A FORTIORI LOGIC
CHAPTER 25 – Abraham, Gabbay and Schild
Michael Abraham (or Avraham), Dov Gabbay and Uri Schild have, in 2009, together written two long papers relating to a fortiori argument, namely: “Analysis of the Talmudic Argumentum A Fortiori Inference Rule (Kal Vachomer) using Matrix Abduction” and “מידות הדרש ההגיוניות כאבני הבסיס להיסקים לא דדוקטיביים” – the latter presented as the Hebrew companion of the English version, covering Talmudic analysis in more detail. I have copies of both them as I write the present review, but I will mainly refer to the English one, because my Hebrew is too elementary to deal with the companion piece in a serious manner.
Naturally, the first thing I looked for in these papers is any mention of my past work on a fortiori argument – not out of personal vanity, but because I know that my findings are crucial to any discussion on this subject. There is no mention of it in the English paper. In the Hebrew paper, they mention it in a footnote, saying: “See also Avi Sion, Judaic Logic, Editions Slatkine, Geneva 1995,” after which they state within the main text:
“These sources offer a logical formalism that translates the inference into formal logical language, but it is not much more than a translation. Therefore it is difficult to see in these attempts a logical or mathematical model of the real” (p. 8).
This very summary and quite unfair treatment suggests to me that they made no effort to actually study my work. On top of that they have the gall to declare sweepingly, without providing any proof, in their English paper:
“The rule [of Kal-Vachomer] has never been properly formulated, though there have been many attempts” (p. 46).
They do go on to very briefly describe ideas on the subject by Louis Jacobs and Adolf Schwarz. Of the latter they say: “The simple Kal-Vachomer was analysed as an Aristotelian syllogism by A. Schwarz,” and they “compare” a famous syllogism (All men are mortal; Socrates is a man; therefore, Socrates is mortal) and the a fortiori argument in Exodus 6:12 (Behold, the children of Israel have not hearkened unto me; how then shall Pharaoh hear me, who am of uncircumcised lips?), by casting both arguments in the same symbolic straightjacket, thus presumably demonstrating how ‘modern’ and ‘scientific’ their approach is. Of course, if anything is “not much more than a translation” it is surely this inane rewriting of ordinary words into esoteric symbols.
In plain English, what they have done is rewrite Moses’ above argument as: “For all cases, if ListenIsrael is denied, then ListenPharaoh is denied; in the present case, ListenIsrael is denied, therefore ListenPharaoh is denied.” What does ListenIsrael or ListenPharaoh mean? They do not say. Presumably the former expression means “the children of Israel hearken to Moses,” and the latter means “Pharaoh does so.” But these writers prefer to conceal than reveal, it seems. In any case, what is this argument of theirs? Is it a credible translation of a fortiori argument? Not at all! It is not even a syllogism! It is merely a modus ponens apodosis. So this alleged symbolization is mere smoke in our eyes.
The authors do not, however, intend thereby to subscribe to Schwarz’s theory, for they add: “Louis Jacobs refutes the similarity” (notice the word “refutes,” connoting utter disproof), and then, in an authoritative tone: “Certainly the more complex cases of Kal-Vachomer are not syllogisms at all” (without any demonstrations to that effect). The latter remark seems to imply that they consider that some simple a fortiori arguments may well be syllogisms. Regarding Jacobs’ approach, which they seem to have more faith in, they tell us only that he “distinguishes two types of Kal-Vachomer. The simple one and the complex one.” Their structures are respectively: “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.”
They give as examples in the Torah of the simple type, the one in Exodus 6:12 just quoted and Deuteronomy 31:27, but as regards the complex type they state categorically (simply passing on an erroneous belief of Jacobs’) that: “The Bible does not contain instances of the complex case. This has emerged later, after the Bible.” They make no effort to assess Jacobs’ theory, not even bothering to “translate” his two forms in symbolic terms as they did for Schwarz. They curiously do not wonder why A having x should imply B to have x (in the ‘simple’ type), or A lacking y but having x should imply B which has y to have x (in the ‘complex’ type). They apparently do not realize that proposing a formal representation is only half the work of a logician, the other half being to validate that form. If they disagreed with Jacobs, they should have shown his conclusions do not logically follow from his premises.
That’s it: Abraham, Gabbay and Schild mention no other theories on the subject other than their own, and their criticism of the theories they do mention is minimal, because stated in very vague terms. I will not bother here to analyze and criticize the work of Schwarz and Jacobs, as I have done so in considerable detail in separate chapters (14 and 16), showing there that a fortiori argument cannot be equated to syllogism and that Jacobs’ simple and complex forms (though a bit better than Schwarz’s apparent syllogistic proposal) are inaccurate non-sequiturs. My aim here is only to show the paucity and conventionality of the approach of our three authors in relation to past work, on which their above mentioned general negative statements are apparently exclusively based.
One thing is evident: they did not get acquainted with my past work on the subject, and therefore missed the opportunity to learn the various forms of a fortiori argument and how they may be logically validated. When they say: “It is difficult to see in these attempts a logical or mathematical model of the real” and “The rule [of Kal-Vachomer] has never been properly formulated, though there have been many attempts” – they are only referring to the work of Schwarz and Jacobs, and clearly not to that of Avi Sion. The latter does in fact offer a “logical model of the real” because it systematically presents, validates and explains the varieties of the argument.
We shall presently try and analyze the three authors’ own contribution, but first I wish to highlight the error of its basic premise. This is immediately apparent in the opening sentence of the Abstract to their English paper:
“We motivate and introduce a new method of abduction, Matrix Abduction, and apply it to modelling the use of non-deductive inferences in the Talmud such as Analogy and the rule of Argumentum A Fortiori” (p. 1).
The error here, at the outset, is to regard “the rule of Argumentum A Fortiori,” i.e. Kal Vachomer, as a “non-deductive inference.” This starting point of theirs naturally colors and skews their whole approach. They understandably base this assumption on the failure of Schwarz and Jacobs to convincingly formalize, validate and explain a fortiori argument; but it is merely a hasty generalization from a limited sample.
In truth, as my own analyses show indubitably, a fortiori argument is a strictly deductive inference, although – as with all deductive inference, including syllogism – its premises may be open to discussion due to their inductive origins. Also, of course, we must take into consideration that many people reason incorrectly, in arguments which though labeled as “a fortiori” can in truth only be characterized as “pseudo a fortiori.”
Looking at Talmudic debates or other documents, involving a fortiori arguments, and in some cases fake a fortiori arguments, people are easily driven, into believing that it is not an essentially deductive form of inference. The shifting ground of intuitive insights and objections to them, of opinions and counter-opinions, of facts and conflicting facts, give the overall impression that this kind of reasoning is largely inductive. But this is a misleading impression, as closer analysis using formal tools shows.
Keep in mind that the process of inference is the relationship between given premises and putative conclusion. If given the premises the putative conclusion necessarily follows, then the process is indeed deductive, whether the premises given are true or false. If given the premises two or more alternative conclusions are logically conceivable, then each of these conclusions is by itself a deductive non-sequitur, and the selection of one of these conclusions in preference to the other(s) – for a host of other reasons, whatever they be – is indeed an inductive act.
To claim as do our three authors that a fortiori argument is “non-deductive inference” is just to admit being unaware of the formal validation processes for such arguments. If one looks at the work of Schwarz or Jacobs, one will of course not find formal validation of a fortiori argument. But if one takes the time to study my Judaic Logic, not to mention the present work, one will indeed find such formal validation. So the basic premise of Abraham, Gabbay and Schild is mistaken.
For them, then, a fortiori argument is inductive, and this motivates them to look for a way to find appropriate conclusions to such arguments. Having no formal understanding of the argument, they must instead look approximate solutions to a fortiori type problems. The way they propose to do this is “matrix abduction.” In their view, “the Talmud is trying to abduce” (p. 9) a conclusion from certain premises. Further on, they explain:
“The Kal-Vachomer rule (and the algorithms supporting it) are non-monotonic rules of induction. This means they are not absolute deductive rules but defeasible common sense rules. So we may use these rules to obtain a conclusion A, but further information and further arguments may force us to doubt A or even come to accept ~A [i.e. non-A].” (P. 12.)
In other words, the conclusions drawn from a fortiori arguments are always open to doubt, though one might at a given stage seem more credible to “common sense” than another. “Matrix abduction,” the method they propose to use, visualizes the problem at hand as a matrix (a table with multiple rows and columns) in which most but not all the cells are filled in with given yes or no information (marked “1” or “0” respectively), and the solution as some mechanical (as against purely ‘intuitive’) way to predict the content of the empty cell(s), i.e. those for which yes or no information is not given (marked with a “?”). The result will be, effectively, a ‘best guess’ even if it is not absolutely sure.
Now, I have nothing against abduction (or adduction) or the use of matrices (matricial analysis). I have used them for many years, and written much about them, and consider them essential to logic research. Nor, note well, do I doubt the general value of the particular “matrix abduction” method proposed by our three authors, or (offhand) the accuracy of the complex diagrams and calculations it involves. What I do doubt, right from the start, is that this really has anything to do with a fortiori argument as such!
But we should anyway question the wisdom of resorting to the method of tabulation to resolve the complex issues posed by a fortiori argument, or even mere argument by analogy. Such argument is always based on some generalizations, and it is very difficult to represent this fact in tables. A table is a simplified picture of events, not to say a simplistic representation of them. The names of the rows and columns seem clear and definite, but this is only a superficial impression, concealing underlying complexities. To say that a cell (the intersection of a row and a column) has either a value “1” or “0,” or eventually “?” is accurate only when dealing with individual events, which are either present, absent or uncertain. When dealing with general propositions, however, a fourth value must be introduced.
If the row label signifies a subject (or antecedent) and the column label a predicate (or consequent), then the conjunction of these two terms (or theses) may be true ‘in all cases’ (symbol 1), ‘in no cases’ (symbol 0) or ‘in some cases yes and in others no’ (say symbol *), or uncertainly (symbol ?). There are here clearly four and not just three alternatives: always yes, always no, sometimes yes and sometimes no, and unsure which of the preceding three is applicable. To put it another way, you cannot represent the family of propositions ‘all A are B’ (or ‘if A, then B’), ‘no A is B’ (or ‘if A, then not B’), ‘some A are B and some are not B’ (or ‘if A, not-then B and not-then not-B’) and ‘unsure’ by means of only three symbols. And things get still more complicated when natural modalities are taken into consideration!
Our authors do not realize this and cheerfully plunge into their analysis armed with only three alternatives. This is bound to lead them into error (and indeed, as we shall see, errors do arise), though the errors may remain well hidden below the surface of their discourse. Moreover, much depends on how accurately one selects the concepts to be put into rows and columns. As we shall see, it is not always easy to design an appropriate table. If one does not correctly perceive the data at hand, one may misconstrue the rows and columns needed for them. The truth of the matter is that, for all its usefulness, the tabular method of representation of facts is simplistic and awkward. It should only be used with full awareness of the dangers of misrepresentation it involves, treading carefully.
Moreover, I would like to draw attention to the overall complexity and consequent limited practical utility of the “matrix abduction” method. Humans have been reasoning a fortiori for millennia without any reference to the intricate techniques that Abraham, Gabbay and Schild here prescribe or even merely propose. The authors’ approach, even if theoretically applicable and potentially usable with computers (which is open to doubt, as we shall see), has nothing to do with actual human practice. They do not reflect ordinary human ways of thinking a fortiori and therefore cannot effect improvements in these ways. The science of logic should not be considered as a purely theoretical exploration for purposes of computer programming, let alone as a game or a display of virtuosity; it is primarily aimed specifically at improving the art of logic, i.e. at helping real people to think straight.
The authors claim that “Matrix abduction is a new form of induction, arising from the Talmud, which can solve problems currently in the scientific community” (p. 60). As just mentioned, I do not regard the method they propose as entirely new or essentially related to qal vachomer in particular or, I might add, Talmudic reasoning in general. They may have gotten the idea while studying the Talmud, but I very much doubt that it can be said to have arisen from it, in the sense that the Talmudic rabbis used or discussed such a method. Tabulation of empirical data as a means of predicting missing information has been in use since the dawn of modern science. This is often accompanied by graphical representations, where a line is drawn in between all the points marked on the graph. Such a line is used to predict values otherwise not known, the assumption being that the variables compared are subject to concomitant variation.
Regarding the authors’ statement that “rabbinic thinking is very similar to that of western science,” as Menachem Fisch claimed (1997) and contrary to what Jacob Neusner claimed (1987), I would to some extent but not entirely agree. However, I would not go so far as to say as they do that “the logic of the Talmud is far richer and complex than currently available western logic.” They do not say just what these “epistemological comments” of theirs are based on; it is more a hopeful credo of theirs than a demonstrated thesis. The truth is, Talmudic logic and Western logic have some things in common, and each has things to teach the other and things to learn from the other (see my detailed remarks in the conclusions of my Judaic Logic on this issue, which they do not mention).
It is not my wish to dismiss offhand or belittle the work of Abraham, Gabbay and Schild, but only to assess its relevance to the issue at hand, i.e. a fortiori argument. If they have not discovered and understood what a fortiori argument is and how it works, i.e. its form(s) and validity – how can they possibly claim to find appropriate conclusions to premises? Think about that. Is what they do in their paper really pertinent to the issue of a fortiori argument, or are they just demonstrating impressive but unrelated skills? We shall have to answer this question with reference to the concrete cases they actually deal with.
Abraham, Gabbay and Schild deal, in their English paper (I do not here take into account their Hebrew paper, to repeat), with only two Talmudic examples. One, Baba Qama 25a-b, very briefly (on pp. 8-10); and another, viz. Kiddushin 5a-b, in some detail (on pp. 48-59). Let us now take a look at their treatment of these texts.
With regard to Baba Qama 25a-b, I have written much already in previous chapters (7-8) and will here assume the reader has already read them. Our authors present this as a “small example” of Talmudic qal vachomer, though they forgot to state the Talmudic pages concerned. As a “preview of the model,” i.e. of their “matrix abduction” approach, Abraham, Gabbay and Schild represent the problem as a matrix with four cells, of which three have known content and one has unknown content, as follows:
x = ?
They conceive the problem as one which can be approached from two viewpoints (intuitively or by maths, as they put it), each in two different directions (along rows or columns). According to them, it proceeds as follows.
- From the “intuitive” point of view. If we read the matrix horizontally, given x is to ½ as 1 is to 0, then x is more than or equal to ½ (“pay at least half”). If we read the matrix vertically, given x is to 1 as ½ is to 0, then x is equal to 1 (“pay in full”).
- From the “maths” point of view. Horizontally, the answer must either be x = 1 or x ≤ ½. If we assume the latter (i.e. ≤ ½), the two columns are “incomparable.” Whereas, if we assume the former (i.e. 1), we obtain an “ordering” result, which is “nicer.” Vertically, the answer must either be x = 0 or x ≥ ½. If we assume the former (i.e. 0), the two columns “are not comparable.” Whereas, if we assume the latter (i.e. ≥ ½), we obtain an “ordering” result, which is again “nicer.” (The concept of niceness here used is intuitive – but to be defined more rigorously later, the authors promise us.)
Thus, the solution of the problem, by our three authors, whether by intuition or maths, is that x = ≥ ½ or 1 (or vice versa) – which means, more briefly put, that x = ≥ ½. However, they note without further comment: “The actual case is decided in Jewish law as x = ½.” It is surely amazing that the discrepancy between their own expectation and the preferred result in the Talmud does not arouse any reconsideration on their part! Their theory is put in doubt, but they just blithely move on. This is anyway a very superficial treatment of the case…
Note, for a start, with reference to the Mishna, that the proposed horizontal reading corresponds to R. Tarfon’s first argument and the proposed vertical reading corresponds to R. Tarfon’s second argument. However, R. Tarfon concludes with a “1” in both cases, and the Sages conclude with a “½” in both cases. Neither party concludes anywhere with x = ≥ ½ as do our authors overall, and neither party concludes with x = ≥ ½ or 1 as in the intuitive point of view, or with x = 1 or ≥ ½ as in the maths point of view. Thus, the authors’ conception of this debate does not correspond to, or precisely explain, the participants’ conceptions, let alone arrive at the same results.
What their overall conclusion means, knowing that only three alternative penalties are allowed in the Talmudic conception (no fractions other than half, and no amount greater than full compensation), is that x ≠ 0 and x = ½ or 1. In one respect they agree with the rabbinical conclusions – in denying that x = 0. But they are unable to decide whether x = 1 as R. Tarfon claims, or x = ½ as the Sages claim. This suggests (assuming all three methods viable) that their method is less precise than theirs – it yields a vaguer result. Their solution x = ≥ ½ simply means “either R. Tarfon or the Sages may be right.” In other words, they have found no way to decide the issue treated in the Mishna! But let us look at their reasoning again, more critically.
As regards the ‘intuitive’ approach, their vertical conclusion that “x equals 1” is credible, since just as ½ is more than 0, so x must be more than 1, and as 1 is the maximum allowed then x must be equal to 1. But their horizontal conclusion that “x is greater than or equal to ½” is unjustified, for just as 1 is more than 0, so x must be more than ½, i.e. x must equal 1, as R. Tarfon claimed. As regards the ‘maths’ approach, their horizontal conclusion that “x equals 1” is credible, because both “x equals 0” and “x equals ½” would be lopsided. But their vertical conclusion that “x is greater than or equal to ½” is again unwarranted, because here again both “x equals 0” and “x equals ½” would be asymmetrical; so the conclusion can only be “x equals 1” as R. Tarfon claimed.
From this review, it becomes clear that our authors reasoned incorrectly twice out of four. Judging by their own standard of “comparable ordering,” their answer every which way should have been uniformly x = 1, like the answer of R. Tarfon. This result implies that their thinking was essentially argument by analogy (i.e. merely pro rata), rather than a fortiori (of whatever sort). Had they truly reasoned a fortiori, they would have at least been able to offer a formal explanation for the Sages’ dayo objection in relation to R. Tarfon’s first argument (their horizontal reading). They would also had been intrigued by the Sages’ dayo objection in relation to R. Tarfon’s second argument (their vertical reading), albeit the latter’s conformity with their first objection.
Moreover, if we look again at their ‘intuitive’ and ‘maths’ methods, we see that when they are both correctly applied (as distinct from how they applied them), they yield exactly the same conclusion that x = 1, both horizontally and vertically. The reason for that is simply that they are one and the same technique! In their minds, they looked different; but in reality there is no difference between them. Both are simply pro rata argument, i.e. thinking based on ratios. The ‘maths’ method looks more sophisticated because it appeals explicitly to disjunctive reasoning; but in fact the ‘intuitive’ method is implicitly the same. So, the reference to two methods is just evidence of confusion; and it sows confusion. The authors got different solutions from them because they subconsciously wanted to. They had foregone conclusions in mind that they sought to justify. They were not entirely objective.
I would say Abraham, Gabbay and Schild made the two mistakes they made out of a desire to keep the option of x = ½ in the game, so as to somewhat resemble and not exclude the Sages’ position. Had they not done so, their posture would have been evidently for all to see only representative of R. Tarfon’s and therefore insufficient to explain the debate. In the ‘intuitive’ horizontal case, they erred by suggesting that the x = ½ option has a comparable relation to ½ as 1 has to 0. In the ‘maths’ vertical case, they erred by lumping the x = ½ option with the 1 option in the expression ≥ ½, whereas if they had treated it as a separate alternative they would have seen that it too can be eliminated. There was evidently, therefore, a subconscious attempt at manipulation of premises to obtain a desired final result (viz. “x is greater than or equal to ½”). But, fellows, you must be more careful, because there’s always someone out there who will spot such maneuvers!
Moreover, as I show elsewhere in detail, there are various ways to view the Mishna debate between R. Tarfon and the unnamed Sages confronting him. As regards the first exchange, we can say that R. Tarfon’s argument is either analogical (i.e. pro rata), where A is to B as C is to D, or a crescendo (i.e. a fortiori with a pro rata additional premise, and therefore a ‘proportional’ conclusion). Here, the Sages’ dayo objection acts as an arbitrarily imposed (i.e. extra-logically, for moral reasons, albeit by rabbinical inference from the Torah), limitation on ‘proportional’ reasoning. This means that, if R. Tarfon’s argument was a crescendo, the Sages confronted it with a purely a fortiori argument, and it explains why the former concludes with a “1” whereas the latter conclude with a “½.”
With regard to R. Tarfon’s second argument, whether we construe it as analogical (pro rata), a crescendo (a fortiori cum pro rata) or purely a fortiori, the conclusion is the same. So the Sages’ new dayo objection, though worded like their preceding objection, must be interpreted differently (since the previous explanation is inapplicable). The only available explanation is that it is a limitation on the generalization preceding this second argument (whether taken as analogical or a fortiori). Here, then, the Sages are not confronting R. Tarfon with an alternative argument, which generates a conclusion of its own, but merely inhibiting him from inductively forming a major premise and thus drawing the conclusion he sought. Again, this explains their different “conclusions.”
What is noteworthy, to repeat, is that the conception and results of Abraham, Gabbay and Schild do not square with any of these scenarios. As we have just demonstrated, their method of reasoning is essentially analogical (pro rata) throughout, and the reason why they do not get the same results as the traditional analogies is simply that they made errors of calculation. They have not found any other way to bring the Sages’ opinion into the discussion than through such error. Their approach has in fact nothing to do with a fortiori argument, whereas only through understanding a fortiori argument (including the difference between the a crescendo and purely a fortiori forms) can the difference of opinion between the Mishna participants be convincingly explained (as just done above).
The authors make no mention of the distinction between mere analogy (pro rata) and a fortiori argument, or (more importantly) between a crescendo and purely a fortiori argument, which implies they are not even aware of these distinctions. They assume from the start, without reflection, the universal validity of ‘proportionality’. Their method can only produce ‘proportional’ inferences. This means that it can only at best assimilate merely pro rata and a crescendo (a fortiori cum pro rata) arguments. There is no room in their logical universe for purely a fortiori argument; they have not intellectually assimilated this concept.
Also, they nowhere mention or discuss the Sages’ dayo principle, and why and how it ought to apply in the present arguments. The authors do confront this issue in their Hebrew paper, but as far as I can tell their explanation is that the rabbis prefer the lowest value out of caution. This somewhat answers the why but not the how: they do not perceive where exactly the dayo principle operates. They cannot do so, really, because they lack the distinction between mere analogy and a fortiori argument, and between a crescendo and purely a fortiori argument. As well, they show no awareness of the generalizations through which the major premises of both of R. Tarfon’s arguments are made possible, but take the results of such induction for granted. Yet such awareness is necessary to fully understand the Sages’ dayo principle. This failure of awareness is due in part to their resort to tabular representation. This method is pretty well bound to lead to oversimplification and rigidity.
To top it all, the authors remain unfazed by the divergence between their vague final conclusion (viz. x = ≥ ½) and the two precise ones proposed in the Mishna (viz. x =1 for R. Tarfon and x = ½ for the Sages). They apparently imagine that, since their conclusion (which is anyhow, as we saw, in fact fabricated) logically includes, by way of disjunction, both of the Mishna conclusions, they have succeeded in proposing an interpretation of the Mishna. But in fact, all their indefinite conclusion shows (if anything) is that one of these finite conclusions must be correct; but it is incapable of sorting out which of the two is right. So it is as if nothing was said at all!
Thus, it appears from their treatment here (in the essay reviewed) that the claim by Abraham, Gabbay and Schild that their “matrix abduction” approach has something to do with the a fortiori thinking apparently used in the Mishna under scrutiny is not correct. A fortiori argument is a sophisticated deductive process with specific formal rules, yielding exact and reliable results. If the argument is well formed, its conclusion follows necessarily and entirely from the given premises. It is not as the authors assume an inductive, merely analogical process producing a tentative, educated guess, like their proposed “matrix abduction” method. Their method, however valuable in itself, is not relevant to a fortiori reasoning, since the latter is can be formally validated.
Their calling their essay “Analysis of the Talmudic Argumentum A Fortiori Inference Rule (Kal Vachomer) using Matrix Abduction” is therefore misleading advertising.
Abraham, Gabbay and Schild claim that the a fortiori argument used in Kiddushin 5a-b is “one of the most complicated uses of this rule.” Certainly, they here deal with only a part of the whole discussion, which they have subdivided into 16 or 17 “steps.” The opening move in their selection is the following a fortiori argument:
Huppa refers to the wedding canopy, signifying the Jewish marriage ceremony. “Acquisition” here refers to kiddushin, the betrothal of a woman, i.e. the legal status of being engaged to be married. “Money” here refers to a gift of value, such as a ring. Teruma refers to a food offering to priests, and a woman being allowed to eat it signifies that she is indeed married (to a Kohen). This passage would be rewritten by me in standard a fortiori format as follows:
Huppa (P) allows teruma eating (R) more than money (Q),
and, money (Q) allows teruma eating (R = 0) enough to acquire (S);
therefore, huppa (P) allows teruma eating (R > 0) enough to acquire (S).
In the case of money (the minor term, Q), the allowance of eating teruma is nil (R = 0), whereas for huppa (the major term, P), the allowance exists (R > 0). Even so, note well, the middle term R, “allows teruma eating” (i.e. signifying, more broadly, being married, to repeat) can be expressed as ranging from zero up (i.e. ‘0’ is effectively a positive value of R in this range). On this basis, we can construct our major premise, and justify the conclusion from the minor premise, as shown. The argument is thus positive subjectal, since it goes from minor to major. Our three authors, on the other hand, conceive the argument in tabular form, as follows:
x = ?
We can, of course, guess that the column regarding “marriage” is tacitly inferred from the original middle term “allowance of teruma eating” (money does not suffice to allow teruma eating, whereas huppa does) and the column for “engagement” refers to the original subsidiary term “acquisition” (money does acquire, and the question is whether huppa also does so), though the authors do not explicitly tell us all that. It seems, then, that the row labels (money and huppa) refer to causes and the column labels (marriage, engagement) refer to effects. Thus, the table tells us that money produces engagement but does not produce marriage, whereas huppa produces marriage, and it leaves open the question as to whether huppa results in engagement or not.
They explain that this argument (based on R. Huna’s) is designed “to prove that the ceremony itself can do the job of the ring, i.e. it can do the engagement,” and after drawing two “graphs” for x= 0 and x = 1, they consider that the latter is “better” and “more connected,” so their conclusion is that huppa causes engagement. As can be seen, their thinking is based entirely on considerations of symmetry. It is an argument by analogy – but the distinctive a fortiori underpinning of it has been completely hidden away or lost.
In passing, I would propose to them, instead, the following modified table. This would in my opinion be a more accurate rendition of the operative a fortiori argument. If we symbolize the four items involved, viz. huppa, money, permission to eat teruma (i.e. married status), and engagement status, by means of the four standard terms, P, Q, R and S, respectively, we get the following representative table:
Minor term, Q
Major term, P
The logic of it is that since Q has enough R (even though 0) to imply S present, then P has more than enough R (since more than 0) to imply S present. Although this tabular schema is an improvement over the one proposed by our authors, it too is imperfect in that the major premise, which tells us that P has more R than Q, is still left tacit. It has to be artificially signaled by calling Q the minor term and P the major term, but it cannot be expressed within the matrix itself. Consequently, using this representation, we cannot see exactly how the conclusion is derived from the premises.
Note that my first column has two entries ‘1’ (meaning: ‘yes, enough of R’ for S), whereas the authors have a ‘0’ and a ‘1’ (meaning: ‘not married’ and ‘married’). My second column has two entries ‘1’ (meaning: ‘yes, S present’ due to R enough), whereas the authors have a ‘0’ and a ‘?’ (meaning: ‘not engaged’ and ‘engagement status unknown’). What all this signifies is merely a difference in perspective: ‘not married’ is indeed a ‘no’ with regard to marriage (signified by allowance of teruma eating), but I call it a ‘yes’ since it is still ‘sufficient to effect’ engagement (acquisition). As for my having a ‘1’ where they have a ‘?’ – this is again not a disagreement between us, but only due to my saying the conclusion follows necessarily, whereas they consider it uncertain and needing to be “abduced.”
Before reviewing and criticizing how our authors assimilate the Talmudic discussion that follows R. Huna’s opening qal vachomer, I shall now briefly sketch the way I perceive that discussion, based on the very clear translation and commentary of the Art Scroll edition of this tractate. Whereas our authors present the discussion as a debate between a “proponent” (R. Huna) and an “opponent” (unnamed), the Art Scroll edition presents it as a pro and con reflection of “the Gemara,” and I will do the same (finding this approach more realistic). However, I will number each “step” of the discussion as our authors have done, so as to maintain a reference to their work. But I will go through only about half the steps, just enough to prove the main point I will be making. If you do not want to read all the details of this exposé, you may skip it (the indented segment), and just proceed to the summary and conclusions further down.
The opening a fortiori argument by R. Huna, already described above, is labeled step 1a. In step 1b, “the Gemara calls into question the premise of this kal vachomer” by objecting: “And is it really true that the giving of money for kiddushin does not enable a woman to eat teruma?” This objection, note well, is clearly not itself a qal vachomer, but merely intends to put in doubt a premise of R. Huna’s argument. He claimed that although money does not allow the woman to eat teruma, it still effects acquisition. This objection suggests that maybe the woman is allowed to eat, and goes on to argue that this is so according to Biblical law, although Rabbinic law forbids her to do so. In step 2, the Gemara “proposes a new explanation of this kal vachomer,” which I would write as follows:
Huppa (P) completes marriage (R) more than money (Q),
and, money (Q) completes marriage (R = 0) enough to acquire (S);
therefore, huppa (P) completes marriage (R > 0) enough to acquire (S).
This is actually a new qal vachomer, resembling the first in all respects except that the middle term has changed from “allowance of teruma eating” (a special indicative of marriage) to “completes marriage” (by effecting nissuin). From here on, this is rather the qal vachomer that seems to be referred to. In step 3, the Gemara then turns around and “suggests a refutation of this” second argument, by pointing out that “money is unique in that it can be used to redeem consecrated items [hekdeshot] and second tithe [maaser sheni],” whereas huppa cannot so redeem. This objection does not propose a third qal vachomer, note well, but only suggests something which puts the major premise of the preceding argument in doubt, so that the conclusion can no longer be confidently drawn from the minor premise.
In step 4, the Gemara responds to the latter objection with the remark that “cohabitation effects kiddushin” so as to “demonstrate that the ability to effect kiddushin is not limited to those methods of acquisition which can also redeem” – i.e. though cohabitation cannot redeem, it can still effect kiddushin; from which it follows that huppa’s “inability to redeem should therefore not impede its capacity to effect kiddushin.” In other words, the previous attempted refutation of the qal vachomer is denied, which allows the second argument to be maintained, at least temporarily.
In step 5, the Gemara then counters the preceding resolution with the following objection: cohabitation is “unique in that it acquires the yevamah” (this refers to a wife by levirate marriage) – whence, “it does not follow from the fact that cohabitation effects kiddushin that chuppah should do so as well.” What this counter objection does is annul or weaken the previous objection, so that the second argument is again put in doubt. In step 6, the Gemara responds by arguing: “The fact that money effects kiddushin will demonstrate that the ability to effect kiddushin is not limited to those methods of acquisition which can also acquire a yevamah.” What this does is remove the previously indicated doubt, i.e. it reaffirms the major premise by allowing that kiddushin might be effected otherwise than by money or cohabitation.
In step 7, the Gemara sums up that “neither money nor cohabitation carries the unique ‘strength’ of the other,” so that (as stated in the Notes) “neither of the two methods can serve alone as a source that chuppah effects kiddushin, since each displays a ‘strength’ that chuppah does not share.” The purpose of this comment is suggest that “the ability to effect kiddushin is not tied to either of their unique strengths” – i.e. that there is some unidentified underlying factor, in common to them, that makes them capable of effecting kiddushin. This common ground between them and with huppa is identified in the next step.
In step 8, the Gemara adds: “However, the characteristic common to both of them is that each acquires elsewhere, i.e. each has a capacity to effect a transaction other than kiddushin – and each acquires here, in the case of kiddushin. So too, by means of analogy, I will include chuppah in the class of ways a woman can be acquired, for it acquires elsewhere, and therefore should acquire here, i.e. should effect marriage without the need for prior kiddushin.” In the Notes, this is further explained as: since “chuppah effects nissuin,” which gives the husband certain rights, “it possesses the capacity to effect a transaction” and thus “should be able to effect kiddushin.”
And so it goes, on and on; but I will stop here having accumulated enough data to make my point. My point is simply this: as you can see, most of the listed “steps” of the argument are not a fortiori arguments. The first step (1a) is a fortiori reasoning. The next (1b) simply intends to weaken a premise of that argument. Step 2 proposes a rival qal vachomer, which goes in the same direction but with a more reliable middle term. Step 3 raises a doubt concerning the major premise of this new argument. Step 4 attacks the preceding objection by pointing out an exception to it. Step 5 puts the preceding objection in doubt, which again weakens the proposed argument. Step 6 dissolves the preceding doubt, which again strengthens the (new) a fortiori argument. Steps 7 and 8 effectively abandon the attempt at a fortiori argument, and resort instead to an argument by analogy that I would rather classify as binyan av (a kind of causative argument). Subsequent steps (9-16) present objections and counter-objections to the latter argument, until the analogy is appropriately refined.
What is clear from this sequence of events is that it is totally misleading to refer to the totality of this discourse as “qal vachomer” or “a fortiori argument” as Abraham, Gabbay and Schild do. It involves one or two genuinely qal vachomer statements; but most of the other statements are attempts to either put in doubt one of the premises of a qal vachomer (and thus put the argument in doubt), or to object to the objection (and thus recover some credibility for the qal vachomer). Furthermore, some steps are neither qal vachomer themselves nor have nothing to do with any qal vachomer! Obviously, just because the opening argument in the series is an a fortiori argument, it does not follow that the whole series can be treated under the heading of a fortiori argument.
Yet our authors proceed precisely in this wrong manner, cheerfully lumping together apples and oranges in the same qal vachomer basket. Thus, for instance, the way they integrate step 3 into their tabular schema is simply by adding a third column (signifying redemption, pidion) to it, as follows:
x = ?
Since money redeems but huppa does not, whereas huppa effects marriage but money does not, it is now impossible on the basis of symmetry to decide whether x should be 1, as previously supposed, or 0, instead. In fact, as we have seen, the purpose of the Gemara’s statement about the distinctive redemptive power of money is to weaken the major premise of the proposed a fortiori argument; but since our authors’ tabular schema does not make possible a representation of the major premise, they are forced to express this objection by means of an additional column. This expedient gives the impression that the a fortiori argument has been amplified, although in fact it has been dissolved.
After that, the authors continue representing the ongoing Talmudic discussion by means of additional tables. The third table is like the first (with 4 cells) except that its first row label is changed; the fourth table is like the second (with 6 cells) except that its first row and third column labels have changed; the fifth table has 12 cells, its 3 rows and 4 columns being all those used in the four preceding tables. These first five tables cover steps 1-8. Two more tables, with respectively 15 and 24 cells, cover the remaining steps. As the tables get bigger, their corresponding graphs of course get more complicated.
I will not here repeat their whole presentation in detail, nor make any effort to verify their specific interpretations or results, since (as will presently be made clear) there is no need to do so. Finally, the authors get the answer they were hoping for, which apparently matches that given in the Talmud (viz. x = 1). Whereupon, they conclude:
“We built a matrix abduction model whose components and concepts use only topologically meaningful notions (and hence the model is culturally independent) and we used it to analyse an involved Talmudic argument and we got a perfect and meaningful fit.”
However, if we observe the actual flux of these authors’ matrix analysis of the Kidushin 5a-b discussion, building up representative tables by changing, adding or subtracting variables, i.e. rows and columns – it is clear that what is involved here cannot by any stretch of the imagination be conceived as a mechanical process. It is a process quite dependent on human agency, a process of ad hoc tailoring of overlays to match given data. The accuracy of this process depends primarily on the intelligence of the observer and interpreter of the data. It is not, to repeat, a mechanical process of inference, one that could be fed into a computer and proceed by itself. Therefore, one can well suspect that the authors obtained a final conclusion that perfectly fits (or so they claim) the Talmudic one, because they tailored their discourse to that precise end.
But my main question is, here again, what has all this to do with a fortiori argument? Is it essential, incidental or accidental to the issue at hand? The answer should be clear: purely a fortiori argument formally involves, and can only involve, four terms (the major, the minor, the middle and the subsidiary); a crescendo argument may be said to involve five terms, insofar as the subsidiary term has a different value in the minor premise and in the conclusion. Thus, a fortiori argument can never involve more terms than that, contrary to what the authors suggest when they draw tables with multiple columns and rows.
Therefore, they have missed the essence of a fortiori argument. This does not mean their tables are wrong; it only means that such tables are not representative of a fortiori argument per se. A fortiori argument can apparently (only very roughly, as already explained) be tabulated in the way they have done it; but—this changes the argument from a deductive to an inductive operation. The underlying logical power of a fortiori is lost, and a more diluted form of reasoning (which is mere analogy) is used in its stead. Note this well.
What is manifestly lacking in their account here, notice, is a credible general analysis of the Talmudic processes of objection (pirka or teshuvah) to qal vachomer argument. I have proposed a detailed analysis of this discursive phenomenon in the chapter on Moses Mielziner (13). The latter described such objections as “refutation of inferences,” which may be achieved by attacking a premise or the conclusion. But as I showed there, even Mielziner’s understanding of the process was inaccurate because too vague.
If premises A and B are considered as logically implying conclusion C, ‘refutation of the inference’ would consist in saying that A and B do not together imply C; but this is very rarely what is intended in Talmudic objections. Usually, either premise A or premise B is denied somewhat (or even both A and B are denied, though that is rarer), by some means (e.g. some other Biblical passage is brought to bear on the case at hand), whence it follows that conclusion C can no longer be inferred (which does not mean that other ways might not be found to maintain it, though it could also be denied); this is more prosaically, from a formal point of view, ‘refutation of a premise’.
When a rabbi proposes a qal vachomer and another opposes it, the latter is usually not positing a counter a fortiori argument of his own, but merely putting the proponent’s argument in doubt in some way. Usually, this is achieved by denying the truth of the major premise, or at least its general applicability, i.e. its applicability in the particular case at hand; alternatively (or as well), the proponent’s minor premise might conceivably be attacked. Very often, too, the conclusion is apparently challenged; but closer examination reveals that this is not, as might be thought, a rejection of the proponent’s reasoning process, but simply an explicit denial of his conclusion that implies a tacit denial of one of his premises. In very few cases is the reasoning process as such rejected as fallacious.
If we look at the series of arguments in Kidushin 5a-b that our authors have tabulated, we see that they do not constitute a chain (sorites) of a fortiori arguments and counter-arguments. Rather, what we have before us is an initial a fortiori argument or two; followed by a series of objections and counter-objections which apparently put a premise of an a fortiori argument in doubt to some extent, or recover its credibility to some extent; and then an argument which is not a fortiori at all but really binyan av; and so forth. This is, to repeat, not a chain of competing a fortiori arguments, but a trial and error process of individual reflection (or a give and take process of debate) through which a convincing conclusion is gradually sought and presumably found, which takes into consideration all data at hand in a consistent manner. This process does involve at least one a fortiori argument, but it is certainly not limited to that one form of reasoning.
Therefore, the presentation by Abraham, Gabbay and Schild is mostly beside the point. It is best characterized as tinkering. It may well (as they claim, but I have not checked it) succeed at generating the same final conclusion as the Talmud (at least in the example given), but that does not demonstrate that they understand the nature of a fortiori argument as such. This is evident, given the vagueness of their observations as to what is going on in these Talmudic discourses – i.e. the fact that they do not in their tabular schema reflect the significant differences between actual a fortiori arguments and objections thereto which are not themselves a fortiori arguments, or even arguments which are not at all related to a fortiori. They treat all elements of the dispute indiscriminately as “a fortiori,” simply because the chain of reasoning that they selected started with an a fortiori argument. This is not a “meaningful” model.
Abraham, Gabbay and Schild also go through some non-Talmudic examples – but what is evident there again is that they have nothing or little to do with a fortiori argument specifically. Their applications are just attempts to fill in cells with missing data by extrapolation from information available in other cells of the same table. Quite possibly, identical results would be obtained if the sample arguments involved were mere analogies instead of a fortiori. To label something as “a fortiori” is not proof that it is in fact so; the qualification refers to a distinctive movement of thought that has to first of all be correctly perceived and understood.
We can, therefore, conclude by saying that, albeit their valiant effort, the three authors have not succeeded in (to use their own phrase) “properly formulating” this form of argument. As we have seen with reference to their analysis of the Baba Qama example, their approach is essentially analogical and therefore inductive. We also found that they wrongly processed even these simple arguments by analogy, so that their conclusions were anyway incorrect and for that reason in disagreement with the traditional ones for such form of reasoning. Possibly the same sort of unconscious manipulation and error might be found in their treatment of the Kiddushin example, though I did not take the trouble to look for it. I would certainly recommend to them to give their account a long and hard second look.
Moreover, their “matrix abduction” method (though no doubt valuable in itself as an aid to induction) does not truly represent a fortiori argument, because it is based on a false premise, a false stimulus; namely, the supposition that such argument is constitutionally not deductively certain and therefore must needs be dealt with by inductive means. And the reason they came to adopt that false premise became apparent in their treatment of the Kiddushin example, when they made no formal distinction between a fortiori argument and mere objections thereto or other forms of argument, and expanded the proposed matrix beyond the maximum allowable of four cells, making it a sort of pot pourri of disparate reasoning processes.
Moreover, it should be noted that the two Talmudic examples we have above considered both refer, at least in appearance (though not in reality), to positive subjectal a fortiori argument. The authors show no awareness of the positive predicatal form, and none of the negative forms of the argument. Presumably, these could be fitted in to their schema; but it was up to the authors to demonstrate it. Logicians must always aim for universality, and formally consider all variations on a given theme. I can say from hands-on experience that such further research is often very helpful in deepening one’s understanding of the initially identified form and avoiding mistakes.
Note, too, that the authors nowhere formally locate the various components of a fortiori argument (the terms and relations) in their tables – I had to do it for them (see above, my ‘modified table’ (24.1) with the row and column titles identified as P, Q, R and S). This again shows a lack of formalism on their part, which is unacceptable in scholarly discourse by logicians. Such informality is no doubt the root cause of their mistakenly opting for an inductive solution when deduction was in fact possible.
I should also here repeat that the science of logic itself is not a purely theoretical construct divorced from human practice. The work of logicians is not merely speculative or inventive, but requires empirical observation of actual human thinking. We first observe how human reason actually functions in different discursive situations; then we try to identify the precise conditions of validity of such thought processes. We do not approach the matter academically and impose artificial thought forms on the mind, especially when the mind already has natural ways and means. The “matrix abduction” method proposed by our authors is of course not in itself artificial – but it is artificial in relation to a fortiori reasoning.
I am tempted here to get heavy and quote Isaac Newton on the scientific method:
“For the best and safest method of philosophizing seems to be, first to enquire diligently into the properties of things, and to establish these properties by experiment, and then to proceed more slowly to hypotheses for the explanation of them. For hypotheses should be employed only in explaining the properties of things, but not assumed in determining them, unless so far as they may furnish experiments.”
Newton was of course referring to physical science, not to the science of logic. But the methodological principles are approximately the same for all sciences, i.e. all consciously rigorous pursuits of knowledge. Many logicians nowadays tend to rush to judgment, and seek to develop tabular schemes, graphical models, symbolic formulas, and what have you, before having taken the trouble to carefully observe and examine the concrete data at hand. They aim their intellectual arrows in the approximate direction of the target, hoping to hit the bull’s eye by luck – but that won’t do. The form must exactly fit the content, no less and no more.
Abraham, Gabbay and Schild will of course retort that they used the scientific method, insofar as they tested their “matrix abduction” hypothesis on two actual Talmudic examples. Having in the more complex of the two examples obtained “a perfect and meaningful fit,” they felt justified in claiming an “achievement.” But the truth is, as they themselves admit, they did not get the same answer as the Talmud in their first, simpler example. So that is a success rate of only one out of two, at best!
Moreover, I presume that they got the same answers as the Talmud in the second example because they say so – but not having myself analyzed their every step in detail, I cannot independently confirm that this is indeed the case. I stopped considering it necessary for me to follow our authors’ interpretations of the Talmudic arguments and counter-arguments, remember, the moment I realized that they were lumping together one or two a fortiori arguments with objections and counter-objections thereto, as well as other types of argument, so that effectively they regarded as “a fortiori argument” a portion of discourse capable of containing more than four (or five) terms.
But anyway, even granting that the authors followed the second Talmudic debate correctly, the fact remains that they did not manage to get a likewise positive result for the first debate. As I show above, they actually made errors of calculation. Okay – anyone can make an error. Let us declare their errors are now corrected, so that their tabular approach now produces credible results matching R. Tarfon’s. This would still be an inexact representation of the Mishna debate, since the Sages’ part in it would remain unexplained. Moreover, even if both these examples had been correctly represented by the “matrix abduction” model, it would not follow that other applications of the proposed model would automatically be correct, because this model is far from mechanical, depending greatly on human interference.
And indeed, even generously assuming that both examples, and a few others besides, may be considered as successfully assimilated by their hypothesis, does it follow that their hypothesis is established? I think not. We cannot generalize from two instances, especially not when the ‘empirical data’ at hand is itself tenuous. For the fact is, as I have shown in my Judaic Logic and elsewhere in the present work, the rabbis do occasionally (if not often) engage in fallacious reasoning, including fallacious a fortiori argument. I am sorry to say it, but it is demonstrably true. Therefore, Talmudic discourse is not strictly speaking ‘empirical data’ that can be safely relied on for confirmation of such a hypothesis.
This being the case, what will our authors do with their “matrix abduction” hypothesis when its conclusions perchance do not match those of the Talmud? They cannot declare the Talmud is wrong, for their method is inductive and therefore constitutionally not capable of judging deductive validity or invalidity. Would they then declare their hypothesis wrong? But if the Talmud can be wrong, their hypothesis might still be right! So nothing is settled. Clearly, the authors cannot appeal to conformity with one or two Talmudic results as a gauge of the accuracy of their hypothesis. Such conformity proves nothing much, except by circular argument. If the “matrix abduction” method perchance obtained results that matched some Talmudic arguments that happen to be erroneous, it would wrongly be judged thereby to be scientifically confirmed on the false assumption that the Talmud’s arguments are always reliable.
Finally, let us reiterate that the proposed method rests on mere analogy, and not as its proponents claim on a fortiori argument. It follows from such general methodological considerations that the “matrix abduction” method, whatever its merits as a general inductive technique, is in any case not particularly relevant to the solving the theoretical problem of a fortiori argument. What is needed is a deductive method of formalization and validation and invalidation, and I have already successfully developed that method in my Judaic Logic and further still in the present volume. Unfortunately for them, Abraham, Gabbay and Schild did not study my first work on the subject, though they knew of its existence.
 Studia Logica 92(3): 281-364 (2009).
 (Hebrew paper on Kal-Vachomer.) In the journal: BDD (Bekhol Derakhekha Daehu). Ramat Gan: Bar Ilan University, 2009.
 Note: the page numbering I use here refers to the 84 page pdf version I have of the English paper (#338); so my page 1 is really page 281 in the Studia Logica publication; so add 280 pages to any page number I use. The Hebrew paper (#340) pdf in my possession is 116 pages long.
 I asked Dov Gabbay for a translation of the Hebrew paper, but he did not provide one; in my opinion, if one wants one’s work to be considered internationally, one must provide translations of it into English (today’s lingua franca). These three authors have also more recently together written in Hebrew a paper called “Logical Model for Talmudic Hermeneutics,” which was published in BDD – in two parts (the first in No. 23 in 2010, and the second in No. 24 in 2011). This work, among other things, presents “a mathematical model representing… Argumentum a Fortiori (Kal Vachomer),” and studies “the methods of refuting such inferences.” I have not read it to date.
 My translation, partly based on a machine translation by Google.
 That is, a hypothetical major premise, a minor premise affirming its antecedent and a conclusion affirming its consequent.
 In a perfunctory attempt at wider research, the three authors also give two examples of a fortiori argument in the “New Testament,” as well as an example drawn from “Sanskrit logic (Nyaya),” where it is “known as Kaimatya Nyaya (or Kaimutika Nyaya, the even more so) rule.” These three examples are okay, but they do not analyze them at all. They also give two examples from Islamic jurisprudence where such argument is “known as Qiyas (analogy).” However, while the second example they give is indeed akin to the Jewish technique of “Kal-Vachomer,” the first is not so. Although the original author of the example (it is repeated in many websites, anonymously) correctly analyzes the thinking in it, our three authors do not explicitly identify it as akin to the Jewish technique of “Binyan Abh.” This could be an oversight; but I suspect they lump these two kinds of reasoning by “analogy” together because they cannot tell the difference between them.
 All this concerns the English paper – there may be more detailed analyses on these and other past authors in the Hebrew paper. But I doubt it, seeing the level of knowledge and understanding they display here.
 As regards “Analogy,” we discover later that what the authors mean by that is Binyan Abh (the third hermeneutic rule in R. Ishmael’s list), which does not directly concern us in the present inquiry. Even so, it is worth noting in passing that even binyan av can be shown to be essentially deductive, with reference to the logic of causation.
 I would like to express in passing my antipathy to the word “abduction.” This word was, I gather, rather recently (early 20th century) introduced, by C. S. Pierce, and it commonly used by many modern logicians. I personally prefer the older (15th century) word “adduction.” For me, abduction connotes kidnapping, whereas adduction connotes adding. We adduce information (data, reasons) in support of a hypothesis, we don’t abduct it! If the English language has a perfectly adequate word for something, why introduce a new one? The motive is, I suggest, a sort of snobbery – one seems more intellectual if one uses esoteric language.
 See especially my book The Logic of Causation. A crucial difference between my approach there and the approach of Abraham, Gabbay and Schild in the paper here examined, is that early on in my research (in phase II, 2003) I eschew the unknown “?” value and focus entirely on known “1” and “0” values through grand matrices that analyze all possible outcomes of various situations. This allows for systematic development of deductive arguments, and makes all the difference in the clarity and certainty of the results. They have apparently not realized the superior efficacy of this approach.
 Chapter 15, “Epilogue.”
 Note (as the authors acknowledge in a footnote) that here the alternative to 0, i.e. 1 in the sense of ≥ 0, has two values, viz. 1 (full compensation) and ½ (half compensation). Note (though the authors do not do so) that no value in between or beyond these two is admitted under Jewish law – thus ¼, ¾ or 2, say, are inadmissible.
 They use the label “maths” to suggest a more scientific approach – but in truth, this has nothing to do with mathematics or mathematical logic. It is just another sort of “intuitive” approach, a bit more reasoned since it consists in trying the different possible solutions one by one and seeing if the “ordering” is “comparable.”
 This translation is stated by the authors to be “from the El-Am edition.”
 The Soncino edition explains in a footnote: “If a priest betroths an Israelite’s daughter with money, she may not eat terumah until the huppah.” This is found further on within the text, in R. Ulla’s rejoinder.
 See preceding footnote, where it says: “until the huppah.”
 I take acquisition to mean kiddushin (betrothal), following Tosafot to Yevamot 57b. Other authoritative commentaries, notably Tosafot HaRosh, however consider that R. Huna may have additionally intended nisuin (the final stage of Jewish marriage). See Note 35 in the Art Scrolls Talmud edition, p. 5a.
 This is their Figure 46.
 That would be my proposed table for positive subjectal argument. For positive predicatal, the rows would be transposed, with that for P above that for Q. For the corresponding negative arguments, the first column would contain zeros (meaning: ‘not enough of R’ for S) and the second column would contain question marks (meaning: ‘possibly S, possibly not S’). But in all these cases, the explanation for the a fortiori conclusion remains hidden, note well. Tabulation is thus not a fully revealing way to present such argument.
 Note that when I here quote passages from the Art Scroll edition, I omit the bold type that is used to distinguish literal wording from the translator’s explanatory interpolations.
 This is their Figure 48.
 The same expedient is used some years earlier by Gabriel Abitbol. I show in my critique of the latter’s work why it is not very smart. As I explain there, rather than expand the initial table by adding a column (or a row), it would be wiser to illustrate the objection raised to the initial table by constructing a rival table. But in any case, tables cannot fully illustrate the evolving rabbinic discourse.
 The latter three being referenced as Figures 50, 52 and 54.
 These are referenced as Figures 56 and 58.
 But it happens. A case in point is R. Yehoshua’s objection to R. Eliezer’s a fortiori argument in the Mishna Zebahim 7:4.
 Quoted by Freely, p. 217.
 Maybe they explain the discrepancy in their Hebrew paper, I don’t know – but certainly they nowhere do so in their English paper.
 See for examples my analysis of the qal vachomer arguments in Pesachim 23a-b and Baba Batra 111a-b, in the chapter on Adin Steinsaltz (18.2).