Logic in the Torah

A Thematic Compilation by Avi Sion

3. Qal vaChomer


1.   Background

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, 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)[1]. 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[2], showing that there is a misapprehension still today. Or again, consider the following brief entry in the Encyclopedie Philosophique Universelle[3]: “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.[4]

(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. 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. 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 practice 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 practiced in one culture than any other[5], 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 judgments. 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 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). 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.[6]

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.


2.    The valid moods

Let us begin by listing and naming all the valid moods of a-fortiori argument[7] in abstract form; we shall have occasion in later chapters to consider examples. We shall adopt a terminology which is as close to traditional as possible, but it must be kept in mind that the old names used here may have new senses (in comparison to, say, their senses in syllogistic theory), and that some neologisms are inevitable in view of the novelty of our discoveries.

An explicit a-fortiori argument always involves three propositions, and four terms. We shall call the propositions: the major premise, the minor premise, and the conclusion, and always list them in that order. The terms shall be referred to as: the major term (symbol, P, say), the minor term (Q, say), the middle term (R, say), and the subsidiary term (S, say). In practice, the major premise is very often left unstated; and likewise, the middle term (we shall return to this issue in more detail later).


Table 3.1         Classification of A-Fortiori Arguments





(1) Subjectal

(2) Predicatal


(3) Antecedental

(4) Consequental




(a) Positive

Minor to major

Major to minor

(b) Negative

Major to minor

Minor to major



We shall begin by analyzing “copulative” forms of the argument. There are essentially four valid moods. Two of them subjectal in structure, and two of them predicatal in structure; and for each structure, one of the arguments is positive in polarity and the other is negative.

a.         Subjectal moods.


(i) Positive version. (Minor to major.)

P is more R than Q (is R),

and, Q is R enough to be S;

therefore, all the more, P is R enough to be S.

As we shall see further on, a similar argument with P in the minor premise and Q in the conclusion (“major to minor”) would be invalid.


(ii) Negative version. (Major to minor.)

P is more R than Q (is R),

yet, P is not R enough to be S;

therefore, all the more, Q is not R enough to be S.

As we shall see further on, a similar argument with Q in the minor premise and P in the conclusion (“minor to major”) would be invalid.


b.         Predicatal moods.


(i) Positive version. (Major to minor.)

More R is required to be P than to be Q,

and, S is R enough to be P;

therefore, all the more, S is R enough to be Q.

As we shall see further on, a similar argument with Q in the minor premise and P in the conclusion (“minor to major”) would be invalid.


(ii) Negative version. (Minor to major.)

More R is required to be P than to be Q,

yet, S is not R enough to be Q;

therefore, all the more, S is not R enough to be P.

As we shall see further on, a similar argument with P in the minor premise and Q in the conclusion (“major to minor”) would be invalid.


The expression “all the more” used with the conclusion is intended to connote that the inferred proposition is more “forceful” than the minor premise, as well as suggest the quantitative basis of the inference (i.e. that it is a-fortiori). Note that instead of the words “and” or “yet” used to introduce the minor premise, we could just as well have used the expression “nonetheless”, which seems to balance nicely with the phrase “all the more”.

The role of the major premise is always to relate the major and minor terms (P and Q) to the middle term (R); the middle term serves to place the major and minor terms along a quantitative continuum. The major premise is, then, a kind of comparative proposition of some breadth, which will make possible the inference concerned; note well that it contains three of the terms, and that its polarity is always positive (this will be demonstrated further down). The term which signifies a greater measure or degree (more) within that range, is immediately labeled the major; the term which signifies a smaller measure or degree (less) within that range, is immediately labeled the minor (these are conventions, of course). P and Q may also conveniently be called the “extremes” (without, however, intending that they signify extreme quantities of R).

Note that here, unlike in syllogism, the major premise involves both of the extreme terms and the minor premise may concern either of them; thus, the expressions major and minor terms, here, have a different value than in syllogism, it being the relative content of the terms which determines the appellation, rather than position within the argument as a whole. Furthermore, the middle term appears in all three propositions, not just the two premises.

The function of the minor premise is to positively or negatively relate one of the extreme terms to the middle and subsidiary terms; the conclusion thereby infers a similar relation for the remaining extreme. If the minor premise is positive, so is the conclusion; such moods are labeled positive, or modus ponens in Latin; if the minor premise is negative, so is the conclusion; such moods are labeled negative, or modus tollens. Note well that the minor premise may concern either the major or the minor term, as the case may be. Thus, the inference may be “from major (term, in the minor premise) to minor (term, in the conclusion)” - this is known as inference a majori ad minus; or in the reverse case, “from minor (term, in the minor premise) to major (term, in the conclusion)” - this is called a minori ad majus.

There are notable differences between subjectal and predicatal a-fortiori. In subjectal argument, the extreme terms have the logical role of subjects, in all three propositions; whereas, in predicatal argument, they have the role of predicates. 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).

Implicational” forms of the argument are essentially similar in structure to copulative forms, except that they are more broadly designed to concern theses (propositions), rather than terms. The relationship involved is consequently one of implication, rather than one of predication; that is, we find in them the expression “implies”, rather than the copula “is”.[8]


c.         Antecedental moods.


(i) Positive version. (Minor to major.)

P implies more R than Q (implies R)

and, Q implies enough R to imply S; therefore,

all the more, P implies enough R to imply S.


(ii) Negative version. (Major to minor.)

P implies more R than Q (implies R)

yet, P does not imply enough R to imply S; therefore,

all the more, Q does not imply enough R to imply S.


d.         Consequental moods.


(i) Positive version. (Major to minor.)

More R is required to imply P than to imply Q

and, S implies enough R to imply P; therefore,

all the more, S implies enough R to imply Q.


(ii) Negative version. (Minor to major.)

More R is required to imply P than to imply Q

yet, S does not imply enough R to imply Q; therefore,

all the more, S does not imply enough R to imply P.


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[9].


A couple of comments, which concern all forms of the argument, still need to be made.

The standard form of the major premise is a comparative proposition with the expression “more...than” (superior form). But we could just as well commute such major premises, and put them in the “less...than” form (inferior form), provided we accordingly reverse the order in it of the terms P and Q. Thus, ‘P is more R than Q’ could be written ‘Q is less R than P’, ‘More R is required to be P than to be Q’ as ‘Less R is required to be Q than to be P’, and similarly for implicational forms, without affecting the arguments. 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).

Also, it should also be obvious that the major premise could equally have been an egalitarian one, of the form “as” (e.g. ‘P is as much R as Q (is R)’). The arguments would work equally well (P and Q being equivalent in them). However, in such cases it would not be appropriate to say “all the more” with the conclusion; but rather use the phrase “just as much”. Nevertheless, we must regard such arguments as still, in the limit, a-fortiori in structure. The expression “all the more” is strictly-speaking a redundancy, and serves only to signal that a specifically a-fortiori kind of inference is involved; we could equally well everywhere use the word “therefore”, which signifies for us that an inference is taking place, though it does not specify what kind.

It follows that each of the moods listed above stands for three valid moods: the superior (listed), and corresponding inferior and egalitarian moods (unlisted).

Lastly, it is important to keep in mind, though obvious, that the form ‘P is more R than Q’ means ‘P is more R than Q is R’ (in which Q is as much a subject as P, and R is a common predicate), and should not be interpreted as ‘P is more R than P is Q’ (in which P is the only subject, common to two predicates Q and R, which are commensurable in some unstated way, such as in spatial or temporal frequency, allowing comparison between the degrees to which they apply to P). In the latter case, R cannot serve as middle term, and the argument would not constitute an a-fortiori. The same can be said regarding ‘P implies more R than Q’. Formal ambiguities of this sort can lead to fallacious a-fortiori reasoning[10].


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. of the members of a class X) or singular (refer to an individual, or to a group collectively, by means of a name or an indicative this or these X). A predicate is inevitably a class concept (say, Y), referred to wholly (as in ‘is not Y’) or partly (as in ‘is Y’); even a predicate in which a singular term is encrusted (such as ‘pay Joe’) is a class-concept, in that many subjects may relate to it independently (‘Each of us paid Joe’). 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 subjectal argument, whether positive or negative, since the subjects of the minor premise and conclusion are not one and the same (they are the major and minor terms, P and Q), we can only quantify these propositions if the major premise reads: “for every instance of P there is a corresponding instance of Q, such that: the given P is more R than the given Q”. 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. 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. These are purely deductive issues; but it should be noted that in some cases the a-fortiori argument as a whole is further complicated by a hidden argument by analogy from one term or thesis to another, so that there are, in fact, more than four terms/theses. 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. To avoid losing the reader in minutiae, we will not say any more about such details in the present volume.[11]


3.   Preliminaries

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 [in JL].[12]

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 practice 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 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).


4.   Samples in the Torah

(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). 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? 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.

Major premise:

“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);

Minor premise:

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 (God 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). 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?”

This argument may be may be construed to have run as follows:

Major premise:

The children of Israel (=P) “fear God” (=R) more than Pharaoh (=Q) does;

Minor premise:

yet, they (P) did not fear God (R) enough to hearken unto Moses (=S);


all the more, Pharaoh (Q) will not fear God (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 God”.

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 God 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:

Major premise:

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);

Minor premise:

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 Lord; and how much more after my death?”

We may reword it as follows, for our purpose:

Major premise:

More “self-discipline” (=R) is required to obey God in the absence of His emissary, Moses (=P), than in his presence (=Q);

Minor premise:

the children of Israel (=S) were not sufficiently self-disciplined (R) to obey God 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 [5.5]).

The following is a quick and easy way to classify any Biblical example of qal vachomer:

  1. 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.
  2. 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.
  3. 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.
  4. 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.


Drawn from Judaic Logic (1995), chapters 3:1 and 4:1-2 (part).


[1]           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).

[2]           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.

[3]           Vol. 1, p. 51, my translation.

[4]           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).

[5]           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.

[6]           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).

[7]           Such arguments occur quite often in everyday discourse. I give you a couple of examples: "if he can readily run a mile in 5 minutes, he should certainly be able to get here (1/2 a mile from where he is now) in 15 minutes." Or again: "if my bus pass is transferable to other adults, I am sure it can be used by kids."

[8]           "Implication" is to be understood here in a generic sense, applicable to all types of modality - we shall avoid more specific senses, to keep things clear and simple.

[9]           The logical relationship between "is" and "implies" is well known. X "is" Y, in class-logic terminology, if it is subsumed/included by Y, which does not preclude other things also being Y. X "implies" Y, if it cannot exist/occur without Y also existing/occurring, even if as may happen it is not Y. Thus, if X "is" Y, it also "implies" Y; but if X "implies" Y, it does not follow that it "is" Y. In other words, "is" implies (but is not implied by) "implies"; "implies" is a broader more generic concept, which covers but is not limited to "is", a narrower more specific concept.

[10]          For example: Jane is more good-looking than a nice girl; she is good-looking enough to win a beauty contest; therefore, a nice girl is good-looking enough to win a beauty contest.

[11]          As regards validation of the above arguments, see JL 3:2 and AFL 1:3. In the present volume, see the brief treatment in chapter 6:1-4, which also deals with a crescendo argument.

[12]          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.

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