Q the extreme terms as subject and S the subsidiary term as predicate, whereas in predicatal argument they have S as subject and P or Q as predicate, but otherwise the form remains identical; for this reason, we may deal with all issues using a single paradigm, having X and Y as subject and predicate respectively and R as middle term.

In the broadest perspective, Ry may be an exact magnitude, or a single interval, ranging from an upper bound to a lower bound including the limits , or a disjunction of several intervals. Similarly for Rx. However, very commonly, Ry expresses the threshold of a continuous and open-ended range, as of which, and over and above which or under and below which, the consequent Y occurs; while Rx is often a point for an individual X or a limited range for the class of X.

Since negative suffectives unlike negative commensuratives are used in the primary forms of a-fortiori argument which we identified earlier, they must be given attention too. The strict contradictory of the above conjuncts of two categoricals and one comparative is an inclusive disjunction of their denials:. I will not here say more about such variants, but only wish to give the reader an idea of the complexities involved. In general, absolute precision can only be attained through the explicit listing of the pieces intended, be they positive, negative or unsettled.

Having sufficiently analyzed the propositional forms involved for our purposes here, we can now proceed with reductive work on a-fortiori argument proper. The positive moods here considered are the paradigms of the form; the negative moods are really derivative. The negative moods can always be derived from the positive moods by means of a reductio ad absurdum , just like in the validation of syllogisms or apodoses. Validation : translate the bulk forms into their pieces here, expressed as hypotheticals, for the sake of simplicity; these are, tacitly, of the extensional type, to be precise , and verify that the conclusion is implicit in the premises by well-established hypothetical arguments.

One can see, here, why, if the minor premise was with P rather than Q, no conclusion would be drawable i. Validation : translate the bulk forms into their pieces here, again, expressed as hypotheticals, for the sake of simplicity , and verify that the conclusion is implicit in the premises by standard hypothetical arguments.

One can see, here, why, if the minor premise was with Q rather than P, no conclusion would be drawable i. All the above is applicable equally to copulative and implicational a-fortiori argument, and as already stated the negative moods are easily derived. These dissections make evident the formal similarity and complementarity between subjectal and predicatal arguments. Although on the surface their uniformity is not very obvious, deeper down their essential symmetry becomes clear. And this serves to confirm the exhaustiveness of our treatment.

Also note: our ability to reduce a-fortiori argument to chains known as sorites of already established and more fundamental arguments, signifies that this branch of logic, though of value in itself, is derivative — a corollary which does not call for new basic assumptions. In view of the above and certain additional details mentioned below the formal definition of a-fortiori argument we would propose is, briefly put: a form of inference involving one commensurative and two suffective propositions, sharing four terms or theses.

A larger theory of a-fortiori would require much more sophisticated formal tools — a much more symbolic and mathematical treatment, which is outside of the scope of the present study. I do not want to go into overly picky detail; these are very academic issues. However, we might here succinctly consider the language through which we colloquially express such inhibitions to a-fortiori arguments, signifying thereby that the situation under consideration is abnormal.

The following are examples of such statements; they are not arguments, note well, but statements consisting of three sentences which signal an abnormal situation, inhibiting a-fortiori inference from the first two sentences to a denial of the third. Here, the condition R for S has an upper limit, which Q fits into, yet P surpasses. Similar statements may appear in predicatal form; for example:.

We should, however, note that there are similar statements, which do not inhibit a normally valid mood, but positively join sentences which would normally not be incompatible but merely unable to constitute a valid mood; for example:. In such cases, the relations between P, Q, R, and S might be such that inferences are not possible, or at least not without access to some contorted formulae.

We do not have, in ordinary language, stock phrases for such situations — in practice, if necessary, we switch to mathematical instruments. Let us, to begin with, take the following subjectal merging two valid moods into a compound argument :. If we deny the conclusion and retain the minor premise, we obtain the denial of the major premise.

Thus, the following secondary mood is valid:. Similarly, we can derive the predicatal moods by ad absurdum from the corresponding primaries; note that here the structure resembles third figure syllogism:. Here, as everywhere, the conclusion must be fully guaranteed by the premises.

Furthermore, strictly-speaking, these two predicatal conclusions are more general than they ought to be. They are true at least for cases of S ; assuming them to be true for more would be an unwarranted generalization; one can conceive that in cases other than S, the requirements of R, to be P or Q, are different. In primary a-fortiori, this issue does not arise, insofar as the commensurative proposition is major premise and implicitly given as general; but in secondary a-fortiori, i.

Note that in all valid secondaries, the suffective premises are of unequal polarity — this is what makes possible the drawing of a commensurative conclusion, which is never egalitarian. The proposed conclusion obviously cannot follow from the premises, because the premises are identical in form for the terms P and Q, and therefore there is nothing to justify their distinction in the conclusion.

In short, there is no conclusion of the proposed kind with the given premises. It follows that the primary arguments below, with a negative major premise commensurative and negative conclusion suffective , cannot be valid, either. For if they were, then the secondary argument just considered would have to be valid, too. That is, whether we try major to minor or minor to major form, whether with a superior or inferior shown in brackets or egalitarian similarly, though not shown below major premise, all such moods are invalid:. It follows that primary moods of the kind below are also invalid:.

All this is very understandable, because the negative commensurative propositions, which are the major premises of these invalid primary arguments, are all relatively weak bonds between their terms. The situation is similar to that of first-figure categorical syllogism with a particular or possible major premise, or similarly hypothetical syllogism with a lower-case major premise. One can further explore this issue by translating all the propositions involved from their bulk forms into their pieces; negatives, remember, emerge as disjunctions of hypotheticals and comparatives.

This is not a problem particular to a-fortiori, but may be found in syllogistic logic. We might in principle hope to find certain combinations of premises capable of yielding new valid moods. However, I can report that I have not found any, because the conceivable premises are always incompatible with each other. For example, given the premises:. Similarly, for other atypical combinations of premises. It may be that someone discovers valid derivative moods of this sort that I have not taken into consideration, but I doubt it.

In any event, any encounter with cases of this kind should be treated with great care: they are tricky. Also, keep in mind that, ontologically, R and NotR, viewed as ranges, are very distinct, their values not having a general one-for-one correspondence. The denial of any given value of R say, R1 is an indefinite affirmation in disjunction of all remaining values of R R2, R3, etc.

Some of the primary and secondary valid moods, already dealt with above, involved negative relationships; so that we have incidentally covered part of the ground. Copulative arguments of the sort under consideration are easy to validate, since we merely change predicate, positing a negative instead of negating a positive; for examples:. In contrast, in the corresponding implicational arguments shown below , try as we might to apply the same analytical validation procedure as we used for other implicational arguments translating bulk forms into their pieces , the proposed inferences are found to be illegitimate, because we cannot syllogistically derive the fourth piece needed to construct the concluding bulk form from the given data [59] :.

With regard to the proposed conclusion: we can infer from the given premises that P implies Rp, Rp implies Rs, Rs does-not-imply S; but whether P implies or does-not-imply S remains problematic, so that we cannot infer that P implies R enough not to imply S though note that if we were given as an additional premise that P does not imply S, we could infer the desired bulk conclusion.

With regard to the proposed conclusion: we can infer from the given premises that S implies Rs, Rs implies Rq, Rq does-not-imply Q; but whether S implies or does-not-imply Q remains problematic, so that we cannot infer that S implies R enough not to imply Q though note that if we were given as an additional premise that S does not imply Q, we could infer the desired bulk conclusion. There is therefore no automatic guarantee that permutation is acceptable, in any given field, and we should not be surprised when, as in the present context, we discover its invalidity.

To sum up the research: implicational a-fortiori, whether antecedental or consequental, involving the negative relationships, were found invalid , using the above mentioned and other methods. The above samples are positive; but it follows that negative versions are equally invalid, since otherwise positive moods could be derived from them by reductio ad absurdum.

The same results can be obtained with inferior and egalitarian major premises even though in the latter case more data is implied. To be precise, I did not prove the various irregular a-fortiori arguments to be invalid, but rather did not find any proof that they are valid. It is not inconceivable that someone else finds conclusive paths of inference, but in the absence of such proof of validity, we must consider the proposed moods invalid.

These findings allow us to conclude that, although the analogy between regular copulative and implicational arguments is very close, there are irregular cases where their properties diverge , and copulatives are found valid while analogous implicationals are found invalid. They are significant findings, in that:.

In the previous chapter, we considered the formal, deductive aspects of a-fortiori argument. In the present chapter, we shall relate our findings to past Jewish studies in this field, and also consider certain more inductive and epistemological issues. J ewish logic has long used and explicitly recognized a form of argument called qal vachomer meaning, lenient and stringent. According to Genesis Rabbah , Parashat Miqets , an authoritative Midrashic work, there are ten samples of such of argument in the Tanakh: of which four occur in the Torah which dates from the 13th century BCE, remember, according to Jewish tradition , and another six in the Nakh which spreads over the next eight or so centuries.

Countless more exercises of qal vachomer reasoning appear in the Talmud, usually signaled by use of the expression kol sheken. Hillel and Rabbi Ishmael ben Elisha include this heading in their respective lists of hermeneutic principles, and much has been written about it since then. In English discourse, as we saw in the previous chapter, such arguments are called a-fortiori ratione , Latin; meaning, with stronger reason and are usually signaled by use of the expression all the more. The existence of a Latin, and then English, terminology suggests that Christian scholars, too, eventually found such argument worthy of study influenced no doubt by the Rabbinical precedent [60].

But what is rather interesting, is that modern secular treatises on formal logic all but completely ignore it — which suggests that no decisive progress was ever achieved in analyzing its precise morphology. Their understanding of a-fortiori argument is still today very sketchy; they are far from the formal clarity of syllogistic theory. This is in fact not an example of a-fortiori argument, but merely of syllogism [61] , showing that there is a misapprehension still today.

It is not a logically valid argument, since it depends not on the form but on the content Ed. Many dictionaries and encyclopaedias do not even mention a-fortiori.


Qal vachomer logic was admittedly a hard nut to crack; it took me two or three weeks to break the code. The way I did it, was to painstakingly analyze a dozen concrete Biblical and Talmudic examples, trying out a great many symbolic representations, until I discerned all the factors involved in them. It was not clear, at first, whether all the arguments are structurally identical, or whether there are different varieties.

When a few of the forms became transparent, the rest followed by the demands of symmetry. Validation procedures, formal limitations and derivative arguments could then be analyzed with relatively little difficulty. Although this work was largely independent and original, I am bound to recognize that it was preceded by considerable contributions by past Jewish logicians, and in celebration of this fact, illustrations given here will mainly be drawn from Judaic sources.

The formalities of a-fortiori logic are important, not only to people interested in Talmudic logic, but to logicians in general; for the function of the discipline of logic is to identify, study, and validate, all forms of human thought.

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And it should be evident with little reflection that we commonly use reasoning of this kind in our thinking and conversation; and indeed its essential message is well known and very important to modern science. What seems obvious at the outset, is that a-fortiori logic is in some way concerned with the quantitative and not merely the qualitative description of phenomena. Aristotelian syllogism deals with attributes of various kinds, without effective reference to their measures or degrees ; it serves to classify attributes in a hierarchy of species and genera, but it does not place these attributes in any intrinsically numerical relationships.

This is very interesting, because — as is well known to students of the history of science — modern science arose precisely through the growing awareness of quantitative issues. Before the Renaissance, measurement played a relatively minimal role in the physical sciences; things were observed if at all mainly with regard to their qualitative similarities and differences.

Things were, say, classed as hot or cold, light or heavy, without much further precision. Modern science introduced physical instruments and mathematical tools, which enabled a more fine-tuned pursuit of knowledge in the physical realm. A-fortiori argument may well constitute the formal bridge between these two methodological approaches. Its existence in antiquity, certainly in Biblical and Talmudic times, shows that quantitative analysis was not entirely absent from the thought processes of the precursors of modern science. They may have been relatively inaccurate in their measurements, their linguistic and logical equipment may have been inferior to that provided by mathematical equations, but they surely had some knowledge of quantitative issues.

In the way of a side note, I would like to here make some comments about the history of logic. Historians of logic must in general distinguish between several aspects of the issue. I tend to believe that all forms of reasoning are natural; but it is not inconceivable that anthropologists demonstrate that such and such a form was more commonly practiced in one culture than any other [64] , or first appeared in a certain time and place, or was totally absent in a certain civilization.

Logic can be grasped and discussed in many ways; and not only by the formal-symbolic method, and not only in writing. Also, the question can be posed not only generally, but with regard to specific forms of argument. The question is by definition hard for historians to answer, to the extent that they can only rely on documentary evidence in forming judgments. But orally transmitted traditions or ancient legends may provide acceptable clues.

Some historians of logic seem to equate the subject exclusively with its third, most formal and literary, aspect see, for instance, Windelband, or the Encyclopaedia Britannica article on the subject.


But, even with reference only to Greek logic, this is a very limiting approach. Much use and discussion of logic preceded the Aristotelian breakthrough, according to the reports of later writers including Aristotle. Thus, the Zeno paradoxes were a clear-minded use of Paradoxical logic though not a theory concerning it. Note that granting a-fortiori argument to be a natural movement of thought for human beings, and not a peculiarly Jewish phenomenon, it would not surprise me if documentary evidence of its use were found in Greek literature which dates from the 5th century BCE or its reported oral antecedents since the 8th century ; but, so far as I know, Greek logicians — including Aristotle — never developed a formal and systematic study of it.

The dogma of the Jewish faith that the hermeneutic principles were part of the oral traditions handed down to Moses at Sinai, together with the written Torah — is, in this perspective, quite conceivable. We must keep in mind, first, that the Torah is a complex document which could never be understood without the mental exercise of some logical intuitions. Second, a people who over a thousand years before the Greeks had a written language, could well also have conceived or been given a set of logical guidelines, such as the hermeneutic principles.

They do not, it is true, appear to have been put in writing until Talmudic times; but that does not definitely prove that they were not in use and orally discussed long before. With regard to the suggestion by some historians that the Rabbinic interest in logic was a result of a Greek cultural influence — one could equally argue the reverse, that the Greeks were awakened to the issues of logic by the Jews.

The interactions of people always involve some give and take of information and methods; the question is only who gave what to whom and who got what from whom. The mere existence of a contact does not in itself answer that specific question; it can only be answered with reference to a wider context.

A case in point, which serves to illustrate and prove our contention of the independence of Judaic logic, is precisely the qal vachomer argument. The Torah provides documentary evidence that this form of argument was at least used at the time it was written, indeed two centuries earlier when the story of Joseph and his brothers, which it reports, took place. If we rely only on documentary evidence, the written report in Talmudic literature, the conscious and explicit discussion of such form of argument must be dated to at least the time of Hillel, and be regarded as a ground-breaking discovery.

To my knowledge, the present study is the first ever thorough analysis of qal vachomer argument, using the Aristotelian method of symbolization of terms or theses. The identification of the varieties of the argument, and of the significant differences between subjectal or antecedental and predicatal or consequental forms of it, seems also to be novel. We needed to show that there are legitimate forms of such argument, which are not mere flourishes of rhetoric designed to cunningly mislead, but whose function is to guide the person s they are addressed to through genuinely inferential thought processes.

This we have done in the previous chapter. With regard to Hebrew terminology. The general word for premise is nadon that which legalizes; or melamed , that which teaches , and the word for conclusion is din the legalized; or lamed , the taught. I do not know what the accepted differentiating names of the major and minor premises are in this language; I would suggest the major premise be called nadon gadol great , and the minor premise nadon katan small.

Note also the expressions michomer leqal from major to minor and miqal lechomer from minor to major. This usage could be misleading, and is best avoided. Let us now, with reference to cogent examples, check and see how widely applicable our theory of the qal vachomer argument is thus far, or whether perhaps there are new lessons to be learnt. I will try and make the reasoning involved as transparent as possible, step by step. The reader will see here the beauty and utility of the symbolic method inaugurated by Aristotle. Biblical a-fortiori arguments generally seem to consist of a minor premise and conclusion; they are presented without a major premise.

This was known to Aristotle, and did not prevent him from developing his theory of the syllogism. Such incomplete arguments, by the way, are known as enthymemes the word is of Greek origin. The missing major premise is, in effect, latent in the given minor premise and conclusion; for, granting that they are intended in the way of an argument, rather than merely a statement of fact combined with an independent question, it is easy for any reasonably intelligent person to construct the missing major premise, if only subconsciously.

If the middle term is already explicit in the original text, this process is relatively simple. In some cases, however, no middle term is immediately apparent, and we must provide one however intangible which verifies the argument. In such case, we examine the given major and minor terms, and abstract from them a concept, which seems to be their common factor.

To constitute an appropriate middle term, this underlying concept must be such that it provides a quantitative continuum along which the major and minor terms may be placed. Effectively, we syllogistically substitute two degrees of the postulated middle term, for the received extreme terms.

Note that a similar operation is sometimes required, to standardize a subsidiary term which is somewhat disparate in the original minor premise and conclusion. We are logically free to volunteer any credible middle term; in practice, we often do not even bother to explicitly do so, but just take for granted that one exists. Of course, this does not mean that the matter is entirely arbitrary. In some cases, there may in fact be no appropriate middle term; in which case, the argument is simply fallacious since it lacks a major premise.

But normally, no valid middle term is explicitly provided, on the understanding that one is easy to find — there may indeed be many obvious alternatives to choose from and this is what gives the selection process a certain liberty. It is the third occurrence of the argument in the Chumash , or Pentateuch Numbers, God has just struck Miriam with a sort of leprosy for speaking against her brother, Moses; the latter beseeches God to heal her; and God answers:.

If her father had but spit in her face, should she not hide in shame seven days? If we reword the argument in standard form, and make explicit what seems to be tacit, we obtain the following. Concerning the subsidiary term these propositions have in common, note that it is not exactly identical in the two original sentences; we made it uniform by replacing the differentia hiding and being shut up with their commonalty being in isolation. God tells Moses to go back to Pharaoh, and demand the release of the children of Israel; Moses replies:.

Behold, the children of Israel have not hearkened unto me; how then shall Pharaoh hear me, who am of uncircumcised lips? Here again, we were only originally provided with a minor premise and conclusion; but their structural significance two subjects, a common predicate and polarity were immediately clear.

Concerning our choice of middle term. It is a positive predicatal a-fortiori. Here again, the middle term honesty was only implicit in the original text. The original argument is in fact positive predicatal in form, and it is the fourth and last example of qal vachomer in the Pentateuch:. For I know thy rebellion, and thy stiff neck; behold, while I am yet alive with you this day, ye have been rebellious against the Lord; and how much more after my death? The constructed major premise is common sense. We have thus illustrated all four moods of copulative qal vachomer argument, with the four cases found in the Torah.

For the record, I will now briefly classify the six cases which according to the Midrash occur in the other books of the Bible. The reader should look these up, and try and construct a detailed version of each argument, in the way of an exercise. In every case, the major premise is tacit, and must be made up. Jeremiah, This is a positive antecedental in fact, there are two arguments with the same thrust, here. According to him, there is no undefeatable argu- ment, not even the a fortiori argument.

The a fortiori is then not set apart from the other types of arguments. Instead, they are heuristic reconstructions of real trends within the history of theories of argumentation. Logical theory of a fortiori The logical theory of argumentation deals with validity of argu- ments and not with their persuasiveness. An argument is consid- ered either valid or invalid. There are all sorts of syllogisms divided in two categories depending on whether they are valid and never defeatable or invalid and always defeatable. There is no place, ac- cording to the law of excluded middle, for relatively valid syllogisms or arguments in general.

The a fortiori argument is given several accounts. McCall con- siders a fortiori arguments as both oblique and syllogistic. Usually, being oblique is a weakness in logic, whereas the valid syllogistic form indicates a well-formed logical proposition.

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Avi Sion has devoted many pages to a fortiori arguments within the Bible and the Talmud. We do not claim to provide the reader with such a logical proof — Sion is right. The a fortiori argument is not only a logical but also linguistic device. This is why a logical approach to the a fortiori argument is insufficient to grasp its linguistic specificity.

Two-dimensional theory of a fortiori Two-dimensionalism in argumentation has been sketched in Goltzberg [4]. This theory considers that both defeatable and unde- featable arguments are to be accounted for by a comprehensive view of argumentation. Logical and topical arguments are two di- mensions within argumentation and it would be misguiding to re- duce argumentation either to logics or topics. Our hypothesis is that arguments are not defeatable or unde- featable in themselves but presented as defeatable or undefeatable [2, p.

This by no means leads to a relativism according to which nothing would be sure in itself. An argument is never nude but always accompanied by a commentary, an in- struction on how exactly the argument is to be taken. Most of the time, the argument provides the listener or the reader with instruc- tions as to how to interpret it. If argumentation has to do with presentation of argument, let us ask: how exactly are the arguments presented?

Arguments are structured by two main parameters: orientation and strength. The four types of arguments may be analyzed through the following transitional keywords examples. Keywords may be co-oriented or counter-oriented and stronger or weaker. This dialectical dimension is not sufficiently highlighted in Goltzberg [4]. In order to assess the strength of an argument, one should understand the strength of the various arguments that come into the picture.

Let us now move to the strength parameter: in 1 even if introduces an argument q that is presented as weaker, which makes the main claim p stronger. In 2 unless introduces an argument q presented as stronger, which makes p weaker. In 3 or even introduces an argu- ment q presented as stronger, which makes p weaker. It also makes the entire claim weaker, because part of it — the part or even q — is more risky.

In 4 or at least introduces an argument q presented as weaker, which makes p and the general claim stronger. Not only do transitional keywords make it possible to ascribe a certain strength to one argument; they are also able to distribute the strength to each relevant part of the line of argumentation.

Let us come back to the a fortiori argument: it contains an ar- gument introduced by at least that is presented as weak in the pre- cise sense that the speaker could have afforded to claim more. It is stronger because it demands less than it could. Let us consider these two examples: 4 He can run 10 miles or at least 5 miles. Whereas the claim as to the 10 miles is cancelled in 4 , in 5 , the a fortiori does not cancel or undermine the first part of the sentence: since he can run 10 miles, he can for sure run at least 5 miles. This is why he demands to be heard all the more when his claim is weaker.

The fact that he diminishes his claim makes it stronger if he sticks to the first claim too. Talmudic theory of a fortiori It is sometimes asked whether Talmudic argumentation is different from other types of discourses. When it comes to a fortiori argu- mentation our question is: what is specific about a fortiori in the Talmud? Three potential answers deserve attention: 1 the dayo, 2 the autonomous use of a fortiori and 3 the interdiction of punish- ment on the basis of an a fortiori. Let us recall the meaning of the dayo device: this instruction aims at insisting on the fact that the second situation deserves the judgment applied to the first situa- tion, in a degree that is at least as great but not greater.

The Talmud would have added the dayo device and transformed thereby the very structure and use of a fortiori arguments. This meets a prohibitive objection: dayo, as a claim that the second situation be treated precisely as the former, is not added to the a fortiori argument. It is simply inherent in it. In fact, this issue deserves a closer scrutiny. This point merits wider examination. The freedom of utilization a fortiori probably originates from the fact it is a strong ar- gument, if not undefeatable.

This point is a general feature of the a fortiori argument that has been focused on by the Talmud, even though the next point somehow undermines the force of the a fortiori by limiting its application.