Chapter 4. QAL VACHOMER.
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.
Jewish logic has long used and explicitly recognized a form of argument called qal vachomer (meaning, lenient and stringent). According to Genesis Rabbah (92:7, 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). 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.
Witness for instance the example given in an otherwise quite decent Dictionary of Philosophy: “If all men are mortal, then a fortiori all Englishmen – who constitute a small class of all men – must also be mortal”. This is in fact not an example of a-fortiori argument, but merely of syllogism, showing that there is a misapprehension still today. Or again, consider the following brief entry in the Encyclopedie Philosophique Universelle: “A fortiori argument rests on the following schema: x is y, whereas relatively to the issue at hand z is more than x, therefore a fortiori z is y. It is not a logically valid argument, since it depends not on the form but on the content (Ed.)“. The skeptical evaluation made in this case is clearly only due to their inability to apprehend the exact formalities; yet the key is not far, concealed in the clause “relatively to the issue at hand”. 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. 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. Aristotelean 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. The only “quantity” which concerns it, is the extrinsic count of the instances to which a given relationship applies (which makes a proposition general, singular or particular).
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.
(a) The art or practise of logic: as an act of the human mind, an insight into the relations between things or ideas, logic is part of the natural heritage of all human beings; it would be impossible for us to perform most of our daily tasks or to make decisions without some exercise of this conceptual power. 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 practised in one culture than any other, or first appeared in a certain time and place, or was totally absent in a certain civilization.
(b) The theoretical awareness and teaching of logic: at what point in history did human beings become self-conscious in their use of reasoning, and began to at least orally pass on their thoughts on the subject, is a moot question. 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 judgements. But orally transmitted traditions or ancient legends may provide acceptable clues.
(c) The written science of Logic, as we know it: the documentary evidence (his written works, which are still almost totally extant) points to Aristotle (4th century BCE) as the first man who thought to use symbols in place of terms, for the purpose of analyzing various eductive and syllogistic arguments, involving the main forms of categorical proposition. Since then, the scope of formal logic has of course greatly broadened, thanks in large measure to Aristotle’s admirable example, and findings have been systematized in manifold ways.
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 Aristotelean 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). Or again, Socrates’ discussions (reported by his student Plato) about the process of Definition may be classed as logic theorizing, though not of a formal kind.
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. These were not, admittedly, logic theories as formal as Aristotle’s; but they were still effective. 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 Aristotelean 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.
Our first job was to formalize a-fortiori arguments, to try and express them in symbolic terms, so as to abstract from their specific contents what it is that makes them seem “logical” to us. 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 major, minor and middle terms are called: chomer (stringent), qal (lenient), and, supposedly, emtsa’i (intermediate). 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).
I have noticed that the expression “qal vachomer” is sometimes used in a sense equivalent to “kol sheken” (all the more), and intended to refer to the minor premise and conclusion, respectively, whatever the value of the terms that these propositions involve (i.e. even if the former concerns the major term, and the latter concerns the minor term), because the conclusion always appears more ‘forceful’ than the minor premise. 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. They are worded in typically Jewish fashion, as a question: “this and that, how much more so and so?” The question mark (which is of course absent in written Biblical Hebrew, though presumably expressed in the tone of speech) here serves to signal that no other conclusion than the one suggested could be drawn; the rhetorical question is really “do you think that another conclusion could be drawn? no!”
Concerning the absence of a major premise, it is well known and accepted in logic theorizing that arguments are in practise not always fully explicit (meforash, in Hebrew); either one of the premises and/or the conclusion may be left tacit (satum, in Hebrew). This was known to Aristotle, and did not prevent him from developing his theory of the syllogism. We naturally tend to suppress parts of our discourse to avoid stating “the obvious” or making tiresome repetitions; we consider that the context makes clear what we intend. 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 practise, 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).
(1) Let us begin our analysis with a Biblical sample of the simplest form of qal vachomer, subjectal in structure and of positive polarity. It is the third occurrence of the argument in the Chumash, or Pentateuch (Numbers, 12:14). Gd has just struck Miriam with a sort of leprosy for speaking against her brother, Moses; the latter beseeches Gd to heal her; and Gd answers:
If her father had but spit in her face, should she not hide in shame seven days? let her be shut up without the camp seven days, and after that she shall be brought in again.
If we reword the argument in standard form, and make explicit what seems to be tacit, we obtain the following.
“Divine disapproval (here expressed by the punishment of leprosy)” (=P) is more “serious disapproval” (=R) than “paternal disapproval (signified by a spit in the face)” (=Q);
if paternal disapproval (Q) is serious (R) enough to “cause one to be in isolation (hide) in shame for seven days” (=S),
then Divine disapproval (P) is serious (R) enough to “cause one to be in isolation (be shut up) in shame for seven days” (=S).
Note that the middle term (seriousness of disapproval) was not explicit, but was conceived as the common feature of the given minor term (father’s spitting in the face) and major term (Gd afflicting with leprosy). 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). More will be said about the specification “for seven days” in the subsidiary term (S), later.
(2) A good Biblical sample of negative subjectal qal vachomer is that in Exodus, 6:12 (it is the second in the Pentateuch). Gd 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?
This argument may be may be construed to have run as follows:
The children of Israel (=P) “fear Gd” (=R) more than Pharaoh (=Q) does;
yet, they (P) did not fear Gd (R) enough to hearken unto Moses (=S);
all the more, Pharaoh (Q) will not fear Gd (R) enough to hear Moses (S).
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. The major premise, however, had to be constructed; we used a middle term which seemed appropriate – “fear of Gd”.
Concerning our choice of middle term. The interjection by Moses, “I am of uncircumcised lips”, which refers to his speech problem (he stuttered), does not seem to be the intermediary we needed, for the simple reason that this quality does not differ in degree in the two cases at hand (unless we consider that Moses expected to stutter more with Pharaoh than he did with the children of Israel). Moses’ reference to a speech problem seems to be incidental – a rather lame excuse, motivated by his characteristic humility – since we know that his brother Aaron acted as his mouthpiece in such encounters.
In any case, note in passing that the implicit intent of Moses’ argument was to dissuade Gd from sending him on a mission. Thus, an additional argument is involved here, namely: “since Pharaoh will not hear me, there is no utility in my going to him” – but this is not a qal vachomer.
(3) The first occurrence of qal vachomer in the Torah – and perhaps historically, in any extant written document – is to be found in Genesis, 44:8 (it thus dates from the Patriarchal period, note). It is a positive predicatal a-fortiori. Joseph’s brothers are accused by his steward of stealing a silver goblet, and they retort:
Behold, the money, which we found in our sacks’ mouths, we brought back unto thee out of the land of Canaan; how then should we steal out of thy lord’s house silver or gold?
According to our theory, the argument ran as follows:
You will agree to the general principle that more “honesty” (=R) is required to return found money (=P) than to refrain from stealing a silver goblet (=Q);
and yet, we (=S) were honest (R) enough to return found money (P);
therefore, you can be sure that we (S) were honest (R) enough to not-steal the silver goblet (Q).
Here again, the middle term (honesty) was only implicit in the original text. The major premise may be true because the amount of money involved was greater than the value of the silver goblet, or because the money was found (and might therefore be kept on the principle of “finders keepers”) whereas the goblet was stolen; or because the positive act of returning something is superior to a mere restraint from stealing something.
(4) There is no example of negative predicatal a-fortiori in the Torah; but I will recast the argument in Deuteronomy, 31:27, so as to illustrate this form. 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 Lrd; and how much more after my death?
We may reword it as follows, for our purpose:
More “self-discipline” (=R) is required to obey Gd in the absence of His emissary, Moses (=P), than in his presence (=Q);
the children of Israel (=S) were not sufficiently self-disciplined (R) to obey Gd during Moses’ life (Q);
therefore, they (S) would surely lack the necessary self-discipline (R) after his death (P).
In this case, note, the middle term was effectively given in the text; “self-discipline” is merely the contrary of disobedience, which is implied by “stiff neck and rebelliousness”. 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.
Samuel I, 23:3. This is a positive antecedental.
Jeremiah, 12:5. This is a positive antecedental (in fact, there are two arguments with the same thrust, here).
Ezekiel, 15:5. This is a negative subjectal.
Proverbs, 11:31. This is a positive subjectal.
Esther, 9:12. This is a positive antecedental (if at all an a-fortiori, see discussion in a later chapter).
The following is a quick and easy way to classify any Biblical example of qal vachomer:
(a) What is the polarity of the given sentences? If they are positive, the argument is a modus ponens; if negative, the argument is a modus tollens.
(b) Which of the sentences contains the major term, and which the minor term? If the minor premise has the greater extreme and the conclusion has the lesser extreme, the argument is a majori ad minus; in the reverse case, it is a minori ad majus.
(c) Now, combine the answers to the two previous questions: if the argument is positive and minor to major, or negative and major to minor, it is subjectal or antecedental; if the argument is positive and major to minor, or negative and minor to major, it is predicatal or consequental.
(d) Lastly, decide by closer scrutiny, or trial and error, whether the argument is specifically copulative or implicational. At this stage, one is already constructing a major premise.
I will here only give one example of the more complex, implicational form of qal vachomer. It is described in the Encyclopaedia Judaica (8:367), as follows: “It is stated in Deuteronomy 21:23 that the corpse of a criminal executed by the court must not be left on the gibbet overnight, which R. Meir takes to mean that Gd is distressed by the criminal’s death. Hence, R. Meir argues:… (Sanh. 6:5).”
If Gd is troubled at the shedding of the blood of the ungodly, how much more at the blood of the righteous!
This is evidently a positive antecedental argument; verbalized more fully, it would be stated as follows:
“The shedding of the blood of the righteous” (=P) is generally more “troubling” (=R) than “the shedding of the blood of the ungodly” (=Q);
if “the blood of the ungodly is shed” (Q), then Gd is to some extent “troubled” (R), specifically to the extent of enacting the law in Deuteronomy 21:23 (=S);
therefore, if “the blood of the righteous is shed” (P), then Gd is to some extent “troubled” (R), to an extent not here specified but at least similar to the previous (S).
The middle term (trouble) is in this case given in the original text. It is not expressed identically in the major premise (troubling) and the other propositions (troubled), but this is a turn of language which is easily remedied. The major premise could have been expressed more elliptically as “P implies, for any subject, that he will be troubled, more than Q implies”. Note the absence of an explicit consequent (subsidiary thesis) in the conclusion, and our use of the clause “to some extent”; more will we said about this later.
We will have occasion to discuss other examples of implicational qal vachomer, drawn from the Gemara, in a later section.
Rabbinical logicians raised an important question in relation to certain qal vachomer arguments. For instance, in the argument about Miriam (which we analyzed in the previous section), the minor premise posits a punishment of seven days for a relatively lesser crime, and the conclusion likewise posits a punishment of seven days for a relatively greater crime. Why only seven days? they wondered; should not the punishment be more, proportionately to the severity of the crime? A reasonable question.
Since the sample argument is of Divine origin, some Rabbis postulated that it suggests a universal logical rule, namely that the conclusion of a qal vachomer can never go further than the minor premise, in the specification of the measure or degree of the terms involved. They called this, the dayo (sufficiency) principle (see Baba Qama, 2:5). Other Rabbis, like R. Tarphon (in Baba Qama, 25a), did not concur, but regarded a proportionate inference as permissible, at least in some cases. For my part, I would like to say the following.
In the argument concerning Miriam, it can easily be countered that Gd sentenced her in the conclusion to only seven days incarceration out of sheer mercy, though she might have been strictly-speaking subject to infinitely more; and that in any case, the seven days mentioned in the minor premise are not known through natural human insight, but equally through Divine fiat. Thus, this example does not by itself resolve the issue incontrovertibly.
If we compare, for instance, the argument made by R. Meir (also previously mentioned), we see that going beyond the given quantity is intuitively quite reasonable. Here, the minor premise is that Gd is (to some unspecified degree) troubled by the blood of a criminal, but the intended conclusion is that He is troubled even more (to a greater, though also unspecified degree) by the blood of an innocent. It has to be so, because the concrete expression of the distress of Gd, in the first case, is that the court must remove the criminal’s corpse before nightfall; the implied obligation, in the second case, cannot be the same, since the court would not execute an innocent – it is rather a general prescription that good people be treated still better than bad people.
Note, however, that in both our examples, the quantitative factor at issue may be made to stand somewhat outside the regular terms of the a-fortiori argument as such. In both cases, it is not the quantitative difference between the major and minor terms which is at issue; that is already given (or taken for granted) in the major premise. What is at issue is a quantitative evaluation of the remaining terms, the middle term and the subsidiary term, as they appear in the minor premise and conclusion.
According to our theory, the outward uniformity of these terms in those propositions is a formal feature of a-fortiori argument. But this feature does not in itself exclude variety at a deeper level. Such specific differences are side-issues which the a-fortiori argument itself cannot prejudge. It takes supplementary propositions, in a separate argument, which is not a-fortiori but purely mathematical in form, to make inferences about the precise quantitative ramifications of the a-fortiori conclusion.
Thus, we may acknowledge the dayo principle as correct, provided it is understood as being a minimal position. It does not insist on the quantitative equality of the subsidiary or middle term (as the case may be) in the conclusion and minor premise, nor does it interdict an inequality; it merely leaves the matter open for further research. A-fortiori argument per se does not answer the question; it is from a formal point of view as compatible with equality as with inequality. To answer the question, additional information and other arguments must be sought. This is a reasonable solution.
Generally speaking, what is needed ideally is some mathematical formula which captures the concomitant variation between a term external to the a-fortiori argument as such (e.g. amount of punishment), and a term of variable value implicit in the a-fortiori (e.g. severity of the sin). This formula then stands as the major premise in a distinct argument, whose minor premise and conclusion contain the indefinite term at issue in the a-fortiori argument (the middle or subsidiary term, as the case may be, to repeat) as their common subject, and the said external term’s values as their respective predicates.
There is no guarantee, note well, that the variation in the major premise will be an arithmetical proportionality; it could just as well be an inverse proportionality or a much more complex mathematical relationship, even one involving other variables. This is why the a-fortiori argument as such cannot predict the result; its premises lack the information required for a more refined conclusion. In some cases, the concomitance is simple and well known, and for this reason seems to be an integral part of the a-fortiori; but this is an illusion, the proof being that it does not always work, and in more complex cases a separate judgement must be made.
Let us now analyze the issue underlying the dayo principle in more formal terms. Consider a positive subjectal a-fortiori, whose subsidiary term (S) is a conjunction of two factors, a constant (say, K) and a variable (say, V); and suppose V is a function (f) of the middle term (R), i.e. that V = f(R) in mathematical language. On a superficial level, the argument is simply as follows:
P is more R than Q,
and, Q is R enough to be S;
therefore, P is R enough to be S.
But “R enough” is a threshold, it is not a fixed quantity. In the case of the minor premise, involving Q, the value of R is Rq, say; whereas, in the case of the conclusion, involving P, the value of R is Rp, say; and we know from the major premise that Rp is greater than Rq. Looking now at S, it is evident that if it consists only of a constant (K), it will be identical in the minor premise and the conclusion. But, if S involves a variable V, where V is a function of R, then S is not necessarily exactly the same in both propositions. If V = f(R) represents a straightforward linear relationship, then Vp = f(Rp) will predictably be proportionately greater than Vq = f(Rq); but if V = f(R) represents a more complicated relationship, then Vp = f(Rp) may be more or less than Vq = f(Rq), or equal to it, depending on the specifics of the formula.
Similar comments can be made with regard to the other valid moods of qal vachomer. Note in any case that all this is well and good in principle; but in practise, we may not be able to provide an appropriate and accurate mathematical equation. Some phenomena are difficult and even impossible to measure; we may know that they somehow vary, but we may have no instruments with which to determine the variations, precisely or at all.
The physical sciences acquired enormous prestige, because they concentrated their efforts on accessible phenomena (at least until the advent of sub-atomic physics, where according to the Heisenberg Principle precise determinations become in principle impossible, in view of the influence of available experimental means on the matter observed). Measurement is currently more difficult in biological or psychological contexts. In the still more abstract realm of ethical and legal discussions, not to mention purely spiritual issues, objective means are well-nigh non-existent, and we have to refer to Biblical hints or intuitive conventions to establish scales.
Nevertheless, if that is a consolation, what is of interest to us here is the essential similarity in principle – with regard to the formal logic involved – between all human endeavors in the pursuit of knowledge. The proverbial superiority of modern physical sciences, in view of their powers of measurement, is relative and incidental. Their epistemological tools are no different than those of any other discipline. Other disciplines may be equally “scientific”, in the root sense of the word, which refers to knowledge acquired through the strictest methodology; they are not totally incapacitated by the strictures which their peculiar subject-matter imposes on them.
The subject-matter of physics is relatively easy of access, so that it can measure more and achieve greater precision; other domains are progressively more difficult to deal with, and so the (in the widest sense) scientific endeavors which concern them are bound to be accordingly limited. But the requirements of objectivity of attitude, open-mindedness to new data, carefulness in reasoning, and honesty, are the same throughout; and this is what counts in evaluating any body of knowledge.
The formalization of a-fortiori argument has been found difficult by past logicians for various reasons. (a) The complexity and variety of the propositional forms involved. (b) There are many varieties of the argument. (c) Known samples are usually incompletely formulated. (d) Known samples often intertwine a mixture of purely a-fortiori and other forms of deductive inference. (e) The deductive and inductive issues were not adequately separated. We will clarify these matters in the present section.
Thus far, our goal has been to discover the essential form(s) of a-fortiori argument. We found the various kinds of premises and conclusion which ideally constitute such movements of thought. As in all formal logic, the conclusion follows from the premises; if the premises are true, then the conclusion is true. The presentation of a form of argument as valid does not in itself guarantee the truth of the premises. If any or all of the premises are not true, then the conclusion does not follow; the conclusion may happen to be false too, or it may be true for other reasons, but it is in any case a non sequitur.
This understanding of the relationship of premises and conclusion is not a special dispensation granted to our theory of a-fortiori, but applies equally well to all inference, be it eductive, syllogistic or otherwise deductive, or even inductive. In all cases, the question arises: how are the premises themselves known? And the answer is always: by any of the means legitimatized by the science of logic. A premise may be derived from experience by inductive arguments of various kinds, or be a logical axiom in the sense that their contradictories are self-denying, or even be Divinely revealed; or it may be deductively inferred in one way or another from such relatively primary propositions (whether they are a posteriori or a priori, to use the language of philosophers).
This issue has been acknowledged in the literature on Talmudic logic, through the doctrine of objection (in Hebrew, teshuvah; in Aramaic, pirka). A given a-fortiori argument, indeed any argument, may be criticized on formal grounds, if it is shown not to constitute a valid mood of reasoning. But it may also be objected to on material grounds, by demonstrating one or both of its premises is/are wholly or partly false, or at least open to serious doubt. The deduction as such may be valid, but its inductive backing (in the widest sense) may be open to doubt.
Consider for examples the Biblical samples of qal vachomer we have used as our illustrations.
In the argument concerning Miriam, we were given two sentences, neither of which is in itself obvious. Assuming that the Biblical verse as a whole is indeed intended as an argument, and not as two unrelated assertions, we may regard the first as a Divinely guaranteed truth and use it as our minor premise, but the second must somehow emerge as a conclusion. However, the major premise, which we ourselves construct to complete the argument, is in principle not indubitable. The one we postulated happens to seem reasonable (i.e. appears to be consistent with the rest of our knowledge); but it is conceivable that some objection could eventually be raised concerning it (say, that Gd attaches more importance to sins against parents than to sins against Himself).
In the next argument, by Moses, the major and minor premises are both known by empirical means. The former is a generalization, based on the past behavior patterns of the children of Israel and Pharaoh; and the latter is a statement concerning more recent events. These propositions happen to be true, so that the conclusion is justified, but they might conceivably have been factually inaccurate, in which case an objection could have been raised.
The argument made by Joseph’s brothers is much more open to debate. The steward might have argued that they returned the money they found out of some motive(s) other than the sheer compulsion of their honest natures: (a) to liberate their brother Simeon, which had been kept hostage (see Genesis, 42:24 and 43:23); or (b) because the famine in Canaan forced them to come back to Egypt (see 43:1); or even (c) because they feared eventual pursuit and retaliation; or simply (d) because the silver cup, being a tool for divining purposes, had more value than the sacks full of money, and thus tempted them to take more risks.
We accept the brothers’ argument, because we believe that their honesty proceeded from their exceptional fear of Gd (irrespective of any more down to earth concerns), but it is not unassailable. Clearly, the empirical foundations of the major premise are rather complex, and an additional complication is the rather abstract psycho-ethical concept (namely, honesty) it involves. With regard to the minor premise, about the restitution of money – that was a straightforward observation of a singular physical event. In any case, this example well illustrates the inductive issues which may underlie an a-fortiori argument.
In the case of the argument by Moses concerning the stiff-neck and rebellion of the children of Israel, the major premise might be construed as a generalization from common experience. We know that children are less well behaved in the presence of their parents or school-teachers than in their absence, and similarly that people follow their leaders more strictly when their leaders’ backs are not turned – and on this basis, the postulated major premise seems reasonable. But it might well be argued that though this is more often than not true, it is not always true (the children of Israel are indeed requested by Moses to make it untrue!) – and thus put the whole argument in doubt, or at least make it probable rather than necessary. As for the minor premise, it could be viewed as an overly severe evaluation of the behavior of the children of Israel – there is a subjective aspect to it.
Similar comments can be made with respect to Rabbi Meir’s argument, demonstrating its possible weaknesses. We need not belabor the matter further. All this goes to prove, not as some logicians have claimed that a-fortiori argument is in principle without formal validity, but that it is often difficult to find solid material grounds for its effective exercise. It is thus understandable why Rabbinical legislators have usually regarded qal vachomer arguments as insufficient in themselves to justify a law, unless supported by the authority of tradition.
I would like now, in the way of a final illustration and test of our theory, to analyze an a-fortiori argument given in the Encyclopaedia Judaica. It is drawn from the Talmud (Chulin, 24a), which bases the argument on certain passages of the Torah (Leviticus, 21:16-21; and Numbers, 8:24-25). The argument seems complicated, but it is simply, as we shall see, positive antecedental in form; I quote:
If priests who are not disqualified for service in the Temple by age, are disqualified by bodily blemishes; then Levites, who are disqualified by age, should certainly be disqualified by bodily blemishes.
The clue to a solution is in the verb involved; we notice that the central issue under discussion is the threshold of disqualification from Temple service. Our middle term (R), then, must be a concept with many different degrees (say, “unfitness”), such that there is a cut-off point along it, which signifies the occurrence of disqualification; this is effectively the subsidiary term (S), which will be the consequent of our minor premise and conclusion.
“Having bodily blemishes” (=P) implies more “unfitness for Temple service” (=R) than “being past a certain age” (=Q);
if a Levite reaches that age (Q), he is sufficiently unfit (R) that “he is disqualified” (=S);
therefore, all the more, if a Levite has bodily blemishes (P), he is sufficiently unfit (R) to be disqualified (S).
Note that the antecedents of the minor premise and conclusion, respectively, contain the minor and major terms, which cause the requisite degree of unfitness for disqualification. We see that this argument is identical in form to that of R. Meir, which we previously analyzed. What distinguishes it, however, is the way we construct the major premise. In the R. Meir argument, no explicit source is given for the major premise; but in the present example, we do have some additional data with which to justify our major premise.
The a-fortiori argument as such makes no mention of the priests; it only concerns the Levites. The logical utility of the statements in the original text about priests, is to serve as a springboard from which we can leap to the needed major premise. The two propositions “priests are not disqualified by old age” and “priests are disqualified by bodily blemishes”, provide us with the Scriptural grounds for an inductive generalization to the proposition “Bodily blemishes more easily disqualify than old age”, which in turn becomes our major premise.
An argument by analogy is involved, when we move from the case of priests, to all cases (all Temple servers), including eventually the case of Levites. This argument is not formally unassailable; the Torah might well have made a fine distinction, and allowed Levites with bodily blemishes to serve in the Temple (in view of their distinctive functions there). Two subjects can always have opposite predicates, without doing violence to logic. However, since the Torah does not in fact make such a distinction, we may reasonably generalize as the Rabbis did.
Thus, to summarize, not all of the Talmudic passage under discussion constituted an a-fortiori argument. The first section, concerning priests, was not an inherent part of the qal vachomer inference per se, but served as the premise of a preliminary inductive argument (namely, a generalization) which established the major premise of the qal vachomer as such. Only the second section, about Levites, belongs within the qal vachomer process proper.
In this context, I would like to criticize and reject the theory of qal vachomer arguments proposed by the author, L. Jacobs (presumably), of the aforementioned Encyclopaedia Judaica article. He rightly (together with Kunst) dismisses the claim by some researchers (notably, A. Schwartz), that they may be identified with syllogistic reasoning; for the latter serve only the eventual purposes of subsumption of individuals in classes, or classes in classes-of-classes. However, Jacobs’ own analysis of the topic is also faulty.
Jacobs’ effort at formalization is not only an inadequate oversimplification, but also contrary to reason. He claims that the (above mentioned) argument of R. Meir (which he labels “simple”) can be formalized as “if A has x, then certainly B has x”; but this explains nothing, it does not tell us why the inference is at all possible, because it is too vague. Similarly, he formalizes the argument about the priests and Levites (which he contrasts as “complex”) as follows: “if A, which lacks y, has x, then B, which has y, certainly has x”; but this is absurd! The arrow is pointing in opposite directions in the antecedents (in one case against y, in the other case towards y), and then it flips over and points in the same direction in the consequents (toward x)!
Clearly, more precise formal tools, more careful logic and more perspicacious linguistic analyses, were needed to solve the mystery of qal vachomer. I believe that the theory I have proposed offers a definitive solution.
An important test of our general forms of qal vachomer, is their applicability to the formulation of a-fortiori argument traditionally made in the Rabbinic literature. Some logicians, like R. Luzatto (also known as the Ramchal), have a pretty large concept of qal vachomer, which includes any kind of scale of comparison as the effective middle term. However, most authors seem to limit their concept to one specific kind of middle term, namely the concept of ‘legal restriction’. Thus, for instance, R. Chavel describes the argument as follows:
A form of reasoning by which a certain stricture applying to a minor matter is established as applying all the more to a major matter. Conversely, if a certain leniency applies to a major matter, it must apply all the more to the minor matter.
R. Feigenbaum’s description is even clearer, as the following quotation shows. (Note that we are effectively dealing with a scale of modality, and with nesting of modalities within modalities.)
a) Any stringent ruling with regard to the lenient issue must be true of the stringent issue as well;
b) any lenient ruling regarding the stringent issue must be true with regard to the lenient matter as well.
These special formulations are easily assimilated by our general theory of qal vachomer argument, as follows:
a) P generally implies more ‘stringency for the practitioner’ (=R) than Q implies,
nonetheless, Q is stringent (R) enough to imply ‘the practitioner subject to a certain restriction (or not-subject to a certain liberty)’ (=S),
all the more, P is stringent (R) enough for this same ruling to apply (S).
b) P generally implies more ‘stringency for the practitioner’ (=R) than Q implies,
nonetheless, P is not stringent (R) enough to imply ‘the practitioner subject to a certain restriction (or not-subject to a certain liberty)’ (=S),
all the more, Q is not stringent (R) enough for this same ruling to apply (S).
Note that both arguments are antecedental in form, and one is expressed positively and the other negatively. The extreme theses (P, Q) are legal rulings; their middle thesis (R) is the magnitude of burden, let us say, they impose on a practitioner, and their subsidiary thesis (S) is a third legal clause, itself evaluated as burdensome to a certain degree. If the smaller burden (Rq) includes the subsidiary (Rs), then so does the larger (Rp); and by contraposition, if the greater burden excludes the subsidiary, then so does the lesser. Note, for the sake of symmetry, that we could conceive of similar formulas in which the middle thesis (R) is ‘leniency for the practitioner’, provided the subsidiary thesis (S) likewise changes in polarity, becoming ‘the practitioner is subject to a certain liberty (or not-subject to a certain restriction)’.
Such formulas may be objected to, firstly, on the ground of their limited concept: they are conceived specifically in relation to the severity or laxity of ethical propositions (legal rulings, in Rabbinical terminology), whereas a-fortiori is a much wider process, applicable to non-ethical propositions. Secondly, and more radically, these formulas involve a middle thesis (‘burdensomeness’, say) too vague and diffuse to enable a sure conclusion: the major premise must be general, and such generality can only be known by generalization or enumeration. If by generalization, the conclusion is at best probable; if by enumeration, we are begging the question (i.e. we had to know the desired conclusion beforehand).
For a law P may be burdensome in many respects and another law Q may be burdensome in many respects, and P may well be burdensome in numerically more respects than Q is burdensome; even so, the burdens of P may or may not include all the burdens of Q, and indeed the burdens of P and Q may not overlap at all! In other words, in principle (i.e. formally), the inference is not necessary without further specifications which somehow guarantee that the burdens of P include all those of Q. That is, the laws under discussion here, P and Q, have certain implicit material relations which must be brought out into the open.
Thus the above mentioned Rabbinical formulations of a-fortiori argument, are not only limited in scope (to ethical theses), but they cannot be considered as having formal validity (i.e. invariably guarantee inference). They are at best broad guidelines, which may occasionally be found inapplicable. Indeed, the Rabbis were aware of this problem, and did occasionally object to attempted such inferences by one of their colleagues, and claim that a stringency of Q did not necessarily apply to P or a leniency of P did not necessarily apply to Q. Effectively, they invalidated the major premise, denying it to be general and making it at best probable, by apposition of an acknowledged exception; and by this means, they inhibited application of qal vachomer reasoning to S, the new case under consideration.
 There are already, in the Christian Bible, examples of a-fortiori, some of which are analyzed by H. Maccoby in The Mythmaker: Paul and the Invention of Christianity. The author mentions Paul’s fondness for the argument, but shows him to have lacked knowledge of the ‘dayo principle’ (see further on), concluding that his use of the form was more akin to the rhetoric of Hellenistic Stoic preachers (pp. 64-67).
 It could be said that there is an a-fortiori movement of thought inherent in syllogism, inasmuch as we pass from a larger quantity (all) to a lesser (some). But in syllogism, the transition is made possible by means of the relatively incidental extension of the middle term, whereas, as we have seen, in a-fortiori proper, it is the range of values inherent to the middle term which make it possible.
 Vol. 1, p. 51, my translation.
 I must report that near the end of writing this book, I uncovered a much better definition of a-fortiori argument by Lalande, in the Vocabulaire technique et critique de la philosophie. He writes (my translation): “Inference from one quantity to another quantity of similar nature, larger or smaller, and such that the first cannot be reached or passed without the second being [reached] also.” Note, however, that this definition fails to specify that the positive movement from large to small is predicatal, while that from small to large is subjectal; and it ignores negative moods altogether, as well as differences between copulative and implicational forms. Lalande adds that the argument is of legal origin, quoting the Latin rule “Non debet, cui plus licet, quod minus est non licere” (p. 32).
 I have an impression, for instance, that modern French discourse involves more use of a-fortiori than modern English discourse. To what extent that is true, and why it should be so, I cannot venture to say.
 It is interesting to note in any case, that Josephus Flavius claims that a disciple of Aristotle, called Clearchus, wrote a book, which is no longer extant, in which he reports a meeting between Aristotle and a Jew, during which presumably ideas were exchanged. What ideas were exchanged, and whether this story is fact or legend, I do not know (see Bentwich).
 I wish to make an acknowledgement at this stage. My special interest in a-fortiori argument was aroused back in 1990 by a Vancouver, B.C., lawyer, Mr. Daniel Goldsmith. I had written an article on “Jewish Logic” which was gradually published in a local Jewish paper called “World of Chabad“. One reader, Mr. Goldsmith, wrote to me suggesting that I pay special attention to a-fortiori argument, as a form of reasoning which was particularly Jewish and which had not so far received much formal treatment. I resolved at the time to follow this suggestion, and the present essays on the subject are the result.
 The principle is stated as din leba min hadin lihiot benadon. Note that J.E. translates this as “the conclusion of an argument is satisfied when it is like the major premise”; but what they mean by ‘major premise’ is what we here, more precisely, name ‘minor premise’.
 We can now mention the Rabbinical theory that the dayo principle has subdivisions, and may be applied either ‘to the lenient case’ or ‘to the strict case’. This idea was based on a limited comprehension of a-fortiori argument, due precisely to the Rabbis’ failure to distinguish between the deductive process itself and its inductive precedents. An example provided by Bergman (p. 128) is the following “If shen and regel, which cause no liability when committed on public property, nevertheless give rise to full liability if committed on the injured party’s property, then keren, for which there is [half] liability if it occurred on public property, certainly should cause full liability if committed on the property of the injured party” (B.Q., 25a): on the basis of the dayo principle, one cannot so argue, he says, but must conclude on half liability, for keren on private property, i.e. no more than keren invokes on public property. I agree with his conclusion, but for other reasons: we have 3 givens: (1) relatively unintentional damages by animals (shen and regel) in the public domain imply owner’s liability to pay none of the damages; whereas, (2) the same on private property imply his liability to pay all of the damages; and (3) relatively intentional damages by animals (keren) in the public domain imply liability to pay half of the damages. The first two givens serve to induce the major premise of our actual a-fortiori: make the comparison ‘for unintentional acts, acts committed on private property imply more liability than those committed in the public domain’, then generalize to ‘for all acts, the same’, then educe the new particular ‘for intentional acts, the same’. This result, combined with the third given, which serves as minor premise, form the a-fortiori argument proper, whose conclusion is ‘intentional acts on private property imply liability to pay half the damages’. The application of the dayo principle involved in this last stage (subsidiary term: ‘having to pay half damages’) is perfectly regular, and requires no special new division, note well.
 We might also mention a description proposed by Maccoby, “if something is known about one thing which has a certain quality in relatively ‘light’ form, then it must be true ‘all the more so’ of some other thing that has the same quality in a relatively ‘heavy’ form”. This description is incomplete in various ways, but at least does not limit itself to legal issues.
 P. 27, n. 106.
 P. 88.
 The indefinition and apparent subjectivity of the concepts of ‘lenient’ and ‘stringent’ (or synonyms to the same effect) is important to note. They seem to refer to subjective/emotional reactions to laws; i.e. whether a law is felt by people as a further hardship or as a release from duty. If we suppose more formal definitions (see ch. 13), and regard every law – positive or negative, i.e. an imperative or a prohibition – as “stringent”, and every absence of law – i.e. ethical contingency, permission and exemption – as a “leniency”, then we must be very careful in this context, as modal logic is involved, which has special syllogistic behaviour-patterns (notably, one cannot draw a conclusion from a first-figure major premise which is not positively or negatively necessary). This matter requires further study, in relation to Rabbinical formulations of a-fortiori argument concerning “leniency”.
 Perhaps a word should be said about this difference in terminology. In philosophy, a distinction is made between ethical and legal propositions as follows. Ethics is the broader concept, which includes law. Law refers specifically to ethical propositions enforced by society (or the ruling segment thereof); some ethical propositions are not considered so enforceable (though supposedly inferred from nature). The Rabbis are clearly aware of this functional distinction, but tend to regard all ethical propositions as “laws”, because they view them as ultimately enforced by Gd if not by society.