A FORTIORI LOGIC
CHAPTER 22 – Hyam Maccoby
Hyam Maccoby (Britain, 1924-2004) wrote some interesting comments regarding a fortiori argumentation in the Talmud in one of his last works, The Philosophy of the Talmud (2002). The relevant parts of this – namely, chapter 14, “Talmudic Logic,” and Appendix A, “Qal va-chomer in the Aggadah,” were collected (presumably by himself) in an essay called “Some Problems in the Rabbinic Use of the Qal Va-Homer Argument” and published online. We will in the present chapter review this essay.
My interest in Hyam Maccoby dates from the time I started writing my book Judaic Logic in the early 1990s, when I learned the importance of the rabbinic dayo principle through his critique of the use of the a fortiori argument by Paul of Tarsus (1986). I will not, however, here examine this critique, since I do so in an earlier chapter of the present work (viz. 10.3).
Maccoby’s more recent essay, on rabbinic use of a fortiori argument, is a mix of true insights and unfortunate errors. I refer the reader to the earlier chapters of the present volume (viz. 7-8) where I present my own detailed analysis of the Talmud’s treatment of qal vachomer and the dayo principle; I take it for granted here that the reader has indeed studied that analysis, so as to avoid unnecessary lengthy repetitions.
Hyam Maccoby nowhere formally defines qal vachomer in general and nowhere identifies its varieties (viz. positive or negative, copulative: subjectal or predicatal, or implicational: antecedental or consequental, etc.). He makes no attempt at formalization, and consequently cannot engage in formal validation. His approach is therefore, overall, rather vague and, as the saying goes, ‘intuitive’; in other words, he refers to “common sense.” Nevertheless, Maccoby is outstanding in his understanding of a fortiori argument and steadfast adherence to the logical rule (which I have much later labeled ‘the principle of deduction’) that its conclusion cannot be more informative than its premises, which he identifies with the rabbinical dayo (sufficiency) principle. Thanks to this lucid and uncompromising posture, he is able to find fault with many a fallacious argument billed as a fortiori. In the abstract of his essay, he writes:
“The qal va-homer (a fortiori) argument is a logic of analogy, not of classes or sets (the subject-matter of Aristotelian logic), and this makes it suitable for legal, rather than scientific, argument.”
In this general statement concerning the nature of qal vachomer, Maccoby shows his independence from the many commentators who claim this form of argument to be syllogistic. He rightly refers it to the more general category of argument by analogy. However, he fails to explicitly mention its quantitative aspect, which distinguishes it from qualitative analogy. Moreover, it is inaccurate to oppose analogy to classification, since the latter is also based on perceived or conceived similarities and differences. Also, that a fortiori argument is particularly suitable for legal thinking is true; but this does not make it any the less suitable for scientific purposes, if properly used.
Concerning the dayo principle, which note well (to repeat) he effectively equates to the principle of deduction, he writes:
“What makes [qal va-homer] an exact reasoning is a special rule (unknown to Greek rhetorical use of a fortiori), namely the rule of dayyo, which lays down that the conclusion must not contain anything that was not present in the premises.”
Here, we should note Maccoby’s observation – which may be original with him – that the dayo principle formulated by the Mishnaic rabbis was unknown to the Greeks. I confirm that I have not so far found any comparable rule in Greek, Hellenistic or Roman logic or rhetoric. These cultures may conceivably have occasionally or even usually reasoned in unconscious accord with the principle of deduction, but in any case they apparently never explicitly said that the conclusion must mirror the minor premise and should not be made ‘proportional’ to it. As regards the dayo principle in its truly rabbinical sense, this concerns Jewish law in particular; so, though it is not inconceivable to find a similar restriction in some other law system, it is not surprising not to find one.
In truth, as I have shown in an earlier chapter (7), the rabbinical dayo principle correctly understood is not commensurate with the principle of deduction. Looking at the discussions Talmudic tractate Baba Qama, pp. 25a, we might at first make such an equation. That is because a possible and very common interpretation of the Mishna has it that R. Tarfon’s first argument was an attempt at ‘proportional’ qal vachomer and the Sages’ the dayo principle was intended to remind or inform him that the logical conclusion cannot surpass the given minor premise. However, this simplistic interpretation, which seems to be that favored by Maccoby, does not in fact correspond to the Gemara’s viewpoint and subsequent tradition.
Furthermore, irrespective of the Gemara’s viewpoint, the said interpretative hypothesis is definitely belied by the second exchange in the Mishna, where R. Tarfon proposes a modified qal vachomer, which is in perfect accord with the principle of deduction, and yet is struck down by the Sages using the exact same dayo objection! On this basis, the dayo principle has to be seen as different from the principle of deduction. Even if the two principles happen to have the same result in a particular context (viz. the first argument), they do not coincide in all contexts (notably not in relation to the second argument).
In fact, as I explain in my detailed analysis, the rabbinical dayo principle is not a logical principle at all, but a moral one, related to the principle of measure for measure (midah keneged midah) and other extra-logical considerations. As such, it need not apply to all a fortiori argument and may even apply to argument forms other than a fortiori (as indeed may the principle of deduction, of course).
Thus, Maccoby’s understanding of the rabbinical dayo principle is not entirely correct, although his understanding of a fortiori argument is essentially correct. What he calls the dayo principle is really the principle of deduction, a purely logical rule known by rational insight. The dayo principle, in its true sense, is actually a rabbinical majority ruling presented as a Divine decree (based on Numbers 12:14-15, as the Gemara teaches). In fact, these two principles ought not to be confused; we may nevertheless verbally equate them in the present analysis so as to continue using Maccoby’s own terminology.
Maccoby describes qal vachomer reasoning in general terms by saying: “If a conclusion is true in a weak situation, it is true ‘all the more so’ in a strong situation.” This description, note well, only points to the most common form of the argument, viz. the positive subjectal mood; and he does not point to other moods elsewhere. For him, as already said, the dayo principle is what makes qal vachomer an exact form of reasoning. The example he proposes for it is very clear and accurate:
“If a moderately good child deserves one sweet, [then,] all the more so:
– a very good child deserves one sweet (correct);
– a very good child deserves two sweets (incorrect).”
However, to repeat, Maccoby does not formally explain why the first putative conclusion is correct and why the second is incorrect, other than to point to the fact that “the conclusion must not contain anything that was not present in the premises,” which is true of all deductive processes. The structure of a fortiori argument is lacking in his treatment – he fails to distinguish the major premise from the minor premise and conclusion, and the middle term (R) from the major (P), minor (Q) and subsidiary terms (S); moreover, he does not have the idea of a threshold value of R as being “enough for” S.
If he had analyzed structural features, he would have formulated his argument more explicitly as follows. When an a fortiori argument is so precisely structured (in this case as a positive subjectal), it becomes easier to formally validate it and understand why the conclusion could not have been ‘proportional’ (namely, because S must be identical in the minor premise and conclusion):
A very good child (P) is more deserving (R) than a moderately good child (Q),
and, a moderately good child (Q) is deserving (R) enough to get one sweet (S);
therefore, a very good child (P) is deserving (R) enough to get one sweet (S).
Because Maccoby failed to do this (though he could have done it if he had studied my earlier work on the subject, Judaic Logic), he does not correctly perceive why, as he puts it: “a qal va-homer argument is not as unchallengeable as a syllogism, and the rabbis recognised various grounds of challengeablity.” He explains this feature as follows:
“The qal va-homer reasoning is open-ended, in that it depends on a distinction between ‘light’ and ‘heavy’ that is always open to question. This aspect, however, does not invalidate this type of reasoning, but differentiates it from the mathematical or logical kind of reasoning, where intuition or grasp of human values play no part.”
That is to say, he thinks this form of argument is intrinsically open to debate, because it is concerned with human values. But that is not always the case: a fortiori argument is possible without reference to valuations. He is thus wrong to differentiate it so radically. The truth of the matter is that given the truth of the premises, if we reason in accord with validated forms, the argument is unassailable. In a fortiori argument, there may be debate regarding the premises (and in particular, often, the major premise) and/or debate regarding the validity of the attempted process – but, as in all logical argumentation, once these issues are settled, the argument is mechanical and not open to further doubt. In other words, a fortiori argument is deductive and not merely inductive.
All that has been said so far only concerns purely a fortiori argument; that is, argument with two appropriate premises (the major and the minor): in such case, the conclusion can only be as Maccoby has it ‘non-proportional’. However, contrary to what Maccoby believed, a ‘proportional’ conclusion from those very same premises is indeed logically permissible, provided a third premise is introduced that informs us of the general ‘proportionality’ between the subsidiary term and the middle term (I put the word proportionality in inverted commas to remind us that the intended proportions are in practice often rough rather than exact).
If such an additional premise is involved, a ‘proportional’ conclusion is in full accord with the principle of deduction, and therefore with the dayo principle as Maccoby effectively conceives it. In other words, Maccoby’s presentation of qal vachomer reasoning and the dayo principle is all very well, so long as what we are referring to is specifically non-proportional a fortiori premises. However, if in a given case we admit, as well as the two premises of pure a fortiori argument, an additional premise about ‘proportionality’, the ‘proportional’ conclusion becomes quite valid and Maccoby’s rejection of it cannot apply. Maccoby’s sample argument would in such case look like this:
A very good child (P) is more deserving (R) than a moderately good child (Q),
and, a moderately good child (Q) is deserving (R) enough to get one sweet (S);
and, reward in sweets to children (S) are to be proportioned to their deserts (R);
therefore, a very good child (P) is deserving (R) enough to get two sweets (S).
Actually, unless we can provide a precise mathematical formula for the concomitant variation between S and R, the conclusion should rather read, more indefinitely, “more than one sweet;” or we should admit that the exact quantity is somewhat subjectively assessed.
Maccoby’s error was to base his model of a fortiori argument, and consequently of the dayo principle, on only the first exchange between R. Tarfon and the Sages in the Mishna Baba Qama 2:5. Like many commentators before him and after him, including the Gemara no less – he completely failed to take into consideration the very different second exchange between R. Tarfon and the Sages. He did notice the latter (which is more than can be said of the Gemara), but he did not realize its significance. If one tailors one’s hypothesis to fit only part of the data at hand, one is pretty well bound to go astray. Thus, though Maccoby did rightly conceive purely a fortiori argument, his ideas are too restrictive because he altogether missed out on what I have lately named a crescendo argument.
A crescendo argument, as a combination of purely a fortiori argument and pro rata argument, is (as I have formally demonstrated in an earlier chapter (2)) a fully legitimate form of deductive reasoning. A crescendo argument is a species of a fortiori argument, since it consists of purely a fortiori argument combined with pro rata argument. Moreover, many examples of its use can be found in everyday discourse, as well as in the Torah and in rabbinical literature. So, in the last analysis, Maccoby evidently strayed considerably in this matter.
The principal part of Maccoby’s commentary naturally revolves around the famous Talmudic discussion on page 25a of tractate Baba Qama. Maccoby’s position, briefly put, is that the Mishna and Gemara are at odds. He begins by detailing the Mishna, then remarks:
“Both Rabbi Tarfon and his opponents the Sages accept not only the validity of qal va-homer, but also the validity of its limiting principle of dayyo. Where they differ, in this instance, is how to draw up the list of terms involved in the reasoning.”
Actually, Maccoby’s position here is logically untenable, since R. Tarfon’s first qal vachomer argument is definitely not purely a fortiori argument and definitely not in accord with the Sages’ first dayo objection interpreted as the principle of deduction. Funnily enough, in imagining that the Mishna opponents are in essential agreement, Maccoby mimics the Gemara, the very Gemara that he goes on to reject on the basis that it advocates ‘proportional’ a fortiori argument.
Maccoby is right to reject the Gemara, insofar as it appears to claim that all a fortiori argument is ‘proportional’ (i.e. a crescendo). But he is wrong to reject it, if all that the Gemara claims is that this particular case should be read as ‘proportional’. As we have just seen, Maccoby thinks that valid qal vachomer is necessarily purely a fortiori; but as we have shown elsewhere a crescendo argument is quite possible, if an additional premise is granted. The Gemara here goes to the opposite extreme, seeming to claim that valid qal vachomer is necessarily ‘proportional’, thus totally ignoring pure a fortiori argument (even though the Mishna it comments on does contain such argument).
Looking at the two arguments of R. Tarfon, Maccoby cannot consistently claim them to be both a fortiori by his definition, since the first of them is definitely not purely a fortiori (though the second could be so interpreted, and ultimately must be). Whereas, the Gemara can consistently claim both to be a fortiori by its definition, since both can be read as a crescendo (even if the second one could also be read as purely a fortiori, and ultimately must be). In either case, it cannot be said that R. Tarfon was in essential agreement with the Sages; their two objections were indicative of significant difference of opinion.
In Maccoby’s view, R. Tarfon and the Sages are merely in disagreement about “the terms” of the arguments proposed. This is a traditional posture found in Tosafot, which is based on R. Tarfon’s quick reformulation of his argument in a bid to satisfy the Sages’ first dayo objection. Maccoby then explains the Sages’ reticence to R. Tarfon’s second argument by saying, somewhat lamely: “The Sages, however, see something illegitimate about this move, since, in the process, ‘horn’ has been surreptitiously promoted from ‘half-damages’ to ‘whole-damages’, which appears to be a breach of the principle of dayyo.” Clearly, Maccoby does not understand exactly why R. Tarfon’s change of terms does not change the Sages’ resistance to his thesis. This forces Maccoby to conclude, rather mystically if I may say so:
“We see from this that in a qal va-homer argument there may sometimes be an uncertainty arising from the choice of appropriate terms. This choice of terms may be a matter of intuition, rather than strict logic, and thus one person’s valid qal va-homer may be another’s fallacy. This does not mean that this method of argument should be condemned as subjective, but only that it belongs to the area of rationality rather than strict logic… a region of logic that transcends the usual parameters of Western logic.”
In truth, from the perspective of a fortiori logic, R. Tarfon’s second argument is logically stronger than his first, since the first is only valid as a crescendo (or even simply as pro rata) argument whereas the second is also valid as purely a fortiori argument. The issue is not therefore one of “choice of terms,” as Maccoby vaguely suggests; it goes much deeper. The fact is that R. Tarfon’s second try is logically immune to the Sages’ first dayo objection; from which it follows that the Sages’ second dayo objection must be directed at the inductive formation of the new major premise (with reshuffled terms) rather than at the additional premise of ‘proportionality’.
Maccoby’s difficulty in unraveling the problem is due to his simplistic identification of the rabbinical dayo principle with the principle of deduction. Regarding the first argument, this identification works out without apparent difficulty because there the two principles just happen to converge. But regarding the second argument, the two principles clearly diverge. This serves to show us that the Sages’ dayo principle is a rejection of any extrapolation of legal penalties from Scriptural data, even if such inference is in accord with the principle of deduction. In other words, their second objection has to extend dayo restriction to the inductive preliminaries of R. Tarfon’s second argument, since the subsequent deduction does not contravene their first objection. The Sages’ dayo principle therefore cannot be limited, as Maccoby takes it to be, to qal vachomer inference of the kind used by R. Tarfon in his first argument, i.e. a crescendo. The Sages’ dayo principle has to be broadened in such a way as to also interdict R. Tarfon’s second argument, which is purely a fortiori.
Nevertheless, there is a bit of truth in Maccoby’s insight concerning “choice of terms” – in the sense that it is interesting to observe that by judiciously reshuffling the given data R. Tarfon was somehow able to pass from an a fortiori argument not in accord with the Sages’ first dayo objection to one in accord with it. But on closer analysis, this reconstruction is not as freewheeling as it seems. What differentiates the two arguments is the different directions of the preliminary generalizations which their respective major premises are based on. In this new context, we can view the Sages’ dayo principle as a decree that when a possible generalization leads to a more presumptive conclusion, the safest course inductively and morally is to avoid it. Their principle is therefore, in this larger context, only incidentally related to a fortiori argument.
Maccoby’s overly narrow reading of the Mishna explains his firm belief that qal vachomer reasoning is identical with purely a fortiori argument, and its attendant dayo principle is identical with the principle of deduction. His approach to both processes is evidently that they are natural and rational, even though not perfectly logical (as we just saw him say). Given this rationalistic attitude, Maccoby’s strongly negative reaction to the Gemara is quite understandable and laudable. For him, the Gemara’s explication of the Mishna is a manifest error:
“The Amoraic discussion of the Mishnah (b. Bava Qamma, 25a) must be discounted, since it shows no comprehension of the logical force of the dayyo principle. Instead, it imagines that the rule is an arbitrary fiat of the Torah….
It seems that in the Amoraic period the rationale of the dayyo rule, perfectly understood in earlier times, had been lost. In earlier times, too, the derivation of the rule from Scripture (if made, which is doubtful) was not intended to give it the status of an arbitrary fiat, but to give authoritative approval to a deliverance of reason.”
However, things are not so simple. Maccoby apparently did not know, or at least fails to mention, that the statement in the Gemara on which this position is based is there presented as a baraita – i.e. as a Tannaic statement not recorded in the Mishna but having an authority close to Mishnaic. Maccoby is thus, consciously or not, either denying the reliability of this particular baraita, or claiming that it is a fabrication, or perhaps even claiming most or all baraitot to be unreliable or fabricated. Maccoby is not, however, doing this arbitrarily, note well. For him, the Gemara’s claim (with regard to Numbers 12:14) that Miriam deserved fourteen days instead of just seven days isolation is contrary to natural logic, for only pure a fortiori argument can be valid. It follows that, in his eyes, the Gemara must be wrong.
When I set out to analyze this Talmudic passage, before I studied Maccoby’s article, my own first reaction was to, much like Maccoby, discount the Gemara’s explication as a later misunderstanding. But when I realized that the Gemara’s position was based on a baraita, I realized that a more subtle approach would be necessary to resolve the difficulty. In truth, the Gemara is only wrong if it thinks that all a fortiori argument is necessarily ‘proportional’; but it is logically quite possible for it to claim that a particular a fortiori argument is ‘proportional’.
In truth, if the dayo principle is none other than the principle of deduction as Maccoby thought, it is a redundancy. That the conclusion cannot go beyond what is given in the premises is true of both purely a fortiori argument and a crescendo argument, as indeed of all deductive argument, without any need to state it as a special principle; it is the very definition of deduction, as against induction or fallacious thought, and so the subtext of any deductive act. So, I thought, it is possible that the dayo principle concerns qal vachomer not per se, but only per accidens. The rabbis may well have thought it concerns qal vachomer as such, but judging from their reading of Numbers 12:14 it would seem that they were unconsciously mentally referring to the principle of justice.
In the Miriam story, we do have an intuitive sense that Miriam deserves more punishment for her act of lèse majesté towards God (in saying nasty things about her brother Moses) than a daughter who angered her father (a mere human being, after all) would deserve. So the Gemara does have a leg to stand on. What needs to be understood, however, is the source of this intuitive sense. It cannot, as the Gemara suggests, be due to qal vachomer necessarily having a ‘proportional’ conclusion (i.e. 14 days instead of 7), for this explanation can be proved indubitably wrong by means of formal logic. Therefore, it must be due to something else – viz. (I suggested) to the ‘sense of justice’, i.e. the belief that the punishment should be commensurate with the crime, or more intellectually put, the principle of ‘measure for measure’ (midah keneged midah). In that case, the dayo principle becomes an important tool, used to block excessively strict justice and impose some mercy.
Thus, the dayo principle can be viewed as essentially different from the principle of deduction, and the Gemara’s claim that it is a Divine fiat can be justified. Maccoby, in his above quoted remark, expresses doubt as to the dayo principle’s derivability from Scripture (i.e. from Numbers 12:14-15), though he grants it might at best be taken as confirming what is rationally obvious. He also excoriates the Amoraim for effectively denying the logical force of the dayo principle, since they effectively view it as an arbitrary fiat. These postures are simply due to Maccoby’s inaccurate identification of the dayo principle with the principle of deduction.
Maccoby is, of course, quite justified in forcefully rejecting the thesis that the dayo principle, if it is taken as identical with the principle of deduction, can occasionally be bypassed. He presents the Gemara’s suggestion that R. Tarfon claimed such exceptions as follows:
“The Gemara then explains that Rabbi Tarfon, while acknowledging the rule of dayyo, had a variant view of it which would [have] excluded the present case (he considered that a qal va-homer that, by the application of dayyo, yields a result already derivable from other sources is not subject to the rule of dayyo). This implausible account ignores totally the plain reason which Rabbi Tarfon himself gives in the Mishnah: that he was using different terms as the basis for his reasoning from those used by the Sages. It ignores also the fact that Rabbi Tarfon’s expressions in the Mishnah show that he is not arguing that this instance is exempt from the rule of dayyo but, on the contrary, that he is bringing it into the rule.”
I agree with Maccoby’s last sentence here to some extent: R. Tarfon was not, in his second argument, seeking exemption from the Sages’s objection to his first argument, but on the contrary was seeking to conform to it. But Maccoby evidently did not realize that since the Sages countered this with a renewed objection, the dayo principle was thereby extended. I agree, too, with Maccoby’s characterization of the Gemara’s claim that R. Tarfon had a variant view of the dayo principle as “implausible.” The variant view the Gemara imputes to R. Tarfon does not stand up to detailed logical scrutiny.
Nevertheless, if we take the dayo principle as referring to something other than a merely logical rule, namely to refraining from excess in the application of just ‘proportionality’ in retaliation, then occasional exception to the rule is not inconceivable or logically reprehensible. If the principle is a Divine fiat, then exceptions to it might also have been decreed. So the Gemara’s position is not as indefensible as Maccoby thinks.
Although I have considerably criticized some of Maccoby’s views, please do not get me wrong: I consider his contribution to the discussion as valuable. Although he inaccurately equates the dayo principle to the principle of deduction, his unswerving commitment to the latter and his courage to denounce those who indulge in deviations from that logical norm, makes him an inspiring figure.
As already said, Maccoby had a good intuitive understanding of a fortiori argument. But he lacked the formal tools needed to fully support and express this understanding. As a result, he misperceived some parodies of qal vachomer reported in the Talmud as indicative of “vulnerability” in this form of reasoning. He cites the following as an example of parody:
“In b. Sanh. 17a… it is said that one of the qualifications of a member of the Sanhedrin was the ability to prove that the body of a reptile (sheretz) was clean, i.e. did not convey ritual impurity. This would seem an impossible undertaking, since the Torah says explicitly that it is unclean. However, in response to the assertion, Rav offered to prove this impossible proposition by means of a qal va-homer argument. It ran as follows: the dead body of a snake does not convey impurity. Yet a snake is the means of spreading impurity, for it causes many deaths. How much more so should a reptile (which is harmless) be regarded as not causing impurity! This same argument is attributed to Ravina in b. Eruvin 13b. The Gemara immediately refutes this argument by denying the ‘heaviness’ of the heavy term. Being a cause of impurity indirectly by causing death has nothing to do with causing impurity by direct contact. Otherwise, we would have to regard a thorn as a cause of impurity by contact, since it may cause death to someone who becomes impaled on it.”
However, if we examine this argument closely, we see that the problem with it is nothing to do with the mechanics of a fortiori inference; it is rather an obvious case of equivocation, the term “impurity” having a different sense in the premise and in the conclusion. Purely a fortiori argument can readily be validated with reference to four terms (the major, minor, middle and subsidiary) – but it cannot be validated with reference to five or more terms. This is not a weakness or flaw in the argument’s logic – it is a rule in its logic. Just as in syllogism there is a “fallacy of four terms” – so with regard to purely a fortiori argument there is a “fallacy of five terms.”
Maccoby admits as much, saying: “Of course, it is possible to produce parodies even of syllogistic arguments,” adding: “perhaps we should attribute such parodies merely to high spirits, not to any flaw in the type of argument itself.” His deficiency of formal equipment can also be seen in the other example of parody he gives, drawn from Derekh Eretz Rabba, 1. As it happens, I have already dealt with this sophistical argument in the chapter on Mielziner (13). As we saw there, the problem with it is not the a fortiori inference as such, but the overgeneralization used to inductively formulate its major premise. In fact, the absurdity of the a fortiori conclusion is logic’s way of telling us that there is a problem in the premises (in this instance, the major). The parody therefore reveals, not a weakness in the argument form, but its strength as an instrument for ensuring internal consistency in our knowledge.
Nevertheless, Maccoby rejection of the first a fortiori argument in Mishnah Makkot 3:15 makes clear that his logical instincts are basically sound. This argument reads: “R. Hananiah ben Gamaliel said: If he that commits one transgression thereby forfeits his life, how much more, if he performs one religious duty, shall his life be given to him!” According to Maccoby, this involves an “apparently glaring infringement of the rule of dayo;” i.e. it constitutes an illicit process, since the predicate in the premise, “forfeits his life,” is not identical with, but indeed contrary to, that in the conclusion “his life shall be given to him.”
However, although Maccoby is right in rejecting this argument, his reason for its rejection is not correct. The problem is not essentially in the fact that the two predicates (subsidiary terms) are different, or even that they are antithetical. The problem lies in the a contrario form of the argument, i.e. in the fact that not only are the predicates antithetical, but also the subjects – viz. someone who “transgresses the law” and someone who “obeys the law” – are contrary, and these two contrarieties occur in lockstep.
The argument seems to have the following basic form: “If X, then Y; therefore, if not X, then not Y” – i.e. it refers to inversion. This is, of course, formally wrong – we cannot inverse a hypothetical proposition at will; i.e. the given premise does not formally imply the putative conclusion. It is true that, judging by the speaker’s use of the expression “how much more,” his intent is not immediate inversion, but something more, a sort of a fortiori argument. Of course, although this expression is conventionally understood to signal a fortiori inference, its use does not certify that a fortiori reasoning is actually involved.
But let us suppose an a fortiori argument is indeed intended. In that case, to draw the conclusion from the given (minor) premise, we would need a (major) premise that places the terms X and notX as minor and major in some scale (i.e. identifies them as bigger or smaller in some respect, say Z). Moreover, we must suppose that these subjects have enough of Z to receive predicate Y or notY, respectively. Furthermore, since the subsidiary terms Y and notY are not identical, we need some third premise about proportionality to justify an a crescendo inference from the one to the other.
But even if all these conditions are met, we would still not be able to draw the putative conclusion! For this argument is not just one with antithetical subjects or just one with antithetical predicates – it is one with both these features together. This means that the change from X to notX and the change from Y to notY must happen simultaneously. Yet the premises we have adopted are not sufficient to guarantee this simultaneity. We would need more information to guarantee it.
This sounds feasible offhand; but if we reflect we see that it is not easy to provide such additional information capable of justifying the desired inference. An attempt to do that would seem to be bound to end in circular argument. And this is the reason why a contrario argument must be considered as invalid. Thus, R. Hanahiah’s above argument is indeed invalid, but not for the reasons Maccoby gives. The problem is not that it constitutes an a crescendo argument; such arguments may well be valid. The problem is that it is an argument so demanding of further information that it forces us to beg the question.
Maccoby does not conceive the issue in such formal terms, but he does analyze this example at length and propose the following resolution of the problem as he sees it:
“I suggest a very simple solution… The expression notel nafsho has been wrongly translated by all commentators as ‘forfeits his life’, but is much better translated as ‘receives his life’… With this correction, the difficulties of continuity and logic disappear. The… saying now becomes: ‘If he that commits one transgression (i.e. the transgressor of a karet law) receives his life (i.e. has his sentence commuted from death by the hand of God to flogging), how much more so will one who performs a commandment be given his life!’”
In this way, Maccoby harmonizes the subsidiary terms in the minor premise and conclusion, and changes the apparent a crescendo argument into a purely a fortiori argument. This example is, Maccoby admits, an extreme case. He grants, as most commentators do, that “the rules of the qal va-homer, and especially the rule of dayyo” ought perhaps not to be “so strictly applied in aggadic as opposed to halakhic reasonings.” I am personally, I must say, not very tolerant of breaches in logic, even for homiletic purposes. But Maccoby is willing to grant some poetic license. He gives another example drawn from the same Mishna that seems less flagrant:
“R. Shimon ben Rabbi reasons: abstaining from blood, which causes revulsion, brings reward; all the more, abstaining from robbery and incest, which the soul longs for, should bring reward for all generations to come up to the end of the world. The last part of this reasoning constitutes a breach of dayyo, for nothing was said in the premise about all generations to come. Yet in this reasoning, faulty as it is by halakhic standards, the actual terms are not changed from premise to conclusion. There is only an intensification of the conclusion, in an enthusiastic, homiletic style.”
This could be a correct reading. But of course, we could alternatively say that “abstaining from blood… brings reward” tacitly intends “for all generations to come up to the end of the world,” in which case there would be no breach of the dayo principle! Very often, we read things unstated in the premise by means of the conclusion, as well as vice versa. For example, take Proverbs 21:27: “If [even brought with a ‘sincere’ intent] the sacrifice of the wicked is an abomination, how much more: brought with a wicked intent [is it abomination]?” The phrases in square brackets were added by me; but they are both clearly tacitly intended.
But in any case, this second example from Makkot 3:15 is not formally identical to the first. The second argument, the one by R. Shimon, is that abstinence from blood is psychologically less taxing, because blood is revolting, whereas abstinence from robbery and incest is more onerous, because they arouse desire; consequently, while the former (which is demanding enough) deserves some reward, the latter (which is more difficult) deserves a greater reward. While this argument resembles the first, in that both involve parallel continua, one for the subjects and one for the predicates, the subsidiary terms in the second argument are not in opposition but two degrees of the same thing (viz. “reward” and “greater reward”). Thus, in the latter case, there is no a contrario intent, and the argument is much easier to validate.
Maccoby does notice this important formal difference between the two arguments, but he does not formally explain why that difference should make one more invalid than the other. To him, both remain essentially invalid; and they are so for the same reason, viz. that they are ‘proportional’. Thus, we see here again that Maccoby’s judgments are negatively affected by his narrow view of a fortiori argument as necessarily non-proportional, and his failure to grasp the logical possibility of a crescendo argument if appropriate additional information is provided.
Nevertheless, to repeat, in view of his emphasis on ‘rule of law’ in a fortiori logic, he deserves recognition as one of the good guys in this field.
 London: Routledge Curzon, 2002. Some excerpts of this book may be read online at: books.google.com/books?id=0uHwGt1kypEC&printsec=frontcover&dq=Hyam+Maccoby+The+Philosophy+of+the+Talmud+%282002%29&hl=en&src=bmrr&ei=GZl4Tee7LoKV4gaMsNj6BQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCwQ6AEwAA#v=onepage&q&f=false.
 I was gratified to see that Maccoby cited my book Judaic Logic in a footnote to this essay, though disappointed that he apparently did not realize the full extent of my contribution there – namely, the formalization and validation of a fortiori argument. I assume from that that he did not actually read the book.
 Maccoby reads this rule as: “dayyo lav’o min ha-din lihyot ke-nidon” and adds: “note that here din means ‘conclusion’ while nidon means premise; the transliteration, often found, nadon is incorrect, as this is Biblical, not Mishnaic, Hebrew, see Jastrow.”
 Note in passing that Maccoby would have been more accurate, as regards Talmudic logic (though not generic logic) to have given a negative example, say: “a moderately bad child deserves one slap; therefore, a very bad child deserves one slap too (not two slaps).” Why? Because, generally, the rabbinic dayo principle relates to penalties, rather than rewards. The rabbis were against proportionality of punishment, not of recompense. Their purpose was to limit the former, not the latter. Dayo perhaps applies more broadly to all negatives; i.e. not only to penalties but to the ‘burdens’ of mitzvoth. An inferred mitzvah should likewise not be made more burdensome than necessary on the basis of some misplaced ‘proportionality’.
 Whereas, it is true, the rabbinical dayo principle is always concerned with human values.
 E.g. Inferring a conclusion of 14 instead of 7 days penalty for Miriam in the story at Numbers 12:14-15.
 Unless we consider the statement “In earlier times, too, the derivation of the rule from Scripture (if made, which is doubtful)…” to be intended as a reference to the baraita.
 It is not unthinkable that a baraita might be fabricated. Louis Jacobs, in his Rabbinic Thought in the Talmud, devotes a chapter to this issue in general. He mentions the thesis advanced by I. H. Weiss (in Dor Dor ve Dorshav) that many of the baraitot in the Babylonian Talmud may be “fictitious, i.e…. not authentic transmissions of tannaitic opinion but… invented by the Babylonian Amoraim as alleged support for their views.” Though this view is later opposed by Abraham Weiss (in Le-heqer ha-Talmud), Jacobs considers it to have some credibility, and explains why with various examples. It does not follow, of course, that the particular baraita of concern to us here is fabricated. But, to repeat, it is not unthinkable that it might be.
 However, Maccoby is not right when he ascribes to the Gemara that “the conclusion of a qal va-homer reasoning is cut in half (why by precisely a half is not explained)” by the dayo principle. First, because nowhere is the proportion of “half” literally mentioned as a general rule, even if 7 happens to be half of 14 in the Miriam example. Second, because later commentators have indeed proposed explanations (however flimsy) for this particular proportion in this particular case.
 Valid ‘proportional’ a fortiori (i.e. a crescendo) argument is, of course, exceptionally not subject to that rule, since it contains two versions of the subsidiary term (one bigger than the other) and so effectively five terms.
 See more detailed analysis of this issue in Appendix 2.