A FORTIORI LOGIC
CHAPTER 16 – Louis Jacobs
Louis Jacobs was an ordained orthodox Rabbi who was repudiated by his more orthodox colleagues for his view that the Torah is partly of human origin, and later founded the British Masorti (equivalent to the American Conservative) branch of Judaism. He wrote many thoughtful and interesting books on Judaism. Always knowledgeable, intelligent, honest and courageous, he tries in them to bring Jewish doctrine in phase with modern science, history and good sense. His theory of a fortiori reasoning appears in the first chapter of his Studies in Talmudic Logic and Methodology. I will here detail and critically analyze it.
Let us go straight to the main issue in this work – Jacobs’s attempt at formalization of a fortiori argument. He distinguishes two types of qal vachomer:
“Simple: If A has x then B certainly has x.
Complex: If A, which lacks y, has x, then B which has y certainly has x.”
Jacobs’ “simple” qal vachomer has the form: “If A has x, then B certainly has x.” He apparently considers all ten of the Biblical examples given in Genesis Rabbah 92:7 to fit into this form, rather than in the complex form. His simple form only refers to subjectal argument, since the subjects (A, B) differ, while the predicate (x) is the same, in the premise (A has x) and conclusion (B has x). The premise mentioned is of course the minor premise – there is no mention of the major premise. The subjects A, B can be identified as the minor and major terms (Q, P), respectively; and the predicate x as the subsidiary term (S). There is no mention of the middle term (R), which would be required in the major premise, and in the minor premise and conclusion. Put in standard form this argument would look like this:
B (P) is more ? (R) than A (Q) is,
and A (Q) is ? (R) enough to be x (S);
therefore, B (P) is ? (R) enough to be x (S).
Jacobs’ “complex” qal vachomer has the form: “If A, which lacks y, has x, then B, which has y, certainly has x.” His prime example of this type is a baraita found in Hullin 24a, viz.: “If priests (A) who are not disqualified for service in the Temple by age (lack y) are yet disqualified by bodily blemishes (have x), [then] the Levites (B) who are disqualified by age (have y) should certainly be disqualified by bodily blemishes (have x)”. Here, an additional term is brought into play, viz. ‘disqualification for Temple service due to age’ (labeled “y”), which could serve as the middle term (R), if we consider y as referring to a range of values including zero. In that event, “lacks y” means ‘y = 0’ and “has y” means ‘y > 0’, so that B can be said to have (or be) ‘more y’ than A does (or is). We can then put this argument in standard form as follows:
B (P) is more y (R) than A (Q) is,
and A (Q) is y (R) enough (albeit no more than zero y) to be x (S);
therefore, B (P) is y (R) enough (being more than zero y) to be x (S).
We see then that, logically, the only difference between the simple and complex arguments is that the former has an unstated middle term we have marked as ‘?’ (R), whereas the latter has an explicit one labeled ‘y’ (R). In all other respects, the two arguments are formally identical. It follows that Jacobs’ distinction is not one of logical form, but merely one of degree of explicitness. His forms are two discursive forms, i.e. two ways (among many more) that a fortiori argument may be formulated in practice. They are not logical forms, i.e. not formulas that reveal all the elements underlying and relevant to the discourse. They describe only the surface appearance of the discourse; they do not plunge into the deeper aspects needed for true understanding of it. Logically, the two forms are one and the same.
But Jacobs conceives these two arguments as more radically distinct, as his following statements (in pp. 3-4) make clear:
“The simple qal wa-homer has a long history – its use is traced by the Midrash (Gen R. 92.7) to the Bible itself…. The complex qal wa-homer is of later Halakhic origin and is found in a Baraitha for example as follows (Hull., 24a): ‘If priests who are not disqualified for service in the Temple by age are yet disqualified by bodily blemishes the Levites who are disqualified by age should certainly be disqualified by bodily blemishes.”
We see here that Jacobs regards all Biblical samples as falling under the category of simple a fortiori argument, and complex a fortiori argument as a later development in Judaic logic – as of Talmudic times. He repeats this conviction further on (p. 5): “The complex qal wa-homer is, as we have noted of late origin and appears to be an halakhic-methodological development from the simple one found in the Bible.” There is some truth in this, insofar as it is difficult to find Biblical examples that are explicit enough to be readily fitted into Jacobs’ complex formula; but it is not strictly correct.
Most Biblical examples leave out the middle term. For instance, in Exodus 6:12, Moses says: “Behold, the children of Israel have not hearkened unto me; then how shall Pharaoh hear me?” This means: if the Israelites, who have much faith, have not had enough of it to listen to Moses, then the chief of the Egyptians, who has far less faith (if any), will not have enough of it to do so. The middle term here is ‘having faith’, or something of the same sort by which we can explain the difference between the behavior of the Israelites and that of Pharaoh. But this term is left tacit, presumably because it is obvious enough.
However, some Biblical examples, even among those listed in Genesis Rabbah (and thus known to Jacobs), do hint at the middle term. For instance, in Ezekiel 15:5, God says: “Behold, when it [the vine-tree] was whole, it was not meet for any work; how much less when the fire hath devoured it and it is burned, shall it then yet be meet for any work?” This means: if when whole the vine-tree was not meet (i.e. in good enough condition) for any work (i.e. to be useful); then now, when severely damaged, it is certainly not meet for any work. Here, note well, the intended middle term (“meet”) is explicitly given.
Clearly, this argument can be cast in Jacobs’ “complex” form, since the major, minor and subsidiary terms are also clearly given, being respectively “the whole vine tree,” “the damaged vine tree,” and “for work.” However, we must change the polarities involved, because this argument happens to be negative rather than positive, and thus goes from the major (B) to the minor (A). The result is as follows: “If the vine tree when whole (B), which has good condition (has y), lacks utility (lacks x), then same when damaged (A), which lacks good condition (lacks y), certainly lacks utility (lacks x).”
It is evident from this example and should be stressed that Jacobs’ formulae for qal vachomer did not take into consideration negative forms of the argument. These would obviously be as follows. The negative simple form would be: “If B lacks x, then A certainly lacks x;” and the negative complex form would be: If B, which has y, lacks x, then A, which lacks y, certainly lacks x.” These arguments can, of course, be derived from the preceding by reductio ad absurdum. The example we have adduced for the complex type clearly fits into the latter form, without any artifice.
We have thus demonstrated, empirically, that Jacobs’ claim that “the complex qal wa-homer is of later Halakhic origin” is not correct. This is not to deny that a fortiori arguments in later Jewish literature, notably in the Mishna and the Gemara, are not very often more complex in form than the Biblical ones. However, the fact remains that Jacobs’ complex type is found in the Bible, as well as his simple type. He should have examined Biblical a fortiori argument more closely before making generalizations. Moreover, no doubt, the simple type is also found in Talmudic contexts.
Clearly, Jacobs is misleading himself and others when he says that the use of simple a fortiori argument “is traced by the Midrash to the Bible.” All the Midrash does is point out ten cases of qal vachomer in the Bible, without referring to any distinction resembling Jacobs’ between simple and complex forms. He makes the same error again when he says that the complex variety is “of later Halakhic origin and is found in a Baraitha,” since that later source too makes no mention of such distinction but refers to qal vachomer alone. Jacobs is unconsciously projecting his interpretation into the facts.
The underlying methodological error here is to confuse use of a form of argument with awareness and discussion of its use. A fortiori argument is used in the Bible (not only ten times, but at least thirty times according to my enumeration in Judaic Logic) and in the much later Talmudic and post-Talmudic literature (umpteen times, yet to be enumerated). The Midrash shows awareness of this kind of argument by naming it and giving examples of its use, and the baraita shows additional awareness of it by listing it as a halakhic hermeneutic principle, although neither of these sources (or any other) theoretically describes or validates it. Such more logical work occurs much later in history.
None of these facts in themselves justify the proposed simple-complex distinction: it is not a traditional doctrine but Jacobs’ interpolation. There is nothing wrong, of course, with making a new distinction – unless of course that distinction happens not to be accurate.
Missing forms. We have also shown that the two argument types Jacobs formulated, being positive, do not suffice to represent Biblical qal vachomer; he should also have formulated two corresponding negative forms (as we just did for him). Another criticism we must level against Jacobs’ alleged formalization of a fortiori argument is that he fails to take into consideration predicatal argument. That is, he ignores arguments in which the minor premise and conclusion have the same subject but different predicates. His forms, “If A has x, then B certainly has x,” and “If A, which lacks y, has x, then B, which has y certainly, has x,” are inherently subjectal in orientation.
To be thorough, he should also have proposed the forms: “If x has B, then x certainly has A,” and “If x has B, which requires y, then x certainly has A, which does not require y,” in which x is the subject, and A, B are the predicates. Note that, here, we must refer to y being required or not-required (rather than to having or lacking y), due to the distinctive structure of predicatal argument. The negative equivalents of these two predicatal forms would therefore be: “If x lacks A, then x certainly lacks B,” and “If x lacks A, which does not require y, then x certainly lacks B, which requires y.”
Note that here, in the positive form of predicatal argument, the movement is from B to A (from major to minor), whereas in positive subjectal argument it is from A to B (from minor to major). Accordingly, in the negative form of predicatal argument, the movement would be from A to B, just as in negative subjectal argument it is from B to A. This is not a mere theoretical prediction. There are arguments of all four kinds (positive and negative subjectal and predicatal) in the Bible, and they must all be taken into account in any theory. As we saw earlier, the Ramchal was fully aware of these four forms back in 1741.
Also lacking in Jacobs’ treatment is a consideration of a crescendo argument, i.e. ‘proportional’ a fortiori argument. His two (subjectal) forms are clearly purely a fortiori, since the predicate in them (x) remains constant. In subjectal a crescendo argument, the predicates in the minor premise and conclusion differ in magnitude (in accord with a ratio determined in an additional premise). In predicatal a crescendo argument, of course, it is the subjects in the minor premise and conclusion that differ in magnitude (again, as per a ratio specified in an additional premise).
There are several examples of a crescendo argument in the Tanakh, and several in the Mishna, and no doubt many more in the Gemara, Midrash and later rabbinic writings. No doubt it was due to misunderstanding of the dayo principle that Jacobs ignored a crescendo argument; the Mishna Baba Qama 2:5 seems to reject such argument. But the truth is that the Gemara Baba Qama 25a clearly accepts it, and Jacobs for some reason did not pay attention to that.
Furthermore, Jacobs’ formulae only refer to copulative a fortiori argument and do not take into consideration implicational a fortiori argument. Some Biblical arguments, and many more arguments in subsequent Talmudic and rabbinic literature, are more precisely implicational in form. We might suppose that he viewed his two types as prototypes for all the missing forms we have enumerated; but there is no hint to that effect in his text: he seems to consider his treatment as exhaustive.
Jacobs’ verbal formula for “complex” a fortiori argument bears comparison to the symbolic formula H. S. Hirschfeld proposed in 1840, viz. “A – a = A + x :: B + a = B + x”. Jacobs may have been inspired by it, or may have produced his own independently; but in any event Jacobs’ formula is much clearer and less ambiguous.
Do not confuse. As we have seen, Jacobs’ simple form of a fortiori argument is essentially identical with his complex one, except that some information found explicitly in the latter is left tacit in the former. To understand Jacobs’ view of a fortiori argument, therefore, we must especially focus on the complex form, “If A, which lacks y, has x, then B, which has y, certainly has x.” It should, for one thing, be seen that this form implies that “A is (always) not-y” and “B is (always) y;” We might well specify ‘always’ (or ‘in all cases’), so as to make clear that these implications are categorical and not conditional.
We have seen that there are Biblical examples of this form (e.g. Ezek. 15:5), and moreover that it is found frequently used in later Judaic literature (e.g. Hull. 24a); and indeed, I might add, in non-Jewish literature. However, this form should not be confused with the following form, which is very superficially similar and also exemplified in the Bible and Talmud: “If z, when it lacks y, has x, then z, when it has y, certainly has x.” The latter form is really a special case of Jacobs’ simple form, viz. “If A has x, then B certainly has x.” For here, A = ‘z when not y’ and B = ‘z when y’; i.e. A and B are compounds, which though different have two elements in common, ‘z’ (the same in both, note well) and ‘y’ (denied in the one and affirmed in the other).
An example of the special simple form would be Deuteronomy 31:27, where Moses says: “Behold, while I am yet alive with you this day, ye have been rebellious against the Lord; how much more after my death [ye will be rebellious]!” This means, in more standard form: if the people (z) during Moses’ lifetime (not-y) are unfaithful (R) enough to rebel (x), then they (z) after his death (y) will be unfaithful (R) enough to rebel (x). Here, the minor (z + not-y), major (z + y) and subsidiary (x) terms are all explicit; but the middle term (R) is not specified and must eventually be added in.
Note well how this form significantly differs from Jacobs’ complex form. In the latter, as we saw, A and B are distinct terms, one of which always lacks y while the other always has y; therefore, not-y and y essentially stand outside A and B, since the latter items can be thought of without referring to the former. In the special simple form we are now considering, on the other hand, the relation of z to not-y or y is conditional or occasional; therefore, A and B cannot be thought of without reference to not-y and y, because the latter are differentia of the former (which, for the rest, share element z).
That is to say, one subject, A, is about an individual z in circumstances when it is not-y, or about instances of z that are not-y; while the other subject, B, is about an individual z in circumstances when it is y, or about instances of z that are y. There is here, obviously, no implication that “A is (always) not-y” and “B is (always) y,” as this would result in a contradiction, since A and B have z in common. So the form here considered is clearly very different from Jacobs’ complex form, even though their wording might seem similar at first sight. Another way to state the difference is to point out that in Jacobs’ complex form, the elements y and not-y point to the middle term, whereas in the special simple form they point (in conjunction with a recurring element z) to the major and minor terms.
It should also be emphasized that Jacobs does not, in his attempt at formalization, explicitly acknowledge the major premise, which is necessary if we are to characterize one term as the major and the other as the minor. Moreover, the major premise has to include a middle term, which tells us in respect of what the major term is ‘more’ and the minor term is ‘less’. Furthermore, the middle term must appear in the minor premise and conclusion, because it is only due to a subject having enough of it that a predicate can be predicated of it. Jacobs does not at all mention the middle term and the crucial roles it plays in the deduction.
Jacobs is certainly to be congratulated for having attempted to formalize a fortiori argument, by using abstract symbols like A, B, x, y, in lieu of concrete terms. But his formalization was certainly deficient. He identified a minor premise (“A has x” or “A, which lacks y, has x”) and a conclusion (“B has x” or “B, which has y, has x”), but he failed to identify the major premise involved. Without a major premise, how would we know which term to call the ‘major’ and which to call the ‘minor’? Obviously, there has to be a proposition at the back of our minds which makes possible that distinction! It is not “self-evident.”
At the outset, Jacobs explains his simple-complex division as follows (p. 3):
“The simple qal wa-homer is a plain argument de minore ad majus, in which the severity of the major over the minor is self evident. The complex qal wa-homer is one in which this severity has to be proved by reference to external factors.”
Jacobs likewise tries, in the Encyclopaedia Judaica article on Hermeneutics that he authored, to explain his distinction by saying that in the simple form of a fortiori argument “the ‘minor’ and ‘major’ are readily apparent,” whereas in the complex form “an extraneous element has to be adduced to indicate which is the ‘minor’ and which the ‘major’.” These explanations are very vague indeed, and he does not manage to make them more precise.
I am not denying the value and validity of these remarks by Jacobs, i.e. that in some cases we can readily see which of the two terms (A and B) refers to the lesser quantity and which to the greater quantity, whereas in other cases we have to make a marked intellectual effort to pinpoint some specific difference (such as lacking y versus having y) before we can clearly discern which term is the lesser and which is the greater quantity. This insight of his can be taken as an often useful ‘rule of thumb’ (a heuristic guideline). But it cannot be understood and justified without reference to an underlying quantity comparison (which is to be brought out in the major premise). Jacobs nowhere explicitly does that. Though he uses the traditional terms ‘major’ and ‘minor’, he does not emphasize their quantitative significance.
In truth, as we have already shown, the difference between Jacobs’ simple and complex forms is merely one of explicitness. That is to say, they are essentially one and the same logical form, but in one case less is said out loud than in the other. Therefore, the complex form – “If A, which lacks y, has x, then B, which has y, certainly has x.” – may be taken as Jacobs principal contribution to the field of a fortiori logic.
Now, this form does give a lot of the information needed to reason a fortiori, but the problem with it is that it does not give all the information needed. If we examine it carefully, we see that it contains the four terms needed to draw the conclusion. But it does not interrelate these four terms with the necessary subtlety, in the major and minor premises and thence in the conclusion.
Jacobs no doubt induced this form by looking at actual cases. There is no doubt that “If A, which lacks y, has x, then B, which has y, certainly has x.” is a recurring form of discourse with a fortiori intent, not only in the Bible and in the Talmud, and in other Jewish literature, but in non-Jewish literature and indeed in current oral expression. However, this is in fact a special case of a wider form, which is also (note well) widely used, namely: “If A, which has less y, has x, then B, which has more y, certainly has x.” The clauses “lacks y” and “has y” are just special cases of the clauses “has less y” and “has more y.”
That A lacks y is indicative of A being the minor term, and that B has y is indicative of B being the major term, is due to the possibility of placing ‘zero y’ and ‘more than zero y’ along the same quantitative continuum. This scale of comparison is what the so far tacit major premise, which says “A is more y than B,” contributes to the argument. Clearly, the term ‘y’ here plays the formal role of middle term. For this reason, we can say that Jacobs’ form does indeed contain the middle term. However, we can also say that Jacobs did not realize that this was the role played by his term ‘y’.
This, incidentally, explains why Jacobs (quite rightly) specified that the minor term A lacks y whereas the major term B has y, rather than specifying that A has y whereas B lacks y. Of course, the latter combination can be contrived by substituting a negative term for a positive one and vice versa. But the underlying meaning has to be that lacking y (or some ‘positive’ equivalent) is in some significant way less than having y (or some ‘negative’ equivalent). For example, lack of intelligence and stupidity are equally ‘less capable of understanding than’ intelligence and lack of stupidity.
Furthermore, Jacobs’ form is deficient in that it does not make clear the other function of the middle term (y), which is to set a threshold as of which a subject (here, the minor and major terms, A and then B) may gain access to a predicate (here, the subsidiary term x). It is because A (which still, though lacking any y) is y enough to be x, that B (which having some y, has more y) is likewise y enough to be x. This is another crucial feature of a fortiori argument, without which the conclusion just cannot logically be drawn from the minor premise.
Thus, a fortiori argument is essentially an argument about magnitudes or degrees. Jacobs’ form does not bring out this quantitative nature of the argument (less y, more y), but makes it seem to concern a contrast of opposite qualities (lacks y, has y). Had he reflected as bit longer, and examined more examples, he might well have realized this clearly. But he stopped too soon in his reflection. Without sufficient training in formal logic, it is difficult for someone to know just what is being sought in the process of formalization.
It is also difficult for someone insufficiently trained in logic to develop a broad, exhaustive theory. Notice, in the above quoted statement by Jacobs, his reference to “argument de minore ad majus.” Having mentioned inference from minor to major, he should have immediately asked himself: “what of inference from major to minor – is there such a thing and what shape would it have?” He might then have at least discussed negative subjectal argument, and maybe even with further effort come upon the also important predicatal forms of the argument. But he nowhere does that.
This is surprising, considering that Talmudic logic (as of precisely when in history, I still do not know) does acknowledge michomer leqal argument as well as miqal lechomer argument, and that there are examples of both types throughout Jewish and non-Jewish literature. Jacobs must have known that, but somehow did not make the connection (at least in this book). Perhaps he was too focused on the general Hebrew term for a fortiori argument, viz. qal vachomer, and forgot the variations on that theme.
It should also be said that Jacobs, through his narrow vision of a fortiori, effectively leaves many arguments encountered in practice without a model to refer to. For instance, some a fortiori arguments are manifestly about degrees of a common property, and cannot readily be made to fit into Jacob’s yes-or-no model. Take for example 1 Kings 8:27, where Solomon prays: “heaven and the heaven of heavens cannot contain thee; how much less the house that I have builded?” Here, the volumes of the heavens and of a mere building on earth are compared. It is not said or implied that one item has volume and the other lacks it – the intent is clearly that both have volume though in different degrees. The argument is that just as the larger volume (of the heavens) cannot contain God, so the smaller volume (of the First Temple) cannot do so.
To conclude: there is no viable substitute for the analysis of a fortiori argument that I proposed in my Judaic Logic (and reproduced and developed in the present work), which by introducing the middle term into the equation demystifies all such inference in a uniform manner. Whether an a fortiori argument looks simple or complex to us, we must in fact always look for the underlying middle term. If we refer as Jacobs does, and indeed almost all traditional commentators do, only to the ‘major’ and ‘minor’ terms, without consciously specifying a common ground relative to which they are compared as greater or lesser (in a range of values that may stretch from minus infinity through zero to plus infinity), we have not fully grasped this form of reasoning and are therefore unable to justify it.
Comparison to syllogism. Jacobs’ comparisons and contrasts between a fortiori argument and syllogism are largely correct, but not entirely. He rightly insists that “there is no connection between the two forms of reasoning” (p. 3) – but at the same time he seems to allow for some equation between them. Consider his following remark (p. 4-5):
“Of the two types of qal wa-homer it is the simple one which has affinities with the Syllogism in that both the simple qal wa-homer and the Syllogism draw their conclusion from a major premise without having recourse to any external factors.”
According to this statement, the simple and complex forms of qal vachomer differ so radically that the former has “affinities” with syllogistic reasoning, whereas the latter does not. It is not clear what affinities he has in mind when he says this. The statement that they “draw their conclusion from a major premise without having recourse to any external factors” is far from clear, not to say confused.
In syllogism, typically, the major premise ‘B is C’ would allow us to draw the conclusion ‘A is C’, given the minor premise ‘A is B’. But Jacobs’ simple a fortiori argument is claimed to be immediate inference from ‘A is x’ to ‘B is x’ – i.e. as a direct insight, somehow, of the conclusion in the single premise. Presumably he regards ‘A is x’ (which I would call the minor premise) as the ‘major premise’ here, since there is no other premise on hand in his model. It is doubtful that Jacobs thought of syllogism as likewise an immediate inference (without the help of the minor premise ‘A is B’) from ‘B is C’ to ‘A is C’, since it is clear from an example he adduces further on that he is well aware that syllogism (as its name implies) involves two premises.
Perhaps the clue is his reference to “external factors.” By this expression he presumably intends the clauses “lacks y” and “has y” which occur in his complex form of a fortiori. It may be that he realizes subconsciously that these clauses provide the middle term needed to construct major premise of complex a fortiori argument. Evidently, he does not realize that his simple a fortiori argument is just as much in need of a major premise, even if the middle term in their case is not suggested. No, the only “affinities” that I can imagine he had in mind is the fact that syllogism and simple a fortiori involve only three terms each (ABC and ABx, respectively) whereas “complex” a fortiori has a fourth term (y/not-y).
His above statement is therefore based on very superficial resemblance, and can be ignored. As we have already shown, Jacobs’ simple and complex forms cannot in fact be logically differentiated, even if they are differently worded. There is certainly no conceivable way to draw the conclusion ‘B has x’ if all the information we are given is that ‘A has x’. Logic admits of no such magical shortcuts; it demands a plausible explanation for any inference, a formal array that can prove the inference.
We can thus move on, and consider his more interesting discussion of the differences between complex qal vachomer and syllogism. He rightly stresses that the former has an element of “how much more so” that is lacking in the latter. He rightly explains the difference as follows (p. 6):
“…in the Syllogism the inference concerns the relationship between genus and species; we are saying that since Socrates belongs to the class man then he must share the characteristics of that class. Whereas in the qal wa-homer inference we do not say that a weighty precept belongs to the class light precepts; it obviously does not. We say that what is true of light precepts is true of weighty precepts.”
However, he does not manage to formally explain why, for instance, “what is true of light precepts” should be “true of weighty precepts” – he does not realize he is appealing to a tacit major premise that quantitatively orders precepts into lighter and weightier, and that this tacit premise (which should also tell us what lighter and weightier mean more precisely, i.e. should reveal the operative middle term) is the missing link between the minor premise and the conclusion.
The probable reason why he failed to see this is hinted at in his attempted formalization of complex a fortiori argument as “If A, which lacks y, has x, then B which has y certainly has x.” Jacobs apparently conceived the argument as a sort of a contrario (or more precisely, obverted inversion), rather than a fortiori, movement of thought, with one term (B) having something (y) that the other (A) lacks.
Somehow, but he does not explain how, the fact that the one (A) who lacks y has x is supposed to convince us that the one (B) who has y also has x. But if we reflect, it is obvious that this cannot always be true. Generally put, any two subjects A and B may have in common any number of things and yet also differ in any number of respects. There is no formal reason why some particular difference (with regard to y, for instance) should imply some particular sameness (e.g. with regard to x). We must be given some precise additional information that allows such inference; the information may well in practice be implicit, but we must be able to make it explicit to justify the argument. In other words, the three propositions “B is y and A is not y and A is x” do not together formally imply the proposition “B is x”: there is no way to validate such a general claim.
In short, Jacobs’ formalization was incomplete. Although he deserves credit for his effort to formalize, and for noticing the feature of difference in properties between the subjects (no mean feat), the fact remains that he did not succeed in disclosing the whole underlying discourse. Had he persevered in his effort of formalization he might have discovered that the difference between having and lacking something is not the essence of the arguments he labeled complex. The essence is, rather, an implicit or explicit quantitative comparison of some sort between those two distinct qualifications – or any others for that matter. Once this is understood, we see that there is no essential difference between so-called simple and complex a fortiori argument – and that a fortiori argument cannot be clarified by such a distinction.
The importance of validation. Many contemporary writers continue to refer to Jacobs’ theory of a fortiori argument as their standard of judgment. Some of these may be excused, since they have not heard of or read my work on the subject. But those who are apparently acquainted with my work (e.g. since they mention it in their bibliography) and yet persist in referring to Jacobs’ theory are clearly not being scientific. I greatly respect Jacobs as a theologian and Talmudist; but in the matter of a fortiori logic he is less than expert. The authority of a writer is not based on how widely known he is, but on rational considerations. The nature of a fortiori argument is not my word against Jacobs’ – it is an objective matter to be decided by reason. There is no ongoing ‘debate’ – so that any commentator can opt for one or the other contestant at will. The matter was settled already over fifteen years ago, as of when I published my Judaic Logic.
Why so? Because I there formally validated my proposed a fortiori forms, whereas Jacobs has only given a couple of examples of his. What is validation? It is to show by formal processes that the given premises do indeed (together, if they are more than one) logically imply the putative conclusion. The given premises and putative conclusion may well have been correctly formalized by looking at a number of examples; but this is only the first step in the logician’s work of validation. The second step is to show that the assumed premises are sufficient to imply the putative conclusion. It may well be that, though the assumed premise(s) is/are among the propositions needed to obtain the conclusion, some additional unstated premise(s) is/are also required to obtain it. Of course, in some cases the proposed premises, or at least parts or aspects of them, are completely useless to the putative conclusion; but what some people forget is that they may be useful but insufficient.
The third step in the formalization and validation procedure is to verify that the form(s) of argument one has postulated do indeed cover the whole field being studied. A theory has to be exhaustive in its applications. It is not enough to explain some practical examples; one has to make sure one has succeeded in explaining every sort of practical example. For instance, if one has only taken note of positive forms, but ignored negative ones, one’s theory is inadequate, its scope being too limited. I can tell you through large experience in logic theorizing that one can never be sure of one’s theory of some argument till one has actually investigated all aspects of the question. A theory of argument is always in flux until it grows to full maturity; an initial insight or two are never enough to guarantee its correctness or completeness.
As I have argued above, although Jacob’s proposed forms, the simple and the complex, may seem reasonable at first sight, they are upon closer scrutiny found wanting in many respects. Jacob’s two forms cannot be validated because, as already shown, their stated premises on their own just do not imply their putative conclusions. Furthermore, although his forms superficially cover some examples, they leave many examples without explanation. That being the case, Jacob’s two forms cannot be upheld by later writers as if they were valid and exhaustive. Such writers are satisfied with a ‘quick fix’ and make little effort to verify the matter. In this way, untruths and partial truths are perpetuated.
I do not say any of this out of disrespect for Jacobs (I have read a number of his books and found myself happily in agreement with most of the things he says) or out of egotistic competitiveness – my sole interest is in the truth of the matter at hand. If Jacobs’ theory (to which my attention was drawn only recently, years after my own theory was completed) was better than mine, I would frankly admit it. I am rich enough in logic discoveries not to be avaricious. I have often made independent discoveries and then found them already made before me – and in such cases have modestly named my predecessors as the true discoverers. Occasionally, a reader finds an error in my work – in such cases I thank him profusely, grateful to have been saved from seeming foolish to other readers. I try to be scientific, in the best sense of the term.
At the time (1961) Jacobs presented his theory of a fortiori argument, it was without doubt a notable contribution and a valuable reference. But the fact is, since publication of my theory in Judaic Logic (1995), for the reasons above given, its significance is only historical and people should not continue to refer to it as if it were the final word on the subject.
Some time after writing the above lines, I discovered the work of Moshe Chaim Luzzatto on a fortiori argument. In his Sepher haHigayon (1741), he succeeds in defining the four moods of a fortiori that I much later labeled as positive and negative subjectal, and positive and negative predicatal. Therefore, if anyone should be mentioned as the first to have clearly formulated a fortiori argument in formal terms it is in truth R. Luzzatto (also known as the Ramchal), rather than myself or Louis Jacobs or any other later writers. Note however that R. Luzzatto did not fully formalize nor attempt to validate a fortiori argument; for a detailed analysis of his contribution, see the earlier section (9.10) devoted to him of the present work. I hope that in the future due regard will be given to this historical finding.
Allow me a few more comments, while on the subject of Louis Jacobs’ Studies in Talmudic Logic and Methodology. This is a very interesting work, but it is not intended as a systematic, let alone exhaustive, analysis of Talmudic logic. Jacobs devotes only four chapters (about a quarter of the book) to the latter subject. In the first chapter, he deals with a fortiori argument (the first of the thirteen hermeneutic rules of R. Ishmael), as we saw above. In the second chapter, he examines the binyan av (the third rule); more on that topic below. In the third chapter, he deals with svara (or sebhara, in his transliteration), which refers to arguments based on the rabbi’s “common sense” or “reason,” in contrast to arguments based more intentionally on Scriptural givens or traditional transmissions. The fourth chapter deals with rabbinical use of “reductio ad absurdum,” which (by the way) here refers not only to arguments leading to a contradiction, but also to arguments leading to a result obviously contrary to experience, or to common knowledge or opinion. That’s it for logic.
The rest of the work is devoted to methodology, by which Jacobs means modern literary and historic analysis of Talmudic sugyot (segments dealing with specific topics) and their arrangements in tractates. Jacobs’ detailed study, in this book and others (notably in his 1991 work, Structure and form in the Babylonian Talmud), of the way the Talmud must have been redacted is fascinating and worth reading. He manages to show, by consideration of the dates when the various rabbis involved were active and by consideration of the progressive presentation of arguments, and occasionally by comparisons to parallel material in the Jerusalem Talmud, how the redactors produced somewhat artificial constructs designed to retransmit information in the most dramatic and memorable way. This is as against the traditional notion that the Talmud is a verbatim transcript of rabbinical discussions in the order they historically occurred.
What is manifest, incidentally, when reading Jacobs’ descriptions of various sugyot, is how frequently and how competently the rabbis use hypothetical logic in very complex knots. There are complicated comparisons of the implications of different theses (e.g. the theories of different rabbis on some issue of law): If thesis A, then such and such are the implications and the implications of the implications; if thesis B, then so and so; if thesis C, then thus and thus. There is ample use of nesting, whose intricacy is sometimes mind-blowing. Apodoses are used to eliminate theses with materially false or rationally absurd implications (reductio ad absurdum), or to confirm theses with true and consistent implications. Disjunctive arguments and dilemmas are also found in ample use, to decide between theses.
We should also mention here Jacobs’ essay “The Talmudic Argument” (1984), where he masterfully identifies nineteen “formal types or patterns” of Talmudic arguments (see also the corresponding footnotes, where he gives the Hebrew or Aramaic phrases that distinguish these arguments), namely:
“Argument from authority; argument by comparison; argument by differentiation; either/or argument; on the contrary argument; acceptance of an argument in part; argument based on an opponent’s position; argument exposing the flaws in an opponent’s argument; argument based on historical or geographical conditions; argument based on the analysis of states of mind; readmission of an argument that has been previously rejected; argument against a statement of the obvious; argument to resolve a contradiction between sources; argument by textual emendation; argument from the principle of literary economy; different versions of an argument; argument presented by different teachers; consequences of different arguments; limited application of an argument.”
All this is significant – because if we want to talk about Judaic logic, we must look not only at the explicit principles of rabbinical hermeneutics, but at the implicit practices of the rabbis. The latter greatly expands the field, and shows rabbinical logic in action to be much broader a field than rabbinic logic in theory. The distinction (stressed in my Judaic Logic) between the art of logic and the awareness and discussion of logic is of course very pertinent in this context.
A digression on binyan av. Returning to logic, a word now on Jacob’s treatment of binyan av inference. I was very pleased to discover that our views on this topic are very close. In his Studies (chapter 2, pp. 9-10), Jacobs writes:
“… a more fruitful way of explaining the principle [of binyan ‘abh] is to compare it with John Stuart Mill’s method of agreement, to which it bears a striking resemblance. It goes without saying that we are not suggesting any kind of anachronistic ‘anticipation’ of Mill by the Rabbis. All we suggest is that the Rabbis, in their attempt to discover general principles behind the laws of the Torah, used, apparently, a method similar in form to that classified by Mill as a means of discovering the laws of nature.”
After which Jacobs goes on to give some Talmudic examples of the inference and tries to formalize it. I independently made a similar comparison to Mill’s method of agreement, many years after him, in my Judaic Logic (chapter 10.2), saying:
“Inferences of the binyan av type (Rule No. 3) seem to be a Rabbinical attempt at causal inference.… Causal inference has been much clarified in more recent times by John Stuart Mill, who identified the ‘methods of agreement and difference’…. However, the Rabbinical attempts at formulation of this natural principle stressed more the side of ‘agreement’ than that of ‘difference’.”
After which, I analyzed binyan av reasoning in some detail, again in ways comparable to Jacobs’. However, certain differences in our approaches must be stressed. What is evident here, as with his treatment of a fortiori argument, is that though Jacobs makes some effort to formalize rabbinic thinking, he does not make sufficient effort to validate his forms. Not really being a formal logician (though he no doubt could have been had he tried to be), he does not realize the importance of validation – not only for the purpose of justification, but even for the purpose of increased accuracy in formalization.
We could say that Jacobs’ comparison of rabbinical binyan av reasoning to Mill’s method of agreement, is effectively putting forward the latter as a justification (albeit ex post facto) of the former; i.e. he uses this analogy as a validation argument of sorts. In his formal descriptions of the rabbinic thought processes, Jacobs’ emphasis is on the positive aspect of causal relations, just like that of the rabbis. The negative aspect of the reasoning is mostly left unsaid, effectively ignored. This explains why Jacobs refers specifically to Mill’s method of agreement and no other.
In my above mentioned study of binyan av, on the other hand, I refer rather to Mill’s methods of agreement and difference. I was well aware, even then, that reasoning about causation requires consideration of both positive and negative aspects. Especially, to fully understand a causal relation, we must clarify not only what the effects are when the putative cause is present, but also what they are (or are not) when it is absent. Rabbinical emphasis in their practice of binyan av reasoning is, to repeat, rather on the positive side (though, to be sure, not always: sometimes they do take pains to ensure the negative side) – and Jacobs unconsciously follows suit in that respect in his more formal treatment.
Consider his proposed formalization of the binyan av argument:
AB results in ‘a’, and AC results in ‘a’;
therefore, A is the cause of ‘a’.
Whence, if A is found in some new context, ‘a’ may logically be applied there.
This argument proceeds by comparing the components of different contexts, such as AB and AC (say), where a certain law ‘a’ is known to apply. Suppose it is found that these contexts have property A in common (and no other), and they differ in that one has B (and lacks C) while the other has C (and lacks B). We can infer from this information that the common property A is the source of the law ‘a’ relative to them. If now some new context (say, AN) is found to also have the property A, we may reasonably apply law ‘a’ to that context too.
This formula is initially convincing, but it is incompletely formulated and therefore difficult to validate as it stands. We may assume from the wording that AB lacks C and AC lacks B, and moreover that there is nothing besides A common to contexts AB and AC (the latter condition is a tall order, to be sure, but let us suppose it). Even so, we cannot prove that A implies ‘a’ (and not-A does not imply ‘a’) from these premises, for A may not be a complete cause of ‘a’ but only a partial cause in conjunction with either B or C (or some other conjunct like N). Therefore, the inference that “A is the cause of a” is inductive rather than deductive. It involves a tacit generalization from the given contexts AB and AC to ‘all contexts involving A’. Thereafter, on this basis, we may infer by simple syllogism that a new context with A (such as AN) is subject to law ‘a’. Thus, although the proposed argument looks superficially deductive, it is in fact essentially inductive.
The same reflections apply to the more complex form of binyan av, which the rabbis routinely use and Jacobs here also formalizes. He describes this process as follows: we start with knowing only that ‘ABD results in a’ and ‘ACD results in a’. Since these two implications have not only one but two factors in common, viz. A and D, we are in a quandary and cannot decide which of these two items is ‘the cause’ of a. We must therefore look for and find a third implication, say that ‘AE results in a’, which has A but lacks D. In that event, we can conclude that ‘A implies a’. In truth, this argument does not formally differ from the preceding, except in the number of contexts in which law ‘a’ is known to apply. We still need to generalize from the given contexts (ABD, ACD and AE) to all contexts involving A, to be able to apply law ‘a’ to any new context with A (such as AN).
However, we must here first realize that item D, contrary to initial appearances (in contexts ABD and ACD), plays no role in the causation, and this is achieved through the discovery of context AE, which lacks D, where law ‘a’ nonetheless applies. Jacobs rightly refers to the process as “the method of elimination,” meaning that having eliminated hypothesis D we are only left with alternative hypothesis A, which may therefore be considered ‘the cause’ of ‘a’. But note that the elimination is a preliminary to the binyan av inference, and not really part of it. If we did not find a context AE devoid of D, we would simply conclude that the cause of ‘a’ is the joint feature AD! This is not mentioned by Jacobs, because he develops the argument in relation to a specific Talmudic example, where a single differentia is sought by the rabbis. Clearly then, there is no significant difference between Jacobs’ simpler and more complex versions.
I should add that whereas Jacobs refers to A as “the cause” of law ‘a’, I would prefer to label A as “a complete cause” of law ‘a’. I would do that due to awareness that there may be more than one complete cause: there may be two or more parallel complete causes, provided they all imply each other and thus behave in the same ways. Also, by specifying complete (i.e. sufficient) causation, I mean to remind us that there are also the formal possibilities of partial causation, and of necessary and contingent causation. Jacobs does not mention these in his formalization, possibly because he has not noticed them occurring in the Talmud, but more probably simply out of unawareness of these alternative forms of causation. I do not affirm it as established fact, but I would be very surprised if we did not indeed find in the Talmud use of these other determinations of causation. Of course, I refer here to de facto use of such forms; I do not expect them to be discussed by the rabbis (though they may well do so somewhere), much less formalized and validated.
Certainly, in view of its frequency in human thought, the sine qua non (without which not) form must occur often. This may not seem like binyan av reasoning at first sight, because instead of looking for a common property (a positive) we would be looking for a common privation (a negative). But apart from the changed polarity of the terms, the form of necessary causation is essentially similar to that of complete causation. Very likely, too, partial and likewise contingent causations occur at least occasionally. Such occurrences may be difficult to find, as the form of reasoning is more complex, since it refers to the composite action of two or more factors which cannot produce the effect separately. Jacobs mentions (pp. 13-15) examples from the Gemara where binyan av inference is attempted (without unanimous rabbinical approval, however) when the cases compared have no apparent common factor! I suggest as a possible solution to the problem they pose that such cases be closely examined with a view to find partial (or alternatively, contingent) causation in them.
To my mind, some of the common factors that the rabbis come up with are very contrived. The alleged common factor may be a somewhat subjective take on the subject, or something quite incidental to the subject. One would think, looking at the formal presentation, that the common factor (A) is an essential element of the contexts compared – but I do not think that is always the case. Rather, I think, the rabbis want to apply law ‘a’ to the new context: if they cannot readily find a reason that naturally fits the bill, they try to make one up, i.e. they look for a pretext however flimsy for the preconceived desired result. It is only when even such last ditch effort fails that they admit there seems to be no common factor; and even then, they might still assume one exists though admittedly they cannot pinpoint it explicitly. My point here is that the common factor, in Talmudic reasoning, need not be a particularly significant property; it may be something otherwise irrelevant to the matter at hand.
Before closing this parenthesis, I would like to briefly discuss the issue as to whether binyan av is induction or deduction. Jacobs begins his chapter on this hermeneutic principle by remarking:
“In the nature of things there can have been little correspondence in the Talmudic literature with modern inductive logic. Schwarz, in his work on the principle of binyan ‘abh, remarks on this and seeks to explain this principle as a form of analogy…”
Jacobs goes on “to use his [Schwarz’s] own terminology, as a type of ‘Species Induction Reference’ or ‘Genus Induction Reference’” – but since I do not know what Schwarz means by these terms, I cannot comment on them. I should add here that, as already pointed out in the chapter devoted to him (14), Schwartz seems to confuse historical and logical questions. He thinks that Talmudic arguments can be referred to Aristotelian logic because the latter preceded them in time, and cannot be referred to modern inductive logic because the latter came later. But this viewpoint absurdly assumes that forms of human thinking are not and cannot be used before they are discovered by logicians! The truth is, it does not matter when in history (and indeed where in geography) syllogism or induction were intellectually discovered – being natural human means of knowledge they were doubtless (though to varying degrees) used long before (and everywhere else), at least as of when (and where) modern man biologically evolved from a less rational species. Logical comparisons are feasible independently of historical or geographical issues.
As I have shown at length in my Judaic Logic, Talmudic thinking is very frequently inductive; i.e. it involves ‘trial and error’. To say so is not to claim that the rabbis were in advance of our time (though they may well have been in some or even many respects). For the practice of logic, whether deductive or inductive, is natural to all human beings since our species appeared on Earth, and considerably independent of theoretical reflection on logic. Theory may and usually does improve practice, but practice can proceed apace without theory. Indeed, the art of logic is ultimately more important than the theorizing on logic, for we cannot arrive at true theories without competent practices. Thus, it is quite consistent to say that the rabbis were practicing induction as well as deduction very ably, without implying that they were in theoretical terms on a level comparable to modern thinking or even to ancient Greek thought.
I would agree with Schwarz that the binyan av is (as Jacobs puts it) “a form of analogy,” inasmuch as the rabbis proceed essentially by arguing as follows (using Jacobs’ above symbols): since all contexts where law ‘a’ is known to be applicable have factor A and only factor A in common, it follows that if a new comparable context where law ‘a’ is not known to be applicable also has factor A, we may reasonably assume that law ‘a’ is likewise applicable to it. This is analogy, insofar as we have copied the law in question from areas where it is given and pasted it in areas where it is not given. Although this process may to some people look like deduction, it is clearly induction since it involves a tacit generalization. If generalization was forbidden, we could not pass information from known cases to unknown cases however much the contexts resembled each other.
Additionally, the premise of the analogical argument is somewhat inductive. It involves an assumption that all relevant cases in the text have indeed been considered and thoroughly analyzed. This depends on human perceptiveness and to some extent on human judgment. Note that in Judaism it is assumed that the rabbis know all the Scriptures by heart, but it is not supposed that they individually know and have fully assimilated all the traditions and judgments handed down by earlier teachers – whence arise discussions and disagreements among them, till a consensus is established. As Jacobs point out, the argument “is not infallible. There is always the possibility that a common factor [besides A] has been overlooked.” Of course, once a rabbinical decision is handed down it is thereafter treated as well-nigh infallible.
So, all things considered, we should rather regard binyan av as inductive, or more precisely as a logical process with both inductive and deductive stages. It should be added that this conclusion of mine is not peculiar to binyan av. As I explain in my Phenomenology, even syllogism, the paradigm of deductive inference, is in fact only partly deductive; i.e. it is partly inductive. When we infer from “All men are mortal, and Caius is a man” that “Caius is mortal,” we can claim the conclusion to be new knowledge because the major premise was based on generalization rather than on an enumeration of all cases including that of Caius. The conclusion is a prediction from the hypothesis of general mortality; we cannot in fact be absolutely sure Caius is mortal till he actually dies; moreover, when he is found empirically dead, his case becomes a further confirmation of the truth of the major premise.
Several months after I wrote most of the above, I discovered that a more recent work by Louis Jacobs, Rabbinic Thought in the Talmud (2005), contains an essay devoted to “The Qal Va-Homer Argument in the Old Testament”. We shall here analyze the contributions made in this late essay of his (Jacobs z”l passed away in 2006). Having by then read many of his works, and developed a true admiration for this scholar, I was very pleased to see more input from him. One thing that saddened me about it, though, was that Jacobs makes no mention in it of my contributions to the same subject in my Judaic Logic (1995), even though it was published about ten years before his essay. I am sure he would have been stimulated by it, had he read that work.
Enumeration. We shall start by comparing the list of Biblical qal vachomer drawn up by Jacobs in his latest essay with the list in my Judaic Logic and later findings. I feel obliged to engage in this accounting, so as to give everyone his due. If we merge the two lists together, we obtain a grand total of at least 46 instances (all presented in Appendix 1). Jacobs’ list is apparently based, largely if not entirely, on an early 19th century work by Wolf Einhorn of Grodno, which I have not seen, but which reportedly contains 40 instances. Jacobs presumably rejected some of the latter, since he only lists a total of 35 instances; but he does not say which instances he rejected or even just why he did so. Nor does Jacobs tell us whether any of the instances he lists are his own findings, or all are included in Einhorn’s list.
We both have
I have, he lacks
He has, I lack
Jacobs and I have 24 instances in common. These of course include the famous ten instances given in Genesis Rabbah 92:7; namely, Genesis 44:8, Exodus 6:12, Numbers 12:14-15, Deuteronomy 31:27, 1 Samuel 23:3, Jeremiah 12:5 (2 cases), Ezekiel 15:5, Proverbs 11:31, and Esther, 9:12. Interestingly, probably because the Genesis 44:8 instance is spelled out first and then R. Ishmael says: “This is one of the ten instances of qal va-homer in the Torah,” Jacobs suggests that only the first of these ten instances was originally in the Midrash, saying: “In what is in all probability an editorial, or even later, gloss, the Midrash gives the other nine” after R. Ishmael’s remark.
Jacobs uncovers another two instances mentioned in the same Midrash but not listed among the ten, namely: Genesis 4:24 (“If Cain shall be avenged sevenfold, truly Lamekh seventy and sevenfold.”), which I already knew of thanks to Rashi; and Genesis 17:20-21, which I did not know about, and so have included under the category of “He has, I lack” further down. It is interesting that the Midrash lists only ten instances of a fortiori argument in a later page, even though the very same volume mentions another two earlier on! Such inconsistency certainly suggests that there was successive editing of the work.
The failure of the Midrash to list its own 12 instances at once is otherwise inexplicable – unless its author used some unspecified selection criteria. Probably, the Gen. 4:24 case was left out as an “evil” case and Gen. 17:20-21 was left out as an “implicit” case; another possible explanation is that there was later addition of these two cases. The question posed here is of course part of a larger one, which I already asked in my Judaic Logic: how is it that the author of the Midrash, who presumably knew the Tanakh by heart and was not half asleep, missed out on the numerous other a fortiori arguments that we have lately found there? This is a mystery. Jacobs acknowledges this mystery, saying: “the commentators to the Midrash and other scholars are puzzled by R. Ishmael’s reference to only ten Scriptural cases.”
Note also in this context that I do not consider Esther 9:12, which the Midrash list includes, as a credible, sufficiently explicit instance of a fortiori argument. This example, which reads: “The Jews have slain and destroyed five hundred men in Shushan the castle, and the ten sons of Haman; what (meh) then have they done in the rest of the king’s provinces!” uses language that is to my knowledge nowhere else connected with a fortiori argument. The interpretation of the word meh as meaning “how much more” therefore seems a bit forced to me. I would at best consider this as an “implicit” a fortiori argument, in the sense of one read into the text (more on such arguments later), since no number greater than 500 is actually specified in the conclusion (which has the form of a question). Nevertheless, because this argument is so universally accepted as a fortiori, just because it is one of the main ten listed in the Midrash, I do exceptionally count it as an explicit a fortiori.
The remaining 13 instances we have in common are: 1 Samuel 14:29-30, 2 Samuel 12:18, 2 Samuel 16:11, 1 Kings 8:27, 2 Kings 10:4, Jonah 4:10-11, Proverbs 15:11, Proverbs 19:7, Proverbs 19:10, Proverbs 21:27, Job 4:18-19, Job 15:15-16, Job 25:5-6. That these instances were found independently by two or more parties is of course no surprise. Anyone looking out for arguments of a certain kind, who has some idea as to how they go about, will notice them as he reads through the Bible. In my case, the research was more systematic. I looked at the wording of known instances of a fortiori discourse, and then sought other Biblical passages with the same wording using a concordance.
As regards Jonah 4:10-11, where God says: “Thou hast had pity on the gourd, for which thou hast not laboured, neither madest it grow, which came up in a night, and perished in a night; and should not I (vaani lo) have pity on Nineveh, that great city, wherein are more than sixscore thousand persons that cannot discern between their right hand and their left hand, and also much cattle?” Although this case lacks distinctive a fortiori language, it clearly has a fortiori intent. I did not have this case in the early editions of my Judaic Logic, but after finding it by chance added it on (as a final footnote to chapter 6) as of June 1998.
Because of my use of a concordance, no doubt, I found numerous cases apparently previously unknown. The 14 instances I have but Jacobs lacks are: 1 Samuel 17:37, 1 Samuel 21:6, 2 Samuel 4:10-11, 2 Kings 5:13, 2 Kings 18:23-24 and its repetition in Isaiah 36:8-9, Ezekiel 14:13-21, Psalms 78:20, Psalms 94:9-10 (3 instances), Daniel 2:9, 2 Chronicles 6:18, 2 Chronicles 32:15. Note that since Jacobs only mentions 35 cases and Einhorn enumerates 40, it may well be for all I know that some of these 14 cases were known to the latter and rejected by the former. But it seems unlikely – why would Jacobs reject any of these cases, which are all pretty clear and explicit?
Note that 1 Samuel 17:37 was publicized and analyzed in Addendum 4 of my Judaic Logic (as of 2001); I did not myself discover it, but had my attention drawn to it by a reader named Mark Leroux (from South Africa). Ezekiel 14:13-21 was not mentioned in my Judaic Logic: I only recently discovered it (in 2012). It may be paraphrased as saying: “More spiritual credit is required to stop more numerous negative Divine decrees than fewer ones; therefore, if holy men, like Noah, Daniel or Job, lack sufficient spiritual credit to prevent the execution of the four separate decrees of the sword, famine, evil beasts, and pestilence, then they lack enough credit to stop all four of these decrees together”. The latter case was not easy to spot, because it is spread out over several verses; what helped me find it was the key phrase “How much more” (af ki, in Heb.) used in it.
As regards 2 Kings 18:23-24 and its word-for-word repetition in Isaiah 36:8-9, they are mentioned in my Judaic Logic, but I had considerable skepticism concerning them and so did not at the time count them as sure cases. However, reviewing the argument involved at a later date, its a fortiori intent became clearer to me. Rab-shakeh (emissary of the king of Assyria) says: “Now therefore, I pray thee, make a wager with my master the king of Assyria, and I will give thee two thousand horses, if thou be able on thy part to set riders upon them. How then (ve-ekh) canst thou turn away the face of one captain, even of the least of my masters servants? and yet thou puttest thy trust on Egypt for chariots and for horsemen!” The Assyrian spokesman thinks that king Hezekiah is hoping for Egyptian chariots and horsemen; so he says to him: ‘even if your force was increased by 2000 horses (which I am willing to give to you), you could not find warriors to ride them and therefore could not hope to defeat the invaders; all the more so, without such additional force, you cannot hope to defeat the invading force, even the least fraction of it’. I do, therefore, henceforth class these two identical cases as surely a fortiori.
Let us now look at the cases Jacobs has but I lack. Since I have never before analyzed these, I will do so now. I will first list the 8 instances I accept as explicit a fortiori argument, and thereafter deal with the instances he mentions that I consider as only at best implicit.
Judges 14:16. “And he (Samson) said unto her (his wife): Behold (hine), I have not told it my father nor my mother, and (ve) shall I tell thee?” This is a clear case of qal vachomer, using keywords (hine/ve) found elsewhere. So much so that I am surprised I missed it! The reason I did so was probably that the word hine is very often used in contexts where there is no a fortiori intent, so I did not closely examine every occurrence of it.
Isaiah 66:1. “The heaven is My throne, and the earth is My footstool; where (eizeh) is the house that ye may build unto Me? And where (eizeh) is the place that may be My resting-place?” This passage obviously echoes the message of 1 Kings 8:27 and 2 Chronicles 6:18, though the wording differs somewhat; viz. that God is too great to be housed in an earthly abode. I perhaps missed it because the Hebrew operator used in it, eizeh (זֶה-אֵי), meaning what? (or which? rather than where? as this JPS translation has it) does not to my knowledge occur in other a fortiori contexts. Nevertheless, it is quite credible as a case of a fortiori discourse.
Jeremiah 25:29. “For, lo (hine), I begin to bring evil on the city whereupon My name is called, and (ve) should ye be utterly unpunished? Ye shall not be unpunished; for I will call for a sword upon all the inhabitants of the earth.” Here again, we have the keywords hine/ve, sometimes indicative of qal vachomer intent. The a fortiori argument is that if God is willing to bring evil on the city whereupon His name is called, he is certainly willing to utterly punish less important kingdoms.
Jeremiah 45:4-5. “Behold (hine), that which I have built will I break down, and that which I have planted I will pluck up; and this in the whole land. And (ve) seekest thou great things for thyself? seek them not; for, behold, I will bring evil upon all flesh.” Here again, we find the keywords hine/ve used. The a fortiori argument is that if God is willing to break down what He has built, etc., he is certainly willing to prevent the success of endeavors by Baruch ben Neriah.
Jeremiah 49:12. “Behold (hine), they to whom it pertained not to drink of the cup shall assuredly drink; and (ve) art thou he that shall altogether go unpunished? thou shalt not go unpunished, but thou shalt surely drink.” Here again, note use of the keywords hine/ve. The a fortiori argument is that if God is willing to punish those who do not deserve it, he is certainly willing to punish those who do. Note the similar form of the three a fortiori arguments of Jeremiah mentioned here – it is indicative of their common authorship.
Ezekiel 33:24. “They that inhabit those waste places in the land of Israel speak, saying: Abraham was one, and he inherited the land; but (ve) we are many; the land is given us for inheritance.” In this case, there is no keyword indicative of a fortiori intent; but that happens. There clearly is an a fortiori intent, even if the argument is logically rather weak. Why should ‘many’ be more assured of inheritance than just ‘one’? Indeed, this is precisely what the next two verses (25-26), which are spoken by God, contend – that it is not quantity but moral quality that determines ownership of that land:
“Ye eat with the blood, and lift up your eyes unto your idols, and shed blood; and (ve) shall ye possess the land? Ye stand upon your sword, ye work abomination, and ye defile every one his neighbour’s wife; and (ve) shall ye possess the land?”
This case is very interesting, because it provides a Biblical example of rebuttal of a weak a fortiori argument by attacking the major premise. Note well that God’s reply is not itself an a fortiori argument, but an objection to such argument. This form of counter-argument is later practiced routinely by the rabbis of the Talmud, under the heading of pirka (in Aramaic) or teshuvah (in Hebrew). No doubt there are many such counter-arguments in the Bible, which we should henceforth lookout for and register. I have not looked for or noticed such rebuttals in the past.
Job 9:13-14. “God will not withdraw His anger; the helpers of Rahab did stoop under Him. How much less (af ki) shall I answer Him, and choose out my arguments with Him?” Job considers himself as less worthy than “the helpers of Rahab,” therefore he is more than them bound to incline before God’s judgment. This is a clear case of qal vachomer, using keywords (af ki) found elsewhere. I am very surprised I did not spot it!
Nehemiah 13:26-27. “Did not Solomon king of Israel sin by these things? yet among many nations was there no king like him, and he was beloved of his God, and God made him king over all Israel; nevertheless even (gam) him did the foreign women cause to sin. Shall we then (ve) hearken unto you to do all this great evil, to break faith with our God in marrying foreign women?” The gist of the argument is: If even Solomon could be caused to sin by foreign women, will not the lesser men of today be likewise caused to sin? This is clearly a fortiori argument, even if the operators (gam, ve) are rarely used.
We have thus drawn eight new, credible and pretty explicit, a fortiori arguments from Jacobs’ (or Einhorn’s) list. Six use known operators: four use hine/ve, one uses af ki, one uses gam/ve; one involves only the ubiquitous conjunction ve; and one involves the previously unheard of operator eizeh.
Jacobs lists in his paper another three Biblical passages that in his opinion involve a fortiori arguments. The first of these, which he has found mentioned in Genesis Rabbah, is:
Genesis 17:20-21. “And as for Ishmael, I have heard thee; behold, I have blessed him, and will make him fruitful, and will multiply him exceedingly; twelve princes shall he beget, and I will make him a great nation. But (ve) My covenant will I establish with Isaac, whom Sarah shall bear unto thee at this set time in the next year.” Jacobs casts this in a fortiori form as follows: “If Ishmael, the son of the handmaiden will be blessed in this way then all the more will Isaac, the son of Sarah, be blessed.”
Although I can see that such an a fortiori argument can certainly be read into the text, I do not agree that it is the only way the passage can be read. God may simply be saying to Abraham: I have blessed Ishmael thus and thus, but My covenant I will not establish with Ishmael but only with Isaac. The emphasis in this alternative reading is clearly different, and not a fortiori. Note moreover, that whereas Jacobs’ a fortiori interpretation makes no mention of the covenant, it is central to my reading. For this reason, I would say that the proposed a fortiori argument qualifies as implicit rather than explicit. This is using the word “implicit” in the sense Jacobs uses it when presenting the next two cases. These instances are mentioned in the so-called Baraita of R. Eliezer b. R. Jose the Galilean:
Psalms 15:4. “He that sweareth to his own hurt, and changeth not.” This verse could be read, as Jacobs has it, as saying that if he (i.e. the good man that the psalmist is describing) doesn’t go back on his word when it is hurtful not to, then he certainly won’t do so when it is for his good. But we could simply read this as saying that when the good man utters an oath, he sticks to it no matter how strong the pressure to break it increases. There is no necessity for the a fortiori interpretation; it is read into the text, rather than drawn from it. There is no call for it, because if the man utters an oath which causes him pleasure rather than pain, he obviously will be under no pressure to break it. This is not a conclusion obtained by a fortiori inference, but something everyone can confirm by introspection. So, really, this a fortiori reading is rather artificial. No doubt, it was concocted simply because its author needed some examples for teaching purposes.
Psalms 15:5. “Nor taketh a bribe concerning (al) the innocent.” Two translations of this verse are possible, the Hebrew word al (meaning: on) being a bit equivocal (even in English).
Let us first consider the translation used by Jacobs: “Nor taketh a bribe to side with (al) the innocent.” This verse can be read, as Jacobs has it, as saying that if he (i.e. the good man) won’t take bribe to rule in favor of an innocent person, then he certainly won’t do so regarding a guilty party. But we could simply read this as saying that the good man would not take a bribe even if he was being bribed to judge a matter as he would without being bribed, i.e. in favor of the innocent. It is true that, in this case (in contrast to the previous one), the a fortiori argument just indicated can additionally be constructed, and it makes a valuable prediction. So here we are justified in referring to an “implicit” a fortiori argument. It is not explicit, because the text can be read in a first phase without an a fortiori thought. But it is implicit, in that, if we dig deeper into it, we can indeed use it to make a useful a fortiori inference.
Let us now consider the alternative translation of the same verse given in the JPS 1917 edition: “Nor taketh a bribe against (al) the innocent.” We can interpret this translation like the previous one, albeit with an interpolation: if he (i.e. the good man) won’t take a bribe not to rule against an innocent person, then he certainly won’t do so regarding a guilty party; and we can say more about it as before. However, a quite different, more literal approach to this translation is also possible: we could simply read it as saying that the good man would not take a bribe against an innocent person, i.e. in favor of a guilty one. We might now attempt the following a fortiori argument: if he (i.e. the good man) won’t take a bribe to rule against an innocent person, then he certainly won’t do so regarding a guilty party. But this argument is a non sequitur, since someone might well refuse to rule against the innocent for a bribe, but accept to rule against the guilty for a bribe, thinking that since he intended to rule against the guilty anyway, no harm is done by taking a bribe for it. Therefore, in this translation and reading there is no a fortiori adjunct, whether explicit or implicit.
Thus, if we qualify as “explicitly” and “implicitly” a fortiori argument, respectively, “a text that can only be read as a fortiori” and “a text that can be read as a fortiori but can also readily be read otherwise (i.e. more simply)” – we would have to say that Gen. 17:20-21 is implicit, that Ps. 15:4 is not a fortiori at all, and that Ps. 15:5 is implicit if read one way and not a fortiori at all if read another way. This is contrary to Jacobs, or rather to the rabbinic sources he refers to, who read these arguments as respectively explicit, implicit and implicit. As for the two examples of explicit a fortiori, which Jacobs mentions as given in the said Baraita, namely Jeremiah 12:5 and Esther 9:12 – we would for our part agree that the Jeremiah instances are explicit, but insist (for reasons already put forward) that the Esther example is (if at all a fortiori) at best implicit.
Clearly, we have here a serious divergence of views. I have to say that I have in the past, until I read Jacobs’ present article, assumed that the distinction made by Eliezer ben Jose, between a fortiori arguments that are meforash (explicit) and those that are satum (implicit), was referring to how much of the argument’s elements are laid out in the text at hand. If the a fortiori premises and conclusions, with all their terms or theses, are all fully laid out in the given text – then that text is a fully explicit a fortiori argument. If one premise or the conclusion are missing, or some of the terms or theses involved are missing – then that text is partly implicit to varying degrees. By that standard, of course, most if not all arguments in Scripture are partly implicit.
But Jacobs’ article suggests that the rabbis’ “explicit” means sufficiently explicit that there can be no interpretation other than an a fortiori one, while their “implicit” means not so explicit that there can be no interpretation other than an a fortiori one. At least, this is how I now understand these expressions. It could be that the rabbis do not draw the lines so clearly, and understand them sometimes this way, sometimes that way. In any case, to conclude this discussion, the three arguments above listed are – as far as I am concerned, in the light of the above analyses – not to be listed among the explicit a fortiori arguments. They are possible interpretations of the texts, or artificially read into the texts, but the texts in themselves allow of readings that are not a fortiori.
Why is this issue important? Because it relates to attribution and dating. When a Biblical text clearly has an a fortiori intent, we may regard it as an explicit instance of Biblical a fortiori argument. If, however, the a fortiori intent of the Biblical text is not so obvious, and has only been brought out later in time by a rabbinical or other commentator, we must count it as only implicitly a fortiori, and attribute the a fortiori argument as such to the historically later commentator. It is not an issue of who discovers the a fortiori argument, note well, but of whether or not the author of that passage of the Bible worded it with a manifest a fortiori intent. If the a fortiori argument has later been read into the text, rather than found in it, then its author is really the person who proposed the interpretation. This is commonsense hermeneutics.
Of course, the rabbis consider that whatever they read into a Biblical text was indeed intended by that text, since God – its ultimate author – is all-knowing. But, even granting their premises, their conclusion does not follow. That is to say, God may well have foreseen the rabbinical interpretation, but that does not make it any the less an interpretation. Such foreknowledge is not indicative of an actual or direct intent, but only at best of a potential or indirect one. The explicit text constitutes the primary message; other information that can eventually be derived from that message is not strictly part of it, but at best an implicit adjunct to it. Sometimes, of course, it is highly debatable that the original text allows for a certain interpretation; and in such case, the interpretation must be characterized as forced rather than implicit. It would be irrational to accept unquestioningly whatever the rabbis claim; they are, after all, just human beings.
Jacobs additionally mentions (in an endnote) three Biblical passages presented as a fortiori arguments by Chaim Hirschensohn. Jacobs rejects these examples as “extremely doubtful,” and I incline to agree with him. To my mind, they are at best implicit a fortiori arguments, but certainly not explicit ones. The first two texts in question are the following:
Genesis 3:22. “Behold (hen), the man is become as one of us, to know good and evil; and now (ve-atah), lest he put forth his hand, and take also of the tree of life, and eat, and live for ever. Therefore (ve), [He] sent him forth from the garden of Eden, to till the ground from whence he was taken.”
Genesis 11:6-7. “Behold (hen), they are one people, and they have all one language; and this is what they begin to do; and now (ve-atah) nothing will be withholden from them, which they purpose to do. Come (habah), let us go down, and there confound their language, that they may not understand one another’s speech.”
It is interesting that the two verses, though chapters apart, use the same language (hen/ve-atah) and have similar form. Also note that in both cases, one of the operators used (hen) is sometimes indicative of a fortiori discourse, and the argument is concerned with increasing magnitudes of something. The argument involved can be paraphrased as follows: This event is bad enough, therefore to avoid an even worse event we had better take certain precautions. This is an interesting form of reasoning in itself, but it is clearly causal and ethical, rather than a fortiori as Hirschensohn reportedly claims.
We could admittedly formulate an a fortiori argument from it as follows: P is worse (R) than Q, and Q is bad (R) enough to be combated (S); therefore, P is bad (R) enough to be combated (S). But where in the text does it say that Q was fought against? It only says that P is to be fought against. So this a fortiori argument, if at all implied, must be characterized as implicit. It is anyway not the essence of the explicit discourse facing us, which has it that a minor problem (the “bad enough” clause) could well eventually develop into a major problem (the “even worse” clause), and for that reason some preemptive action against the latter is called for before it happens. The said a fortiori argument is perhaps implied by the text, but the text evidently tries to communicate considerably more than just that.
The third text is: Genesis 17:17. “Then Abraham fell upon his face, and laughed, and said in his heart: Shall a child be born unto him that is a hundred years old? and shall Sarah, that is ninety years old, bear?” It is difficult to perceive the a fortiori argument Hirschensohn had in mind here. Perhaps his thought was that it is unlikely enough for a hundred year old man to have a child, and therefore even more unlikely for a ninety year old woman to do so. But frankly, was that Abraham’s thought? No, he was simply saying that it is unlikely for both a hundred year old man and a ninety year old woman to have a child. It is a statement, not a process of inference.
Certainly, in this case, the literal reading is not a fortiori, so that if an a fortiori argument be read into the text, it is at best implicit. But moreover, the a fortiori reading seems rather forced. Since we can fully understand the text without it, it serves no purpose other than to inflate the list of Biblical a fortiori arguments. Therefore, I would not even include this case as an example of implicit a fortiori. Thus, while the first two examples could conceivably qualify as involving an a fortiori discourse implicitly, the third is much less credible. In any event, none of these three cases is explicit.
General observations. Let us now take a closer look at various general observations in Jacobs’ essay. To begin with, it is evident that even in 2005 he had not yet grasped the actual form of a fortiori argument, since he here still describes it very superficially as “If A is so then B must surely be so; if the ‘minor’ has this or that property, then the ‘major’ must undoubtedly have it.” This is what he has in the past called ‘simple’ a fortiori argument. He still, note also, fails to detect the use of what he has called ‘complex’ a fortiori argument in the Bible. He rightly remarks that the Rabbis learnt this form of inference from its occurrences in the Bible itself, and then used it “as one of their hermeneutical principles by means of which they expand and elaborate on the Biblical teachings.” And this fact stimulates his present research into the actual examples of the argument in the Bible.
However, it is very surprising to see Jacobs assert (in an endnote) that “There does not appear to be, in fact, any real parallel to the qal va-homer in Greek thought.” This is, as demonstrated in the present volume, quite off the mark – a fortiori argument is quite present, and consciously so, in Greek (and then Roman) literature, even if not as frequently as in rabbinical literature. He is here going further than he has in the past, where he only (and rightly) contended, against the apparent claims of Adolf Schwarz, that the identification of this form of argument with Aristotelian syllogism is “untenable.” That a fortiori argument is not syllogistic does not imply that it was not used by the Greeks! The latter did not only think syllogistically, any more than the rabbis only thought by means of a fortiori argument. Moreover, to admit that the Greeks used a fortiori argument is of course not the same as to claim that the rabbis learnt it from them. So Jacobs’ position in this matter, even if expressed offhandedly, is very surprising.
It is a pity that Jacobs did not push his analysis of the a fortiori arguments he lists more deeply. While he acknowledges the rabbis’ debt to the Biblical occurrences of qal vachomer, he does not sufficiently examine how they actually interpreted those arguments. Notably, while he reads the qal vachomer in Numbers 12:14 correctly, saying: “if when [Miriam’s] human father showed his disapproval of her actions she would hide herself in shame for seven days then when the Lord shows His disapproval all the more should she be shut away for seven days,” he does not look further into the matter and discover the significantly different interpretation given by the Gemara in Baba Qama 25a, and the Pandora’s Box of interesting problems (and opportunities) that the latter creates.
Nevertheless, Jacobs makes some valuable general observations:
“From all that has been said it is surely well established that the argument from the minor to the major is used frequently throughout the Old Testament. Its use is not limited to any single phase in Israel’s history but, it would appear, was employed in all periods. Neither is the usage confined to any single book of the Old Testament nor to any particular document, stratum, and trend… Moreover, as in many of the examples quoted, its use is generally of a formal nature, beginning with hien or hinneh and concluding with ‘eykh or ‘aph.”
I came to similar conclusion in my Judaic Logic. But Jacobs takes the reflection further, raising “important questions, hitherto barely considered by Old Testament scholarship, regarding the use of rhetoric in ancient Israel.” He cites O. Eissfeldt, who suggested that there were men and women “specially skilled in speech,” using argumentative techniques acquired through “tradition and ‘training’,” resorting to “certain fixed forms for speech” and rhetorical devices such as “first obtaining from the person addressed an admission which does not appear to be relevant to the matter in hand,” which “then compels him to grant the request which is really involved.” Jacobs concurs, in view of the evidence provided by his listing of Biblical qal vachomer.
Such tradition and training is of course evident in the rabbinic period, Jacobs adds, when “formal argument was consciously and extensively cultivated” and “there are certain stereotyped rules” of argumentation. He then asks: “Was there anything like this in the Old Testament period?” and replies: “there seems to be no doubt that the answer should be in the affirmative.” He admits, however, that “it is hard to find anything like an explicit reference anywhere in the Old Testament to schools in which rhetoric was taught.” He could have buttressed his case by adducing that the rabbis did and do believe that such schools (yeshivot) existed throughout the past. The patriarch Jacob is said to have studied in the tents of Shem and Eber (Genesis Rabbah 63:10); king David is said to have studied with his counselor Ahithophel (Pirqe Avot 6:2); and so on.
In my view, to be honest, such claims are largely anachronistic, projecting later mores onto earlier times. It is not inconceivable that there were, very early on, educational institutions of sorts that organized common study of and reflection on knowledge inherited from the past. The issue is, as of when such institutions can be credibly claimed, and what it is that was studied in them. While transmission of knowledge and skills by village adults and elders to children and youth can be classified under the heading of education and is as old as mankind, it is less certain as of when in history the formal study of Torah and related argumentative skills began in Israel. I would say it developed apace in the period after the Return from Babylon after the First Exile, i.e. the formative period of the rabbinical doctrine and class. This is suggested, for instance, in the Mishnaic Pirqe Avot, which refers (1:1) to the Knesset Hagedolah (the Great Assembly).
This hypothesis seems most likely, in view of what was happening at the same time in other nations near and far. What is evident when we study world history is that cultural developments tend to be (increasingly over time) worldwide rather than local. Many major developments occurred as of the middle of the first millennium BCE, as if a new phase in human intellectual evolution was taking place. Suddenly, it seems, existing civilizations burst with newfound energy, producing religious and philosophical thoughts more sophisticated (at least on the surface) than ever before. Though scattered, they awoke simultaneously, in various directions, but also somewhat comparably.
In India, the ancient Vedic religion began its transformation into Hinduism, and Buddhism was founded. In China, Confucianism and Taoism emerged. In Greece, philosophy flowered in earnest, with the advent of Socrates, Plato, Aristotle, and many others. In Israel, Judaism came increasingly under the authority of Torah scholars. This ‘rabbinic’ Judaism was apparently planted at the beginning of the Second Temple period and gradually grew and took shape in the following generations, till it fully flowered in the Mishna (which then stimulated the Gemara and subsequent rabbinic works). The intellectual growth in Israel, involving increasingly legalistic thinking, and therefore to some extent logical reflection, was thus rather typical of that epoch, and can only with difficulty be projected backwards into earlier ones.
Be that as it may, Jacobs ends his reflections with an interesting suggestion, also drawn from Eissfeldt, that the Hebrew root dbr, used to refer to ‘speech’, may in fact often be used with the intention to mean ‘argument’. An example he gives is Gen. 44:18, where Judah begins his plea before Joseph by saying: “O my lord, let thy servant, I pray thee, speak a word in my lord’s ears.” Jacobs comments: “Since the expression yedabber dabhar is used, should it be translated as ‘present an argument’?” Similarly, in other passages, ‘speak rightly’ might be taken to mean ‘argue convincingly’, ‘these are the words’ might be taken to mean ‘these are the arguments’, and so forth. This insight seems credible to me. All this goes to show in what directions and how far Jacobs’ research into Biblical use of qal vachomer drove his reflections.
 England, 1920-2006.
 See his book: Beyond Reasonable Doubt (London: Littman, 1999).
 London: Vallentine Mitchell, 1961. (Republished as paperback, 2006.)
 The present essay was written in 2010-13. I must have read Jacobs’ Hermeneutics article in the Encyclopaedia Judaica, which contains the same two formulas, when I was writing my Judaic Logic in the early 1990’s. But at that time I was searching for a formalization of a fortiori argument convincing to me, rather than intent on evaluating other authors’ theories; so I did not mention them there.
 The material in brackets is here added by me, so as to clarify the relation of the example to the proposed form.
 This is further confirmed in a 1972 paper of Jacobs’, viz. “The Qal Va-Ḥomer Argument in the Old Testament” (Bulletin of the School of Oriental and African Studies, 35:221-227. Cambridge University Press), where he, presumably speaking of all Biblical a fortiori, says: “The argument runs: if A is so then B must surely be so; if the ‘minor’ has this or that property then the ‘major’ must undoubtedly have it.”
 This is not, note well, something arbitrary. We must ask what tacitly differentiates the children of Israel and Pharaoh. The difference between them is obviously that the Israelites are spiritually close to Moses, whereas Pharaoh is not – so we would expect Pharaoh to ignore Moses’ advice more readily than the Israelites do. Precisely how we word this difference is not important; but we must verbalize it.
 Presumably he is referring to the Baraita of R. Ishmael given at the beginning of Sifra, which lists the thirteen rules of hermeneutics of R. Ishmael mostly derived from the seven rules of Hillel.
 I was pleased to learn from Jacobs in his footnote 4 (p. 3) that “The commentaries state that far more than ten are to be found.” He there refers us to H. Hirschensohn’s Berure HaMiddoth and H.L. Strack’s ‘Introduction to the Talmud and Midrash’. More will be said on this issue in a later section of the present chapter (16.4).
 See Appendix 1 for a full list.
 In Halachische Exegese (Berlin, 1840), p. 227 (cited by Samely, p. 177). See the section devoted to Hirschfeld in a later chapter (30.1) for discussion of this formula.
 Another example would be Prov. 21:27, where Solomon says: “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]?” That is: if the sacrifice of the wicked (z) brought with a ‘sincere’ intent (not-y) is abominable (R) enough to be rejected (x), then the sacrifice of the wicked (z) brought with a wicked intent (y) is abominable (R) enough to be rejected (x). Here, the middle term (R) is explicit, while the subsidiary term (x) must be added on.
 Actually, in my opinion, it would be more accurate to characterize many svara as propositions or premises, rather than as arguments. For instance, “one who is in pain visits the house of the doctor” is (to R. Ashi) a reasonable proposition. Such a svara is used within an argument, but it is not itself an argument. But as Jacobs makes clear, the term svara is also often taken in the larger sense of human reasoning without direct Scriptural authority such as qal vachomer or binyan av inferences.
 Thus, only two of the 13 hermeneutic principles of R. Ishmael are dealt with. As for svara and reductio ad absurdum, although not among the 13 explicit principles, they are rabbinical practices and so implicit principles, so it is legitimate and desirable to analyze them. But here again, these are only two of many logical practices unlisted by R. Ishmael or others. Clearly, Jacobs made no attempt to be exhaustive here.
 Some of this book may be read online at: books.google.ch/books?id=Q7oXP0dPelgC&printsec=frontcover&dq=louis+jacobs&hl=en&sa=X&ei=F8KMUbGbOceI4AS1jYHIDw&redir_esc=y.
 It is easy enough to follow a Talmudic discussion under the guidance of a modern master like Louis Jacobs. The experience is very different if one tries to do so alone or even under the guidance of purely traditional guides, for the convictions and ways of thinking of the ancient rabbis are very different from ours today, and their speech is telegraphic and dense and often obscure. They unfortunately apparently never discovered or used flowcharts!
 In: Essential Papers on the Talmud. Ed. Michael Chernick. New York: NYU Press, 1994. Pp. 429-460.
 I should mention that a few years later, after developing my own theory of causation in my book The Logic of Causation, I analyzed Mill’s methods more carefully and became much more critical of his work. See the Appendix to Phase I of that work; actually, though first written in 2003, it was largely rewritten in 2005, and so really belongs to Phase II.
 I have substituted plain English for his relational symbols. He uses a long arrow for ‘results in’ and a long dash for ‘is the cause of’ – but he does clearly at the outset define his symbols in these words. I would not use a causal phrase like ‘results in’ in the premises (unless it is specified in the source text), because the neutral phrase ‘occurs in conjunction with’ suffices (if the argument is valid) to establish the causal conclusion (‘is the cause of’). The whole point of causal argument is to construct a causal proposition from non-causal ones!
 In my Judaic Logic, I formulate the essentially same argument in different terms. Jacob’s AB, AC, A and ‘a’ are the same as my X, Y, A and B, respectively. Thus, our A symbols correspond, but his ‘a’ symbol is symbol B in my book. The new context, which Jacobs does not name (but which I here call AN), I there call Z.
 Jacobs does show his awareness of this negative condition in his presentation of a material example and when he cites Mill’s method of agreement and stresses: “Mill is careful to write ‘have only one circumstance in common’” – but he (Jacobs) unfortunately does not explicitly integrate this knowledge into his forms.
 It should be noted that there is a paradoxical aspect to binyan av. The source text contains at least three contexts with A (viz. AB, AC and AN) of which two occur in conjunction with law ‘a’ (viz. AB and AC) and one (viz. AN) does not do so. Taking all three of these contexts into consideration, we cannot logically conclude that A is the cause of ‘a’, since we have evidence that A is sometimes not textually associated with ‘a’! The only way we can arrive at the desired conclusion is to momentarily turn a blind eye to the context AN (which lacks ‘a’) and base our conclusion about causation of ‘a’ by A on generalization from the remaining contexts (AB and AC); thereafter we return to AN and apply the conclusion to it. Thus, our reasoning is based on the (debatable) premise that ‘not textually associated’ is not the same as ‘textually dissociated’.
 I refer you at this point to my book The Logic of Causation (chapter 2), where these various concepts are fully defined and discussed. Note that even Mill was not entirely clear about these different concepts, so there is no reason to expect Jacobs to have adequately dealt with them.
 The binyan av argument formalized by Jacobs (from two or more contexts) is the main one used by the rabbis. But, as I point out in my Judaic Logic (chapter 10.2), there is another version of binyan av (from only one context), which some rabbis advocate and use. This may be expressed as “if a context with feature A prescribes law ‘a’, then another context with feature A may also be assumed to prescribe law ‘a’ (or, more precisely, may be used to justify such a ruling), and is called chada mechada (Aramaic, meaning ‘one from one’). It should be clear that this reasoning is far more tenuous, since (a) it does not establish that there is no common factor other than A between the two contexts concerned, and (b) even if A were indeed their only common factor, A is not thereby proven to be the reason why law ‘a’ is given in the first context, and so we cannot be sure that law ‘a’ applies in the second context. There may be some other factor, say B, which is found in the first context but not in the second, and which is the true reason for law ‘a’. Indeed, the chada mechada reasoning can be shown fallacious by pointing out that if it is used with regard to B in relation to some third context, which has B but not A, we would be saying that the law ‘a’ has two distinct reasons. That is, we would be applying the same law to contexts A+B, A without B, and B without A. It is not unthinkable that a law might be given with such various applications – but the claim of inference from case AB to the other two cases is very dubious. I suggest that rabbis use such weak reasoning as a last resort, when they have difficulty constructing a proper binyan av, but still want to justify a certain foregone conclusion. It occurs to me that such inference might best be classified as gezerah shavah (hermeneutic rule #2 of R. Ishmael), rather than as binyan av.
 The rabbis sometimes use frequency of occurrence as a predicate. For instance in B.Q. 2b: “‘Falling and Kicking’; are not these derivatives of Foot? — No; the damage of foot occurs frequently while the damage of these does not occur frequently.” Sometimes, the differentiation is made with reference to subjective perceptions (i.e. how people commonly view the matter or imagine it). These are clearly incidental properties, which would not be regarded as significant in scientific discourse.
 See chapter 7.4 there.
 London: Vallentine Mitchell, 2005. See chapter 12, pp. 109-116.
 Namely, 1 Samuel 17:37, the case given in Addendum 4, which was pointed out to me orally by Mark Leroux (from South Africa, a colleague at an office where I worked) in 2001. Before that (in 1998), I found a further case, namely: Jonah 4:10-11, by happenstance. More recently (in Aug. 2012), I found yet another case, Ezekiel 14:13-21, by means of a search for key phrases at www.mechon-mamre.org/p/pt/pt0.htm. Incidentally, the latter search only yielded a total of 19 cases: 13 cases with ‘how much more and 6 cases with ‘how much less’; there were no cases with the key phrases ‘all the more/less’ and ‘(how/so) much the more/less’.
 Zeev Wolf Einhorn (Maharzav), Grodno, Lithuania, 1813- 1862. Sefer Midrash Tannaim, 1838.
 Jacobs adds that “other commentators [have] come up with similar results” – but he does not say which commentators, nor what these results were nor compare them.
 He does tell us that some of the instances proposed by various researchers “must be rejected as far-fetched and dubious,” but he unfortunately does not perform his triage in public, and all too confidently declares that his list “contains all the definite references.”
 Goltzberg, in his 2010 essay “The A Fortiori Argument In The Talmud” (which I review further on), mentions “the forgotten a fortiori arguments,” without however saying how many he thinks there are or listing them. Apparently, he draws this information from Moshe Koppel’s Meta-Halakha. Logic, Intuition And The Unfolding Of Jewish Law (Northvale, NJ, Jason Aronson, 1987). Not having seen the latter work, I cannot say if it is any more informative than that. Note that Jacobs does not mention Koppel’s book, which is earlier than mine, either.
 It is interesting that R. Ishmael does not here rather mention Num. 12:14-15, which plays such important role in the Gemara explication of Mishna Baba Qama 2:5.
 But only, as it turns out as an implicit case; not as an explicit case.
 Jacob’s comment is based on a note by Theodor-Albeck, the editor of the Genesis Rabbah edition that he refers to. In that edition, 92:7 is on pp. 1145-6, and 4:24 and 17:20-21 are on p. 225. Jacobs also informs us (in endnotes, p. 116) that Yalkut, 1 Sam. 132 “refers to ten but lists only nine” (he does not say which one is left out); and that Gen. 4:24 is also mentioned as a qal vachomer in Avot de-Rabbi Nathan (version B) 44 and in JT Sanh. 10:1 (27d).
 Wiseman claims “evil” qal vachomer to be a rabbinical category, pp. 174-6. As regards the “implicit” case, see further on.
 In this context Jacobs mentions (in an endnote) A. Schwarz, Ch. Hirschensohn, Jofe Ashkenazi (in Yephe toar) and H. Strack.
 This quotation, as indeed all those from the Bible in the present section, is taken from the Mechon Mamre website at www.mechon-mamre.org/p/pt/pt0.htm, which is based on the 1917 edition of the Jewish Publication Society. Note that the word “then” is an interpolation by the translator; it is not found in the original Hebrew. Likewise, the exclamation mark is an addition; a question mark may have been more appropriate. Obviously, the translator was influenced by the tradition that this statement is a fortiori.
 As I explain in my Judaic Logic (6:3), although the given text of Daniel 2:9 is not directly a fortiori argument, since it consists of an order by the king coupled with a reflection as to its utility, the reasoning used by the king is indubitably a fortiori argument, so much so that it can be counted as effectively explicitly so.
 This is a repetition of 1 Kings 8:27, which Jacobs does have.
 The original wording of this argument is given in Appendix 1.
 Such copy-and-paste repetition is surely useful for purposes of “higher criticism.”
 In chapter 6, where I write: “Note also: 2 Kings 18:23-24, and its repetition in Isaiah 36:8-9, might at first glance be construed as a-fortiori in style. But try as I might, I have not been able to make a clear a-fortiori argument out of it, however artificial and logically improbable.”
 Although, to repeat, I am still counting the Esther example as explicit, so as not to go against this too well-established tradition.
 An important case in point that we have seen is the Gemara (Baba Qama 25a) interpretation of Numbers 12:14-15.
 Israel, 1857 – 1935. The work cited is Berure Ha-Middot (Jerusalem, 1929), pp. 40-45.
 In The Old Testament: An Introduction (Oxford, 1965), pp. 12-15.
 Perhaps starting with “Also we made ordinances for us,” in Nehemiah 10:33. The Hebrew word used is mitsvot, which is usually translated as commandments. The laws gradually enacted by Jewish lawmakers were, it is worth noting, distinctively based on Torah law. They were not arbitrary, but guided and circumscribed by the strong moral standards already instituted by that document. The ‘legalism’ involved here is of a very different sort than that found in the same period of history in, say, China.