Logic in the Torah

A Thematic Compilation by Avi Sion

6. Miriam’s A Fortiori Argument


The a fortiori argument made by Miriam, the sister of Moses, in Numbers 12:14-15, plays an important role in rabbinical discussions relating to such arguments, notably in the Gemara (the commentary on the Mishna). For this reason, I have had to analyze it in considerable detail.


1.   Formal validation of a fortiori argument

The paradigm of a fortiori argument, the simplest and most commonly used form of it, is the positive subjectal mood[1], in which the major and minor terms (here always labeled P and Q, respectively) are subjects and the middle and subsidiary terms (here always labeled R and S, respectively) are predicates. It proceeds as follows[2]:

P is R more than Q is R (major premise).

Q is R enough to be S (minor premise).

Therefore, P is R enough to be S (conclusion).

An example of such argument would be: “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.” (Num. 12:14). This can be read as: if offending one’s father (Q) is bad (R) enough to deserve seven days isolation (S), then surely offending God (P) is bad (R) enough to deserve seven days isolation (S); the tacit major premise being: offending God (P) is worse (R) than offending one’s father (Q).

This form of argument can be logically validated (briefly put) as follows. The major premise tells us that P and Q are both R, though to different measures or degrees. Let us suppose the measure or degree of R in P is Rp and that of R in Q is Rq – then the major premise tells us that: if P then Rp, and if Q then Rp, and Rp is greater than Rq (which in turn implies: if something is Rp then it is also Rq, since a larger number includes all numbers below it[3]). Similarly, the minor premise tells us that nothing can be S unless it has at least a certain measure or degree of R, call it Rs; this can be stated more formally as: if Rs then S and if not Rs then not S. Obviously, since Q is R, Q has the quantity Rq of R, i.e. if Q, then Rq; but here we learn additionally (from the “enough” clause) that Rq is greater than or equal to Rs, so that if Rq then Rs; whence, the minor premise tells us that if Q then S. The putative conclusion simply brings some of the preceding elements together in a new compound proposition, namely: if P then Rp (from the major premise) and if Rs then S and if not Rs then not S (from the minor premise), and Rp is greater than Rs (since Rp > Rq in the major premise and Rq ≥ Rs in the minor premise), so that if Rp then Rs; whence, if P then S. The conclusion is thus proved by the two premises (together, not separately, as you can see). So, the argument as a whole is valid – i.e. it cannot logically be contested.

Having thus validated the positive subjectal mood of a fortiori argument, it is easy to validate the negative subjectal mood by reductio ad absurdum to the former. That is, keeping the former’s major premise: “P is R more than Q is R,” and denying its putative conclusion, i.e. saying: “P is R not enough to be S,” we must now conclude with a denial of its minor premise, i.e. with: “Q is R not enough to be S.” For, if we did not so conclude the negative argument, we would be denying the validity of the positive argument.

We can similarly demonstrate the validity of the positive, and then the negative, predicatal moods of a fortiori argument. In this form, the major, minor and middle terms (P, Q and R) are predicates and the subsidiary term (S) is a subject.

More R is required to be P than to be Q (major premise).

S is R enough to be P (minor premise).

Therefore, S is R enough to be Q (conclusion).

An example of such argument would be: “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?” (Gen. 44:8). This can be read as: if we (S) are honest (R) enough to return found valuables (P), then surely we (S) are honest (R) enough to not-steal (Q); the tacit major premise being: more honesty (R) is required to return found valuables (P) than to refrain from stealing (Q).

Here the validation proceeds (again briefly put) as follows. The major premise tells us that iff (i.e. if only if) Rp then P, and iff Rq then Q, and Rp is greater than Rq (whence if Rp then Rq). The minor premise tells us additionally that if S then Rs, and (since it is “enough”) Rs is greater than or equal to Rp (whence if Rs then Rp), from which it follows that if S then Rp; and since iff Rp then P, it follows that if S then P. From the preceding givens, we can construct the putative conclusion, using if S then Rs (from the minor premise), and Rs is greater than Rq (from both premises, whence if Rs then Rq); these together imply if S then Rq, and this together with iff Rq then Q (from the major premise) imply if S then Q. The conclusion is thus here again incontrovertibly proved by the two premises jointly. The negative predicatal mood can in turn be validated, using as before the method of reductio ad absurdum. That is, if the major premise remains unchanged and the putative conclusion is denied, then the minor premise will necessarily be denied; but since the minor premise is given and so cannot be denied, it follows that the conclusion cannot be denied.

Notice that the reasoning proceeds from minor to major (i.e. from the minor term (Q) in the minor premise, to the major term (P) in the conclusion) in the positive subjectal mood; from major to minor in the negative subjectal mood; from major to minor in the positive predicatal mood; and from minor to major in the negative predicatal mood. These are valid forms of reasoning. If, on the other hand, we proceeded from major to minor in the positive subjectal mood, from minor to major in the negative subjectal mood; from minor to major in the positive predicatal mood; or from major to minor in the negative predicatal mood – we would be engaged in fallacious reasoning. That is, in the latter four cases, the arguments cannot be validated, and their putative conclusions do not logically follow from their given premises. To reason fallaciously is to invite immediate or eventual contradiction.

Note well that each of the four arguments we have just validated contains only four terms, here labeled P, Q, R, and S. Each of these terms appears two or more times in the argument. P and Q appear in the major premise, and in either the minor premise or the conclusion. R appears in both premises and in the conclusion. And S appears in the minor premise and in the conclusion. The argument as a whole may be said to be properly constructed if it has one of these four validated forms and it contains only four terms. Obviously, if any one (or more) of the terms has even slightly different meanings in its various appearances in the argument, the argument cannot truly be said to be properly constructed. It may give the illusion of being a valid a fortiori, but it is not really one. It is fallacious reasoning.

The above described a fortiori arguments, labeled subjectal or predicatal, relate to terms, and may thus be called ‘copulative’. There are similar ‘implicational’ arguments, which relate to theses instead of terms, and so are labeled antecedental or consequental. To give one example of the latter, a positive antecedental argument might look like this:

Ap (A being p) implies Cr (r in C) more than Bq (B being q) does,

and Bq implies Cr enough for Ds (for D to be s);

therefore, Ap implies Cr enough for Ds.

Notice the use of ‘implies’ instead of ‘is’ to correlate the items concerned. I have here presented the theses as explicit propositions ‘A is p’, ‘B is q’, ‘C is r’ and ‘D is s’, although they could equally well be symbolized simply as P, Q, R, and S, respectively. The rules of inference are essentially the same in implicational argument as in copulative argument.


2.   The principle of deduction

This forewarning concerning the uniformity throughout an argument of the terms used may be expressed as a law of logic. It is true not just of a fortiori argument, but of all deductive argument (for instances, syllogism or apodosis). We can call this fundamental rule ‘the principle of deduction’, and state it as: no information may be claimed as a deductive conclusion which is not already given, explicitly or implicitly, in the premise(s). This is a very important principle, which helps us avoid fallacious reasoning. It may be viewed as an aspect of the law of identity, since it enjoins us to acknowledge the information we have, as it is, without fanciful additions. It may also be considered as the fifth law of thought, to underscore the contrast between it and the principle of induction[4], which is the fourth law of thought.

Deduction must never be confused with induction. In inductive reasoning, the conclusion can indeed contain more information than the premises make available; for instance, when we generalize from some cases to all cases, the conclusion is inductively valid provided and so long as no cases are found that belie it. In deductive reasoning, on the other hand, the conclusion must be formally implied by the given premise(s), and no extrapolation from the given data is logically permitted. In induction, the conclusion is tentative, subject to change if additional information is found, even if such new data does not contradict the initial premise(s)[5]. In deduction, on the other hand, the conclusion is sure and immutable, so long as no new data contradicts the initial premise(s).

As regards the terms, if a term used in the conclusion of a deductive argument (such as a fortiori) differs however slightly in meaning or in scope from its meaning or scope in a premise, the conclusion is invalid. No equivocation or ambiguity is allowed. No creativity or extrapolation is allowed. If the terms are not exactly identical throughout the argument, it might still have some inductive value, but as regards its deductive value it has none. This rule of logic, then, we shall here refer to as ‘the principle of deduction’.


The error of ‘proportional’ a fortiori argument. An error many people make when attempting to reason a fortiori is to suppose that the subsidiary term (S) is generally changed in magnitude in proportion (roughly) to the comparison between the major and minor terms (P and Q). The error of such proportional’ a fortiori argument, as we shall henceforth call it, can be formally demonstrated as follows.

Consider the positive subjectal mood we have described above. Suppose instead of arguing as we just did above, we now argue as do the proponents of such fallacious reasoning that: just as ‘P is more R than S’ (major premise), so S in the conclusion (which is about P) should be greater than it is in the minor premise (which is about Q). If we adhered to this ‘reasoning’, we would have two different subsidiary terms, say S1 for the minor premise and S2 for the conclusion, with S2 > S1, perhaps in the same proportion as P is to Q, or more precisely as the R value for P (Rp) is to the R value for Q (Rq), so that S1 and S2 could be referred to more specifically as Sq and Sp. In that case, our argument would read as follows:

P is R more than Q is R (major premise).

Q is R enough to be S1 (minor premise).

Therefore, P is R enough to be S2 (conclusion).

The problem now is that this argument would be difficult to validate, since it contains five terms instead of only four as before. Previously, the value of R sufficient to qualify as S was the same (viz. R ≥ Rs) in the conclusion (for P) as in the minor premise (for Q). Now, we have two threshold values of R for S, say Rs1 (in the minor premise, for Q) and Rs2 (in the conclusion, for P). Clearly, if Rs2 is assumed to be greater than Rs1 (just as Rp is greater than Rq), we cannot conclude that Rp > Rs2, for although we still know that Rp > Rq and Rq ≥ Rs1, we now have: Rp > Rs1 < Rs2, so that the relative sizes of Rp and Rs2 remain undecidable. Furthermore, although previously we inferred the “If Rs then S” component of the conclusion from the minor premise, now we have no basis for the “If Rs2 then S2” component of the conclusion, since our minor premise has a different component “If Rs1 then S1” (and the latter proposition certainly does not formally imply the former).[6]

It follows that the desired conclusion “P is R enough to be S2” of the proposed ‘proportional’ version of a fortiori argument is simply invalid[7]. That is to say, its putative conclusion does not logically follow from its premises. The reason, to repeat, is that we have effectively a new term (S2) in the conclusion that is not explicitly or implicitly given in the premises (where only S1 appears, in the minor premise). Yet deduction can never produce new information of any sort, as we have already emphasized. Many people find this result unpalatable. They refuse to accept that the subsidiary term S has to remain unchanged in the conclusion. They insist on seeing in a fortiori argument a profitable argument, where the value of S (and the underlying Rs) is greater for P than it is for Q. They want to ‘quantify’ the argument more thoroughly than the standard version allows.

We can similarly show that ‘proportionality’ cannot be inferred by positive predicatal a fortiori argument. In such case, the subsidiary term (S) is the subject (instead of the predicate) of the minor premise and conclusion. If that term is different (as S1 and S2) in these two propositions, we again obviously do not have a valid a fortiori argument, since our argument effectively involves five terms instead of four as required. We might have reason to believe or just imagine that the subject (S) is diminished in some sense in proportion to its predicates (greater with P, lesser with Q), but such change real or imagined has nothing to do with the a fortiori argument as such. S may well vary in meaning or scope, but if it does so it is not due to a fortiori argument as such. Formal logic teaches generalities, but this does not mean that it teaches uniformity; it allows for variations in particular cases, even as it identifies properties common to all cases.

People who believe in ‘proportional’ a fortiori argument do not grasp the difference between knowledge by a specific deductive means and knowledge by other means. By purely a fortiori deduction, we can only conclude that P relates to precisely S, just as Q relates to S in the minor premise. But this does not exclude the possibility that by other means, such as observation or induction, or even a subsequent deductive act, we may find out and prove that the value of S relative to Q (S1) and the value of S relative to P (S2) are different. If it so happens that we separately know for a fact that S varies in proportion to the comparison of P and Q through R, we can after the a fortiori deduction further process its conclusion in accord with such additional knowledge[8]. But we cannot claim such further process as part and parcel of the a fortiori argument as such – it simply is not, as already demonstrated in quite formal terms.

Formal logic cuts up our long chains of reasoning into distinguishable units – called arguments – each of which has a particular logic, particular rules it has to abide by. Syllogism has certain rules, a fortiori argument has certain rules, generalization has certain rules, adduction has certain rules, and so on. When such arguments, whether deductive or inductive, and of whatever diverse forms, are joined together to constitute a chain of reasoning (the technical term for which is enthymeme), it may look like the final conclusion is the product of all preceding stages, but in fact it is the product of only the last stage. Each stage has its own conclusion, which then becomes a premise in the next stage. The stages never blend, but remain logically distinct. In this way, we can clearly distinguish the conclusion of a purely a fortiori argument from that of any other argument that may be constructed subsequently using the a fortiori conclusion as a premise.

Some of the people who believe that a fortiori argument yields a ‘proportional’ conclusion are misled by the wording of such conclusion. We say: “since so and so, therefore, all the more, this and that.” The expression “all the more” seems to imply that the conclusion (if it concerns the major term) is quantitatively more than the minor premise (concerning the minor term). Otherwise, what is “more” about it? But the fact is, we use that expression in cases of major to minor, as well as minor to major. Although we can say “how much more” and “how much less,” we rarely use the expression “all the less”[9] to balance “all the more” – the latter is usually used in both contexts. Thus, “all the more” is rather perhaps to be viewed as a statement that the conclusion is more certain than the minor premise[10]. But even though this is often our intention, it is not logically correct. In truth, the conclusion is always (if valid) as certain as the minor premise, neither more nor less. Therefore, we should not take this expression “all the more” too literally – it in fact adds nothing to the usual signals of conclusion like “therefore” or “so.” It is just rhetorical emphasis, or a signal that the form of reasoning is ‘a fortiori’.


3.   The argument a crescendo

Although ‘proportional’ a fortiori argument is not formally valid, it is in truth sometimes valid. It is valid under certain conditions, which we will now proceed to specify. When these conditions are indeed satisfied, we should (I suggest) name the argument differently, and rather speak of ‘a crescendo’ argument’[11], so as to distinguish it from strict ‘a fortiori’ argument. We could also say (based on the common form of the conclusions of both arguments) that ‘a crescendo’ argument is a particular type of a fortiori argument, to be contrasted to the ‘purely a fortiori’ species of a fortiori argument. More precisely, a crescendo argument is a compound of strictly a fortiori argument and ‘pro rata’ argument. It combines premises of both arguments, to yield a special, ‘proportional’ conclusion.

The positive subjectal mood of a crescendo argument has three premises and five terms:

P is more R than Q is R (major premise);

and Q is R enough to be Sq (minor premise);

and S varies in proportion to R (additional premise).

Therefore, P is R enough to be Sp (a crescendo conclusion).

The ‘additional premise’ tells us there is proportionality between S and R. Note that the subsidiary term (Sp) in the conclusion differs from that (Sq) given in the minor premise, although they are two measures or degrees of one thing (S). This mood can be validated as follows:

The purely a fortiori element is:

P is more R than Q is R,

and Q is R enough to be Sq.

(Therefore, P is R enough to be Sq.)

To this must be added on the pro rata element:

Moreover, if we are given that S varies in direct proportion to R, then:

since the above minor premise implies that: if R = Rq, then S = Sq,

it follows that: if R = more than Rq = Rp, then S = more than Sq = Sp.

Whence the a crescendo conclusion is:

Therefore, P is R enough to be Sp.

If the proportion of S to R is direct, then Sp > Sq; but if S is inversely proportional to R, then Sp < Sq. The negative subjectal mood is similar, having the same major and additional premise, except that it has as minor premise “P is R not enough to be Sp” and as a crescendo conclusion “Q is R not enough to be Sq.”

The positive predicatal mood of a crescendo argument has three premises and five terms:

More R is required to be P than to be Q (major premise);

and Sp is R enough to be P (minor premise);

and S varies in proportion to R (additional premise).

Therefore, Sq is R enough to be Q (a crescendo conclusion).

As before, the ‘additional premise’ tells us there is proportionality between S and R. Note that the subsidiary term (Sq) in the conclusion differs from that (Sp) given in the minor premise, although they are two measures or degrees of one thing (S). This mood can be validated as follows:

The purely a fortiori element is:

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

and Sp is R enough to be P.

(Therefore, Sp is R enough to be Q.)

To this must be added on the pro rata element:

Moreover, if we are given that R varies in direct proportion to S, then:

since the above minor premise implies that: if S = Sp, then R = Rp,

it follows that: if S = less than Sp = Sq, then R = less than Rp = Rq.

Whence the a crescendo conclusion is:

therefore, Sq is R enough to be Q.

If the proportion of R to S is direct, then Rq < Rp; but if R inversely proportional to S, then Rq > Rp. The negative predicatal mood is similar, having the same major and additional premise, except that it has as minor premise “Sq is R not enough to be Q” and as a crescendo conclusion “Sp is R not enough to be P.”

In practice, we are more likely to encounter subjectal than predicatal a crescendo arguments, since the subsidiary terms in the former are predicates, whereas those in the latter are subjects, and subjects are difficult to quantify. We can similarly construct four implicational moods of a crescendo argument, although things get more complicated in such cases, because it is not really the middle and subsidiary theses which are being compared but terms within them. These matters are dealt with more thoroughly in earlier chapters [of AFL], and therefore will not be treated here.

From this formal presentation, we see that purely a fortiori argument and a crescendo argument are quite distinct forms of reasoning. The latter has the same premises as the former, plus an additional premise about proportion, which makes possible the ‘proportional’ conclusion. Without the said ‘additional premise’, i.e. with only the two premises (the major and the minor) of a fortiori argument, we cannot legitimately draw the a crescendo conclusion.

Thus, people who claim to draw a ‘proportional’ conclusion from merely a fortiori premises are engaged in fallacy. They are of course justified to do so, if they explicitly acknowledge, or at least tacitly have in mind, the required additional premise about proportion. But if they are unaware of the need for such additional information, they are definitely reasoning incorrectly. The issue here is not one of names, i.e. whether an argument is called a fortiori or a crescendo or whatever, but one of information on which the inference is based.

To summarize: Formal logic can indubitably validate properly constructed a fortiori argument. The concluding predication (more precisely, the subsidiary item, S) in such cases is identical to that given in the minor premise. It is not some larger or lesser quantity, reflecting the direct or inverse proportion between the major and minor items. Such ‘proportional’ conclusion is formally invalid, if all it is based on are the two premises of a fortiori argument. To draw an a crescendo conclusion, it is necessary to have an additional premise regarding proportionality between the subsidiary and middle items.


4.   The rabbis’ dayo (sufficiency) principle

It is evident from what we have just seen and said that there is no formal need for a “dayo (sufficiency) principle” to justify a fortiori argument as distinct from a crescendo argument. It is incorrect to conceive, as some commentators do (notably the Gemara, as we shall see [in AFL]), a fortiori argument as a crescendo argument artificially circumvented by the dayo principle; for this would imply that the natural conclusion from the two premises of a fortiori is a crescendo, whereas the truth is that a fortiori premises can only logically yield an a fortiori conclusion. The rule to adopt is that to draw an a crescendo conclusion an additional (i.e. third) premise about proportionality is needed – it is not that proportionality may be assumed (from two premises) unless the proportionality is specifically denied by a dayo objection.

In fact, the dayo principle can conceivably ‘artificially’ (i.e. by Divine fiat or rabbinic convention) restrain only a crescendo argument. In such case, the additional premise about proportion is disregarded, and the conclusion is limited to its a fortiori dimension (where the subsidiary term is identical in the minor premise and conclusion) and denied its a crescendo dimension (where the subsidiary term is greater or lesser in the minor premise than in the conclusion). Obviously, if the premise about proportionality is a natural fact, it cannot logically ever be disregarded; but if that premise is already ‘artificial’ (i.e. a Divine fiat or rabbinic convention), then it can indeed conceivably be disregarded in selected cases. For example, though reward and punishment are usually subject to the principle of ‘measure for measure’, the strict justice of that law might conceivably be discarded in exceptional circumstances in the interest of mercy, and the reward might be greater than it anticipates or the punishment less than it anticipates.

Some commentators (for instance, Maccoby) have equated the dayo principle to the principle of deduction. However, this is inaccurate, for several reasons. For a start, according to logic, as we have seen, an a fortiori argument whose conclusion can be formally validated is necessarily in accord with the principle of deduction. In truth, there is no need to refer to the principle of deduction in order to validate the conclusion – the conclusion is validated by formal means, and the principle of deduction is just an ex post facto observation, a statement of something found in common to all valid arguments. Although useful as a philosophical abstraction and as a teaching tool, it is not necessary for validation purposes.

Nevertheless, if a conclusion was found not to be in accord with the principle of deduction, it could of course be forthwith declared invalid. For the principle of deduction is also reasonable by itself: we obviously cannot produce new information by purely rational means; we must needs get that information from somewhere else, either by deduction from some already established premise(s) or by induction from some empirical data or, perhaps, by more mystical means like revelation, prophecy or meditative insight. So obvious is this caveat that we do not really need to express it as a maxim, though there is no harm in doing so.

For the science of logic, and more broadly for epistemology and ontology, then, a fortiori argument and the ‘limitation’ set upon it by the principle of deduction are (abstract) natural phenomena. The emphasis here is on the word natural. They are neither Divinely-ordained (except insofar as all natural phenomena may be considered by believers to be Divine creations), nor imposed by individual or collective authority, whether religious or secular, rabbinical or academic, nor commonly agreed artificial constructs or arbitrary choices. They are universal rational insights, apodictic tools of pure reason, in accord with the ‘laws of thought’ which serve to optimize our knowledge.

The first three of these laws are that we admit facts as they are (the law of identity), in a consistent manner (the law of non-contradiction) and without leaving out relevant data pro or con (the law of the excluded middle); the fourth is the principle of induction and the fifth is that of deduction.

To repeat: for logic as an independent and impartial scientific enterprise, there is no ambiguity or doubt that an a fortiori argument that is indeed properly constructed, with a conclusion that exactly mirrors the minor premise, is valid reasoning. Given its two premises, its (non-‘proportional’) conclusion follows of necessity; that is to say, if the two premises are admitted as true, the said conclusion must also be admitted as true. Moreover, to obtain an a crescendo conclusion additional information is required; without such information a ‘proportional’ conclusion would be fallacious. A principle of deduction can be formulated to remind people that such new information is not producible ex nihilo; but such a principle is not really needed by the cognoscenti.

This may all seem obvious to many people, but Talmudists or students of the Talmud trained exclusively in the traditional manner may not be aware of it. That is why it was necessary for us here to first clarify the purely logical issues, before we take a look at what the Talmud says. To understand the full significance of what it says and to be able to evaluate its claims, the reader has to have a certain baggage of logical knowledge.

The understanding of qal vachomer as a natural phenomenon of logic seems, explicitly or implicitly, accepted by most commentators. Rabbi Adin Steinsaltz, for instance, in his lexicon of Talmudic hermeneutic principles, describes qal vachomer as “essentially logical reasoning”[12]. Rabbi J. Immanuel Schochet says it more forcefully: “Qal vachomer is a self-evident logical argument”[13]. The equation of the dayo principle to the principle of deduction is also adopted by many commentators, especially logicians. For instance, after quoting the rabbinical statement “it is sufficient if the law in respect of the thing inferred be equivalent to that from which it is derived,” Ventura writes very explicitly: “We are resting here within the limits of formal logic, according to which the conclusion of a syllogism must not be more extensive than its premises”[14].

However, as we shall discover further on [in AFL], the main reason the proposed equation of the dayo principle to the principle of deduction is ill-advised is that it incorrect. There are indeed applications where the dayo imperative happens to correspond to the principle of deduction; but there are also applications where the two diverge in meaning. Commentators who thought of them as equal only had the former cases in mind when they did so; when we consider the latter cases, we must admit that the two principles are very different.


5.   Analysis of Numbers 12:14-15

[Numbers 12:14-15 is a Torah passage which plays an important role in the Gemara, in Baba Qama 25a, where it is used as an illustration of the rabbinical hermeneutic rule of qal vachomer (a fortiori argument) and as a justification of its attendant dayo (sufficiency) principle which is formulated in the Mishna BQ 2:5.]

The reason why this Torah passage was specifically focused on by the Gemara should be obvious. This is the only a fortiori argument in the whole Tanakh that is both spoken by God and has to do with inferring a penalty for a specific crime. None of the other four a fortiori arguments in the Torah are spoken by God[15]. And of the nine other a fortiori arguments in the Tanakh spoken by God, two do concern punishment for sins but not specifically enough to guide legal judgment[16]. Clearly, the Mishna BQ 2:5 could only be grounded in the Torah through Num. 12:14-15. This passage reads:


“14. 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. 15. And Miriam was shut up without the camp seven days; and the people journeyed not till she was brought in again.”


Verse 14 may be construed as a qal vachomer as follows:

Causing Divine disapproval (P) is a greater offense (R) than causing paternal disapproval (Q). (Major premise.)

Causing paternal disapproval (Q) is offensive (R) enough to merit isolation for seven days (S). (Minor premise.)

Therefore, causing Divine disapproval (P) is offensive (R) enough to merit isolation for seven days (S). (Conclusion.)

This argument, as I have here rephrased it a bit, is a valid purely a fortiori of the positive subjectal type (minor to major)[17]. Some interpretation on my part was necessary to formulate it in this standard format[18]. I took the image of her father spitting in her face (12:14) as indicative of “paternal disapproval” caused presumably, by analogy to the context, by some hypothetical misbehavior on her part[19]. Nothing is said here about “Divine disapproval;” this too is inferred by me from the context, viz. Miriam being suddenly afflicted with “leprosy” (12:10) by God, visibly angered (12:9) by her speaking ill of Moses (12:1). The latter is her “offense” in the present situation, this term (or another like it) being needed as middle term of the argument.

The major premise, about causing Divine disapproval being a “more serious” offense than causing paternal disapproval, is an interpolation – it is obviously not given in the text. It is constructed in accord with available materials with the express purpose of making possible the inference of the conclusion from the minor premise. The sentence in the minor premise of “isolation” for seven days due to causing paternal disapproval may be inferred from the phrase “should she not hide in shame seven days?” The corresponding sentence in the putative conclusion of “isolation” for seven days due to causing Divine disapproval may be viewed as an inference made possible by a fortiori reasoning.

With regard to the term “isolation,” the reason I have chosen it is because it is the conceptual common ground between “hiding in shame” and “being shut up without the camp.” But a more critical approach would question this term, because “hiding in shame” is a voluntary act that can be done within the camp, whereas “being shut up without the camp” seems to refer to involuntary imprisonment by the authorities outside the camp. If, however, we stick to the significant distinctions between those two consequences, we cannot claim the alleged purely a fortiori argument to be valid. For, according to strict logic, we cannot have more information in the conclusion of a deductive argument (be it a fortiori, syllogistic or whatever) than was already given in its premise(s).

That is to say, although we can, logically, from “hiding in shame” infer “isolation” (since the former is a species the latter), we cannot thereafter from “isolation” infer “being shut up without the camp” (since the former is a genus of the latter). To do so would be illicit process according to the rules of syllogistic reasoning, i.e. it would be fallacious. It follows that the strictly correct purely a fortiori conclusion is either specifically “she shall hide in shame seven days” or more generically put “she shall suffer isolation seven days.” In any case, then, the sentence “she shall be shut up without the camp seven days” cannot logically be claimed as an a fortiori conclusion, but must be regarded as a separate and additional Divine decree that even if she does not voluntarily hide away, she should be made to do so against her will (i.e. imprisoned).

We might of course alternatively claim that the argument is intended as a crescendo rather than purely a fortiori. That is to say, it may be that the conclusion of “she should be shut up without the camp seven days” is indeed inferred from the minor premise “she would hide in shame seven days” – in ‘proportion’ to the severity of the wrongdoing, comparing that against a father and that against God. For this to be admitted, we must assume a tacit additional premise that enjoins a pro rata relationship between the importance of the victim of wrongdoing (a father, God) and the ensuing punishment on the culprit (voluntary isolation, forced banishment and incarceration).

Another point worth highlighting is the punishment of leprosy. Everyone focuses on Miriam’s punishment of expulsion from the community for a week, but that is surely not her only punishment. She is in the meantime afflicted by God with a frightening disease, whereas the hypothetical daughter who has angered her father does not have an analogous affliction. So, the two punishments are not as close to identical as they may seem judging only with reference to the seven days of isolation. Here again, we may doubt the validity of the strictly a fortiori argument. This objection could be countered by pointing out that the father’s spit is the required analogue of leprosy. But of course, the two afflictions are of different orders of magnitude; so, a doubt remains.

We must therefore here again admit that this difference of punishment between the two cases is not established by the purely a fortiori argument, but by a separate and additional Divine decree. Or, alternatively, by an appropriate a crescendo argument, to which no dayo is thereafter applied. We may also deal with this difficulty by saying that the punishment of leprosy was already a fact, produced by God’s hand, before the a fortiori argument is formulated; whereas the latter only concerns the punishment that is yet to be applied, by human intervention – namely, the seven days’ isolation. Thus, the argument intentionally concerns only the later part of Miriam’s punishment, and cannot be faulted for ignoring the earlier part.

It is perhaps possible to deny that an a fortiori argument of any sort is intended here. We could equally well view the sentence “Let her be shut up without the camp seven days” as an independent decree. But, if so, of what use is the rhetorical exclamation “If her father had but spit in her face, should she not hide in shame seven days?” and moreover how to explain to coincidence of “seven days” isolation in both cases? Some sort of analogy between those two clauses is clearly intended, and the a fortiori or a crescendo argument serves to bind them together convincingly. Thus, although various objections can be raised regarding the a fortiori format or validity of the Torah argument, we can say that all things considered the traditional reading of the text as a qal vachomer is reasonable. This reading can be further justified if it is taken as in some respects a crescendo, and not purely a fortiori.

What, then, is the utility of the clause: “And after that she shall be brought in again”? Notice that it is not mentioned in my above a fortiori construct. Should we simply read it as making explicit something implied in the words “Let her be shut up without the camp seven days”? Well, these words do not strictly imply that after seven days she should be brought back into the camp; it could be that after seven days she is to be released from prison (where she has been “shut up”), but not necessarily brought back from “without the camp.” So, the clause in question adds information. At the end of seven days, Miriam is to be both released from jail and from banishment from the tribal camp.

Another possible interpretation of these clauses is to read “Let her be shut up without the camp seven days” as signifying a sentence of at least seven days, while “And after that she shall be brought in again” means that the sentence should not exceed seven days (i.e. “after that” is taken to mean “immediately after that”). They respectively set a minimum and a maximum, so that exactly seven days is imposed. What is clear in any case is that “seven days isolation” is stated and implied in both the proposed minor premise and conclusion; no other quantity, such as fourteen days, is at all mentioned, note well. This is a positive indication that we are indeed dealing essentially with a purely a fortiori argument, since the logical rule of the continuity between the given and inferred information is (to that extent) obeyed.

As we shall see when we turn to the Gemara’s treatment [in AFL] although there is no explicit mention of fourteen days in the Torah conclusion, it is not unthinkable that fourteen days were implicitly intended (implying an a crescendo argument from seven to fourteen days) but that this harsher sentence was subsequently mitigated (brought back to seven days) by means of an additional Divine decree (the dayo principle, to be exact) which is also left tacit in the Torah. In other words, while the Torah apparently concludes with a seven-day sentence, this could well be a final conclusion (with unreported things happening in between) rather than an immediate one. Nothing stated in the Torah implies this a crescendo reading, but nothing denies it either. So much for our analysis of verse 14.

Let us now briefly look at verse 15: “And Miriam was shut up without the camp seven days; and the people journeyed not till she was brought in again.” The obvious reading of this verse is that it tells us that the sentence in verse 14 was duly executed – Miriam was indeed shut away outside the camp for exactly seven days, after which she was released and returned to the camp, as prescribed. We can also view it as a confirmation of the reasoning in the previous verse – i.e. as a way to tell us that the apparent conclusion was the conclusion Moses’ court adopted and carried out. We shall presently move on [in AFL], and see how the Gemara variously interpreted or used all this material.

But first let us summarize our findings. Num. 12:14-15 may, with some interpolation and manipulation, be construed as an a fortiori argument of some sort. If this passage of the Torah is indeed a qal vachomer, it is not an entirely explicit (meforash) one, but partly implicit (satum). In some respects, it would be more appropriate to take it as a crescendo, rather than purely a fortiori. It could even be read as not a qal vachomer at all; but some elements of the text would then be difficult to explain.

It is therefore reasonable to read an a fortiori argument into the text, as we have done above and as traditionally done in Judaism. It must however still be stressed that this reading is somewhat forced if taken too strictly, because there are asymmetrical elements in the minor premise and conclusion. We cannot produce a valid purely a fortiori inference without glossing over these technical difficulties. Nevertheless, there is enough underlying symmetry between these elements to suggest a significant overriding a fortiori argument that accords with the logical requirement of continuity (i.e. with the principle of deduction). The elements not explained by a fortiori argument can and must be regarded as separate and additional decrees. Alternatively, they can be explained by means of a crescendo arguments.


In the present section, we have engaged in a frank and free textual analysis of Num. 12:14-15. This was intentionally done from a secular logician’s perspective. We sought to determine objectively (irrespective of its religious charge) just what the text under scrutiny is saying, what its parts are and how they relate to each other, what role they play in the whole statement. Moreover, most importantly, the purpose of this analysis was to find out what relation this passage of the Torah might have to a fortiori argument and the principle of dayo: does the text clearly and indubitably contain that form of argument and its attendant principle, or are we reading them into it? Is the proposed reasoning valid, or is it somewhat forced?

We answered the questions as truthfully as we could, without prejudice pro or con, concluding that, albeit various difficulties, a case could reasonably be made for reading a valid a fortiori argument into the text. These questions all had to be asked and answered before we consider and discuss the Gemara’s exegesis of Num. 12:14-15, because the latter is in some respects surprisingly different from the simple reading. We cannot appreciate the full implications of what it says if we do not have a more impartial, scientific viewpoint to compare it to. What we have been doing so far, then, is just preparing the ground, so as to facilitate and deepen our understanding of the Gemara approach to the qal vachomer argument and the dayo principle when we get to it.

One more point needs to be made here. As earlier said, the reason why the Gemara drew attention in particular to Num. 12:14-15 is simply that this passage is the only one that could possibly be used to ground the Mishna BQ 2:5 in the Torah. However, though as we have been showing Num. 12:14-15 can indeed be used for this purpose, the analogy is not perfect. For whereas the Mishnaic dayo principle concerns inference by a rabbinical court from a law (a penalty for a crime, to be precise) explicit in the Torah to a law not explicit in the Torah (sticking to the same penalty, rather than deciding a proportional penalty), the dayo principle implied (according to most readings) in Num. 12:14-15 relates to an argument whose premises and conclusion are all in the Torah, and moreover it infers the penalty (for Miriam’s lèse-majesté) for the court to execute by derivation from a penalty (for a daughter offending her father) which may be characterized as intuitively-obvious morality or more sociologically as a pre-Torah cultural tradition.

For if we regard (as we could) both penalties (for a daughter and for Miriam) mentioned in Num. 12:14-15 as Divinely decreed, we could not credibly also say that the latter (for Miriam) is inferred a fortiori from the former (for a daughter). So, the premise in the Miriam case is not as inherently authoritative as it would need to be to serve as a perfect analogy for the Torah premise in the Mishnaic case. For the essence of the Mishnaic sufficiency principle is that the court must be content with condemning a greater culprit with the same penalty as the Torah condemns a lesser culprit, rather than a proportionately greater penalty, on the grounds that the only penalty explicitly justified in the Torah and thus inferable with certainty is the same penalty. That is, the point of the Mishnaic dayo is that the premise is more authoritative than the conclusion, whereas in the Num. 12:14-15 example this is not exactly the case. What this means is that although the Mishnaic dayo can be somewhat grounded on Num. 12:14-15, such grounding depends on our reading certain aspects of the Mishna into the Torah example. That is to say, the conceptual dependence of the two is mutual rather than unidirectional.


Drawn from.A Fortiori Logic (2013), chapter 7:2,4 (parts).


[1]           Note in passing: the Hebrew name of a fortiori argument, viz. qal vachomer (i.e. ‘minor and major’, suggesting minor to major, since the word ‘minor’ precedes the word ‘major’), is indicative that the rabbis likewise viewed this mood as the primary and most typical one. Otherwise, they might have called it chomer veqal!

[2]           I leave out a pari or egalitarian a fortiori argument here for the sake of simplicity. This has been mentioned and dealt with in an earlier chapter (1). But briefly put, this deals with cases where Rp = Rq.

[3]           This is known as the Talmudic rule of bichlal maasaim maneh, although I do not know who first formulated it, nor when and where he did so.

[4]           In its most general form, this principle may be stated as: what in a given context of information appears to be true, may be taken to be effectively true, unless or until new information is found that puts in doubt the initial appearance. In the latter event, the changed context of information may generate a new appearance as to what is true; or it may result in some uncertainty until additional data comes into play.

[5]           For example, having generalized from “some X are Y” to “all X are Y” – if it is thereafter discovered that “some X are not Y,” the premise “some X are Y” is not contradicted, but the conclusion “all X are Y” is indeed contradicted and must be abandoned.

[6]           Of course, if Rs1 was assumed as greater than Rs2, we would be able to infer that Rp > Rs2. But this is not the thrust of those who try to “quantify” a fortiori argument, since the proportion between P and Q would be inversed between Rs1 and Rs2. Moreover, the next objection, viz. that “If Rs2 then S2” cannot be deduced from “If Rs1 then S1,” would still be pertinent.

[7]           I put the adjective ‘proportional’ in inverted commas because the proportion of S2 to S1 is usually not exactly equal to that of P to Q. But whether this expression is intended literally or roughly makes no difference to the invalidity of the argument, note well. If it is invalid when exact, as here demonstrated, then it is all the more so when approximate!

[8]           A neutral example would be: suppose we know that product A is more expensive than product B; knowing a certain quantity of product B to cost $1000, we could only predict by purely a fortiori argument that the same quantity of product A will cost ‘at least $1000’. But this would not prevent us from looking at a price list and finding the actual price of that quantity of product A to be $1250. However, such price adjustment would be an after the fact calculation based on the price list rates, and not an inference based on the a fortiori argument. In fact, once we obtained the price list we would not need the a fortiori argument at all.

[9]           Not to be confused with “none the less”.

[10]          This is evident in the Latin expression a fortiori ratione, meaning ‘with stronger reason’.

[11]          The term is of Italian origin, and used in musicology to denote gradual increase in volume.

[12]          P. 139. My translation from the French (unfortunately, I only have a French edition on hand at time of writing).

[13]          In a video lecture online at:; note, however, that he accepts the Gemara’s idea that the argument in Num. 12:14 would logically yield the conclusion of “fourteen days” instead of “seven days,” were it not for the dayo principle. Another online commentary states: “Unlike a Gezeirah Shavah, the Kal va'Chomer inference need not be received as a tradition from one's teacher, since it is based upon logic;” see this at:

[14]          In the Appendix to chapter 8 of Terminologie Logique (Maimonides’ book on logic, p. 77). Ventura is translator and commentator (in French). The translation into English is mine. He is obviously using the word syllogism in a general sense (i.e. as representative of any sort of deduction, not just the syllogistic form).

[15]          One is by Lemekh (Gen. 4:24), one is by Joseph’s brothers (Gen. 44:8), and two are by Moses (Ex. 6:12 and Deut. 31:27). The argument by Lemekh could be construed as concerning a penalty, but the speaker is morally reprehensible and his statement is more of a hopeful boast than a reliable legal dictum.

[16]          The two arguments are in Jeremiah 25:29 and 49:12. The tenor of both is: if the relatively innocent are bad enough to be punished, then the relatively guilty are bad enough to be punished. The other seven a fortiori arguments in the Nakh spoken by God are: Isaiah 66:1, Jer. 12:5 (2 inst.) and 45:4-5, Ezek. 14:13-21 and 15:5, Jonah 4:10-11. Note that, though Ezek. 33:24 is also spoken by God, the (fallacious) argument He describes is not His own – He is merely quoting certain people.

[17]          Actually, it would be more accurate to classify this argument as positive antecedental, since the predicate S (meriting isolation for seven days) is not applied to Q or P (causing disapproval), but to the subject of the latter (i.e. the person who caused disapproval). That is, causing disapproval implies meriting isolation. But I leave things as they are here for simplicity’s sake.

[18]          I say ‘on my part’ to acknowledge responsibility – but of course, much of the present reading is not very original.

[19]          The Hebrew text reads ‘and her father, etc.’; the translation to ‘if her father, etc.’ is, apparently, due to Rashi’s interpretation “to indicate that the spitting never actually occurred, but is purely hypothetical” (Metsudah Chumash w/Rashi at:

Go Back


Blog Search

Blog Archive


There are currently no blog comments.