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A FORTIORI LOGIC

© Avi Sion, 2013 All rights reserved.

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A FORTIORI LOGIC

CHAPTER 7 – A fortiori in the Talmud

1. Brief history of a fortiori

2. A brief course in the relevant logic

3. A fresh analysis of the Mishna Baba Qama 2:5

4. A logician’s reading of Numbers 12:14-15

5. A critique of the Gemara in Baba Qama 25a

6. A slightly different reading of the Gemara

1. Brief history of a fortiori

There is credible written evidence that a fortiori argument was in use in very early times thanks to the Jewish Bible. Five instances are apparent in the Torah proper (the Five Books of Moses, or Pentateuch) and about forty more are scattered throughout the Nakh (the other books of the Bible). According to Jewish tradition, the Torah dates from about 1300 BCE (the time of the Exodus from Egypt and wanderings in the Sinai desert)[1], and subsequent Biblical books range in age from that time to about the 4th century BCE (the period of the return from Babylon of some of the captives after the destruction of the first Temple). The oldest apparent a fortiori (actually, a crescendo) argument in the Torah is the one formulated in Gen. 4:24 by Lamekh (before the deluge); while the oldest purely a fortiori argument is the one formulated in Gen. 44:8 by Joseph’s brothers (patriarchal era). A fortiori arguments are also found in some of the latest books of the Bible (first exile period).

Of the 46 or so instances of a fortiori argument in the Tanakh (see Appendix 1), at least 10 were known to (i.e. were consciously identified as such by) the rabbis of the Talmud – so it is not surprising that this form of argument came to play such an important role in the development of Jewish law. The qal vachomer argument, as it is called in Hebrew, is mentioned in several lists of Talmudic hermeneutic principles. It is the first rule in the list of 7 attributed to Hillel (the Elder, Babylonia and Eretz Israel, c. 110 BCE-10 CE) and the first rule in the list of 13 attributed to R. Ishmael (ben Elisha, Eretz Israel, 90-135 CE), both of which are given at the beginning of the Sifra (a halakhic midrash, attributed by many to Rab, i.e. Abba Arika, 175–247 CE). It is also found (as rules 5 and 6) in the slightly later list of 32 rules of R. Eliezer b. Jose ha-Gelili (Eretz Israel, ca. 2nd cent. CE)[2], and among the much later 613 rules of the Malbim (Meïr Leibush ben Yechiel Michel Weiser, Ukraine, 1809-1879) in his work Ayelet haShachar, the introduction to his commentary on the Sifra.[3]

As regards historical source, there can be little doubt that the rabbis learned a fortiori argument from its use in the Tanakh – and not (as some commentators have suggested) from surrounding cultures (Greek, Roman, or whatever). We can be sure of that knowing that the Talmudic rabbis’ attention was wholly turned towards Jewish Scriptures and oral tradition; and a fortiori arguments were clearly in use in these sources; and moreover, everyone agrees that the Torah, at least, antedates by several centuries the historical appearance of a fortiori argument in other cultures. This does not, of course, imply that the Greeks and other early users of a fortiori argument learned this form of reasoning from the Torah or other Jewish sources. There is no doubt that a fortiori argument arose independently in different cultures at different times, simply due to its being a natural form of human reasoning[4].

In the lists of Hillel and R. Ishmael, all that is offered is a title or heading: “qal vachomer,” which is variously translated as light and heavy, easy and difficult, lenient and stringent, or minor and major. It should be said that the language of a fortiori argument in the Tanakh, though very varied (but not always distinctive, i.e. not always specifically reserved for such argument), does not include the words qal vachomer. This expression is presumably therefore of rabbinical origin. Two other expressions indicative of a fortiori discourse are also found in rabbinic literature: kol she ken (which seems to be the Hebrew equivalent of ‘all the more so’) and al achat kama vekama (which seems to be the Hebrew equivalent of ‘how much the more’).[5]

The term qal vachomer is somewhat descriptive, in the way of a hint – but note well that it is certainly not a description of a fortiori argument in formal terms, and it does not validate or even discuss the validity of the argument (but, obviously, takes it for granted). The list of R. Eliezer b. Jose ha-Gelili is not much more informative in that respect than those of its predecessors, since it only adds that qal vachomer may be meforash (i.e. explicit) or satum (i.e. implicit)[6]. Other early rabbinic literature does not go much further in elucidating the definition and more theoretical aspects of qal vachomer; it is all taken for granted.

Rather, the form and operation of a fortiori argument are taught through concrete examples. Ten Biblical examples of the argument, four in the Torah and six elsewhere[7], are listed in Genesis Rabbah (in Heb. Bereshith Rabbah), a midrashic work (closed ca. 400-450 CE) attributed by tradition to R. Oshia Rabba (d. ca. 350 CE). This just says: “R. Ishmael taught: [There are] ten a fortiori arguments recorded in the Torah” (92:7), and lists the ten cases without further comment. But of course, the main teaching of such argumentation is through the practice of the rabbis. There are a great many concrete examples of a fortiori reasoning in the Talmud and other rabbinic literature[8], which incidentally serve to clarify the form for future generations.

There is, however, one passage of the Talmud which is very instructive as to how the rabbis theoretically understood the qal vachomer (a fortiori type) argument and the dayo (sufficiency) principle related to it – and that is pp. 24b-25a and further on pp. 25b-26a of the tractate Baba Qama (meaning: ‘first gate’), which is part of the order of Neziqin (‘damages’). For the time being we shall concentrate on this important passage. We shall have occasion further on in the present volume to consider and explore some other significant Talmudic a fortiori arguments.

However, this book makes no claim to constituting an exhaustive study of this subject. Nonetheless, while I must confess being largely ignorant of the ‘Sea of the Talmud’, I believe the present contribution will be found very valuable due to the considerable extent and depth of new logical insight it contains. We shall in the present chapter, further on, describe in detail just what the said passage of the Talmud reveals. But first permit me to prepare you, the reader, with some background information and analysis, so that you come properly armed to the crux of the matter.

The Talmud (meaning: the teaching) in general consists of a series of rabbinical discussions on various legal and other topics stretching over centuries, roughly from about the 1st century BCE to about the 5th century CE. It has two essential components: the first and historically earliest stratum (ca. 200 CE) is the Mishna (meaning: repetition) and the second and later stratum is the Gemara (meaning: completion)[9]. The Gemara is a commentary (in Aramaic) on the Mishna (which is in Hebrew), clarifying, explaining and amplifying it[10].

The compiling and editing of the Mishna (whose participants are known as Tannaim, teachers) is traditionally attributed to R. Yehudah HaNassi (d. 219 CE), while the redaction of the Gemara (whose participants are known as Amoraim, expounders) took more time and was the work of many (until ca. 500 CE). This refers to the main, Babylonian (Bavli) Talmud, with which we are here concerned; there is an earlier, less authoritative compilation known as the Jerusalem (Yerushalmi) – or more precisely put, the Land of Israel[11] – Talmud (closed ca. 350-400 CE).[12]

The genesis of these various documents is an interesting historical issue, which has received much attention over time and more critical attention in modern times. Their redactors are thought to have been numerous and stretched out over centuries[13]. Some of the individuals involved are known by tradition, others remain anonymous. They should not, of course, be viewed as standing outside looking in on the collective discursive process they describe. Some of them were without doubt active or passive contemporary participants in some of the Talmudic discussions they report. But even those who do not fall in the category of eye-witnesses must be considered as effectively participants, albeit sometimes centuries after the fact, since by their selection, ordering and slanting of scattered material, their paraphrases and explanations, not to mention their outright interpolations, they necessarily affect our perceptions of the presumed original discussions. It would be a grave error to regard such redactors as entirely self-effacing, perfectly objective and impartial, contemporary observers and stenographers.

The Mishna and the Gemara[14]were conceived as written records of past and present oral legal (halakhic) and to a lesser extent, non-legal (haggadic) traditions. The rabbis (as we shall here indifferently call all participants) mentioned or implied in them did not all live at the same time and in the same place, note well. Their discussions were rarely face to face, but were brought together in one continuous document by the redactors, who were therefore perforce (albeit often invisibly) themselves important participants in the discussions, by virtue of their work of selection, structuring and commentary. Keep in mind this scattering in time and place of participants, and also the constant presence of the redactors in the background of all discourse[15]. Too often, traditional students of the Talmud approach it naïvely and idealistically as an essentially indivisible unit, somehow transcending time and space, perfectly harmonious.

There were perforce long periods of time when the traditions that were eventually put down in writing were transmitted by word of mouth. It must be considered whether such transmission was always perfect, or whether some elements were lost, transformed or added along the way. While it is true that people in those days were more used to memorizing things than we are today, and that they used various mnemonic devices to do so, one may still reasonably assume that some change in the information occurred over time if only unwittingly. Also, as Louis Jacobs has pointed out[16], in the name of I. H. Weiss, with reference to modern day scholars who are able to recite the whole of the Talmud by heart, it is surely easier to memorize a document one has read than to memorize information never seen in written form. It should also be considered that people naturally vary in intelligence, and students often do not understand all that their teachers do, and indeed sometimes students understand more than their teachers do. In short, the oral tradition should never be looked upon as some static solid phenomenon, but rather as a living mass subject to some change over time.

2. A brief course in the relevant logic

Before we examine any Talmudic text in detail, we need to briefly clarify the logical point of view on a fortiori argument. This clarification is a necessary propaedeutic, because many of the Talmudists and students of the Talmud who may choose to read this essay are probably not acquainted with any objective analysis of the underlying logic, having only been trained in rabbinical ways, which are rarely very formal. The treatment proposed in the present section is of course minimal – much more can be learned about the a fortiori argument in other chapters of the present volume and in my past work called Judaic Logic.

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[17], 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[18]:

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[19]). 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.

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[20], 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)[21]. 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).[22]

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[23]. 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[24]. 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”[25] 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[26]. 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’.

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’[27], 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, 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.

Regarding 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), 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”[28]. Rabbi J. Immanuel Schochet says it more forcefully: “Qal vachomer is a self-evident logical argument”[29]. 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”[30].

However, as we shall discover further on, 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.

3. A fresh analysis of the Mishna Baba Qama 2:5

In the Mishna Baba Qama 2:5, there is a debate between the Sages and R. Tarfon about the concrete issue of the financial liability of the owner of an ox which causes damages by goring on private property. This debate has logical importance, in that it reveals to a considerable extent skills and views of Talmudic rabbis with regard to the a fortiori argument. The Sages consider that he must pay for half the damages, whereas R. Tarfon advocates payment for all the damages[31].

The Sages (hachakhamim) are unnamed rabbis of Mishnaic times (Tannaim) and R. Tarfon is one of their colleagues (of the 3rd generation), who lived in Eretz Israel roughly in the late 1st – early 2nd century CE. We are not told how many were the Sages referred to in this Mishna (presumably there were at least two), nor who they were. The contemporaries of R. Tarfon include R. Eleazar b. Azariah, R. Ishmael b. Elisha, R. Akiva, and R. Jose haGelili; it is conceivable that these are the Sages involved in this debate. They are all big names, note; the latter three, as we have seen, produced hermeneutic principles. R. Tarfon, too, was an important and respected figure. So the debate between them should be viewed as one between equals.[32]

The Mishna (BQ 2:5) is as follows[33]:

“What is meant by ‘ox doing damage on the plaintiff’s premises’? In case of goring, pushing, biting, lying down or kicking, if on public ground the payment is half, but if on the plaintiff’s premises R. Tarfon orders payment in full whereas the Sages order only half damages.

R. Tarfon there upon said to them: seeing that, while the law was lenient to tooth and foot in the case of public ground allowing total exemption, it was nevertheless strict with them regarding [damage done on] the plaintiff’s premises where it imposed payment in full, in the case of horn, where the law was strict regarding [damage done on] public ground imposing at least the payment of half damages, does it not stand to reason that we should make it equally strict with reference to the plaintiffs premises so as to require compensation in full?

Their answer was: it is quite sufficient that the law in respect of the thing inferred should be equivalent to that from which it is derived: just as for damage done on public ground the compensation [in the case of horn] is half, so also for damage done on the plaintiff’s premises the compensation should not be more than half.

R. Tarfon, however, rejoined: but neither do I infer horn [doing damage on the plaintiff’s premises] from horn [doing damage on public ground]; I infer horn from foot: seeing that in the case of public ground the law, though lenient with reference to tooth and foot, is nevertheless strict regarding horn, in the case of the plaintiff’s premises, where the law is strict with reference to tooth and foot, does it not stand to reason that we should apply the same strictness to horn?

They, however, still argued: it is quite sufficient if the law in respect of the thing inferred is equivalent to that from which it is derived. Just as for damage done on public ground the compensation [in the case of horn] is half, so also for damage done on the plaintiff’s premises, the compensation should not be more than half.”

This discussion may be paraphrased as follows. Note that only three amounts of compensation for damages are considered as relevant in the present context: nil, half or full; there are no amounts in between or beyond these three, because the Torah never mentions any such other amounts.

(a) R. Tarfon argues that in the case of damages caused by “tooth and foot,” the (Torah based) law was lenient (requiring no payment) if they occurred on public ground and strict (requiring full payment) if they occurred on private ground – “does it not stand to reason that” in the case of damages caused by “horn,” since the (Torah based) law is median (requiring half payment) if they occurred on public ground, then the law (i.e. the rabbis’ ruling in this case) ought to likewise be strict (requiring full payment) if they occurred on private ground? Presented more briefly, and in a nested manner, this first argument reads as follows:

If tooth & foot, then:

if public then lenient, and

if private then strict.

If horn, then:

if public then median, and

if private then strict (R. Tarfon’s putative conclusion).

R. Tarfon thus advocates full payment for damage on private property. The Sages disagree with him, advocating half payment only, saying “dayo—it is enough.”

(b) R. Tarfon then tries another tack, using the same data in a different order, this time starting from the laws relating to public ground, where that concerning “tooth and foot” is lenient (requiring no payment) and that concerning “horn” is median (requiring half payment), and continuing: “does it not stand to reason that” with regard to private ground, since the law for “tooth and foot” damage is strict (requiring full payment), the law (i.e. the rabbis’ ruling in this case) for “horn” damage ought to likewise be strict (requiring full payment)? Presented more briefly and in a nested manner, this second argument reads as follows:

If public, then:

if tooth & foot then lenient, and

if horn then median.

If private, then:

if tooth & foot then strict, and

if horn then strict (R. Tarfon’s putative conclusion).

R. Tarfon thus advocates full payment for damage on private property. The Sages disagree with him again, advocating half payment only, saying “dayo—it is enough.”

More precisely, they reply to him both times: “it is quite sufficient that the law in respect of the thing inferred should be equivalent to that from which it is derived” – meaning that only half payment should be required in the case under consideration (viz. damages by “horn” on private grounds). In Hebrew, their words are: כנדון להיות הדין מן לבא דיו (dayo lavo min hadin lihiot kenidon) – whence the name dayo principle[34].

Now, the first thing to notice is that these two arguments of R. Tarfon’s contain the exact same given premises and aim at the exact same conclusion, so that to present them both might seem like mere rhetoric (either to mislead or out of incomprehension). The two sets of four propositions derived from the above two arguments (by removing the nesting) are obviously identical. All he has done is to switch the positions of the terms in the antecedents and transpose premises (ii) and (iii). The logical outcome seems bound to be the same:

(a) If tooth & foot and public, then lenient (i).

If tooth & foot and private, then strict (ii).

If horn and public, then median (iii).

If horn and private, then strict (R. Tarfon’s putative conclusion).

(b) If public and tooth & foot, then lenient (same as (i)).

If public and horn, then median (same as (iii)).

If private and tooth & foot, then strict (same as (ii)).

If private and horn, then strict (same putative conclusion).

However, as we shall soon realize, the ordering of the terms and propositions does make a significant difference. And we shall see precisely why that is so.

(a) What is R. Tarfon’s logic in the first argument? Well, it seems obvious that he is making some sort of argument by analogy; he is saying (note the identity of the two sentences in italics):

Just as, in one case (that of tooth & foot), damage in the private domain implies more legal liability than damage in the public domain (since strict is more stringent than lenient).

So, in the other case (viz. horn), we can likewise say that damage in the private domain implies more legal liability than damage in the public domain (i.e. given median in the latter, conclude with strict, i.e. full payment, in the former, since strict is more stringent than median).



Just as in one case we pass from lenient to strict, so in the other case we may well pass from median to strict[35]. Of course, as with all analogy, a generalization is involved here from the first case (tooth & foot being more stringent for private than for public) up to “all cases” (i.e. the generality in italics), and then an application of that generality to the second case (horn, thusly concluded to be more stringent for private than for public). But of course, this is an inductive act, since it is not inconceivable that there might be specific reasons why the two cases should behave differently. Nevertheless, if no such specific reasons are found, we might well reason that way. That is to say, R. Tarfon does have a point, because his proposed reasoning can well be upheld as an ordinary analogical argument. This might even be classified under the heading of gezerah shavah or maybe binyan av (the second or third rule in R. Ishmael’s list of thirteen)[36].

The above is a rather intuitive representation of R. Tarfon’s first argument by analogy. Upon reflection, this argument should be classified more precisely as a quantitative analogy or pro rata argument:

The degree of legal liability for damage is ‘proportional’ to the status of the property the damage is made on, with damage in the private domain implying more legal liability than damage in the public domain.

This is true of tooth and foot damage, for which liability is known to be nil (lenient) in the public domain and full (strict) in the private domain.

Therefore, with regard to horn damage, for which liability is known to be half (median) in the public domain, liability may be inferred to be full (strict) in the private domain.



This argument, as can be seen, consists of three propositions: a general major premise, a particular (to tooth and foot) minor premise and a particular (to horn) conclusion. The major premise is, in fact, known by induction – a generalization of the minor premise, for all damage in relation to property status. But once obtained, it serves to justify drawing the conclusion from the minor premise. The pro rata argument as such is essentially deductive, note, even though its major premise is based on an inductive act. But its conclusion is nevertheless a mere rough estimate, since the ‘proportionality’ it is based on is very loosely formulated. Notice how the minor premise goes from zero to 100%, whereas the conclusion goes from 50% to 100%[37].

The Sages, on the other hand, seem to have in mind, instead of this ordinary argument by analogy or pro rata argument, a more elaborate and subtle a fortiori argument of positive subjectal form. They do not explicitly present this argument, note well; but it is suggested in their reactions to their colleague’s challenge. Their thinking can be construed as follows:

Private domain damage (P) implies more legal liability (R) than public domain damage (Q) [as we know by extrapolation from the case of tooth & foot].

For horn, public domain damage (Q) implies legal liability (R) enough to make the payment half (median) (S).

Therefore, for horn, private domain damage (P) implies legal liability (R) enough to make the payment half (median) (S).



We see that the subsidiary term (S) is the same (viz. ‘median’, i.e. half payment) in the Sages’ minor premise and conclusion, in accord with a fortiori logic; and they stress that conclusion in reply to R. Tarfon’s counterarguments by formulating their dayo principle, viz. “it is quite sufficient that the law in respect of the thing inferred should be equivalent to that from which it is derived,” to which they add: “just as for damage [by horn] done on public ground the compensation is half, so also for damage [by horn] done on the plaintiff’s premises the compensation should not be more than half.”[38]

We see also that the major premise of the Sages’ qal vachomer is identical to the statements in italics of R. Tarfon’s argument by analogy, i.e. to the major premise of his pro rata argument. In both R. Tarfon and the Sages’ arguments, this sentence “private damage implies more legal liability than public damage” is based on the same generalization (from tooth & foot, in original premises (i) and (ii), as already seen) and thence applicable to the case under scrutiny (horn, for which proposition (iii) is already given)[39]. So both their arguments are equally based on induction (they disagreeing only as to whether to draw the conclusion (iv) or its contrary).

But the most important thing to note here is that the same premises (viz. (i), (ii) and (iii)) can be used to draw contrary conclusions (viz. full payment vs. half payment, respectively, for damage by horn on private grounds), according as we use a mere analogical or pro rata argument, like R. Tarfon, or a more sophisticated strictly a fortiori argument, like the Sages. This discrepancy obviously requires explanation. Since both arguments are built on the same major premise, produced by the same inductive act of generalization, we cannot explain the difference by referring to the inductive preliminaries.

The way to rationalize the difference is rather to say that the argument by analogy or pro rata is more approximate, being a mere projection of the likely conclusion; whereas the a fortiori argument is more accurate, distilling the precise conclusion inherent in the premises. That is to say, though both arguments use the same preliminary induction, the argument of R. Tarfon is in itself effectively a further act of induction, whereas the argument of the Sages is in itself an act of pure deduction. Thus, the Sages’ conclusion is to be logically preferred to the conclusion proposed by R. Tarfon.

Note well that we have here assumed that R. Tarfon’s first argument was merely analogical/pro rata, and that the Sages proposed a purely a fortiori argument in response to it. It is also possible to imagine that R. Tarfon intended a purely a fortiori argument, but erroneously drew a ‘proportional’ conclusion from it; in which case, the Sages’ dayo objection would have been to reprove him for not knowing or forgetting (or even maybe deliberately ignoring) the principle of deduction, i.e. that such argument can only yield a conclusion of the same magnitude as the minor premise. However, I would not support this alternative hypothesis, which supposes R. Tarfon to have made a serious error of reasoning (or even intentionally engaged in fallacy), because it is too far-fetched. For a start, R. Tarfon is an important player throughout the Mishna, someone with in general proven logical skills; moreover, more favorable readings of this particular argument are available, so we have no reason to assume the worst.

Another possible reading is that R. Tarfon’s first argument was not merely analogical/pro rata but was intended as a crescendo, i.e. as a combination of a fortiori argument with pro rata argument, which can be briefly presented as follows:

Private domain damage (P) implies more legal liability (R) than public domain damage (Q) [as we know by extrapolation from the case of tooth & foot].

For horn, public domain damage (Q) implies legal liability (Rq) enough to make the payment half (median) (Sq).

The payment due (S) is ‘proportional’ to the degree of legal liability (R).

Therefore, for horn, private domain damage (P) implies legal liability (Rp) enough to make the payment full (strict) (Sp = more than Sq).



In that case, the dayo statement by the Sages may be viewed as a rejection of the additional premise about ‘proportionality’ between S (the subsidiary term) and R (the middle term) in the case at hand. That would represent them as saying: while proportionality might seem reasonable in other contexts, in the present situation it ought not to be appealed to, and we must rest content with a purely a fortiori argument. The advantage of this reading is that it conceives R. Tarfon as from the start of the debate resorting to the more sophisticated a fortiori type of argument, even though he conceives it as specifically a crescendo (i.e. as combined with a pro rata premise). The Sages prefer a purely a fortiori conclusion to his more ambitious a crescendo one, perhaps because it is easier to defend (i.e. relies on less assumptions), but more probably for some other motive (as we shall see).

(b) So much for the first argument; now let us examine the second argument. This, as many later commentators noticed, and as we shall now demonstrate, differs significantly from the preceding. The most important difference is that, here, the mere argument by analogy (or argument pro rata, to be more precise), the purely a fortiori argument and the a crescendo argument (i.e. a fortiori and pro rata combo), all three yield the same conclusion. Note this well – it is crucial. The second analogical argument proceeds as follows:

Just as, in one case (that of the public domain), damage by horn implies more legal liability than damage by tooth & foot (since median is more stringent than lenient).

So, in the other case (viz. the private domain), we can likewise say that damage by horn implies more legal liability than damage by tooth & foot (i.e. given strict in the latter, conclude with strict, i.e. full payment, in the former, since strict is ‘more stringent than’ [here, as stringent as[40]] strict).



This argument is, as before, more accurately represented as a pro rata argument:

The degree of legal liability for damage is ‘proportional’ to the intentionality of the cause of damage, with damage by horn implying more legal liability than damage by tooth & foot.

This is true of the public domain, for which liability is known to be nil (lenient) for damage by tooth and foot and half (median) for damage by horn.

Therefore, with regard to the private domain, for which liability is known to be full (strict) for damage by tooth and foot, liability may be inferred to be full (strict) for damage by horn.



This argument visibly consists of three propositions: a general major premise, a particular (to the public domain) minor premise and a particular (to the private domain) conclusion. The major premise is, in fact, inductive – a generalization of the minor premise, for all damage in relation to intentionality (in horn damage the ox intends to hurt or destroy, whereas in tooth and foot damage the negative consequences are incidental or accidental). But once obtained, the major premise serves to justify drawing the conclusion from the minor premise. Here again, the ‘proportionality’ is only rough; but in a different way. Notice how the minor premise goes from 0% to 50%, whereas the conclusion goes from 100% to 100%.

The purely a fortiori reading of this second argument would be as follows:

Horn damage (P) implies more legal liability (R) than tooth & foot damage (Q) [as we know by extrapolation from the case of public domain].

For private domain, tooth & foot damage (Q) implies legal liability (R) enough to make the payment full (strict) (S).

Therefore, for private domain, horn damage (P) implies legal liability (R) enough to make the payment full (strict) (S).



Note that the conclusion would be the same if this argument was constructed as a more elaborate a crescendo argument, i.e. with the additional pro rata premise “The payment due (S) is ‘proportional’ to the degree of legal liability (R).” The latter specification makes no difference here (unlike in the previous case), because (as we are told in the minor premise) the minimum payment is full and (as regards the conclusion) no payment greater than full is admitted (by the Torah or rabbis) as in the realm of possibility anyway. Thus, whether we conceive R. Tarfon’s second argument as purely a fortiori or as a crescendo, its conclusion is the same. Which means that the argument, if it is not analogical/pro rata, is essentially a fortiori rather than a crescendo.

Observe here the great logical skill of R. Tarfon. His initial proposal, as we have seen, was an argument by analogy or pro rata, which the Sages managed to neutralize by means of a logically more powerful a fortiori argument; or alternatively, it was an a crescendo argument that the Sages (for reasons to be determined) limited to purely a fortiori. This time, R. Tarfon takes no chances, as it were, and after judicious reshuffling of the given premises offers an argument which yields the same strict conclusion whether it is read as an argument by analogy (pro rata) or a more elaborate a crescendo – or as a purely a fortiori argument. A brilliant move! It looks like he has now won the debate; but, surprisingly, the Sages again reject his conclusion and insist on a lighter sentence.

Note well why R. Tarfon tried a second argument. Here, the stringency of the target law (viz. horn in the private domain) is equal to (and not, as in his first argument, greater than) the stringency of the source law (viz. tooth & foot in the private domain); i.e. both are here ‘strict’. This makes R. Tarfon’s second argument consistent with a fortiori logic and with the dayo principle that the Sages previously appealed to, since now “the law in respect of the thing inferred” is apparently “equivalent to that from which it is derived.” Yet, the Sages reiterate the dayo principle and thus reject his second try. How can they do so?

What is odd, moreover, is that the Sages answer both of R. Tarfon arguments in exactly the same words, as if they did not notice or grasp the evident differences in his arguments. The following is their identical full reply in both cases:

“It is quite sufficient that the law in respect of the thing inferred should be equivalent to that from which it is derived: just as for damage done on public ground the compensation is half, so also for damage done on the plaintiff’s premises the compensation should not be more than half.”

(אמרו לו דיו לבא מן הדין להיות כנדון מה ברהר חצי נזק אף ברשות הניזק חצי נזק)

One might well initially wonder if the Sages did not perchance fail to hear or to understand R. Tarfon’s second argument; or maybe some error occurred during the redaction of the Mishna or some later copying (this sure does look like a ‘copy and paste’ job!). For if the Sages were imputing a failure of dayo to R. Tarfon’s second argument, in the same sense as for the first argument, they would not have again mentioned the previous terms “public ground” for the minor premise and “the plaintiff’s premises” for the conclusion, but instead referred to the new terms “tooth and foot” and “horn.” But of course, we have no reason to distrust the Sages and must therefore assume that they know what they are talking about and mean what they say.

Whence, we must infer that the Sages’ second dayo remark does not mean exactly the same as their first one. In the first instance, their objection to R. Tarfon was apparently that if the argument is construed as strictly a fortiori, the conclusion’s predicate must not surpass the minor premise’s predicate; in this sense, the dayo principle simply corresponds to the principle of deduction, as it naturally applies to purely a fortiori argument. Alternatively, if R. Tarfon’s first argument is construed as pro rata or as a crescendo, the Sages’ first dayo objection can be viewed as rejecting the presumption of ‘proportionality’. However, such readings are obviously inappropriate for the Sages’ dayo objection to R. Tarfon’s second argument, since the latter however construed is fully consistent with the dayo principle in either of these senses.

How the second dayo differs from the first. An explanation we can propose, which seems to correspond to a post-Talmudic traditional explanation[41], is that the Sages are focusing on the generalization that precedes R. Tarfon’s second argument. The major premise of that argument, viz. “Horn damage implies more legal liability than tooth & foot damage” was derived from two propositions, remember, one of which was “In the public domain, horn damage entails half payment” (and the other was “In the public domain, tooth & foot damage entails no payment”). R. Tarfon’s putative conclusion after generalization of this comparison (from the public domain to all domains), and a further deduction (from “In the private domain, tooth & foot damage entails full payment”), was “In the private domain, horn damage entails full payment.” Clearly, in this case, the Sages cannot reject the proposed deduction, since it is faultless however conceived (as analogy/pro rata/a crescendo or even purely a fortiori). What they are saying, rather, is that the predicate of its conclusion cannot exceed the predicate (viz. half payment) of the given premise involving the same subject (viz. horn) on which its major premise was based.

We can test this idea by applying it to R. Tarfon’s first argument. There, the major premise was “Private domain damage implies more legal liability than public domain damage,” and this was based on two propositions, one of which was “For tooth & foot, private domain damage entails full payment” (and the other was “For tooth & foot, public domain damage entails no payment”). R. Tarfon’s putative conclusion after generalization of this comparison (from tooth & foot to all causes), and a further deduction (from “For horn, public domain damage entails half payment”), was “For horn, private domain damage entails full payment.” Clearly, in this case, the Sages cannot object that the predicate of its conclusion exceeds the predicate of the given premise involving the same subject (viz. private domain, though more specifically for tooth & foot) on which its major premise was based, since they are the same (viz. full payment). Their only possible objection is that, conceiving the argument as purely a fortiori, the predicate of the conclusion cannot exceed the predicate (viz. half payment) of the minor premise (i.e. “For horn, public domain damage entails half payment”). Alternatively, conceiving the argument as pro rata or a crescendo, they for some external reason (which we shall look into) reject the implied proportionality.

Thus, the Sages’ second objection may be regarded as introducing an extension of the dayo principle they initially decreed or appealed to, applicable to any generalization preceding purely a fortiori argument (or possibly, pro rata or a crescendo arguments, which as we have seen are preceded by the same generalization). The use and significance of generalization before a fortiori argument (or eventually, other forms of argument) are thereby taken into consideration and emphasized by the Sages. This does not directly concern the a fortiori deduction (or the two other possible arguments), note well, but only concerns an inductive preliminary to such inference. However, without an appropriate major premise, no such argument can be formed; in other words, the argument is effectively blocked from taking shape.

The question arises: how is it possible that by merely reshuffling the given premises we could obtain two different, indeed conflicting, a fortiori (or other) conclusions? The answer is that the two major premises were constructed on the basis of different directions of generalization[42]. In the first argument, the major premise is based entirely on tooth & foot data, and we learn something about horn only in the minor premise. In the second argument, the major premise relies in part on horn data, and the minor premise tells us nothing about horn. Thus, the two preliminary generalizations in fact cover quite different ground. This explains why the two a fortiori processes diverge significantly, even though the original data they were based on was the same.

The first dayo objection by the Sages effectively states that, if R. Tarfon’s first argument is construed as purely a fortiori, the conclusion must logically (i.e. by the principle of deduction) mirror the minor premise; alternatively, construing it as pro rata or a crescendo, the needed ‘proportionality’ is decreed to be forbidden (for some reason yet to be dug up). For the second argument, which has one and the same conclusion however construed (whether a fortiori or other in form), the Sages’ dayo objection cannot in the same manner refer to the minor or additional premise, but must instead refer to the inductive antecedents of the major premise, and constitute a rule that the conclusion cannot exceed in magnitude such antecedents. This explains the Sages’ repetition of the exact same sentence in relation to both of R. Tarfon’s arguments.

A problem and its solution. There is yet one difficulty in our above presentation of the Sages’ second dayo objection that we need to deal with.

As you may recall, the first dialogue between R. Tarfon and the Sages could be described as follows: R. Tarfon proposes an a crescendo argument concluding with full payment for damage by horn on private property, whereas the Sages conclude with half payment through the purely a fortiori argument leftover after his tacit premise of ‘proportionality’ is rejected by their dayo. That is, they effectively say: “The payment due (S) is not ‘proportional’ to the degree of legal liability (R).” Thus, the first exchange remains entirely within the sphere of a fortiori logic, despite the dayo application.

But the second dialogue between these parties cannot likewise be entirely included in the sphere of a fortiori logic, because the final conclusion of the Sages here is not obtained by a fortiori argument. Since the effect of their second dayo objection is to block the formation by generalization of the major premise of R. Tarfon’s second a fortiori argument, it follows that once this objection is admitted his argument cannot proceed at all; for without a general major premise such argument cannot yield, regarding horn damage on private property, a conclusion of half compensation any more than a conclusion of full compensation. Yet the Sages do wish to conclude with half compensation. How can they do so?

The answer to the question is, traditionally, to refer back to the Torah passage on which the argument is based, namely Exodus 21:35: “And if one man’s ox hurt another’s, so that it dieth; then they shall sell the live ox, and divide the price of it; and the dead also they shall divide”. This signifies half compensation for horn damage without specifying the domain (public or private) in which such damage may occur – thus suggesting that the compensation may be the same for both domains. In the above two a fortiori arguments, it has been assumed that the half compensation for horn damage applies to the public domain, and as regards the private domain the compensation is unknown – indeed, the two a fortiori arguments and the objections to them were intended to settle the private domain issue.

This assumption is logically that of R. Tarfon. Although the said Torah passage seems to make no distinction between domains with regard to damage by horn, R. Tarfon suspects that there is a distinction between domains by analogy to the distinction implied by Exodus 22:4 with regard to damage by tooth and foot (since in that context, only the private domain is mentioned[43]). His thinking seems to be that the owner of an ox has additional responsibility if he failed to preempt his animal from trespassing on private property and hurting other animals in there. So he tries to prove this idea using two arguments.

The Sages, for their part, read Exodus 21:35 concerning horn damage as a general statement, which does not distinguish between the public and private domains; and so they resist their colleague’s attempt to particularize it. For them, effectively, what matters is that two oxen belonging to two owners have fought, and one happened to kill the other; it does not matter who started the fight, or where it occurred or which ox killed which – the result is the same: equal division of the remaining assets between the owners, as the Torah prescribes. Effectively, they treat the matter as an accident, where both parties are equally faultless, and the only thing that can be done for them is to divide the leftovers between them.

Clearly, if compensation for horn damage on public grounds could be more than half (i.e. if half meant at least half), R. Tarfon could still (and with more force) obtain his two ‘full compensation’ conclusions (by two purely a fortiori arguments), but the Sages’ two dayo objections would become irrelevant. In that event, the conclusion regarding horn damage would be full compensation on both the public and private domains. But if so, why did the Torah specify half compensation (“division” in two)? Therefore, the compensation must at the outset be only half in at least one domain. That this would be the public domain rather than the private may be supposed by analogy from the case of tooth and foot[44]. This is a role played by the major premise of the first argument. This means that the first argument (or at least, its major premise) is needed before the formulation of the second. They are therefore not independent arguments, but form (in part) a chain of reasoning (a sorites) – and their order of appearance is not as accidental as we might initially have thought.

It should be realized that the assumption that the liability for horn damage on private property is equal to or greater than same on public grounds is not an a priori truth. It is not unthinkable that the liability might be less (i.e. zero) in the former case than in the latter. Someone might, say, have argued that the owner of the private property, whose animal was gored there, was responsible to prevent other people’s oxen from entering his property (e.g. by fencing it off), and therefore does not deserve any compensation! In that case, it would be argued that on public grounds he deserves half compensation because he has no control over the presence of other people’s oxen thereon. In this perspective, the onus would be on the property owner, rather than on the owner of the trespassing ox.

Given this very theoretical scenario, it would no longer be logically acceptable to generalize from the liability for damage by tooth & foot, which is less (zero) on public ground and more (full) on private ground, and to say that liability for damage of any sort (including by horn) is greater in a private domain than in the public domain. However, this scenario is not admitted by the rabbis (I do not know if they even discuss it; probably they do not because it does not look very equitable[45]). Therefore, the said generalization is accepted, and serves to determine the compensation for damage by horn on private property in both arguments. In the first argument, this generalization (from tooth & foot damage to all damage) produces the major premise. In the second argument, it serves only to eliminate in advance the possibility of zero compensation in such circumstance.

Thus, we can interpret the Torah as teaching that compensation for horn damage is generally at least half – and more specifically, no more than half on public grounds and no less than half on private property. Thereafter, the issue debated in the Mishna is whether the latter quantity is, in the last analysis, ‘only half’ or ‘more than half (i.e. full)’ compensation. Both parties in the Mishna take it for granted that the half minimum is a maximum as regards public grounds; but they leave the matter open to debate as regards its value on private property. R. Tarfon tries, in his second argument, to prove that the compensation in such circumstance ought to be full, by comparison to the law relating to tooth & foot damage in the same circumstance. But the Sages, interdict his major premise by saying dayo, in view of the textual data that premise was based on, and thus opt for only half compensation.

Following this dayo, note well, the Sages’ conclusion is not obtained by a modified a fortiori argument, since (as already mentioned) such an argument cannot be formulated without an appropriate major premise, but is obtained by mere elimination. Their form of reasoning here is negative disjunctive apodosis (modus tollens):

The appropriate compensation for horn damage on private property is, according to the Torah, at least (lav davka) half, i.e. either only half or full.

But it cannot be proved to be full (since the major premise of R. Tarfon’s attempt to do so by a fortiori cannot be sustained due to a dayo objection).

Therefore, it must be assumed to be only (davka) half (as the Sages conclude).



It should be said that this reasoning is not purely deductive, but contains an inductive movement of thought – namely, the generalization from the failure to prove full compensation specifically through R. Tarfon’s a fortiori argument in the light of the Sages’ renewed dayo objection to the impossibility henceforth to prove full compensation by any means whatever. This is a reasonable assumption, since we cannot perceive any way that the dayo might be avoided (i.e. a way not based on the given of half compensation for damage by horn on public grounds[46]); but it is still a generalization. Therefore, the apodosis is somewhat inductive; this means that further support for the Sages’ conclusion of only half compensation for damage by horn on private property would be welcome.

Thus, strictly speaking, in the last analysis, although a fortiori argument is attempted in the second dialogue, it is not finally used, but what is instead used and what provides us with the final conclusion is a disjunctive argument.

The essence of the dayo principle. We can thenceforth propose a more inclusive formulation of the Sages’ dayo principle, which merges together the said two different cases, as follows. Whenever (as in the present debate) the same original propositions can, via different directions of preparatory induction and/or via different forms of deduction, construct two or more alternative, equally cogent arguments, the chain of reasoning with the less stringent final result should be preferred. This, I submit, is to date the most accurate, all-inclusive statement of the dayo principle formulated on the basis of this Mishnaic sugya.

In the light of this broader statement of the dayo principle, we can read the two applications given in the present debate as follows. In the first argument, where there was a choice between a pro rata or a crescendo argument with a stringent conclusion, and a purely a fortiori argument with a median conclusion, the Sages chose the latter argument, with the less stringent conclusion, as operative. In the second argument, where all three forms of argument yielded the same stringent conclusion, the Sages referred instead to the preliminary generalization; in this case they found that, since the terms of one of the original propositions generalized into the major premise corresponded to the terms of the putative final conclusion, and the former proposition was less stringent than the latter, one could not, in fact, perform the generalization, but had to rest content with the original proposition’s degree of stringency in the final one.

In the first instance, the dayo principle cannot refer to the inductive antecedent of the argument, because that original proposition does not have the same terms as the final conclusion, however obtained; so we must look at the form of the deductive argument. In the second instance, the dayo principle cannot refer to the deductive argument, since whatever its form it results in the same the final conclusion; so we must look at the preliminary generalization preceding such argument. Thus, one and the same dayo principle guides both of the Sages’ dayo objections. Their teaching can thus be formulated as follows: ‘Given, in a certain context, an array of equally cogent alternative arguments, the one with the less stringent conclusion should be adopted’.

In other words, the dayo principle is a general guideline to opt for the less stringent option whenever inference leaves us a choice. It is a principle of prudence, the underlying motive of which seems to be moral – to avoid any risk of injustice in ethical or legal or religious pronouncements based on inference. We could view this as a guideline of inductive logic, insofar as it is a safeguard against possible human errors of judgment. It is a reasonable injunction, which could be argued (somewhat, though not strictly) to have universal value. But in practice it is probably specific to Judaic logic; it is doubtful that in other religions, let alone in secular ethical or legal contexts, the same restraint on inference is practiced.

An alternative translation of the Sages’ dayo principle that I have seen, “It is sufficient that the derivative equal the source of its derivation,” is to my mind very well put, because it highlights and leaves open the variety of ways that the “derivation” may occur in practice. The dayo principle, as we have seen, does not have one single expression, but is expressed differently in different contexts. The common denominator being apparently an imperative of caution, preventing too ready extrapolation from given Scriptural data. In the last analysis, then, the dayo principle is essentially not a logical principle, but rather a moral one. It is a Torah or rabbinical decree, rather than a law of logic. As such, it may conceivably have other expressions than those here uncovered. For the same reason, it could also be found to have exceptions that do not breach any laws of logic. Traditionally, it is deemed as applicable in particular to qal vachomer argument; but upon reflection, in view of its above stated essential underlying motive or purpose, it is evident that it could equally well in principle apply to other forms of argument. Such issues can only be definitely settled empirically, with reference to the whole Talmudic enterprise and subsequent developments in Jewish law.

Alternative scenarios. Our proposed scenario for the Mishna debate is thus as follows. R. Tarfon starts the discussion by proposing a first argument, whose form may be analogical/pro rata or a crescendo, which concludes with the imperative of full payment in the case of horn damage in the private domain. The Sages, appealing to a dayo principle, interdict the attempted ‘proportionality’ in his argument, thus effectively trumping it with a purely a fortiori argument, which concludes with a ruling of half payment. In response, R. Tarfon proposes a second argument, based on the very same data, which, whether conceived as analogical/pro rata or a crescendo, or as purely a fortiori, yields the very same conclusion, viz. full payment. This time, however, the Sages cannot rebut him by blocking an attempt at ‘proportionality’, since (to repeat) a non-‘proportional’ argument yields the very same conclusion as ‘proportional’ ones. So the Sages are obliged to propose an extension or enlargement of the initial dayo principle that focuses instead on the generalization before deduction. In this way, they again rule half payment.

This scenario is obvious, provided we assume the Sages’ two dayo objections are expressions of a dayo principle. It is also conceivable, however, that they have no such general principle in mind, but merely intend these objections to be ad hoc decisions in the two cases at hand. In that case the dayo principle is a “principle,” not in the strict sense of a universal principle that must be applied in every case of the sort, but in the looser sense of a guiding principle that may on occasion, for a variety of unspecified motives, be applied[47]. In fact, if we look at the Mishna passage in question, we see that nowhere is there any mention of a dayo “principle.” There is just statement “It is quite sufficient that the law in respect of the thing inferred should be equivalent to that from which it is derived,” which was presumably labeled “the dayo principle” by later commentators. This statement could be interpreted equally well as having a general or particular intent.

If we adopt the latter assumption, the scenario for the Mishna debate would be as follows: when R. Tarfon proposes his first argument, whether it is construed as pro rata or a crescendo, the Sages merely refuse his inherent ‘proportional’ premise in this particular case, without implying that they would automatically refuse it in other eventual cases. Similarly, when he proposes his second argument, whether it is construed as pro rata, a crescendo, or purely a fortiori, they merely refuse his preparatory generalization in this particular case, without implying that they would automatically refuse it in other eventual cases. Thus, the Sages might be said to making ‘ad hoc’ dayo objections, rather than appealing to a dayo ‘principle’ in the strict sense. Why would the Sages raise a dayo objection in this particular case, and not raise it in other cases? Conceivably, they perceive some unspecified danger in the present case that may be absent in other cases.

Granting this alternative view of the dayo principle, be it said in passing, there is conceivably no need to mention qal vachomer argument at all in this Mishna debate! In this view, it is possible that neither R. Tarfon nor the Sages intended any genuine a fortiori type of reasoning, but were entirely focused on mere analogy. As we shall see, although the Gemara probably does intend an a crescendo interpretation of the two arguments of R. Tarfon, it is not inconceivable that its author simply had in mind analogical/pro rata argument. Although the expression qal vachomer does appear in the Gemara, it does not necessarily have to be taken as referring to a fortiori or a crescendo argument, but could be read as referring to pro rata. It is anyhow worthwhile stating that another viewpoint is possible, because this allows us to conceptually uncouple the dayo principle from qal vachomer.

But the main value of our proposing alternative scenarios is that these provide us with different explanations of the disagreement between R. Tarfon and the Sages. Where, precisely, did they disagree? Given the primary scenario, where the dayo principle is a hard and fast principle in the eyes of the Sages, the question arises: how come R. Tarfon forgot or did not know or chose to ignore this principle? If the Sages claim it as a Divine decree, i.e. an ancient tradition dating “from the Sinai revelation,” whether inferred from Scripture or orally transmitted, it is unthinkable that a man of R. Tarfon’s caliber would be ignorant of it or refuse to accept it. Thus, the primary scenario contains a difficulty, a kushia.

One possible resolution of this difficulty is to say that the Sages were here legislating, i.e. the dayo principle was here in the process of being decided by the rabbis collectively, there being one dissenting voice, viz. that of R. Tarfon, at least temporarily till the decision was declared law. In that event, the conflict between the two parties dissolves in time. Another possible resolution is to say that the Sages did not intend their dayo statement as a hard and fast principle, but as a loose guideline that they considered ought to be applied in the present context, whereas their colleague R. Tarfon considered it ought not to be applied in the present context. In that event, the two parties agree that the dayo principle is not universal, but merely conditional, and their conflict here is only as to whether or not its actual application is appropriate in the case at hand.

This would explain why R. Tarfon can put forward his first and second arguments failing each time to anticipate that the Sages would disagree with him. He could not offhand be expected to predict what their collective judgment would be, and so proposed his opinion in good faith. That they disagreed with him is not a reflection on his knowledge of Torah or his logical powers; there was place for legitimate dissent. Thus, while the hypothesis that the Sages’ dayo objections signify a hard and fast rule of Sinaitic origin is problematic, there are two viable alternative hypotheses: namely, that the Sages’ dayo objections constituted a general rabbinical ruling in the making; or that they were intended as ad hoc, particular and conditional statements, rather than as reflections of a general unbreakable rule. The problem with the former hypothesis is explaining away R. Tarfon’s implied ignorance or disagreement; this problem is solved satisfactorily with either of the latter two hypotheses.

The Gemara commentary revolves around this issue, since its first and main query is: “Does R. Tarfon really ignore the principle of dayo? Is not dayo of Biblical origin?” The Gemara’s thesis thus seems to be that dayo is a principle of Biblical origin and that therefore R. Tarfon knew about it and essentially agreed with it. We shall presently see where it takes this assumption.

About method. An issue arising from this Mishnaic discussion is whether it is based on revelation or on reason. If we examine R. Tarfon’s discourse, we see that he repeatedly appeals to reason. Twice he says: “does it not stand to reason?” (eino din) and twice he claims to “infer” (edon)[48]. This language (the translations are those in the Soncino edition) suggests he is not appealing to Divine revelation, but to ordinary human reason. And, significantly, the Sages do not oppose him by explicitly claiming that their dayo principle is Divinely-ordained (as the Gemara later claims) and thus overrides his merely rational argument – no, they just affirm and reaffirm it as something intuitively self-evident, on moral if not logical grounds. Thus, from such positive and negative evidence, it is possible to suppose that both R. Tarfon and the Sages regard their methodological means as essentially rational.

Concerning the logical skills of R. Tarfon and the Sages, neither party to the debate commits any error of logic, even though their approaches and opinions differ. All arguments used by them are formally valid. At no stage do the Sages deny R. Tarfon’s reasoning powers or vice versa. The two parties understand each other well and react appropriately. There is no rhetorical manipulation, but logic is used throughout. Nevertheless, a pertinent question to ask is: why did R. Tarfon and the Sages not clarify all the logical issues involved, and leave their successors with unanswered questions? Why, if these people were fully conscious of what they were doing, did they not spell their intentions out clearly to prevent all possible error? The most likely answer is that they functioned ‘intuitively’ (in a pejorative sense of the term), without awareness of all the formalities involved. They were skillful practitioners of logic, but evidently not theoreticians of it. They did not even realize the importance of theory.

4. A logician’s reading of Numbers 12:14-15

We have thus far analyzed the Mishnaic part of Baba Qama 24b-25a. Before we turn to the corresponding Gemara, it is wise for us – in the way of a preparatory study – to look at a Torah passage which plays an important role in that Gemara, as an illustration of the rabbinical hermeneutic rule of qal vachomer (a fortiori argument) and as a justification of its attendant dayo (sufficiency) principle.

The Torah passage in question is Numbers 12:14-15. The reason why this 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[49]. 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[50]. Clearly, the Mishna BQ 2:5 could only be grounded in the Torah through Num. 12:14-15.

Num. 12:14-15 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)[51]. Some interpretation on my part was necessary to formulate it in this standard format[52]. 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[53]. 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, 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, 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.

5. A critique of the Gemara in Baba Qama 25a

As regards the Gemara of the Jerusalem Talmud, all it contains relative to the Mishna Baba Qama 2:5 is a brief comment in the name of R. Yochanan[54] that R. Tarfon advocates full payment for damages in the private domain, whereas the Sages advocate half payment[55]. This is typical of this Talmud, which rarely indulges in discussion[56]. On the other hand, the Gemara of the Babylonian Talmud has quite a bit to say on this topic (see p. 25a there), though perhaps less than could be expected. When exactly that commentary on our Mishna was formulated, and by whom, is not there specified; but keep in mind that the Gemara as a whole was redacted in Babylonia ca. 500 CE, i.e. some three centuries after the Mishna was closed, so these two texts are far from contemporaneous[57]. It begins as follows:

“Does R. Tarfon really ignore the principle of dayo? Is not dayo of Biblical origin? As taught: How does the rule of qal vachomer work? And the Lord said unto Moses: ‘If her father had but spit in her face, should she not be ashamed seven days?’ How much the more so then in the case of divine [reproof] should she be ashamed fourteen days? Yet the number of days remains seven, for it is sufficient if the law in respect of the thing inferred be equivalent to that from which it is derived!”

The a crescendo reading. Reading this passage, it would appear that the Gemara conceives qal vachomer as a crescendo rather than purely a fortiori argument; and the dayo principle as a limitation externally imposed on it. It takes the story of Miriam (i.e. Numbers 12:14-15) as an illustration and justification of its view, claiming that the punishment due to Miriam would be fourteen days by qal vachomer were it not restricted to seven days by the dayo principle. The dayo principle is here formulated exactly as in the Mishna (as “It is sufficient, etc.”); but the rest of the Gemara’s above statement is not found there.

In fact, the Gemara claims that the thesis here presented is a baraita – i.e. a tradition of more authoritative, Tannaic origin, even though it is not part of the Mishna[58]. This is conventionally signaled in the Gemara by the expression ‘as taught’: דתניא (detania)[59]. The baraita may be taken as the Hebrew portion following this, i.e. stretching from “How does the rule of qal vachomer work?” to “…from which it is derived.” Note well that baraita thesis is clearly delimited: the preceding questions posed by the Gemara – viz. “Does R. Tarfon really ignore the principle of dayo? Is not dayo of Biblical origin?” – are not part of it; we shall return to these two questions further on.

As we have shown in our earlier analysis, Num. 12:14-15 could be read as devoid of any argument; but then we would be hard put to explain the function of the first sentence: “If her father had but spit in her face, etc.,” and its relation to the second: “Let her be shut up without the camp, etc.”. It is therefore a reasonable assumption that an argument is indeed intended. This argument can be construed as purely a fortiori; in that event, its conclusion is simply seven days isolation, the same number of days as mentioned in the minor premise; and if the dayo principle have any role to play here it is simply that of the principle of deduction, i.e. a reminder that the conclusion must reflect the minor premise. It is also possible to interpret the argument as a crescendo, as the Gemara proposes to do; in that event, its conclusion is a greater number of days of isolation (say, fourteen days); and the dayo principle plays the crucial role of resetting the number of days to seven.

The latter is a conceivable hypothesis, but by no means a certainty, note well. There is clearly no mention of “fourteen days” in the Torah passage referred to, i.e. no concrete evidence of an a crescendo argument, let alone of a dayo principle which cuts back the fourteen days to seven. The proposed scenario is entirely read into the Biblical text, rather than drawn from it, by the baraita and then the Gemara; it is an interpolation on their part. They are saying: though the Torah does not explicitly mention fourteen days, etc., it tacitly intends them. This is not inconceivable; but it must be admitted to be speculative, since other readings are equally possible.

The baraita apparently proposes to read, not only the particular qal vachomer about Miriam, but qal vachomer in general as a crescendo argument, since it says “How does the rule of qal vachomer work?” rather than “how does the following example of qal vachomer work?” Thus, the Tanna responsible for it may be assumed to believe unconditionally in the ‘proportionality’ of a fortiori argument. Likewise, the Gemara – since it accepts this view without objection or explanation. If it is true that this Gemara (and the baraita it is based on – but I won’t keep mentioning that) regards a fortiori argument to always be a crescendo argument, it is way off course, of course.

As we have seen, as far as formal logic is concerned a fortiori argument is essentially not a crescendo, even though its premises can with the help of an additional premise about proportionality be made to yield an a crescendo conclusion. It is conceivable that the particular argument concerning Miriam is in fact not only a fortiori but a crescendo (assuming the premise of proportionality is tacitly intended, which is a reasonable assumption); but it is certainly not conceivable that all a fortiori arguments are a crescendo. The Gemara’s identification of a fortiori argument with a crescendo is nowhere justified by it. The Gemara has not analyzed a fortiori argument in general and found its logical conclusion to be a crescendo (i.e. ‘proportional’); it merely asserts this to be so in the case at hand and, apparently, in general.

While it is true that, empirically, within the Talmud as well as outside it, convincing examples of seemingly a fortiori argument yielding a (roughly or exactly) proportional conclusion can be adduced, it is also true that examples of a fortiori argument yielding a non-proportional conclusion can be adduced. This needs to be explained – i.e. commentators are duty-bound to account for this variation in behavior, by specifying under what logical conditions a ‘proportional’ conclusion is justified and when it is not justified. The answer to that is (to repeat) that a fortiori argument as such does not have a ‘proportional’ conclusion and that such a conclusion is only logically permissible if an additional premise is put forward that justifies the ‘proportionality’. The Gemara does not demonstrate its awareness of these theoretical conditions, but functions ‘intuitively’. Its thesis is thus essential dogmatic – an argument by authority, rather than through logical justification.

Thus, for the Gemara, or at least this here Gemara, the words “qal vachomer,” or their English equivalent “a fortiori argument,” refer to what we have called a crescendo argument, rather than to purely a fortiori argument. There is nothing wrong with that – except that the Gemara does not demonstrate awareness of alternative hypotheses.

A surprising lacuna. Furthermore, it should imperatively be remarked that the Gemara’s above explanation of the Mishna debate, by means of the Miriam story, is only relevant to the first exchange between R. Tarfon and the Sages; it does not address the issues raised by the second exchange between them.

For in the first exchange, as we have seen, R. Tarfon tries by means of a possible pro rata argument, or alternatively an a crescendo argument (as the Gemara apparently proposes), to justify a ‘proportional’ conclusion (i.e. a conclusion whose predicate is greater than the predicate of the minor premise, in proportion to the relative magnitudes implied in the major premise); and here the Sages’ dayo objection limits the predicate of conclusion to that of the minor premise; so the analogy to the Miriam case is possible. But in the second exchange, the situation is quite different! Here, as we earlier demonstrated, the dayo objection refers, not to the information in the minor premise, but to the information that was generalized into the major premise. That is to say, whereas the first objection is aimed at the attempted pro rata or a crescendo deduction, the second one concerns the inductive preliminary to the attempted pro rata or a fortiori or a crescendo deduction.

The Gemara makes no mention of this crucial distinction between the two cases. It does not anywhere explicitly show that it has noticed that R. Tarfon’s second argument draws the same conclusion whether it is considered as pro rata, a crescendo, or even purely a fortiori, so that it formally does not contravene the Sages’ first objection. The Gemara does not, either, marvel at the fact that the Sages’ second objection is made in exactly the same terms, instead of referring to the actual terms of the new argument of R. Tarfon. It does not remark that the Miriam story (as the Gemara interprets it) is therefore irrelevant to the second case, since it does not resemble it, and some other explanation must be sought for it. This lacuna is of course a serious weakness in the Gemara’s whole hypothesis, since it does not fit in with all the data at hand.

To be sure, the distinction between the two cases does appear in rabbinic literature. This distinction is solidified by means of the labels dayo aresh dina and dayo assof dina given to the two versions of the dayo principle. But I do not think the distinction is Talmudic (certainly, it is absent here, where it is most needed). Rather, it seems to date from much later on (probably to the time of Tosafot). These expressions mean, respectively, applying the dayo “to the first term (or law)” and applying it “to the last term (or law).” In my opinion, assof dina must refer to the dayo used on the first qal vachomer, while aresh dina refers to the dayo used on the second qal vachomer[60].

Be that as it may, what concerns us here is the Gemara, which evidently makes no such distinction (even if later commentators try to ex post facto give the impression that everything they say was tacitly intended in the Gemara). What this inattentiveness of the Gemara means is that even if it manages to prove whatever it is trying to prove (we shall presently see just what) – it will not succeed, since it has not taken into account all the relevant information. Its theory will be too simple, insufficiently broad – inadequate to the task. The Gemara’s failure of observation is of course also not very reassuring.

The claim that dayo is of Biblical origin. Let us now return to the initial questions posed by the Gemara, viz. “Does R. Tarfon really ignore the principle of dayo? Is not dayo of Biblical origin?” (ורט לית ליה דיו והא דיו דאורייתא הוא). As already remarked, it is important to notice that these questions are not part of the baraita. They are therefore the Gemara’s own thesis (or an anonymous thesis it defends as its own) – indeed, as we shall see, they are the crux of its commentary. The baraita with the a crescendo reading is relatively a side-issue. What the Gemara is out to prove is that R. Tarfon “does not ignore” the dayo principle, because “it is of Biblical origin.” What is not of Biblical origin may conceivably be unknown to a rabbi of Tarfon’s level; but what is of Biblical origin must be assumed as known by him.

The question of course arises what does “of Biblical origin” (deoraita) here mean exactly? It cannot literally mean that the principle of dayo is explicitly promulgated and explicated in the Torah. Certainly it is nowhere to be found in the Torah passage here referred to, or anywhere else in that document. Thus, this expression can only truly refer to an implicit presence in the Torah. And indeed the Torah passage about Miriam, brought to bear by the Gemara, seems to be indicated by it as the needed source and justification of the principle, rather than as a mere illustration of it. However, as we shall see further on, there is considerable circularity in such a claim. So claiming the dayo principle to have “Biblical origin” is in the final analysis just say-so, i.e. a hypothesis – it does not solidly ground the principle and make it immune to all challenge, as the Gemara is suggesting.

It could well be thought, reading the Mishna, that R. Tarfon was not previously aware of the Sages’ alleged dayo principle, since he did not preempt their two dayo objections. Had he known their thinking beforehand, he would surely not have wasted his time trying out his two arguments, since he would expect them to be summarily rejected by the Sages. Since he did try, and try again, the Sages must have been, in his view, either unearthing some ancient principle unknown to him, or deciding a new principle, or proposing ad hoc decisions. It is this overall reasonable conclusion from the Mishna that the Gemara seeks to combat, with its claim that the dayo principle was of Biblical origin and therefore R. Tarfon must have known it. Note this well.

I do not know why the Gemara is not content with the perfectly legal possibilities that the dayo principle might be either a tradition not known to R. Tarfon, or a new general or particular decision by the Sages (derabbanan). For some reason, it seeks to impose a more fundamentalist agenda, even though the alternative approaches are considered acceptable in other Talmudic contexts. The Gemara does not say why it is here unacceptable for the Sages to have referred to a relatively esoteric tradition or made a collegial ruling (by majority, rov)[61]. It seems that the Gemara is driven by a desire to establish that R. Tarfon and the Sages are more in harmony than they at first seem; but it is not clear why it has chosen the path it has, which is fraught with difficulties.

The claim that dayo is conditional. The Gemara shifts the debate between R. Tarfon and the Sages from one as to if the dayo principle is applicable to one as to when it is applicable. The two parties, according to the Gemara, agree that the dayo principle is “of Biblical origin,” and thus that there is a dayo principle; but they disagree on whether or not it is applicable unconditionally. In this view, whereas the Sages consider the dayo principle as universally applicable, R. Tarfon considers it as only conditionally applicable. Thus, the parties agree in principle, and their disagreement is only in a matter of detail. The Gemara then proceeds to clarify R. Tarfon’s alleged conditions[62]:

“The principle of dayo is ignored by him [R. Tarfon] only when it would defeat the purpose of the a fortiori, but where it does not defeat the purpose of the a fortiori, even he maintains the principle of dayo. In the instance quoted there is no mention made at all of seven days in the case of divine reproof; nevertheless, by the working of the a fortiori, fourteen days may be suggested: there follows, however, the principle of dayo so that the additional seven days are excluded, whilst the original seven are retained. Whereas in the case before us the payment of not less than half damages has been explicitly ordained [in all kinds of grounds]. When therefore an a fortiori is employed, another half-payment is added [for damage on the plaintiff’s premises], making thus the compensation complete. If [however] you apply the principle of dayo, the sole purpose of the a fortiori would thereby be defeated.”

Let us try and understand what the Gemara is saying here. It is proposing a distinction (allegedly by R. Tarfon) between two obscure conditions: when applying the dayo principle “would defeat the purpose of the qal vachomer,” it is not applied; whereas where applying the dayo principle “would not defeat the purpose of the qal vachomer,” it is applied. What does this “defeating the purpose of the a fortiori argument” condition refer to? The Gemara clarifies it by comparing R. Tarfon’s (alleged) different reactions to two cases: that concerning Miriam and the (first) argument in the Mishna (the Gemara has apparently not noticed the second argument at all, remember).

The Gemara here reaffirms its theory that, although the Torah (“the instance quoted” – i.e. Num. 12:14-15) does not mention an initial or an additional seven days[63], “nevertheless, by the working of the a fortiori” (as conceived by the Gemara, meaning a crescendo) fourteen days in all (i.e. seven plus seven) are intended, and the dayo principle serves after that to “exclude” the additional seven days, admitting only the “original” seven days. In this case, then, the dayo principle is to be applied. The Gemara then turns to R. Tarfon’s (first) argument, claiming that in its case the dayo principle is not to be applied. Why? Because “the payment of not less than half damages has been explicitly ordained [in all kinds of grounds].” This is taken by commentators (Rashi is mentioned) to mean that since the Torah does not make a distinction between public and private property when it specifies half liability for damage by horn[64], it may be considered as intending this penalty to be (the minimum[65]) applicable to both locations.

The Gemara goes on to tell us that through “a fortiori” inference “another half-payment is added, making thus the compensation complete.” The implication is that, whereas the Sages would at this stage apply the dayo principle and conclude with only half payment, R. Tarfon (according to the Gemara) considered that doing so would “defeat the purpose of the a fortiori” and he concluded instead with full payment. In the Miriam case, we go from no information to fourteen days and back to seven; so we still end up with new information (seven) after the dayo application to the qal vachomer increase. Whereas in the Mishna case, we go from half to full payment and back to half; so that dayo application here would altogether cancel out the qal vachomer increase. Thus, R. Tarfon is presented by the Gemara as knowing and accepting the dayo principle, but applying it more conditionally than the Sages do[66].

But I would certainly challenge the underlying claim that the a fortiori argument used by R. Tarfon (which concludes with full payment for damage by horn on private property) is “nullified” by the Sages’ objection to it (which limits the payment to half). What is given in the Torah is that such damage (on whatever domain) is liable to half payment. This “half” is indefinite, and must be interpreted as at least half (i.e. a minimum of half, no less than half), which leaves open whether only half (i.e. a maximum of half, no more than half) or full (i.e. more than half) is intended. R. Tarfon’s argues (through a crescendo, i.e. ‘proportional’ a fortiori argument) in favor of the conclusion “full,” whereas the Sages argue (through dayo, or purely a fortiori argument) in favor of the alternative conclusion “only half.” R. Tarfon’s argument is certainly not made logically useless by the Sages’ dismissal of it, but constitutes a needed acknowledgment of one of the two possible interpretations of “half,” just as the Sages’ dayo duly acknowledges the other possibility. If the Mishna had directly interpreted “half” as “only half,” without regard to the possibility of “full,” the interpretation would have seemed unjustified.[67]

An argument ex machina. But let us dig deeper into the alleged conditionality of dayo application. Why, more precisely, does the Gemara’s R. Tarfon consider that applying the dayo principle in the case of the Miriam argument does not “defeat the sole purpose of the a fortiori,” yet would do so in the case of his formally similar (first) argument? What is the significant difference between these two cases? And what sense are we to make of the Gemara’s further explanations, viz.:

“And the Rabbis? — They argue that also in the case of divine [reproof] the minimum of seven days has been decreed in the words: Let her be shut out from the camp seven days. And R. Tarfon? — He maintains that the ruling in the words, ‘Let her be shut out etc.’, is but the result of the application of the principle of dayo [decreasing the number of days to seven]. And the Rabbis? — They argue that this is expressed in the further verse: And Miriam was shut out from the camp. And R. Tarfon? — He maintains that the additional statement was intended to introduce the principle of dayo for general application so that you should not suggest limiting its working only to that case where the dignity of Moses was involved, excluding thus its acceptance for general application: it has therefore been made known to us [by the additional statement] that this is not the case.”[68]

It seems[69] that R. Tarfon’s thought (still according to the Gemara, note well) is that, with regard to Miriam, no part of the penalty for offence against God is explicitly mentioned in the Torah (Num. 12:14-15), so that all fourteen days must be inferred by “a fortiori” (i.e. a crescendo); after which the dayo principle is used to revoke seven of those days, leaving seven. Whereas, in the case of horn damage on private property, the minimum liability of half payment is already explicitly given in the Torah (Ex. 21:35), so that the “a fortiori” (i.e. a crescendo) argument only serves to add on half payment; in which case, applying the dayo principle here would completely nullify the effect of the qal vachomer.

Thus, it is implied, the dayo principle is applicable in the Miriam case, but inappropriate in the case of a goring ox. The Sages (allegedly) then object that the initial seven days are indeed given in the Torah, in the sentence “Let her be shut out from the camp seven days.” To which R. Tarfon (allegedly) retorts that this sentence refers to the dayo principle’s “decreasing the number of days to seven.” The Sages reply that that function is fulfilled by the sentence “And Miriam was shut out from the camp.” To which R. Tarfon retorts that the latter rather has a generalizing function from the present case to all others. As far as I am concerned, most of this explanation by the Gemara is artificial construct and beside the point. It is chicanery, pilpul (in the most pejorative sense of that term).

The claim it makes (on R. Tarfon’s behalf) that all fourteen days for offence against God must be inferred is untrue – for the fourteen days are not inferred from nothing, as it suggests; they are inferred from the seven days for offence against a father. The inference of the conclusion, whether it is a crescendo or purely a fortiori, depends on this minor premise. The seven days for a father are indeed a given minimum, also applicable to God; otherwise, there would be no a crescendo or a fortiori inference at all. The Gemara is claiming an “a fortiori” (i.e. a crescendo) argument to be present in the text, and yet denying the relevance of the textual indicators for such an assumption. Its alleged “a fortiori” argument is therefore injected into the discussion ex machina, out of the blue, without any textual justification whatsoever. This is not logic, but rhetoric.

The situation in the argument about Miriam is thus in fact technically exactly identical to the (first) argument relating to liability for damages by horn in the Mishna. Both arguments do, in fact, have the minor premise needed to draw the conclusion. Whence the Gemara’s concept of “defeating the sole purpose of the a fortiori” is a red herring; it is just a convenient verbal artifice, to give the impression that there is a difference where there is none. The Gemara has evidently tried to entangle us in an imaginary argument. For, always remember, it is the Gemara’s reading which is at stake here, and not R. Tarfon’s actual position as it appears in the Mishna, which is something quite distinct.

The roles of the verses in Num. 12:14-15. What is evident is that neither of the readings of the said Torah portion that the Gemara attributes to R. Tarfon and the Sages fully corresponds to the simple reading (peshat). They are both awkward inventions[70] designed to justify the Gemara’s own strange thesis. The Gemara’s thesis is not something necessary, without which the Mishna is incomprehensible; on the contrary, it clouds the issues and misleads. Whatever the author’s authority, it is unconvincing.

The simple reading of Num. 12:14-15 is, as we saw earlier[71], that the sentence “If her father had but spit in her face, should she not hide in shame seven days?” (first part of v. 14, call it 14a) provides the minor premise of a possible a fortiori argument (whether strict or a crescendo), while the sentence “Let her be shut up without the camp seven days, and after that she shall be brought in again” (second part of v. 14, call it 14b) provides its immediate conclusion. Note well that it is from these two sentences (i.e. v. 14a & 14b) that we in the first place surmise that there is an a fortiori argument in the text; to speak of an a fortiori argument without referring to both these indices would be concept stealing. The further sentence “And Miriam was shut up without the camp seven days; and the people journeyed not till she was brought in again” (v. 15) plays no part in the a fortiori argument as such, but serves to confirm that the sentence was carried out by Moses’ court as prescribed by God.

The Gemara’s R. Tarfon makes no mention of the role of v. 14a in building a qal vachomer, and regards v. 14b as the final conclusion of the argument, after the operation of an entirely tacit a crescendo inference to fourteen days and an also tacit application of dayo back to seven days; as regards v. 15, it effectively plays no role within the argument in his view, having only the function of confirming that the dayo application is a general principle and not an exceptional favor[72]. The Gemara’s Sages, on the other hand, regard v. 14b (not 14a, note well) as the minor premise of the qal vachomer, and v. 15 its final conclusion, after the operation of an a crescendo inference to fourteen days and an application of dayo back to seven days.

Both parties make serious errors. The first of these is that neither of them accounts for v. 14a – why is it mentioned here if as both parties suppose it plays no role? No a fortiori argument can at all be claimed without reference to this information. The R. Tarfon thesis here is largely imaginary, since he ignores the role of v. 14a in justifying a qal vachomer; there is no trace in the Torah text of the a crescendo argument he claims, other than v. 14b. On the basis of only the latter textual given of seven days, he projects into the text a minor premise of seven days, an intermediate a crescendo conclusion of fourteen days and a dayo principle application, yielding a final conclusion of seven days (v. 14b). But if all the textual evidence we rely on is v. 14b, on what basis can we claim any a crescendo reasoning has at all occurred before it, let alone a dayo application, with this verse as the final conclusion? The whole process becomes a patent fabrication.

Nowhere in the proof text, note well, are the words qal vachomer or dayo used, or any verbal signal to the same effect. And this being so, what credence can be assigned to the Gemara’s central claim, viz. that the dayo principle is “of Biblical origin?” It is surely paradoxical that it is able to support this ambitious claim only by means of a very debatable mental projection of information into the Torah, like a magician pulling a rabbit out of a hat after showing us it was empty. This means that the Gemara’s proposed argument in favor of this claim is circular: it assumes X in order to prove X. This is of course made possible through the use of complicated discourse; but the bottom line is still the same.

The Sages’ thesis is a bit more credible in that, even if they also grant no role to v. 14 a, they at least do propose a minor premise (v. 14b), as well as a final conclusion (v. 15). However, it is hard to see how “Let her be shut up without the camp seven days” (v. 14b) could be the minor premise of qal vachomer yielding the conclusion “And Miriam was shut up without the camp seven days” (v. 15)! These two propositions have the same subject (as well as the same explicit predicates), so where is the qal vachomer? Moreover, the Sages thereby subscribe to R. Tarfon’s strange misconception regarding a fortiori argument.

A fortiori argument with a single subject. I am referring here to the bizarre notion that (in the qal vachomer argument under consideration, which is positive subjectal) the subject of the minor premise must be repeated in the conclusion, while the subsidiary terms (i.e. the predicates of these propositions) go from less to more (implicitly). In fact, positive subjectal argument, whether a fortiori or a crescendo, formally has different subjects (the minor and the major terms, respectively) in the minor premise and conclusion (as for the predicate, i.e. the subsidiary term, it remains constant in pure a fortiori, while it increases in a crescendo). There has to be two subjects for the argument to logically function. The bizarre notion in the Gemara of a single subject argument is the reason why both parties in it ignore v. 14a and look for some other proposition to use as minor premise.

It should be stressed that there is no allusion whatsoever to such an idea in the Mishna. The Mishna’s R. Tarfon and Sages manifestly have an entirely different dialogue than the one the Gemara attributes to them. The discussion in the Mishna is much more credible than that in the Gemara. The Gemara makes up this notion solely in order to create a distinction between the Miriam case and the Mishna’s (first) argument. It needs to do this, remember, in order to justify its theory that R. Tarfon and the Sages agree on the dayo principle, although R. Tarfon applies it conditionally whereas the Sages apply it universally. But as we shall demonstrate formally, this notion is logically untenable. Buying the Gemara’s scenario is like buying Brooklyn Bridge from someone who doesn’t own it.

The thesis of R. Tarfon in the Gemara is that, in the Miriam case, we must have a minor premise that offending God (rather than merely one’s father) justifies a minimum of seven days of punishment, in order to be able to infer qal vachomer (i.e. a crescendo) that offending God justifies fourteen days of punishment – just as with regard to an ox, we (allegedly) reason from half liability for damage done on private (rather than public) property to full liability on private property. The Sages do not object to this claim. But this claim is simply not true – there is no such technical requirement for positive subjectal a crescendo (or a fortiori) inference. We can very well, and normally do, reason with a change of subject, i.e. from the penalty for offence to one’s father to that for offence to God, or from the liability for damage on public grounds to that on private grounds. This is precisely the power and utility of a fortiori (and a crescendo) inference.

Moreover, we in fact can, by purely a fortiori argument, infer the needed minor premise about seven days penalty for offending God (from the same penalty for offending one’s father), and likewise the half liability on private property (from the same liability on public property)[73]. One cannot claim an a crescendo argument to be valid without admitting the validity of the purely a fortiori argument (and pro rata argument) underlying it. Obtaining the minor premise demanded by the Gemara’s R. Tarfon is thus not the issue, in either case. The issue is whether such a minor premise will allow us to draw the desired ‘proportional’ conclusion. And the answer to that, as we show further on, is: No!

Furthermore, if we carefully compare the Gemara’s argument here to the first argument laid out in the Mishna, we notice a significant difference. As we just saw, the Gemara concludes with full liability for horn damage on private property on the basis of half liability for horn damage on private property. As earlier explained, it bases this minor premise on the fact that Ex. 21:35 does not make a distinction between public and private property when it prescribes half liability for damage by horn, so that this may be taken as a minimum in either case. Thus, for the Gemara, half liability for horn damage on private property is a Torah given, which does not need to be deduced. On the other hand, in the Mishna, the minor premise of the first argument refers to the public domain rather than to private property.

In his first argument, R. Tarfon argues thus (italics mine): “…in the case of horn, where the law was strict regarding [damage done on] public ground imposing at least the payment of half damages, does it not stand to reason that we should make it equally strict with reference to the plaintiffs premises so as to require compensation in full?” And to justify his second argument he argues thus: “but neither do I infer horn [doing damage on the plaintiff’s premises] from horn [doing damage on public ground]; I infer horn from foot, etc.”[74] Thus, his first argument is clearly intended as an inference from the penalty for horn damage in the public domain (half) to that in the private domain (full). The Gemara’s construct is thus quite different from the Mishna’s, and cannot be rightly said to represent it.

As regards the rule here apparently proposed by the Gemara (which it attributes to R. Tarfon), viz. that the subject must be the same in minor premise and conclusion, as already stated there is no such rule in formal logic for positive subjectal argument[75]. Such argument generally has the minor and major terms as subjects of the minor premise and conclusion respectively, even if the subsidiary term sometimes (as is the case in a crescendo argument) varies in magnitude ‘proportionately’. In the case of a crescendo argument, where the predicate (subsidiary term) changes, there absolutely must be a change of subject, since otherwise we would have no explanation for the change of predicate. That is, we would have no logical argument, but only a very doubtful ‘if–then’ statement. The proposed rule is therefore fanciful nonsense, a dishonest pretext.

We can examine this issue in more formal terms. A positive subjectal a fortiori argument generally has the form: “P is more R than Q is; and Q is R enough to be S; therefore, P is R enough to be S” (two premises, four terms). If the argument is construed as a crescendo, it has the form: “P is more R than Q is; and Q is R enough to be Sq; and S is ‘proportional’ to R; therefore, P is R enough to be Sp” (three premises, five terms). The argument form attributed by the Gemara to R. Tarfon simply has the form: “If X is S1, then X is S2” (where X is the sole subject, and S1 and S2 the subsidiary terms, S2 being greater than S1); that is, in the Miriam sample: “if offending God merits seven days penalty, then offending Him merits fourteen days penalty,” and again in the Mishna’s first dialogue: “If liability for horn damage on private property is half payment, then liability for same on private property is full payment.” This is manifestly not a fortiori or a crescendo argument, but mere if–then assertion; it could conceivably happen to be true, but it is not a valid inference.

It is clear that the latter inference, proposed by the Gemara in the name of R. Tarfon, has no logical leg to stand on. It has no major premise comparing the subjects (P and Q); and no need or possibility of one, since there is only one subject (X). Having no major premise, it has no middle term (R); and therefore no additional premise in which the subsidiary term (S) is presented as ‘proportional’ to it. Thus, no justification or explanation is given why S should go from Sq in the minor premise to Sp in the conclusion. It is therefore not an a fortiori or a crescendo argument in form, even if it is arbitrarily so labeled by the Gemara. You cannot credibly reason a fortiori or a crescendo, or any other way, if you cannot produce the requisite premises. There is no such animal as “argument” ex nihilo.

The Gemara’s proposed if–then statement is certainly not universal, since that would mean that if any subject X has any predicate Y then it has a greater predicate Y+, and if Y+ then Y++, and so forth ad infinitum – which would be an utter absurdity[76]. From this we see that not only has the Gemara’s argument no textual bases (as we saw earlier), but it has no logical standing. There is in fact no “argument,” just arbitrary assertion on the Gemara’s part. For both the Miriam sample and the (first) Mishna sample, the Gemara starts with the convenient premise that “there is a qal vachomer here,” which it considers as given (since it is traditionally assumed present, on the basis of other readings of these texts), and then draws its desired conclusion without recourse to any other proposition, i.e. without premises![77]

If this requirement for a single subject is not a rule of logic, is it perhaps a hermeneutic principle, i.e. a rule prescribed by religion? If so, where (else) is it mentioned in the oral tradition or what proof-text is it drawn from? Is it practiced in other contexts, or only in the present one, where it happens to be oh-so-convenient for the Gemara’s interpretative hypothesis? If it is an established rule, how come the Sages do not agree to it? The answers to these questions are pretty obvious: there is no such hermeneutic rule and no basis for it. It was unconsciously fabricated by the Gemara author in the process of developing the foolish scenario just discussed. It is not a general necessity (or even a possibility, really), but just an ad hoc palliative.

Unfortunately, when people use complex arguments (such as the a fortiori or the a crescendo) without prior theoretical reflection about them, they are more or less bound to eventually try to arbitrarily tailor them to their discursive needs.

To sum up. We have seen that the Gemara introduces a number of innovations relative to the Mishna it comments on. The first we noted was that the Gemara, in the name of an anonymous Tanna, reads the qal vachomer in Num. 12:14-15, and apparently all a fortiori argument in general, as a crescendo argument. Next we noted a surprising lacuna in the Gemara’s treatment, which was that while it dealt with R. Tarfon’s first argument, it completely ignored his second, and failed to notice the curious verbatim repetition in the Sages’ two dayo objections. Third, we showed that the thesis that dayo is “of Biblical origin,” so that R. Tarfon must have been aware of it, was the Gemara’s main goal in the present sugya. In the attempt to flesh out this viewpoint, the Gemara proceeds to portray R. Tarfon as regarding the dayo principle as being applicable only conditionally, in contrast to the universal dayo principle seemingly advocated by the Sages.

To buttress this thesis, the Gemara is forced to resort to an argument ex machina – that is, although vehemently denying the role of both parts of Num. 12:14 in the formation of a qal vachomer, the Gemara’s R. Tarfon nevertheless assumes one (i.e. a phantom a fortiori argument) to be somehow manifest between the lines of the proof-text. Moreover, in order to make a distinction between the Miriam example and the (first) Mishna argument, so as to present the dayo principle as applicable to the former and inapplicable to the latter, the Gemara’s R. Tarfon invents a preposterous rule of inference for qal vachomer, according to which the subject must be the same in the minor premise and the conclusion. In the Miriam example, the absence of a minor premise with the required subject (offending God) means that dayo is applicable, for applying it would not “defeat the purpose of the qal vachomer;” whereas in the (first) Mishna argument, the presence of a minor premise with the required subject (damage by ox on private property) means that dayo is inapplicable, for applying it would “defeat the purpose of the qal vachomer.”

This all looks well and good, if you happen to be sound asleep as the Gemara dishes it out. For the truth is that at this stage the whole structure proposed by the Gemara comes crashing down.

The trouble is, there is no such thing as an a fortiori argument (or a crescendo argument) that takes you from no information to a conclusion, whether maximal or minimal. If the proposed qal vachomer “argument” has no minor premise (since v. 14a is explicitly not admitted as one) and no major premise (since the subject of the conclusion must, according to this theory, be the same in the minor premise as in the conclusion), then there is no argument. You cannot just declare, arbitrarily, that there is an argument, while cheerfully denying that it has any premises. And if you have no argument with a maximum conclusion, then you have no occasion to apply the dayo principle, anyway.

Moreover, there is no such one-subject rule in a fortiori logic; indeed, if such a rule were instituted, the argument would not function, since it would have no major premise, and no major, minor or middle term; consequently, if it was intended as ‘proportional’ (as the Gemara claims), it would imply an inexplicable and absurd increase in magnitude of the subsidiary term. Thus, even if the Gemara’s textually absent argument about Miriam were generously granted as being at least ‘imaginable’ (in the sense that one might today imagine, without any concrete evidence, Mars to be inhabited by little green men), the subsequent demand that a qal vachomer have only one subject would make the proposed solution formally impossible anyway.

The Gemara’s explanation is thus so much smoke in our eyes, a mere charade; it has no substance. We need not, of course, think of the Gemara as engaging in these shenanigans cynically; we can well just assume that the author of this particular commentary was unconscious. In fine, the Gemara’s scenario, in support of its claim that the dayo principle is “of Biblical origin” and so R. Tarfon did not ignore it—is logically unsustainable.

6. A slightly different reading of the Gemara

As we saw previously, the two arguments featured in Mishna BQ 2:5 may objectively be variously interpreted. R. Tarfon’s first argument may be read as pro rata or as a crescendo, though not as purely a fortiori (since his conclusion is ‘proportional’), while his second argument may be read in all three ways. As regards the Sages’ first dayo objection, if R. Tarfon’s first argument is supposed to be intended as a pure a fortiori, the objection to it would simply be that such argument cannot logically yield a ‘proportional’ conclusion; this reading is very unlikely. Rather, the first dayo objection may be taken as a refusal of the ‘proportionality’ of the pro rata or a crescendo arguments, and possibly the proposal of a purely a fortiori counterargument, i.e. one without a ‘proportional’ conclusion. The Sages’ second dayo objection, on the other hand, cannot have the same intent, since in this case all three forms of argument yield the very same ‘proportional’ conclusion; so it must be aimed at the inductive processes preceding these arguments.

In our above analysis of the corresponding Gemara, we have mostly represented it as conceiving of one possible scenario for both[78] arguments of the Mishna, that of a crescendo argument moderated by a dayo principle. This is the traditional and most probable interpretation, but it should be said that an alternative reading is quite possible. Certainly, the Gemara here does not accept, or even consider, the alternative hypothesis that purely a fortiori argument may be involved in the second argument of R. Tarfon, since it clearly assumes that the conclusion’s predicate is bound to be greater than the minor premise’s predicate. However, it would be quite consistent to suppose that the Gemara is in fact not talking of two a crescendo arguments, but of two analogical/pro rata arguments. There is some uncertainty as to the Gemara’s real intent, since it does not explicitly acknowledge the various alternative hypotheses and eliminate all but one of them for whatever reasons.

Looking at the Mishna and Gemara discourses throughout the Talmud, it is obvious that the people involved use purely a fortiori argument, a crescendo argument, and argument pro rata in various locations. But it is not obvious that there is a clear distinction in their minds between these three forms of argument. It is therefore not impossible that when they say “qal vachomer,” they might indiscriminately mean any of these three forms of argument. It should be clear to the reader that the issue I am raising here is not a verbal one. I am not reproaching the Talmud for using the words “qal vachomer” in a generic or vague sense. I certainly cannot reproach it for not using the expressions ‘a crescendo’ or ‘pro rata’, as against ‘a fortiori’, since these names were not in its vocabulary.

What I am drawing attention to is the Talmud’s failure to demonstrate its theoretical awareness of the difference between the three forms of argument, whatever they are called. How could such awareness be demonstrated? It would have sufficed to state (if only by means of concrete examples, without abstract explanations) that the two premises of a fortiori per se do not allow a ‘proportional’ conclusion to be drawn, but must be combined with a third, pro rata premise for such a conclusion (i.e. a crescendo) to be justified; and that it is also possible to arrive at a ‘proportional’ conclusion without a fortiori reasoning, through merely analogical (i.e. pro rata) reasoning.

That is to say, for instance in the positive subjectal mood, the major premise “P is more R than Q is” and the minor premise “Q is R enough to be S” do not suffice to draw the conclusion “P is R enough to be more than S.” To deduce the latter a crescendo conclusion, an additional premise must be given, which says that “S is proportional to R.” Given all three said premises, we can legitimately conclude that “P is R enough to be (proportionately) more than S;” but without the third one, we can only conclude “P is R enough to be S.” Alternatively, we might infer from “S is in general proportional to R,” combined with “a given value of S is proportional to a given value of R,” that “a greater value of S is proportional to a greater value of R” (this is pro rata without a fortiori).

Thus, although we have taken for granted in our above analysis the traditional view that when the Gemara of Baba Qama 25a speaks of qal vachomer, it is referring to a fortiori argument, i.e. more precisely put to a crescendo argument (since it advocates ‘proportional’ conclusions), it is quite conceivable that it was unconsciously referring to mere pro rata argument. The dayo principle is not something conceptually, even if halakhically, tied to a fortiori (or a crescendo) argument, but could equally well concern pro rata argument (or even other forms of reasoning). And what I have above called the “bizarre notion,” which the Gemara credits to R. Tarfon, that the minor premise and conclusion of a positive subjectal argument must have the same subject for the argument to work, could equally be applied to pro rata argument as to a crescendo, since it is an arbitrary rule of Judaic logic without formal support in generic logic. Therefore, our above analysis of the Gemara would not be greatly affected if we assume it to refer to pro rata instead of to a crescendo argument. This is not a very important issue, but said in passing.



[1] Some historians, on the basis of debatable evidence or lack of evidence, claim the Torah to date from as late as the 8th cent. BCE. Even if this were true, it would signify a very early date for the a fortiori arguments present in it.

[2] R. Eliezer’s list is known indirectly from later texts. Jacobs characterizes it as “a post-Talmudic work” (in his Rabbinic Thought in the Talmud, p. 78, fn. 9). See en.wikipedia.org/wiki/R._Eliezer_ben_Jose_ha-Gelili. It is however a very significant work in that it includes the hermeneutic principles of R. Akiva, which rivaled those of R. Ishmael and yet were not (to my knowledge) collected in a list bearing his name. You can find all three lists in the Appendix to A. Schumann’s Introduction to the Judaic Logic collection he edited.

[3] You can easily find additional information on the various lists in a number of Wikipedia articles. Note that Hillel, R. Ishmael and R. Eliezer b. Jose ha-Gelili were all three Tannaim, i.e. Mishnaic rabbis. (More accurately, Hillel is classified as pre-Tannaic, forming together with Shammai the last of the zugot, i.e. “pairs.”)

[4] It would perhaps be more accurate to postulate that a fortiori argument was first formulated far in prehistory, soon after language and logic first formed in the cognitive apparatus of the human species; but it stood out as a recognizable meme at different times in different cultures during the historic period.

[5] Feigenbaum, in his Understanding the Talmud (pp. 88-90), explains the terminology more precisely as follows. The expression qal vachomer is Tannaic. The premise is introduced by them saying mah or umah and the conclusion is signaled by eino din she, or al achat kama vekama, or lo kol she ken. Amoraim on the other hand, use tashta before the premise and mibaya or tserikha lemeimar before the conclusion. R. Nosson Dovid Rabinovich, in his M. Mielziner’s Talmudic Terminology (Jerusalem: Ahavath Torah, 1988 – pp. 69-70), presents the matter slightly differently.

[6] This is of course an important distinction to note, because it indicates that rabbis were already aware quite early that a fortiori argument is in practice not always as fully verbalized as it could and ought to be. Indeed, the Biblical examples of such argument are typically not fully verbalized (to various degrees), so they did not need to look far to realize the fact.

[7] I list and analyze these ten examples in detail in my Judaic Logic (chapters 4, 5 and 6). I show there that one of the cases listed, viz. Esther 9:12, is doubtfully a fortiori. More important, I show there that there are at least another twenty cases of a fortiori in the Bible, one of which is in the Torah, Genesis 4:24. See summary of these and more recent findings in Appendix 1 to the present volume.

[8] Precisely how many concrete cases of qal vachomer argument there are in the Talmud and related documents has never, to my knowledge, been researched. This gigantic task should imperatively be done by someone – not just anyone, but someone with the needed logical knowhow. Indeed, the precise location and form of all rabbinic use of all explicit and implicit hermeneutic principles needs to be researched, so that a fully scientific assessment of Talmudic logic can be effected. The Babylonian and Jerusalem Talmuds should also be compared in this respect, though the latter contains much less commentary than the former. Although I unfortunately have never learned Hebrew and Aramaic well enough to take up the task in the original languages, I hope one day to at least try and draw up a rough list in English based on perusal of the Soncino Talmud.

[9] In between Mishna and Gemara is the Tosefta (ca. 300 CE), a later supplement to the Mishna that the Gemara sometimes refers to for additional information.

[10] The term Talmud is often taken as equivalent to the term Gemara, for whereas the Mishna is published separately, the Gemara is always published in conjunction with the Mishna since the Gemara’s purpose is to comment on the Mishna. But I think the correct use is to say Talmud when referring to the conjunction, and Gemara when referring specifically to the commentary, as one says Mishna when referring to the older material.

[11] Some call it the Palestinian Talmud, because the Land of Israel was at the time of its formation under Roman rule and the Romans chose to rename Judea “Palestine” (more precisely, the Roman emperor Hadrian so decreed after the Bar Kochba rebellion). But it is wise to stop using this name, because it has nowadays, after intense propaganda efforts by anti-Israeli journalists and revisionist “historians,” become associated with current Arab inhabitants of the Jewish homeland, to make them seem like natives (or even aborigines).

[12] See Neusner for a more detailed exposition of these various documents and their interrelationships. I cannot here, of course, get into discussions about dating that emerge from the different modern theories of Talmudic formation, including those of Abraham Weiss and David Weiss-Halivni. This is not my field, though truly a fascinating one.

[13] According to some commentators, the Talmud, though mainly the work the Tannaim and the Amoraim, may have received some further editing by the hand of some Savoraim (ca. 500-600 CE) and perhaps even some Geonim (ca. 600-1000 CE). Abraham Weiss considers that some editing was done in almost every generation, while David Weiss-Halivni attributes most of this work to those he calls the Stammaim (ca. 427 to 501 or 520 CE). See the interesting essays on these subjects in Essential Papers on the Talmud.

[14] Individual sentences or topics in the Mishna are called mishna in the sing., mishnayot in the pl. Likewise for the Gemara: gemara, gemarot.

[15] Note that when in the coming pages I refer to the Gemara’s “author,” I intend this singular term as very vague. It could be taken to refer to some anonymous Amora(s) whose ideas the Gemara just reports, or it could refer to the later redactor(s) injecting his/their own ideas. It is by no means clear in either case whether one person was involved or many; and if they were many, it is not clear whether they cooperated as a team, or they simply succeeded each other, each modifying or adding to the work of his predecessor. Moreover, keep in mind that the author(s) of one sugya may be different from that/those of other sugyas, for all we know.

[16] In his Studies, in a footnote on p. 60. Moreover, note that Maimonides considers, in the introduction to his Commentary on the Mishnah, that “it is not possible for any person to remember the entire Talmud by heart” (p. 110).

[17] 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!

[18] 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.

[19] 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.

[20] 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.

[21] 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.

[22] 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.

[23] 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!

[24] 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.

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

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

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

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

[29] In a video lecture online at: www.chabad.org/multimedia/media_cdo/aid/1158797/jewish/Rules-One-and-Two-of-Torah-Elucidation.htm; 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: www.dafyomi.shemayisrael.co.il/bkama/backgrnd/bk-in-025.htm.

[30] 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).

[31] R. Tarfon’s pursuit of a more stringent legal conclusion might be imputed to his belonging to the School of Shammai, although he is personally reputed to be inclined to leniency. This said in passing.

[32] Although in some contexts the word “sage” (hakham) is intended to refer to someone of lesser rank than a “rabbi,” I use the terms as equivalent in the present essay.

[33] The extracts from the Talmud quoted in the present chapter were found on the Internet at: www.halakhah.com/pdf/nezikin/Baba_Kama.pdf. I have made minor modifications to the text, such as changing the spelling of Kal wa-homer and Dayyo. All explanations in square brackets in the Gemara are as in the original, unless otherwise stated.

[34] A comparable statement of the dayo principle is found in Pesachim 18b, whence we can say that it is intended as a statement of principle and not just as an ad hoc position.

[35] Indeed, R. Tarfon could buttress his argument by pointing out that the latter transition is only half the distance, as it were, compared to the former. Alternatively, we could insist on ‘proportionality’ and say: from lenient (zero) to strict (full) the change is 100%, therefore from moderate (half) we should infer not just strict (full), which is only 50%, but ‘stricter than strict’, i.e. 150% payment! This is just pointed out by me to show that R. Tarfon’s argument by analogy was more restrained than it could have been. Evidently, 100% is considered the maximum penalty by both parties; no punitive charges are anticipated.

[36] I am here just suggesting a possibility, without any intent to make a big issue out of it. The advantage of this suggestion is that it legitimates R. Tarfon’s line of reasoning as an application of another rabbinic hermeneutic principle. The format would be: ‘just as private is stricter in the known case, so private should be stricter in the case to be determined’.

[37] Because, to repeat, judging by Torah practice, it can go no further – i.e. there is no “150%” penalty.

[38] The words “by horn” in square brackets added by me; but they are in accord with the interpolation in the Soncino edition.

[39] Note that the general major premise of the Sages’ qal vachomer can be stated more specifically as “for horn” – in which case, since the minor premise and conclusion are both specified as “for horn,” the whole a fortiori argument can be considered as conditioned by “for horn” and this condition need not be specified as here done for each proposition in it.

[40] Note that whereas in the first argument by analogy the movement is ‘from median to strict’, in the second argument by analogy the movement is ‘from strict to strict’. Assuming here again that 100% payment is the maximum allowed. Otherwise, if we insisted on ‘proportionality’, arguing that just as the increase from lenient (zero) to median (half) is 50%, so the increase from strict (full) ought to be 50%, we would have to conclude an ‘even stricter’ penalty of 150%!

[41] In the notes in the Artscroll Mishnah Series, Seder Nezikin Vol. I(a), Tractate Bava Kamma (New York: Mesorah, 1986), the following comment is made regarding 2:5 in the name of Rav: “Even in this [second] kal vachomer, we must resort to the fact that keren [i.e.horn] is liable in a public domain; otherwise, we would have no kal vachomer.” Other commentators mentioned in this context are: Tos. Yom Tov, Nemmukei Yosef, Rosh and Rambam.

[42] To give a simpler example, for the reader’s assistance: suppose we are given that ‘Some X are Y’; this is equivalent to ‘Some Y are X’. In such case we have two possible directions of generalization: to ‘All X are Y’, or to ‘All Y are X’. Clearly, while the sources of these two results are logically identical, the two results are quite different.

[43] “If a man cause a field or vineyard to be eaten, and shall let his beast loose, and it feed in another man’s field; of the best of his own field, and of the best of his own vineyard, shall he make restitution.”

[44] Note that although Ex. 22:4 only mentions the private domain, it is taken to imply the opposite penalty for the public domain. That is to say, if we take it to mean that damage by tooth & foot in the private domain must be compensated in full, then we can infer from the non-mention here or elsewhere of the public domain that this level of compensation does not apply. This is called a davka (literal) reading of the text. Although strictly speaking the denial of ‘full’ may mean either ‘only half’ or ‘zero’ compensation, the rabbis here opt for an extreme inversion, i.e. for zero compensation for tooth & foot damage in the public domain. Presumably, their thinking is that if half compensation was intended in this case, the Torah would have said so explicitly, since there is no way to arrive at that precise figure by inference.

[45] Another very theoretical possibility is that the compensation, which as we have argued must be only half in at least one domain (since the Torah specifies equal division of remains), is half in the private domain and either nil or full in public domain. It could be argued that it is nil in the public domain because the owner of the killed ox should have watched over his animal, or that it is full in the public domain because the owner of the killing ox should have watched over his animal. These logical possibilities are also ignored no doubt because they do not look equitable: they make one party seem more responsible than the other.

[46] Actually, I believe I have found such a way. We could use the kol zeh assim argument proposed by Tosafot to put the Sages’ dayo principle in doubt, at least in the present context. See my analysis of this possibility in a later chapter (9.7). Even though I do there decide that the dayo principle trumps the kol zeh assim argument, it remains true that this at least proves the Sages’ conclusion to be inductive rather than deductive.

[47] Thus, for instance, we speak in philosophy of the uniformity principle, not meaning that everything is uniform, but that there is considerable uniformity in the universe. Or again, in physics there is the uncertainty principle, which is applicable not in all systems but only in the subatomic domain.

[48] See the sentences: “does it not stand to reason that we should make it equally strict with reference to the plaintiffs premises?” and “does it not stand to reason that we should apply the same strictness to horn?” Also: “R. Tarfon, however, rejoined: but neither do I infer horn from horn; I infer horn from foot.” (My italics throughout.)

[49] 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.

[50] 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.

[51] 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.

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

[53] 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: www.tachash.org/metsudah/m03n.html#fn342).

[54] I presume offhand this refers to R. Yochanan bar Nafcha, d. ca. 279 CE.

[55] See page 11b, chapter 2, law 7.

[56] This Talmud (closed in Eretz Israel, ca. 400 CE) may of course contain significant comments about qal vachomer and the dayo principle elsewhere; I have not looked into the matter further.

[57] Since R. Tarfon flourished in 70-135 CE, and the Mishna was redacted about 220 CE, the Gemara under examination here must have been developed somewhere in between, i.e. in the interval from c. 220 CE to c. 500 CE. The thesis upheld in this particular anonymous Gemara may have existed some time before the final redaction, or may have been composed at the final redaction (or possibly even later, if some modern scholars are to be believed).

[58] According to a note in Talmud Bavli, this baraita first “appears at the beginning of Toras Kohanim,” by which they presumably mean the introduction to Sifra listing the thirteen hermeneutic principles of R. Ishmael and some Biblical illustrations of them.

[59] According to the Introduction to the Talmud of R. Shmuel Ha-Nagid (Spain, 993-1060 – or maybe Egypt, mid-12th cent.), a tosefta (addition) is a form of baraita (outside material) “usually introduced by the word tanya;” so, the use of this word here could be indicative of a tosefta. Further on in the same work, it is said that “an anonymous statement in the Tosefta is according to R. Nechemia;” so, the statement here cited by the Gemara might have been made by the Tanna R. Nechemia (Israel, fl. c. 150 CE). This is just speculation on my part, note well. An English translation of the book by R. Shmuel Ha-Nagid can be found in Aryeh Carmell’s Aiding Talmud Study (5th ed. Jerusalem: Feldheim, 1991); see there pp. 70, 74.

[60] The reason I say “in my opinion,” is that the text where I found this distinction, namely La mishna (Tome 8, Baba Kama. Tr. Robert Weill. Paris: Keren hasefer ve-halimoud, 1973), posits the reverse, i.e. aresh dina for the first argument and assof dina for the second. But that would not make sense in my view. Either there was a typing error, or (less likely) whoever originally formulated this distinction did not really understand how the two dayo applications differ. For it is clear from the analysis presented in the present volume that, in the first argument dayo is applied to the premise about proportionality (which is relatively downstream, whence “at the end”), while in the second argument it is applied before the formation of the major premise (thus, well upstream, i.e. “at the beginning”). Moreover, my view seems to be confirmed by the following comment in the Artscroll Mishnah mentioned in an earlier footnote: “it is easier to apply the principle of dayyo to the first kal vachomer, because in that instance it applies to the end of the kal vachomer.” It also seems to be confirmed by the article on the dayo principle in Encyclopedia Talmudit (reviewed in a later chapter, viz. 31.3).

[61] And I have found no explanation by later commentators.

[62] In truth, the Gemara’s explanations are not entirely clear; it is only by referring to later commentaries (paraphrased in Talmud Bavli ad loc) that I was personally able to fathom them.

[63] It is not clear which seven days the Gemara intends to refer to, when it says “there is no mention made at all of seven days in the case of divine reproof.” It could be referring to the initial seven days (the minor premise of the a fortiori argument), which as we shall later see the Gemara considers as tacit. Or it could be referring to “the additional seven days” mentioned a bit further on in the same paragraph, i.e. the seven days added on to the presumed initial seven to make a total of fourteen (the a crescendo conclusion of the argument), which the Gemara also takes for granted though absent in the text. In any case, the Gemara’s explicit admission that information is lacking is worth underlining.

[64] Here reference is made to Ex. 21:35, which concerns an ox killing (by goring or other such means) another’s ox, in which case the live ox is sold and the price of it divided between the two owners. And this situation is contrasted to Ex. 22:4, which does specify private property with regard to tooth & foot damage. However, this comparison seems a bit forced to me, because though it is true that there is no mention of where the ox was killed, that is because the damage done has nothing to do with location; whereas in the case of someone’s beast feeding in another’s field, it is the field that has been damaged. In any event, the rabbis are evidently making a generalization, from the case of an ox goring another ox (i.e. Ex. 21:35), to an ox goring or similarly damaging anything found on public or private property. Just as in the first case, the oxen are split between the owners, so the minimum for any other such damage by an ox is half liability. This is at least true for damage on public property, and the question asked is whether more than that can be charged for damage on private property.

[65] If we did not say “the minimum,” and instead interpreted the “half damages” on private property as davka, we would be suggesting that this penalty is Torah-given, and therefore no greater penalty can be inferred. If the latter were assumed, the Sages’ dayo objections would only be ad hoc Scriptural stipulations and not expressions of a broad principle. In that event, R. Tarfon’s two arguments were not rejected by the Sages because of any technical fault in them, but simply because the conclusion was already settled by Scriptural decree, so that there was no sense in his trying to infer anything else. But this does not seem to be the intent of the Mishna or the Gemara.

[66] Obviously, this more specific difference of opinion between the parties does not disturb the Gemara authorship. The implication is that the viewpoint attributed to R. Tarfon (about the conditionality of dayo) is not “of Biblical origin” – or, of course, it would be known to and agreed by the Sages! What credence does it have, then? Why hang on to it, if it is just one man’s opinion? One senses a double standard in the Gemara’s approach.

[67] Thus, the comment in Talmud Bavli that “applying dayyo in this case would leave the kal vachomer teaching us absolutely nothing” is not correct. The Mishna does not go from ‘half’ to ‘full’ and back to ‘half’ – it goes from ‘at least half’ to ‘full’ and thence to ‘only half’. We could similarly interpret the Miriam argument as going from ‘at least 7 days’ to ’14 days’ to ‘only 7 days’, and thus show the two cases are logically quite similar, contrary to the Gemara’s claim.

[68] The Gemara goes on and on, the next sentence being “R. Papa said to Abaye: Behold, there is a Tanna who does not employ the principle of dayo even when the a fortiori would thereby not be defeated…” (note the two negations, implying there may be yet other exceptions to dayo application). But this much later comment (dating from the late 3rd cent. CE) goes somewhat against the theory the Gemara attributes to R. Tarfon. So it is safe to stop where we have. Incidentally, if the sequence of events was really as implied in the Gemara, then the anonymous thesis that R. Tarfon “did not ignore” that the dayo principle “is of Biblical origin” would be dated roughly somewhere in the 3rd cent. CE – that is, one or two centuries after the fact, rather than three or more. But it is also possible that the said anonymous thesis was composed after the “R. Papa said to Abaye” part, the latter being adapted by the redactors to “fit in” – as modern scholars say often happens in the Talmud.

[69] I base this interpretation on explanations given in Talmud Bavli ad loc.

[70] I call this ‘pegging’ – this sort of arbitrary association of rabbinical claims with Torah passages irrespective of content. When meaningful reasons are not available, the rabbis sometimes unfortunately engage in such lame excuses to give the impression that they have some Scriptural basis. The conclusions of such arguments are foregone – there is no process of logical inference. Such interpretations would supposedly be classed as asmakhta by the rabbis.

[71] See the earlier section on Num. 12:14-15 for a fuller exposé.

[72] If Miriam was spared the extra seven days incarceration due to the exceptional circumstance that Moses prayed for her, then it was not due to application of a dayo principle but to an ad hoc special favor. Note that there is nothing in v. 15 that suggests either interpretation – all it says is that Miriam was indeed shut up for seven days.

[73] These two a fortiori arguments are given in full in previous sections of the present chapter.

[74] The explanations in square brackets are given in the Soncino edition.

[75] Perhaps, then, the Gemara’s authorship rather has in mind predicatal argument? For in the latter, the subject is normally constant while the predicates vary. But the difference is that in predicatal argument, the subject of the minor premise and conclusion is the subsidiary term, while the predicates are the major and minor terms; and the major premise differs in form, too. However, this schema does not accord with the form of the Miriam argument, so it is unlikely to be intended by the Gemara for R. Tarfon’s first argument, which it considers formally analogous to the Miriam argument.

[76] It is of course possible that in a specific case of Y, “all Y1 are Y2” is true; so that predicating the value Y1 entails predicating the value Y2. But this cannot be proposed as a general truth without absurd infinite reiteration.

[77] This is very much the mentality of a conventional mind – what Ayn Rand has called a “second-hander” in her novel The Fountainhead. Such a person takes the say-so of ‘authorities’ for granted, and makes no effort at independent verification. It builds buildings without foundations. It disregards the natural order of things.

[78] Although, as already remarked, the Gemara does not in fact pay any heed to the second argument or at all take it into consideration in its theorizing.

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