A FORTIORI LOGIC
Appendix 3 – A fortiori discourse in the two Talmuds
There is a great need for someone to go through all Talmudic literature, and in particular the two Talmuds, looking for all sorts of reasoning in it, and more specifically for applications of the rabbinic hermeneutic principles, and in particular for instances of a fortiori argument. This is a massive job, of course, which ideally ought to be carried out in relation to the original texts, in Hebrew and Aramaic. Many people need to get involved in this project, which is really worthwhile. We can never hope to fully and correctly understand and evaluate Talmudic logic without such thorough empirical research. I have no intention to do this important work, for the simple reason that I do not have the linguistic knowledge needed for it. But I here try and do a small part of it, specifically in relation to a fortiori argument and in English translation.
The Jerusalem Talmud (JT) was the first to have been closed, ca. 400 CE. I have almost no personal experience with this Talmud, but judging from what I have read about it, it is shorter and less disputative, and so we may expect it to contain relatively fewer a fortiori arguments. As Neusner wrote:
“The Yerushalmi speaks about the Mishnah in essentially a single voice, about fundamentally few things…. [It] takes up a program of inquiry that is not very complex or diverse. The Yerushalmi also utilizes a single, rather limited repertoire of exegetical initiatives and rhetorical choices, etc.” (Rabbinic Literature: An Essential Guide, p. 41.)
The Babylonian Talmud (BT), closed ca. 600 CE, is the document that will require the most research work. We can expect hundreds of a fortiori arguments in it, to be listed and eventually analyzed. Blau reports that Schwarz estimates the statistics for the second hermeneutic rule, that of gezerah shavah, as follows:
“…in the Babylonian Talmud alone, there has to be close to four hundred, in the Talmud Yerushalmi about one hundred and fifty and in the Tosefta thirty. If one adds to that the gezerah shavah in halakhic works and other sources, there would be, after deduction of the numerous parallel passages, a total of six hundred” (p. 156).
I do not know if Schwarz made similar estimates for the first hermeneutic rule, qal vachomer, in his book devoted to that subject. But it is a fair guess offhand that the statistics are in the same order of magnitude. Needless to say, we are not counting a fortiori arguments in order to discover who has argued a fortiori the most often; there is no competition to be won! The purpose of our counting them is to accurately determine the number of cases we have to eventually list and analyze.
In any study of a fortiori argument in the Talmud, we must of course distinguish the different sources of cases found in it. The Talmud may be quoting a passage from the Torah (the Five Books of Moses) or the Nakh (the rest of the Jewish Bible), or from the Mishna (redacted ca. 220 CE), or from the Tosefta (compiled at about the same date or soon after), or a baraita (a statement the Gemara claims as Tannaic, not included in the Mishna, though it may be in the Tosefta), or lastly the Gemara (which in turn has many layers). I will not do this here, but obviously it has to be done if we want to obtain an accurate picture of a fortiori use.
As regards English translations of the Talmuds, we have, I think only five sets to choose from. The three most recent are The Schottenstein Edition of the Talmud Bavli as well as The Schottenstein Edition of the Talmud Yerushalmi (New York: ArtScroll, various dates), The Steinsaltz Edition of the Talmud (New York: Random House, 1989-99), and The Talmud of Babylonia. An American Translation (Atlanta: Scholars Press for Brown Judaic Studies, 1984-95). To my knowledge, these editions are not available in a form that allows computer search, although the Steinsaltz edition is at least partly posted in the author’s website.
An older translation is The Soncino Edition of the Talmud (London, 36 volumes, 1935-1952), edited by R. Isidore Epstein (1894–1962). This is freely available online, thanks to Halakhah.com (a Chabad project), in 63 pdf files. This resource is potentially very useful, provided we take the trouble to merge all these files into one document so as to avoid repetitive work; the single file would of course need to be purged of all editorial content, such as introductory material and footnotes.
Still older is The Rodkinson Edition of the Babylonian Talmud (1903). This edition is freely available online thanks to the Internet Sacred Text Archive, in Kindle format. I managed to convert this file into a Word file, from which I removed all extraneous material (i.e. all Rodkinson’s introductions, synopses, footnotes, etc.). This edition contains all of the tractates in the Orders (Sedarim) of Moed (Appointed Seasons: 12 tractates) and Nezikin (Damages: 10 tractates). Thus, four entire Orders are missing in it, namely: Zeraim (Seeds: 11 tractates), Nashim (Women: 7 tractates), Kodashim (Holy Things: 11 tractates), Tohoroth (Cleannesses: 12 tractates). Clearly, the Rodkinson edition does not comprise the whole Talmud, so that any information gathered from it is likely to be incomplete.
But my purpose here is to launch a pilot study, to show the way we may obtain the desired information and statistics. I would have preferred to do this pilot study in relation to the Soncino edition, which is not only more complete but also more generally respected; but I decided to focus on the Rodkinson edition to save time and effort. This should suffice to show the way, even if the results obtained will not be as thorough and reliable. Anyway, even if Rodkinson’s translations are not universally approved, this handicap hardly affects our study because it specifically focuses on a fortiori argument.
Pilot study. Ultimately, we need to actually list all passages of the Talmud that seem to have a fortiori intent, and see whether they can indeed be cast in standard form (whether valid or invalid). This can only be done exhaustively by going through the whole Talmud page by page, which I do not propose to do here. Instead, I propose to search for a number of key phrases which are usually, or even just often, indicative of a fortiori discourse. This is why I needed a single file, purged of all commentary. We cannot find key phrases and count instances in the Rodkinson edition by means of an Index, because it does not have one.
I did in the past, when I wrote Judaic Logic (1995), look into the Index Volume of the Soncino edition (1952), and there found 137 entries apparently indicative of a fortiori argument, which I tabulated as follows:
Soncino BT index entries
A minori ad majus
Deduction, proofs by
Inference from minor to major
Major, inference from minor to
Minor, inference from major to
Total count of a fortiori references
Research by means of search strings is bound to give us a more accurate picture of a fortiori use. The problem with it, of course, is that it allows for overlaps. For example, we might count twice the single argument “Aqiba then drew an a fortiori conclusion. He said: ‘If the soft has so much power over the hard as to bore it (water over stone), how much more power will the Torah, the words of which are as hard as iron, have over my heart, which is flesh and blood?’” – once for the phrase “a fortiori” and once for “how much more.” Such overlaps can only be eliminated at a later stage, when each argument is listed and examined closely. For the time being, we shall ignore this difficulty and aim for a rough estimate.
Incidentally, it is important to keep in mind when searching for such arguments that the relevant quantitative indicator has to be in the putative conclusion – not in a premise. In the above example, for instance, the relevant indicator is not the antecedent “so much” (which merely refers to an unspecified, impressive quantity), but the consequent “how much more” (which serves to signal a fortiori argument). Thus, an expression (such as the “so much” used here) might be counted as indicative of a fortiori argument and yet in fact not be so – because, though it would be indicative were it in the conclusion, it is not in the conclusion.
The first step in our research is to think of key phrases to search for. The expressions possibly indicative of a fortiori discourse are of two kinds. The first groups includes idiomatic markers like ‘all the more’, ‘how much the less’, ‘so much more’, and so on. The second groups descriptive markers such as ‘a fortiori’, ‘from minor to major’, ‘inference’, ‘argue’, ‘logical’ – to name just a few.
With regard to the first kind, we need to decide the order in which our search will proceed, so as to avoid unnecessary repetition. For that purpose, I have developed the hierarchical arrangement shown in the following diagram. The a fortiori phrases are there abbreviated, using the first letters of the words constituting them; for example, ‘smms’ means ‘so much more so’. Note that for every expression with ‘more’, there is a similar expression with ‘less’. The root of all these expressions is the top one, the comparative ‘more’ (or ‘less’, as the case may be); from this we derive ‘much more’, ‘how much more’, ‘so much more’, and also ‘still more’ and ‘even more’, and more specific verbal forms. A similar flowchart may be constructed starting with the subsidiary root ‘the more’, from which we derive ‘much the more’, ‘how much the more’, ‘so much the more’, and also ‘still more’ and ‘all the more’, and less generic verbal forms.
Diagram A3.1 Hierarchy of a fortiori expressions
In the above diagram, the most specific expressions (e.g. smms) are on the right; the more generic (e.g. smm) are in the middle, and the most vague (e.g. sm+) are on the left (examples of the latter are ‘so much greater’ or ‘so much worse’). From this we see that the best way to search through a given document, to ensure a minimum of misses or overlaps, would be in the following order. First, we should look for derivatives of ‘the more’, starting with the most specific ones and ending with the most generic ones; second, we should look for derivatives of ‘more’, starting with the most specific ones and ending with the most generic ones. The full orderly list and the results obtained are given in the following table:
A fortiori wording
A fortiori wording
so much the more so
so much more so
so much the less so
so much less so
so much the more (residue)
so much more (residue)
so much the less (residue)
so much less (residue)
so much the (residue)
so much (residue)
how much the more so
how much more so
how much the less so
how much less so
how much the more (residue)
how much more (residue)
how much the less (residue)
how much less (residue)
how much the (residue)
how much (residue)
much the more so (residue)
much more so (residue)
much the less so (residue)
much less so (residue)
much the more (residue)
much more (residue)
much the less (residue)
much less (residue)
much the (residue)
as much as
still the more so
as little as
still the less so
still the more (residue)
still more so
still the less (residue)
still less so
still the same
still more (residue)
still the (residue)
still less (residue)
all the more so
all the less so
even more so
all the more (residue)
even less so
all the less (residue)
even more (residue)
all the same
even less (residue)
all the (residue)
none the less / nonetheless
more so (residue)
less so (residue)
the more so (residue)
the less so (residue)
the more (residue)
the less (residue)
Please note well that this is almost raw data, yet to be fully processed by detailed analysis case by case. However, I have here made a small effort to narrow the field. As regards idioms that are almost sure to signal a fortiori discourse, I looked at most cases briefly, in an offhand manner, and eliminated obvious ‘duds’, by which I mean letter strings that accidentally resemble a fortiori ones (for example: in ‘a children-teacher who struck too much the children’, the string ‘much the’ is obviously not intended as an a fortiori marker). As a general rule, in cases of doubt I counted possible cases as actual cases, without taking the trouble to closely examine the data further.
As regards the search strings labeled ‘residue’, my policy was to discount all cases but the most likely to be a fortiori discourse. The statistics for such more generic words or phrases exclude the counts for more specific phrases derived from them: whence the label ‘residue’. For example, the count for ‘so much the’ excludes the counts for ‘so much the more’ and ‘so much the less’, which in turn exclude the counts for ‘so much the more so’ and ‘so much the less so’. This allows us to see more precisely the a fortiori wording used, and also facilitates dealing with the vaguest residues. To give an example: the count for ‘so much the more’ (152) excludes the more specific cases of ‘so much the more so’ (10), and the more generic string ‘so much the’ (1 instance) excludes the cases of ‘so much the more’ and of ‘so much the less’.
Similarly with other word strings. Obviously, the count of the residue of ‘all the’ must exclude cases falling under ‘all the same’, as well as those under ‘all the more’ and ‘all the less’. The reason why phrases with ‘the more’ (first column) must be counted before those with just ‘more’ (second column) is that the residue of ‘much’ excludes all cases with ‘much the’; and likewise, ‘still’ must exclude ‘still the’, and ‘more/less’ must exclude ‘the more/less’.
The more vague the search string, the more instances it in fact includes; but the method of residues here used allows us to narrow the field somewhat. In the case of ‘so much the’, only one instance (“so much the firmer”) was leftover, and this happened to indeed be a fortiori. In the case of the residue ‘so much’, only 6 instances out of 75 qualified at first glance as a fortiori (namely, those worded “in a so much larger degree” or “in so much greater a degree”). For ‘how much’, out 80 remaining instances only 1 qualified (worded “how much severer”). For ‘much’, out of 234 instances only 4 qualified (worded “much better” or “a much greater”). For the residue ‘the more’, out of 83 instances only 3 turned out to be apparently a fortiori (“I enjoyed myself the more because I fulfilled two religious duties,” “the more should it be allowed…,” “it applies the more to…”).
Note that all admitted cases involve a comparison (e.g. firmer, larger, greater, severer, better, more enjoyable). In many cases, no potentially a fortiori instances were found (at least in my offhand reading of them). Thus, to illustrate: none of the 99 instances of ‘no less’ or ‘not less’ qualified; likewise, none of the residual 33 instances of ‘still the’ and none of the 1010 instances of ‘all the’ qualified (I looked). In certain residual cases, I did not take the trouble to look at the individual instances at all, expecting negligible results (i.e. close to zero). Thus, for ‘nevertheless’ (498 instances) ‘still’ (777), ‘even’ (3642), ‘more’ (1159) and ‘less’ (1949) – I put a question mark, and counted the results as zero. I should explain that the work of individual verification, even done as roughly as I did it, is extremely time consuming.
It is worth remarking that the ‘so’ of phrases ending in ‘more/less so’ obviously refers to a previously given predication. Phrases with ‘the more/less’ are intended as more emphatic than those with just ‘more/less’; likewise, ‘so much’ is more emphatic than just ‘much’; but these emphases are rhetorical: the logical weight is the same. Similarly, ‘how much’ is a rhetorical question and therefore less emphatic than ‘so much’; but their logical weight is the same. Looking at the above list of commonly used expressions, it occurs to me that, from a purely logical point of view, we could equally well use milder forms, like ‘a bit more’, ‘quite a bit more’, ‘a little more’, ‘somewhat more’, or even ‘some more’ – for it is clear that the amount of ‘more’ is irrelevant here. Our habit is to signal a fortiori intent by means of hyperbole (e.g. ‘all the more’), but we could equally do so by understatement. However, looking for such milder expressions in Rodkinson’s Talmud, I found no cases. Maybe some occur in the Soncino Talmud.
Notice that I add in the above table a number of search strings not included in the preceding diagram, namely: ‘still the same’ (1 out of 2 instances), ‘all the same’ (3/4), ‘none the less’ (18/18), ‘nevertheless’ (?/498), ‘as much/little as’ (2/114 and 0/0), and ‘no/not more/less’ (3/117 and 0/99). These are all expressions which may be (though evidently often are not) indicative of a pari a fortiori argument (i.e. forms with an egalitarian major premise, from which we can equally well reason from minor to major or from major to minor). In any event, when we include these expressions in our listing, we realize that there is a continuity in the wording, ranging from ‘all the more’, through ‘much the more’ and ‘much the less’, to ‘all the same’ and ‘none the less’ (and other expressions possibly indicative of a pari).
To repeat, definitive statistics will only be possible when each and every case is actually listed and examined in detail – a massive job, which I will not here attempt to do. Having in the above table dealt with idiomatic a fortiori indicators, we should next deal with the more descriptive ones. The following table should, I think, cover most of the potential ground.
A fortiori wording
A fortiori wording
it is sufficient
a minori (ad majus)
a majori (ad minus)
from minor (to major)
from major (to minor)
it is enough
infer(s), inferred, inferring
deduce(s), deduced, deducing
prove, proved, disproved
argue(s), argued, arguing
N.B. These counts are raw data.
Please note well that the statistics in this table are even more unprocessed than those in the previous table. I just give the raw numbers dished out by the search engine, without taking the trouble to look at individual cases. The total for this table is 3662, and this is not counting words indicative of inference like ‘therefore’ (1616), ‘hence’ (2270, including 861 ‘whence’ and 42 ‘thence’), then (4729, including ‘thence’), etc. Clearly, a lot of work is necessary to sort through all these.
I include ‘kal vochomer’ in this table, because Rodkinson used this phrase in a note; but as it turned out he did not use it in the text proper. Nor does his translation, unlike the later Soncino translation, ever use the key phrases ‘a minori/majori’, or ‘from minor/major’. His main descriptive term is, thus, ‘a fortiori’; this may be used to signal a fortiori intent or to refer to an already proposed a fortiori argument. In any event, use of this key phrase cannot be indicative of anything other than assumed a fortiori discourse.
Nevertheless, many more a fortiori arguments may be found by means of the other key words listed in this table and others like them. The word ‘infer’ presumably usually corresponds to the Hebrew word din, which is in rabbinic discourse often used to refer to a fortiori; the same may apply to the words ‘deduce’, ‘prove’ and ‘argue’. Note that these words often appear in a rhetorical negative question: ‘is it not an inference that…’, ‘can we not deduce that…’, ‘is it not logical that…’.
Obviously, some of these inferences, deductions, proofs or arguments must refer other hermeneutic principles, such as gezerah shavah and binyan av, since a fortiori is not the only form of reasoning used in the Talmud. I do not at this time have a clear idea as to how such other interpretative forms are actually worded in Rodkinson’s edition, or anywhere else for that matter. Obviously, this question must eventually be answered. When we do that, our investigation will expand from specific concern with a fortiori argument to general concern with all the hermeneutic principles. However, I am not disposed at the present time to look further into this matter.
The main key phrases used by Rodkinson to refer to a fortiori argument are now seen to be the descriptive phrase ‘a fortiori’ (147 instances), and the various idiomatic phrases ‘much the more/less’ (189), ‘much more/less’ (71), ‘even more/less’ (20), ‘still (the) more/less’ (15), ‘all the more/less’ (8), among others (36). The overall result is that the number of a fortiori arguments in the Rodkinson edition of the Talmud may be about 500 (a round number). This is ignoring overlaps in the first and second tables (no doubt many), as well as all possibly a fortiori intents in the remainder of the second table (maybe numerous); perhaps these and those balance each other out somewhat. This is still a very rough and uncertain tally, of course; but it is better than nothing – an educated guess, let’s say. Moreover, keep in mind that Rodkinson’s edition includes only two of the six orders of the Talmud – so the final count may be three times this figure!
It should be emphasized that this statistic lumps together purely a fortiori arguments and a crescendo arguments. It is clear that a future fuller study has to distinguish them, i.e. identify how many cases of each of these two types there are. Moreover, each case must be classified as either positive or negative, and subjectal or predicatal (copulative) or antecedental or consequental (implicational), to be really understood. These various moods should then be counted separately. All this additional precision of course requires more detailed analysis of each individual case than here done.
Regarding the dayo principle. The key phrase ‘it is sufficient’ seems to be our main indicator of appeals to the dayo principle here; surprisingly, this occurs very rarely. From the data found through mechanical search for “it is sufficient” (73 instances) in the Rodkinson edition of the Talmud, there appears to be only six passages that explicitly appeal to the principle of dayo in some form, such as “it is sufficient that the result derived from an inference be equivalent to the law from which it is drawn,” or more briefly as “the rule of ‘It is sufficient,’ etc.” The passages concerned are the following: In tract Baba Kama: the Gemara concerning Mishna 2:1 (1 mention); the Mishna 2:5 (1 mention) and its Gemara (5 mentions); the Gemara concerning Mishna 4:3 (1 mention). In tract Baba Metzia, the Gemara concerning Mishna 3:6 (1 mention). In tract Baba Bathra, the Gemara concerning Mishna 8:1 (2 mentions). And in tract Shebuoth, the Gemara concerning Mishna 4:1 (1 mention).
One question to ask here is: do all these applications concern the inference of a penalty from Biblical law? The answer is clearly yes in cases 1-4, which all concern payment of damages. Case 6 concerns legal liability through making an oath, and so can also be viewed as proper for dayo application. Case 5 is open to debate: I have dealt with it in the chapter on Adin Steinsaltz, in the section called ‘A recurring fallacy’ (18.2), under the heading of ‘On Baba Batra 111a-b’, there pointing out that reference to the dayo principle may be misplaced because while for the daughter the proposed judgment is unfavorable, for the son it is favorable. Thus, judging by the Rodkinson edition, in the 6 cases which explicitly appeal to the dayo principle, it is used to limit a penalty or responsibility or right. This accords with my theory of the intended scope of dayo.
This result, of course, does not exclude the possibility that there are cases other than those here enumerated, where the dayo principle is appealed to explicitly but using other wording than “it is sufficient,” or in a more tacit manner, which might yield a different conclusion regarding the intended scope of the principle. To give an example, we saw in the chapter ‘In the Talmud, continued’, in the section ‘Three additional Gemara arguments’ (8.6), how the dayo principle (in both its versions) may be very present in the background of a discussion without being explicitly mentioned. Moreover, the Rodkinson edition is far from complete; so, some dayo applications may well be missed in it – for instance, the dayo principle is appealed to in Zebachim 43b-44a, but the Rodkinson edition lacks this tractate.
Note lastly that I have not here made an effort to determine the standard form(s) of the six arguments relative to which dayo was used. Cases 2 and 5, having been dealt with elsewhere, we know to be positive subjectal; but the other four cases have yet to be classified. Since the original dayo objections in the Mishna are of two sorts, applicable respectively to purely a fortiori argument or to a crescendo argument, we cannot predict how many of these two sorts occur in the Gemara. Furthermore, we should look and see whether the language used in proportional differs from that in non-proportional arguments. I leave these tasks to others.
We will end our pilot study here, without going into more detail or precision, having set an example of methodology and structure of research, and anticipated and dealt with some of the pitfalls that may be encountered.
 In his Der Hermeneutische Analogie in der Talmudischen Litteratur, pp. 84, 87, 89.
 This refers to BT and apparently only includes Bava Metzia, Ketubot, Ta’anit, and Sanhedrin. See: www.steinsaltz.org/learning.php?pg=Talmud_-_Books&articleId=1424.
 This is translated (or edited?) by Jacob Neusner, Tzvee Zahavy and others. Complete.
 This is found at: www.halakhah.com/indexrst.html. A Kindle edition is also available for a small price, at www.talmudicbooks.blogspot.ch/2012/05/amazon-kindle-oral-torah-in-36-volumes.html.
 Michael Levi Rodkinson, previously Frumkin, was a Jew who emigrated to America (1845-1904).
 It is also available in eBook format which can be read using Adobe Digital Editions (ADE) reader. I should mention that, while the Kindle for PC reader has the advantage that its search facility lists ‘all’ the matching cases at once, it has a maximum limit of 100 hits; the ADE reader, on the other hand, has no maximum limit, but it only takes you to the matching cases one at a time.
 “Plus some additional material related to these Orders;” namely: Ebel Rabbathi / Semahoth; Aboth of R. Nathan; Derech Eretz Rabba and Eretz Zuta.
 As I pointed out at the time, this statistic cannot be taken at face value, “because the references are to page numbers, which may contain more than one argument of the same type; also, not having looked at them, I cannot guarantee that they are all legitimate cases. I would suspect offhand, on the basis of my minimal experience of Talmud study, that this list is incomplete (all the more so if we include the Commentaries).”
 The two apparent a pari in the sentence: “Yea, thou hast occupied thyself as much as R. Hyya, but thou hast not multiplied the Torah as much as he did” are perhaps more implicit than explicit. Paraphrasing: If you occupied yourself with Torah as much as he did, then if his credit is x, your credit would be x (as much as his); and if you spread the Torah as much as he, then your credit would be as much as his.
 Wording: “not more rigorous” (2 instances of 3) or “no more than…” (1 instance of 17).
 Rodkinson’s there (in Vol. 2, Part I) says: “This is a case of where the peculiar Talmudical expression of Kal Vochomer appears in the text. The literal translation is ‘light and heavy’, i.e., from the lighter to the heavier or from minor to major. In the Introduction to the Talmud by Prof. Dr. Mielziner an entire chapter is devoted to the explanation of this term (pp. 130-141). However, no general term can be found to express its meaning, and the expression must be varied according to the demand of the text.” This remark is to my mind rather strange, given that the Hebrew expression qal vachomer has long been known to refer to a fortiori argument, and indeed Rodkinson freely uses the expression a fortiori elsewhere!
 Note that the 365 instances of ‘sufficient’ include 56 ‘not sufficient’ and 5 ‘insufficient’; the 144 instances of ‘suffice’ include 20 ‘not suffice’; and the 171 instances of ‘enough’ include 15 ‘not enough’. Note also that besides the 19 instances of ‘it follows’ and the 7 of ‘not follow’, there are 281 other ‘follows’ and 82 other ‘follow’.
 Note that here under ‘more/less’ I include cases of ‘same’ and other comparatives.
 This may be why the halakhah in this case does not align with the dayo principle.