1.Problems encountered

We stated earlier that, according to Genesis Rabbah, there are ten cases of a fortiori argument in the Bible: four of them in the Books of Moses and the other six in various other locations. This Midrashic work is traditionally said to have been compiled either by Rabbi Oshia Rabba (a late Tana) or by Rabba bar Nachmani (a third generation Amora); in any case, circa 3rd century CE[1].

We have already in earlier chapters analyzed in considerable detail the four cases of a fortiori spotted in the Chumash by this Midrash, namely: Gen. 44:8, Exod. 6:12, Num. 12:14, and Deut. 31:27. The other six cases mentioned by it are: 1-Sam. 23:3, Jer. 12:5 (2 cases), Ezek. 15:5, Prov. 11:31, and Esth. 9:12. Presumably, this is intended to be a full enumeration; i.e. it is not just a list of ten cases among others, but an exhaustive list.

At first, I took this authoritative tradition that there are just these 10 qal vachomer arguments in the Bible for granted. But I must admit that over time, to my surprise (not to say, consternation, for I do not want to excite the ire of my religion’s orthodoxy), I have been forced to revise that article of faith considerably. Closer scrutiny of the evidence makes indubitably clear that there are more likely at least about 30 (thirty) cases in the Bible, and furthermore that one of the cases listed by the Midrash is open to doubt as a genuine case.

My first inkling that something was amiss was the quite fortuitous discovery of an a fortiori argument in Job 4:17-19, while leafing through Maimonides’ Guide[2]. I naturally assumed that the list given in the Encyclopaedia Judaica[3], which was my initial source, was erroneous by accident (this is not as far-fetched as it may sound: I once spotted a confusion between 2nd and 3rd figure hypothetical syllogism in the 1967 Encyclopedia of Philosophy[4]); and that the two cases counted under Jeremiah were really one, while the said argument in Job was perhaps merely omitted by the printers. I resolved to look into the original source, and confirm this assumption (I of course did look into G.R. eventually, but found the E.J. list correct).

Meanwhile, having had my consciousness of the issue of logical arguments in the Bible raised by my preceding research, I happened on a Shabbat, while studying the “haftarah of the week” (Tazriaa), to notice yet another unmentioned case, namely 2-Kings 5:13. Again, my immediate reaction was defensive, conservative; I did not want to belie the tradition. I had early on in my formal researches looked with askance on the argument in Esther (we will return to this detail further on); so I thought, well, if we ignore this doubtful case, we still have a total of only ten a fortiori arguments.

At about that time, as I described to people some of the difficulties I was coming across in my Biblical research, someone mentioned that there may be a case of qal vachomer in Daniel; but I could not find it offhand (as we shall see, I did find a probable case eventually).

Also, leafing through an ArtScrolls commentary on Genesis, I noted to my relief their comment that ‘some editions’ of the Midrash include Gen. 4:24 instead of Ezek. 15:5 in the list of ten. The Rashi commentary on this alternative sample, I then found (see Soncino Chumash[5]), is clearly formulated as a qal vachomer. (Assumably, then, Rashi favoured the special editions of the Midrash, since in his commentary to Gen. 44:8 he does not dispute the claim that there are only ten qal vachomer cases ‘in the Torah’ [in the larger sense of the term, meaning Tanakh] - this is said in passing).

Thus, in fact, in practice, at least eleven sentences in the Bible are recognized as a fortiori by Rabbinical authorities taken collectively, and not just ten (though some say these ten and some say those ten, and they all agree on nine cases). How they reconcile this with the Midrash claim, which they apparently all continue to uphold undaunted, is beyond me: a contradiction is a contradiction. I do not know whether any among them have noted and acknowledged yet other cases of a fortiori in the Tanakh, and if so how they dealt with the issues implied; but the issues are implied even with a joint list of just eleven cases. The simplest solution, it seems to me, would be to regard the Midrash claim as not intended as exhaustive; then there is no problem of doubting the Midrash’s infallibility.

2.The solution found

I tell this story in detail to demonstrate my goodwill, my reluctance to contradict authorities (but also my determination to find the factual truth). By now, it had become obvious that the common tradition on this topic was surely factually inaccurate, and that a systematic reevaluation was called for. But, how, other than by rereading the whole Tanakh carefully with this issue in mind? It was at this point that I had a very felicitous insight...

The a fortiori arguments in the Tanakh are noticeably not signaled by expressions like “kol sheken” or “qal vachomer”!

These expressions are utilized in Talmudic (Mishnah and Gemara) and post-Talmudic (Rabbinic) arguments and exposés, but not so far as I know in the Bible itself. If we actually look at the 10 cases mentioned by the (usual) Midrash, we find exclusively the following language:

I saw almost at once that these various phraseologies might be viewed as signals of an intention to formulate an a fortiori argument. After a while, I realized that these sentences have, indeed were bound to have, conditional form, with an antecedent clause (a minor premise), signaled by an “if” operator (one of the particles hen/hine, ki, ve/u), and a consequent clause (a conclusion), signaled by a “then” operator (one of the expressions ve ekh, ve af ki, af ki, halo, and eventually meh). These key words or phrases were limited in number, some half a dozen, and so could with relative ease be used in a search for other cases, if any, in a Concordance of the Bible (which is effectively a word index). Of course, there might be other significant expressions, besides those, but I left the question open; at least, this was a starting-point.

The following stage was painstaking research: each reference to a keyword in the concordance was looked up in the Bible, to see whether or not it signaled an a fortiori argument. In truth, I did not research all the keywords: I looked up all occurrences of ekh, ve-ekh, af, af-ki, ve-af, ve-af-ki, hen, ve-hen, halo, va-halo; but I did not have the patience to also research the words hine, ki. It was quickly evident that not all occurrences of the keywords signaled a qal vachomer (only about 6 percent did so); on the other hand, I found by this method many new cases of the argument, i.e. cases not mentioned in the Midrash (about twenty). In all, I looked up some 500 references in the Bible; by that time my point was proven, since I had about three times the number of a fortiori arguments I started with, and it did not seem important to pursue the matter further and attempt to be exhaustive.

As already said, I was not immediately conscious of the logical role played by the key words/phrases. At first, my approach was pragmatically philological; but once I grasped that what I had to look for were if/then operators, it became obvious that a more detailed linguistic analysis was called for: this laborious research is presented in the next chapter. In this context, I gradually understood the following (which ex post facto perhaps seems obvious, but was not immediately evident). Whereas in modern Hebrew, im/az are the closest and most commonly used equivalents of if/then, in Biblical Hebrew the language is more varied:

a.         There are various alternative expressions for “if”, such as hen/hine, reu, ki, ve/u, im, be; all these announce an antecedent: behold, see, if, when, because, in, etc. The prefix vav (and) fulfills this function, like the other words, by presenting a context, in which certain later mentioned events occur.

b.         There are various alternative expressions for “then”, such as af, ve, ki, im; all these announce a consequent: all the more/less, therefore, then, so, etc. The word af, often translated as ‘all the more/less’ (its distinctively a fortiori reading), more broadly means ‘also, similarly’. The word ki, which in modern Hebrew usually has the limited meaning of ‘because’, has evidently in Biblical Hebrew a broader range of meaning, including even ‘then’. The use of vav (and) in the sense of ‘then’ is also found in English (e.g. “Press the button and the motor starts”), and therefore needs no explanation.

c.         Antecedents and consequents need not in Biblical Hebrew, anymore than in the modern idiom or in English or French, be signaled by any “if” or “then” operators; they may be tacitly understood by the context, or be left out to avoid repetitions. (Nowadays, we often use a comma to signal a tacit “then” in written texts.) Grammatically, logical operators are merely ‘conjunctions’, they serve to bring sentences together in various ways.

d.         Although initially expressions like hen, af-ki, ve-af-ki, ve-ekh, halo made it possible for me to discover a fortiori arguments, I eventually realized that they were not or not-wholly in fact logically essential factors in these arguments. Ekh (how) and halo (is it not that) are never then-operator of arguments, but always an integral part of the consequent/conclusion in which they appear, serving as rhetorical devices: how will you do this? meaning, you cannot do it; is it not that so and so? meaning, it is so and so. As for af-ki, ve-af-ki, although the af particle serves as then-operator of arguments, the ve and ki may have a role either as if-operator of the argument, or as if or then operator of its premise and/or conclusion.

In this context, I would like to refer the reader to Esra Shereshevsky’s very interesting analysis of Rashi’s interpretative techniques, where some of the fine nuances in the meaning of words like ve and ki are discussed[6].

Apart from that, please note that my use of the operators if/then is here very loose, generic (and not exclusively logical); I do not here push the analysis on down to deeper levels, to distinguish between the different modal types of conditioning: the logical (if), the natural/temporal (when, at such times as), and the extensional (in such instances as). The if/then operators of any logical argument are of course of logical modality, but the conditional premises and conclusions (if any) they enclose may be of other modal types.

3.The data and their analysis

The table below lists the results of these researches, my own proposed list of Biblical a fortiori arguments. I repeat, it is not necessarily exhaustive; and it should be added, some of the arguments are strong, unassailable, some are comparatively weak, open to rebuttal, but I think they are all reasonably clear samples of the form. Opposite each Biblical reference I indicate the apparent if/then logical operators (if any), and parenthetically any of the typical a fortiori expressions hen, hine, lahen, af-ki, ve-af-ki, ve-ekh, halo, which helped me personally find the case in addition to the operators themselves.

Table 4.1         Proposed list of Biblical A fortiori

We see that there are at least 31 cases of a fortiori in the Tanakh, 5 of them in four books of the Torah proper, and 26 more in eleven other books (counting Samuel and Kings as two each). Some of these arguments are repetitive, and perhaps should not be counted as distinct. For instance, 1-Kings 8:27 and 2-Chron. 6:18 are definitely one and the same argument, reported in two different books. The three arguments in Job might be counted as one and the same thought, in spite of small verbal variations; and similarly the two in Jeremiah. The two arguments in Ps. 94:9 have the same major premise, and might be viewed as a compound. On the other hand, Ps. 78:20 might be viewed as two arguments with the same premises but separate conclusions, instead of a single argument with a compound conclusion. Thus, the total number may be as small as 26, or as large as 32, depending on how we count. In any event, the above table may be summarized as follows:

Table 4.2         Frequencies of A fortiori Operators

We note that, broadly speaking, the individual key words/phrases, and more significantly their combinations, seem to be fairly evenly distributed throughout the Bible: the language is on the whole pretty uniform. Some books, such as Leviticus, Joshua, Judges, and others, have no a fortiori arguments to my knowledge; but I see no reason why they should, nor what might be inferred from the fact (perhaps somebody else might eventually). If we pay attention to the traditional dating of the reported speakers in each of the above arguments, we find the following results:

Table 4.3         A fortiori Arguments: By Whom, How Often, When

We see in the above table that apart from 4 of the arguments attributed to God, 21 (68%) of them are spoken by Jews and 6 (19%) by non-Jews. Thus, judging from Biblical sources alone, this form of reasoning seems to be rather predominantly Jewish, though not unknown to non-Jews. I do not intend this remark as racist, but merely wish to arouse interest in historical studies of logic. It would be interesting to know whether a fortiori arguments appear, say, in Sumerian, Babylonian, Egyptian, Canaanite, Assyrian, or Greek epigraphs or documents; and if so, as of when and how often.

Furthermore, out of 31 cases, only 2 are pre-Sinaitic; 9 (29%) are from Mose’s time, meaning about 13th century BCE; 14 (45%) are from the monarchies of Saul, David and Solomon, roughly mid-9th/mid-8th century BCE; and the remaining 6 (19%) are from the period from the splitting of the kingdom to the Babylonian Exile, roughly mid-8th/mid-4th century BCE.

In the course of this research, it occurred to me that the language used in the Bible for a fortiori arguments (and eventually for other types of reasoning) might serve as a dating tool, to resolve issues between Traditionalists and “Higher-Critics” with regard to the ages and authorship of the various books of the Bible. However, looking at the above results, I personally see no firm conclusions possible in this respect (even if the dating proposed by the Critical school is considered in lieu of the traditional).

The only overall conclusion I can suggest is that a fortiori argument was a rather common form of reasoning since early on in the Biblical narrative, and on up to its end, with the greatest frequency occurring in the 9th-8th centuries BCE. Perhaps, after all, the valuable conclusion to draw is that the hypothesis of some of the critics that most of the earlier books of the Bible were composed, or at least compiled, much later than tradition claims, i.e. at about the same time as most of the later books, is if not eliminated at least not justified by this data, since if it were true one might expect more, or as, frequent use of the a fortiori argument in the later books compared to the earlier books. But even this is barely probabilistic and open to debate, of course.

4.Synthesis of Results

Now, let us return to the discussion regarding the number of a fortiori arguments in the Bible. First, let me mention in passing that I doubt seriously that Esth. 9:12 qualifies as a genuine qal vachomer argument; I demonstrate this at length in the next chapter already mentioned. I may add here that although Genesis Rabbah purports to embody the undebatable tradition and final truth on the matter, its apparent error in enumerating only 10 qal vachomer arguments in the Bible, when there are evidently at least some three times that number, allows us to evaluate its statements much more critically, and doubt that this 10th statement really qualifies as a qal vachomer.

I say ‘apparent’ error, because one might always put forward the defense that the ten statements chosen by the Midrash were in fact in some hidden way special, having something the others lack. Indeed, a Rabbi of my acquaintance, R. Alexander Safran of Geneva, upon being told by me of the discovery of qal vachomer arguments other than the Midrashic ten, offered precisely this defense.

Now, it must be stressed that there is evidently no formal or linguistic distinction possible: that is evident from all our discoveries and insights and cannot be contested. Therefore, as always in such situations, the defenders of the faith must fall back onto homiletic or mystical interpretations, and claim these ten statements as having some special ethical, historical, or qabalistic import that the others lack. I leave that job to whoever.

A more intriguing defense was suggested to me by a friend, Sammy Soussan, who studies in a kollel (Talmudic study group) in Aix-les-Bains. He asked me to verify whether the Midrash’s ten qal vachomer arguments might not simply be samples of ten distinct formal types, whose typology and no other would be merely repeated in the other twenty or so cases I found. My immediate response was that such a view was unlikely to be true, because my formal studies have revealed that the number of distinct forms is (according to how counted) two, four, or eight, but not ten (nor five).

As we saw earlier, an a fortiori may be positive or negative, subjectal or predicatal (if categorical) or antecedental or consequental (if conditional). With regard to the ten (or eleven) Midrashic a fortiori, they have the following logical forms (most naturally, though they can be recast into other forms): 2 are positive subjectal, 3 are negative subjectal, 2 are positive predicatal, and 2, 3, or 4 are positive antecedental; more specifically:

Gen. 4:24 is negative subjectal;

Gen. 44:8 is positive predicatal;

Exod. 6:12 is negative subjectal;

Num. 12:14 is positive subjectal;

Deut. 31:27 is positive predicatal;

1-Sam. 23:3 is positive antecedental;

Jer. 12:5 has two positive antecedentals;

Ezek. 15:5 is negative subjectal;

Prov. 11:31 is positive subjectal;

Esth. 9:12 is positive antecedental (if at all a fortiori).

It is interesting to note anyway that Gen. 4:24 and Ezek. 15:5 are both negative subjectal in form, because if (a) only one or the other Midrashic list of qal vachomer arguments is to be adopted, but not a fusion of both, though both must be accepted as equally valid, and (b) the Soussan hypothesis turned out to be correct, then these two a fortiori arguments would have to be of the same form, which they are. Nevertheless, the hypothesis is incorrect, because its main prediction, namely that the Midrashic list of ten includes ten (or five) distinct forms, cannot be upheld.

None of these cases, read simply, are negative predicatal, negative antecedental, or either way consequental, in form; therefore, if at best the Midrash may be said to hint at the formalities of a fortiori, it does not represent them all. Furthermore, it can be shown on a case-by-case basis that all the Biblical a fortiori, recognized as such in the present study, fall neatly into our classification; i.e. that as far as the data at hand is concerned, this classification is exhaustive. This reasoning would seem to preclude the proposed defense: we can predict with confidence that the Midrash is not a taxonomy.

Alternatively, we might consider the possibility that the Midrash list of ten qal vachomer arguments reveals ten types of phraseology. There are various aspects to this linguistic question: we may focus on individual operators or on their combinations or on key words/phrases or on their combinations in turn. Also, we may ask whether the Midrashic list amounts to precisely ten such expressions, and we may ask whether that number is (in view of new discoveries) exhaustive.

Firstly, we must admit that the Midrashic list does not cover all the individual operators or combinations thereof found in Biblical a fortiori. With regard to if-operators, it includes hen, hine, ki, u, but ignores reu, im, lahen; with regard to then-operators, it includes ve, af, meh, and ignores gam, im. These oversights are somewhat open to debate: the sentences concerned could be constructed or understood without interpreting these words as operators; but in any case the total number is not ten (it is 7 in the Midrash list, and 11 in mine).

With regard to combinations of operators, while the list spots ki/ve, u/ve, hen/ve, hen/af, hine/af, -/meh, -/-, it misses the most frequent combination ki/af, as well as hine/ve, lahen/ve, reu/af, hen/gam,im, im/-; and in any case, again, the total number is not ten (but 6-7 in the Midrash, and 12-13 in my view).

As for the number of individual key words/phrases presented by the Midrash, it is also nine; hen, hine, ki, u/ve, ve-ekh, halo, ve-af-ki, af-ki, and meh, however we organize our list. Unless, that is, we regard the u signaling the antecedent of the second part of Jer. 12:5, and the ve which flags the consequent of Gen. 4:24, as two distinct terms, which they are in meaning (u=if, ve=then) though not in spelling (vav). In that case, and retaining Esth. 9:12, we obtain the desired number of ten distinct key words/phrases in the Midrash. However, the Midrash is not exhaustive in this respect; since, in a larger perspective, 4-5 expressions are missing here, namely: reu, im (as “if” or as “then”), gam, lahen.

With regard to key words/phrases in combination, since two of the cases the Midrash lists use the same language (hen/ve-ekh in Gen. 44:8 and Exod. 6:12), there are only nine combinations, even if we like Rashi include Gen. 4:24 (ki/ve) in the list instead of Ezek. 15:5 (since its hine/af-ki is then excluded). However, if we both count Jer. 12:5 as one qal vachomer instead of two, but one which reveals two phraseologies, and include Gen. 4:24 in the list without excluding Ezek 15:5, and of course (contrary to my recommendation) do not leave out Esth. 9:12, we obtain the desired number of ten distinct combinations of key expressions. But here again, this number is not exhaustive, ignoring as it does combinations like reu/af-ki, ki/ve-af-ki, and so on.

To sum up: to its credit, the Midrash list reveals crucial expressions like ve-ekh, halo, ve-af-ki, etc., which signal qal vachomer arguments (though not invariably). It includes ten (or eleven) Biblical samples (I say 9-10) of qal vachomer; and these samples can be acknowledged to display ten key expressions and ten combinations thereof. However, the Midrash listing of 10 cases is certainly incomplete, whether regarded statistically, logically or linguistically.

Thus, we have found no scientific justification of the Midrashic listing of only ten qal vachomer arguments. It must be viewed as intended, in the said respects, to be at best a partial and random set of examples. If the author of the list intended it to be complete or systematic with reference to the number of samples or to logical formalities or to language forms, he failed: his research was sloppy. The only possible way out of these conclusions is, following the Safran hypothesis, to presume that the author had homiletic or mystical motives for his selection.

Drawn from Judaic Logic (1995), chapter 5:1-4, and addendum 4.