A FORTIORI LOGIC
CHAPTER 20 – Michael Avraham
We shall again encounter Michael Avraham (or Abraham), as co-author together with Dov Gabbay and Uri Schild of the 2009 paper “Analysis of the Talmudic Argumentum A Fortiori Inference Rule (Kal Vachomer) using Matrix Abduction” and a companion paper in Hebrew focusing in more detail on Talmudic a fortiori argument. But Avraham was evidently an old hand in the field by then, since he long before authored a paper called “The ‘Kal Vachomer’ as a Syllogism – Arithmetic Model” that was published in the 1992 issue of Higayon (vol. 2. Pp. 29-46).
I have a copy of this older article, which is in Hebrew. The article is summarized in English as follows:
“A mathematical and conceptual analysis of the Talmudic Kal Vachomer is presented using the concept of ‘rotation’ of the Kal Vachomer. Several possible models which describe differing interpretations by the commentators are considered. It is argued that the Kal Vachomer – like the rest of the 13 ‘Middot’ – is not a syllogism.”
My Hebrew is simply not good enough to try to review it and assess all its contents independently, although I gather from a brief perusal of the Hebrew text that Avraham’s wording for a fortiori argument (qal vachomer) is as follows:
“If A is light in ‘a’ and heavy in ‘b’, then B which is heavy in ‘a’ will obviously be heavy in ‘b’.”
Note that, as far as I can tell, he fails to stress the relativity of the weights: he does not say lighter/heavier, but light/heavy (or also –/+). Judging from the word ‘in’ (be) he uses, I would say that ‘a’ and ‘b’ refer to subjects, and A and B refer to predicates. If so, then the argument would be positive predicatal, and the inference would be invalid since it is ‘minor to major’ (whereas it should then be ‘major to minor’). Moreover, the number and functions of the terms involved is not correct. Although we could point to ‘weight’ (heavy or light) as the middle term (R), his construct has no subsidiary term (S) in common to the premise and conclusion.
Avraham’s wording posits mere analogy between A and B, in ‘a’ and ‘b’: Just as A is heavier in ‘b’ than it is in ‘a’, so B is heavier in ‘b’ than it is in ‘a’. So really, what we have here is just the argument: Just as A is X, so B is X, where A and B are two subjects and predicate X happens to refer to the comparative term “is heavier in ‘b’ than it is in ‘a’”. For this reason, the argument looks at first sight like a positive subjectal one, and thus like a valid ‘minor to major’ process, since a further given is that A is light in ‘a’ and B is heavy in ‘a’. But on closer scrutiny it clearly is not even an a fortiori argument, let alone a valid one! It cannot be rendered in standard (PQRS) form. Although Avraham uses language suggestive of such argument, he does not correctly formulate it.
The above was my first reaction, but returning to the enigma later, I thought of the following standardization. If we take B and A to be respectively the major and minor terms (P and Q), and “weight in ‘a’” to be the middle term (R) and “heavy in ‘b’” to be the subsidiary term (S), then we can construct the following valid, positive subjectal a fortiori argument:
B (P) is heavier in‘a’(R) than A (Q) is; and
A (Q) is heavy in‘a’(R) enough to be heavy in‘b’(S);
therefore, B (P) is heavy in ‘a’ (R) enough to be heavy in‘b’(S).
Maybe Avraham had this thought at the back of his mind. Just maybe. But in that case, what need had he for the subsidiary term “heavy in ‘b’”? Why this, rather than “light in ‘b’” or “balanced in ‘b’”? Indeed, any predicate would have fitted in here (e.g. “in ‘b’” or just “b”) – why did he specifically use a predicate to do with “weight”? He does not propose a ‘proportional’ conclusion, so that cannot be the reason why. Obviously, the specification “heavy in” relative to ‘b’ is a formal redundancy. This peculiarity is indicative of a lack of clarity on Avraham’s part as to the roles of the terms – which is why I suspect my preceding analysis of his thinking to be more likely. Thus, I maintain that he misunderstood how a fortiori argument is constituted.
I would have left the matter at that, were it not for the fact that Allen Wiseman devotes considerable attention to Avraham’s paper in his own 2010 treatise. So we can briefly examine it further on this basis. As can be expected, Avraham presents in the 1992 paper the same basic theory of a fortiori argument as he (together with Gabbay and Schild) uses in the 2009 paper. The older paper may be said to lay the theoretical foundation for the later one. We might suppose Avraham’s thought on the subject to have evolved over the years, but obviously we cannot suppose that his thinking was more accurate in the earlier paper than it was in the later one. Therefore, we should not expect to find any insights in the former that radically overturn our critical assessment of the latter. That said, let us anyway take a look at Avraham’s 1992 ideas as presented (presumably fairly enough) by Wiseman.
To begin with, he tells us, Avraham “writes the generic, Ostrovsky/Schwarz argument in predicate logic.” I take this symbolic formulation to be Avraham’s own interpretation of Ostrovsky and before him Schwarz, since Wiseman does not mention this formula in his earlier expositions of the latter two authors. Avraham proposed formula is the following (I have changed his letter P to F):
For all x, if x is F, then x is G
Specific instance (of x) a, is F
Specific instance (of x) a, is G
This is stated in the language of symbolic logic, which is supposed to be freer from ambiguities than ordinary-language logic. But to my mind, this is an overly rigid format, which complicates and obscures matters unduly, just to give an illusion of exactitude. A simpler and more natural statement of same would be:
If anything is F, it is G (or categorically put: All F are G),
and a certain thing (say, a) is F;
therefore, that thing (a) is G.
We thus see way that what we have before us is nothing more than a modus ponens apodosis; that is, an extensional conditional proposition, combined with a categorical one affirming its antecedent, concluding with a categorical proposition affirming the consequent. We can also look upon it as a categorical syllogism, as Wiseman has it, and as no doubt Avraham thought of it, by formulating the major premise in categorical form (as shown in brackets). It is not very important how exactly we interpret Avraham’s symbolic formula.
What is important is the fact that, as a somewhat perplexed Wiseman immediately remarks, “Avraham does not mention that the above form does not tell us what is more and less severe… Perhaps we are to know that [F] is less and G is more.” In other words, what has the said formula to do with a fortiori argument if it does not even rank the terms involved as minor and major? It would seem that in Avraham’s mind, the antecedent predicate F is the lesser quantity, and the consequent predicate G is the greater – so that the inference is from minor to major. But if he has not expressed that, we could equally well assume that the inference is from major to minor. So his formula is deficient.
Perhaps Avraham considers that the said formula merely underlays a fortiori argument, but does not wholly cover it. This is suggested by his recourse to what Wiseman refers to as “a chart” – actually a tabular representation. This is intended, he tells us, “to rectify matters.” But the truth is that such a table also does not intrinsically bring out the relative ranking of the terms, unless verbal explanations are additionally given. What all this means is that Avraham in fact does not in fact propose any formula, whether in symbolic or ordinary language, capable of entirely capturing and distinguishing the a fortiori movement of thought, or even one mood thereof. This is the main thing to note.
Nevertheless, Avraham does have things to say about a fortiori argument. Wiseman informs us that: “He understands the overall argument as deductive, although it often comes as an induction.” This comes as a surprise to me, knowing that in a later essay, co-authored with Gabbay and Schild, Avraham seems to have a more definitely inductive, as against deductive, view of a fortiori argument. At any rate, Avraham here distinguishes himself from Schwarz, who we are told insisted on the deductive aspect. The way Avraham (or perhaps Wiseman for him) formulates the more inductive a fortiori argument is to qualify the premises as “possible givens,” i.e. as somewhat uncertain; in which case the conclusion is at least equally unsure, “rather than certain or necessary.”
I agree with this viewpoint: a fortiori argument is essentially deductive, although it is occasionally intended more as an inductive indicator. However, I would stress, to avoid all misapprehension, that the uncertainty in the conclusion is due to subsisting doubts in the contents of one or both of the premises, rather than doubts in the process of inference itself. The process of inference as such remains deductive – even if the conclusion has an inductive character in view of the (in most cases) inductive character of the premises it is based on. Moreover, this possibility of doubt is not peculiar to a fortiori argument, but is true of all deductive argument, including categorical syllogism. I agree with Avraham if this is what he meant by “possible givens.”
So both Schwarz and Avraham can be considered right, assuming they indeed hold the opinions here attributed to them. The former is right in insisting that the form of a fortiori argument is deductive, and the latter is right in keeping in mind that the contents of the premises and therefore of the conclusion may still contain some doubt and therefore be merely inductive. All logicians know this distinction; but it is still worth reminding for the sake of novices. In Talmudic argumentation, putting the premises in doubt (either directly or through doubt in the conclusion) is called teshuva (in Hebrew) or pirka (in Aramaic).
Of course, a good question to ask here is: how would either Schwarz or Avraham know that a fortiori argument is essentially deductive, if they did not in fact correctly formalize the argument and validate it? We must, objectively, say that they posit this as an intuitive claim, without really knowing it by having proven it.
Of course, I hasten to add, even if the a fortiori process is in principle deductive and indubitable (once validated), it does not follow that all a fortiori arguments thought, spoken or written by human beings in practice are properly conceived and formulated. People do occasionally, if not often, make mistakes (or even fake things) while reasoning. They maybe do that more often when reasoning a fortiori, since this is more complicated. In this sense, even the process of a fortiori argument may be considered inductive, at least until we establish clearly that it was correctly done in the case under scrutiny. We do find instances in the Talmud where someone’s process of a fortiori reasoning is questioned.
Let us now focus on Avraham’s treatment of the famous debate in the Mishna (Baba Qama 2:5) between R. Tarfon and the Sages, which I have dealt with extensively in an earlier chapter (7.3). What is interesting at the outset is to see that Avraham is aware that there are two arguments and objections, and somewhat aware of the difficulties these present – which is more than can be said for some other commentators. Actually, we have elsewhere seen and criticized Avraham’s approach to this debate, re-enacted in the essay he co-authored with Gabbay and Schild 17 years onwards. So I will here only cursorily touch upon a few additional points.
Avraham approach here is very tedious, due to his excessive use of symbols, not to mention his repeated changes of symbols for the same items. Moreover, one gets the impression throughout that he is feeling his way around, trying to sort things out somehow – rather than presenting a fully matured product. He beats about the bush, and does not get to the essence of things. Many of the items he refers to are extraneous, even if to his mind they seem necessary to ensure the systematic character and generality of his conception of things. All this is complication, which obscures more than it enlightens.
Avraham’s essay is of course a serious effort. But try as I might, I cannot make out just what it is getting at. I fully understand every statement made, whether in symbols or words. But he does not here seem to be advocating some definite result that can be judged true or false. It could of course be that Wiseman’s account of Avraham’s paper is incomplete. That is, although what his account actually contains looks like an accurate rendition, he may have missed out some crucial information or insight that is clear enough in the original. The explanation may also be that Avraham wishes to take nothing for granted and leave everything open; even so, his position is far from clear.
Regarding the 2009 paper that we have already mentioned, we can say that, although the authors have not realized the standard form of a fortiori argument which I proposed in 1995, they have at least effectively identified such reasoning with argument by analogy, or more precisely pro rata argument, since the conclusions they draw from the givens are visibly proportional (even if we found errors in specific cases). In the 1992 paper, however, Avraham does not seem to yet propose any single final conclusion, but merely maps out different possible hypotheses that apparently peter out without a clear result. What stands out, anyway, is that Avraham has not here grasped a fortiori argument in the true sense.
His main inferences are, as the title of his paper implies, “arithmetical” – they concern quantitative comparisons: if one thing is greater than another, and that other is greater than a third, then the first is greater than the third. This is what he apparently regards as the “deductive part” of the argument. Beyond that, there is an “inductive part” of the argument, in which different possibilities have to be considered, before a final “estimate” is made as to the conclusion to draw. Since this process depends on human judgment-calls, it is inherently open to doubt. Its conclusion is therefore open to “refutation” or “disproof,” meaning at least disagreement.
Avraham views the Mishna debate through the prism of the following table, which is perhaps an innovation to his credit:
In public domain
In private domain
For damage by bite or hoof
For damage by goring
The givens are that there is no penalty (–) for damage by ‘bite or hoof’ in the public domain, and some penalty (full damages) for such damage in the private domain, as well as some penalty (half damages) for damage by ‘goring’ in the public domain, and the question asked is: what is the penalty (+) for the latter sort of damage in the private domain? The plus sign in the corresponding cell suggests, of course, that the answer to that question is positive, or maybe full damages.
As I understand it, two directions of argument by analogy are possible. The first argument is “horizontal:” since ‘bite or hoof’ damage is greater in the private domain than in the public domain, it follows that ‘goring’ damage is greater in the private domain than in the public domain; conclusion: full damages (since more than full is excluded). The second argument is “vertical:” since in the public domain ‘goring’ damage is greater than ‘bite or hoof’ damage, it follows that in the private domain ‘goring’ damage is greater than ‘bite or hoof’ damage; conclusion: full damages, again.
Note that both these arguments of R. Tarfon’s yield the same, proportional conclusion (full damages). Actually, R. Tarfon initially puts forward only the first argument; but the Sages reject his conclusion, saying dayo – it is enough to conclude with half damages for damage by ‘goring’ in the private domain, as originally given for damage by ‘goring’ in the public domain. Then R. Tarfon tries another tack, his second argument; but the Sages again reject his conclusion, repeating dayo (in identical words) – it is enough to conclude with half damages for damage by ‘goring’ in the private domain, as originally given for damage by ‘goring’ in the public domain.
R. Tarfon, interpreted the Sages’ first objection as a demand that the quantity in the conclusion be equal to the quantity in the last premise (viz. half damages in this case), and so was surprised to find them objecting to his second conclusion, even though the quantity in it was equal to the quantity in the last premise (viz. full damages in that case). The Sages, on the other hand, were focused not on the technical issue of the last premise but on the content of the original given (about damage by ‘goring’ in the public domain), and for that reason were unfazed by his logical bravado. This means that the two dayo objections, though verbally the same, are technically different somehow. In the first argument, the original given happens to correspond to the minor premise; but in the second argument the original given is not an overt premise but is used to construct the major premise.
The job of any commentator on this topic is to notice this problem and to propose a credible solution for it (as I just did, for example). It seems as if Avraham has noticed the issue somewhat, and tried to grapple with it in some way. This is suggested in Wiseman’s following statements (even if they do not directly concern the same topic). In the main text: “These two claims differ and are not two versions of the same claim as might seem in a cursory view. Thus, a disproof of either the vertical or the horizontal II statements will not apply to the other as before.” And in a footnote:
“The claim that one can escape disproof by changing directions looks very strange, as if the h and v seem to be differing formulations of the same claim, while the logical disproof does not care about the formulation. Both claim types derive from the same 3 givens and conclude with the same logic. So how can the disproof that attacks one not attack the other in principle? As the disproof of the QC [qal vachomer] comes inductively against the given claim and not the deductive part, one can simply disagree with the claimed givens of h and v as if wholly different; so the impression that they are the same claim presented in two ways is just an illusion.”
However, it is difficult to say whether this is Wiseman trying to personally make sense of this issue, or he is reflecting Avraham’s actual thinking. In any case, what seems clear is that the problem is not stated and dealt with by Avraham with the requisite clarity and thoroughness. His presentation is so padded and tortuous that one cannot readily discern just what he is trying to say. I would like to make a fair assessment, and not dismiss Avraham’s thesis offhand. But his presentation seems too unclear and messy to me. When one has nothing concrete to grab hold of, one can only say that one has nothing concrete to grab hold of.
What is certain, anyway, is that Avraham has not mastered a fortiori argument. We saw that from the beginning of his article he misapprehended what might constitute a fortiori discourse. Not surprisingly, he thereafter fails to produce a fitting and distinctive deductive formula for it. He is therefore reduced to looking for some inductive solution to the problem. But even that attempt leads him to an impasse.
 The same Hebrew letter ‘ב’ may be read as ‘v’ or ‘b’.
 Ed. Moshe Koppel and Ely Merzbach. Jerusalem: Aluma, 1992.
 See p. 30, below the first diagram. My translation.
 Note too that no justification is given for the claimed inference of X from A to B.
 Moreover, he seems to be complicating matters considerably by next introducing an additional term C, and engaging in a second, successive act of a fortiori argument. Why he does so, I cannot tell. This is presumably what he means by “‘rotation’ of the Kal Vachomer.” I would guess that what Avraham had in mind here was what I later (in my Judaic Logic) called a fortiori arguments “in tandem.” But if so, sorry to say, he does not manage to correctly formulate this logical phenomenon either. In any case, it is worth reminding that rabbinical hermeneutics does not allow inference of a new ruling from a previously inferred ruling, i.e. use the conclusion of one a fortiori argument as a premise in the next. So if Avraham predicts ‘rotation’ in Judaic logic, he is apparently ignoring this restriction. It is admittedly possible that he is not the first to do so, i.e. that rabbis have often in practice also ignored it.
 If we look at his diagrams for ‘rotation’ (or what I presume this term to refer to, i.e. a chain of a fortiori arguments), we see that the item ‘+b’ remains the same from one argument to the next. This means that we cannot explain his reference to “heavy in ‘b’” as having something to do with ‘rotation’.
 A Contemporary Examination of the A Fortiori Argument Involving Jewish Traditions (Waterloo, Ont.: University of Waterloo, 2010), pp. 75-81.
 Unless, of course, he was not really involved in the writing of the later paper, but was counted as a co-author because some of his ideas were used by the other two co-authors. In that case, it could be that the latter misunderstood or misused his ideas.
 Pp. 75-81. Wiseman informs us, in a footnote: “His article I translate and reproduce here, mostly from his p. 35 onwards, sometimes paraphrased… All the main material is his.” Thus, though he admits to largely skipping the first 6 pages of the paper, he maintains his presentation to be essentially accurate.
 Wiseman says that Avraham “states that Ostrovsky follows Schwarz in an attempt to construct a deductive argument based on the categorical syllogism” (p. 75). I present a brief study of Ostrovsky in a later chapter of the present volume (30.3).
 I do this primarily so as to preempt confusion with my standard lettering (where P refers to any major term). But also, because I think that when Avraham wrote P he meant F, as is evident from his use of G as the next letter. In Hebrew, the same letter can be read as P or F.
 The symbol ‘x’ is to my mind a useless interjection. Notice that the symbolic formula takes for granted, but does not explicitly say, that ‘a is an instance of x’ – yet it is only on that assumption that we infer that ‘a is F’. So, before we can infer the conclusion ‘a is G’, we have to go through a preliminary apodosis. This is an unnecessary complication, which can be avoided simply by using ‘anything’ in lieu of ‘x’. And indeed, the latter is the way we naturally think. The symbolic formula is a fabrication of people who confuse trivial conventions with intellectual breakthroughs.
 As Wiseman puts it us in a footnote: “He says that Schwarz rejected the possibility claim (of “Korban Aharon”) for that of certainty.” Korban Aharon is an older commentary.
 One is tempted to ask him the question: do you think that when people reason a fortiori they have to go through all the rigmarole you have laid out to arrive at a credible conclusion?
 Wiseman refers to this as “quantificational logic.”
 The words in inverted commas are used by Wiseman, at least.
 Note that if this reading is correct, it means that Avraham’s idea of the “inductiveness” of a fortiori argument is not merely to do with uncertainty in the premises, as I earlier suggest, but also with doubts inherent in the process. But “the process” here is, of course, not the standard a fortiori argument, but the vaguer form of argument that Avraham has in mind and labels ‘a fortiori argument’.
 Wiseman, p. 78; with references to Avraham, p. 41-2.
 Even regarding such a simple thing as what Avraham means by “horizontal” and “vertical,” I have doubts. His “horizontal” (meusan) argument includes a statement (in symbols, of course) that the fine for damage by ‘goring’ in the public domain is greater than that for ‘bite or hoof’ in the public domain – yet if we look at his table, these two items are in the same column, not the same row. Accordingly, his “vertical” (meunakh) argument has a statement that the fine for damage by ‘bite or hoof’ in the private domain is greater than that for same in the public domain – yet these are in the same row, not the same column.