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A FORTIORI LOGIC

© Avi Sion, 2013 All rights reserved.

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A FORTIORI LOGIC

CHAPTER 17 – Heinrich Guggenheimer

1. Tout un programme

2. Theory of a fortiori

3. A faulty approach

Heinrich Guggenheimer wrote a paper called “Logical Problems in Jewish Tradition” (1966)[1] which is often referred to by others. Hyam Maccoby in The Philosophy of the Talmud (2002)[2] made the following remarks concerning it:

“Heinrich Guggenheimer (pp. 181-85) gives a cogent account of the dayyo rule in terms of pure logic, saying that, in virtue of this rule, the qal va-homer argument is ‘an admirable solution (the only one known to me) of the problem of making analogy an exact reasoning’. Guggenheimer also gives a rendering of the qal va-homer in the terminology of modern mathematical logic. He does not mention, however, the Amoraic discussion which takes the rule of dayyo out of the realm of logic, or the considerable medieval discussion based on b. Bava Qamma 25a.”

1. Tout un programme

I have not managed to find a copy of Guggenheimer’s essay to fully comment on it here. Actually, I did read it many years ago, though I do not recall seeing in it the clarifications and proofs Maccoby claims to have seen in it. Of course, what you see or don’t see depends on your own level of logical comprehension[3]. I do mention Guggenheimer’s essay in my Judaic Logic (in a long footnote to chapter 8.2) – very critically – in relation to his statements “The inner logic of the Law… is definitely hostile to modalities…. The Talmud avoids all attempts at modal logic” (pp. 179, 193). So inaccurate a statement makes me very skeptical at the outset as to his overall comprehension.

However, I notice that Allen Wiseman[4] quotes Guggenheimer at some length and analyzes his viewpoint in some detail, so we can here look further into the latter’s theory of a fortiori argument. What appears from this third person account is that Guggenheimer has no precise formula and proof for such argument. He makes some general observations and vague speculations, couched in the language of modern logic with liberal use of fancy Greek symbols, but the bottom line is missing.

Guggenheimer begins by conceiving “the sentences of Scripture, as far as they have legal relevance” as constituting “a system of propositions 0.” This is, in a way, stating the obvious: taking any document as a whole, all the sentences in it form an object of study. But do they form a logically coherent “system”? Those who believe the Talmud infallible may think so. But if we look at the Talmud in detail, there are many controversies in it at many levels; so, it can hardly be viewed as a unitary whole.

Be that as it may, Guggenheimer then proposes “for the sake of clarity” that we “assume that all propositions are of the form ‘φ0(a) is true where a is an element of a given set (of relevant subjects) and φ(x) a predicate.” Now, this is less clear than it seems. If by this form we presume that all the propositions have the specific form of predication of φ to a – we might ask on what basis the author makes such a simplistic assumption at the outset. We know that not all sentences in our minds, or even in the Talmud, fall in this category. Therefore, we must take his form to mean more vaguely that predicate φ stands ‘in some relation or other’ to subject a.

He then presents the problem of logical analysis of the Talmud in the following words: “Our problem is to find a set of predicates φ(x) for which the provable propositions ‘φ0(a) is true under the rules of propositional logics and certain other operations generate a logical system which consists of the true statements of Talmudic Law.” He does not say how exactly these propositions are to be “proved,” but simply assumes that means will be found. Presumably from outside the system, since he later says that such systems are “incomplete.” In any event, these propositions will be used to build up a “logical system” that will churn out “the true statements of Talmudic Law” (p. 181).

This ambitious programme is, note well, partly descriptive and partly prescriptive. He apparently aims to order the data found in the Talmud, distinguishing source material (the “provable” propositions) and derived material (“the true statements of Talmudic Law”); but there is no guarantee offhand that the statements he concludes as “true” (or not so) are those that the authors of the Talmud would consider “true” (or not so).

Indeed, Guggenheimer implies a divergence, when he points out that “We know that the system of statement of 0contains contradictory statements and therefore the usual rules of logics cannot be used in it,” and insists that “We shall ask that the system be free of contradictions if the usual rules of logics are applied to its statements.” He predicts that the outcome will be “not a single system but a whole tree of systems.” Maybe because there is likely to be many alternative resolutions of the contradictions?

So far, to my mind, Guggenheimer has said nothing of any great moment. All he has done is to propose the Talmud as a field of study, in a perspective comprehensible to adherents to ‘modern logic’. He has certainly never gone ahead and executed his programme, traversing the whole Talmud with the systematic ordering he has in mind. It is just something he vaguely imagines as possible on purely intellectual grounds. It remains to the end an untested hypothesis, a speculation.

He apparently assumes he knows all there is to know about logic in advance, and does not foresee any snags on the way – other than occasional contradictions, which he does not tell us how he will resolve. But in truth the ‘modern logic’ conception of logic is very naïve. It oversimplifies complex operations, so as to force them into preconceived forms. This programme is doomed to failure from the start.

2. Theory of a fortiori

Turning now to the more specific problem of a fortiori argument, Guggenheimer says: “This is a fundamental procedure, so much so that it is called din, that is, logic….an admirable solution (the only one known to me) of the problem of making analogy an exact reasoning. As such it is valid not only as a rule of transfer, but also as a rule of derivation within the new system. It is the essential extension of Talmudic logics over propositional (Aristotelian) logics. It works because all [such] systems are incomplete.” He is right in the observation that a fortiori argument is a more exact form of reasoning (derivation) than mere analogy (transfer).

I do not know what he means when he says that a fortiori argument is the essential “extension” of Talmudic logics “over” propositional (Aristotelian) logics. Does he mean that Aristotelian logic historically lacked a fortiori argument, or that it is logically incapable of explaining it? Either statement would be inaccurate. Aristotle and other non-Talmudic players were aware of a fortiori argument and did discuss it, even if not as thoroughly as they discussed other forms of argument. And furthermore, as I have shown, a fortiori argument is reducible to previously known forms of argument, namely hypothetical syllogism and inference of quantitative comparisons. I also do not know what he intends when he says: that a fortiori argument “works because” Talmudic systems are incomplete. Why should the incompleteness of Talmudic systems favor the working of a fortiori argument?

These statements of Guggenheimer’s do not clearly ask or answer the essential questions: what is the exact form (or what are the exact forms) of a fortiori argument? And: is a fortiori argument as such, whether used in the Talmud or elsewhere, valid or not (or which of its possible forms are valid and which not)?

But fortunately, Guggenheimer gets more a bit specific. He observes that predicates may be compared and thus ordered; one predicate may be “stronger” than another. For example, “the severity of the penalty for an offence”[5]. He points out that “Mathematically, the main property of an order relationship is its transitivity;” that is, given comparable predicates, if one is stronger than another, and the latter is stronger than a third, then the first is stronger than the last. Such “ordering,” he warns us, concerns predicates, not whole “statements.”

From this[6] we learn that Guggenheimer was aware that a fortiori argument somehow involves quantitative comparisons. But this is not yet, by far, a thorough understanding of a fortiori argument. Finally, he attempts the following definite thesis:

“If there exists a predicate χ(x) and elements a and b such that ‘χ(b) is true’ is a provable proposition in the original system, but neither ‘χ(a) is true’ nor ‘χ(a) is false’ can be proven on the basis of the available data without recurrence to a new rule, and if a is stronger to[7] b, then the rule kal vahomer states that some statement ‘μ(a) is true’ must hold for some predicate μ(x) which is not weaker than χ(x). By the axiom of definiteness, the only possible solution is μ(x) = χ(x).”

What does that signify? In an attempt to decipher what he had in mind, let us try to recast the argument as he sees it in standard form. What is evident is that Guggenheimer is far from clear as to the roles of the terms he mentions. The “elements” a and b are presumably, since a is stronger than b, respectively the major and minor terms (P and Q). And they are subjects, since a predicate χ(x) is applicable to them. There is no mention of a middle term (R), i.e. no statement telling us in respect of what a is stronger than b. Presumably, predicate χ(x) plays that crucial role, somehow. The predicate μ(x) is apparently intended as the subsidiary term (S), since it figures in the conclusion. Granting these equations, we might propose the following positive subjectal a fortiori argument as representative of what Guggenheimer perhaps had in mind:

a is more χ than b (is χ),

and b is χ enough to be μ;

therefore, a is χ enough to be μ.



From this we see that what he subconsciously meant when he says that “a is stronger than b” is that χ(a) is greater than χ(b). This is effectively his major premise. When he says that “‘χ(b) is true’ is a provable proposition in the original system,” he means that ‘b is χ to some degree’ is a given. This is part of his minor premise. When he tells us that “neither ‘χ(a) is true’ nor ‘χ(a) is false’ can be proven on the basis of the available data without recurrence to a new rule,” he means that ‘a is χ to some degree’ is not a given. In fact, it is implied by the sentence “χ(a) is greater than χ(b),” which is the intent of his “a is stronger than b” – but he does not see that. What he is trying to say, anyway, is that the status of χ(a) will be settled as part of the conclusion, which has not yet been determined.

Now, when he refers to “the rule kal vahomer,” he is postulating that there is an inference “rule” – whether in logic generally or in Talmudic logic – which allows us to draw some (yet unspecified) conclusion from the given premises. On what basis he assumes such inference logically permissible he does not say[8]. So his words “the rule kal vahomer states” should not be taken as suggesting that he has already brought to bear some known and authoritative “rule.” For the moment, at least, he is merely describing a hypothetical process; he has not demonstrated its universal or local validity, note well.

The “rule” he proposes is “that some statement ‘μ(a) is true’ must hold for some predicate μ(x) which is not weaker than χ(x).” This is useful to us, not only to flesh out his putative conclusion, but also his minor premise. For as we have seen, his minor premise is still at this stage only partly formulated. It is thanks to the mention of μ here that we can formulate his minor premise, namely that “b is χ(b), and so χ enough to be μ, specifically μ(b).” Although he does not specifically mention μ(b), it is implied in his general term μ(x). This tells us that ‘b is μ to some degree’, and so amplifies the minor premise.

Where he refers to “some predicate μ(x) which is not weaker than χ(x),” he supposedly has it in mind that ‘there is a threshold value of χ that must be attained before a or b (or any x) can be μ’. That is, to be μ to any degree, something x must be χ enough. This is quite a perspicacious insight on his part. His rule especially predicts that ‘μ(a) is true’ will appear in the conclusion – i.e. that the conclusion will include the information that ‘a is μ to some degree’.

Note that, although he forcefully says that this “must hold,” he is still in the realm of speculative insight. Furthermore, his comparison of μ(x) to χ(x) betrays some confusion as to the respective roles played by these two predicates. Strictly speaking, his condition that μ(x) be “not weaker than” χ(x) is an error. These two terms may not be comparable. This gap in his understanding, as we shall next see, makes him formulate the conclusion erroneously.

Thus far, he has approximately identified the components of the putative conclusion, viz. ‘a is χ enough to be μ’; but he has not yet pinpointed a specific value for μ. Is the value to be μ(a), as his statement “‘μ(a) is true’ must hold” seems to imply, or is the value to be μ(b)? This is where his “axiom of definiteness” comes into play, and prescribes the lesser predicate μ(b) to subject a in the conclusion. This is clearly the intent of his formula “the only possible solution is μ(x) = χ(x).” He instinctively opts for the known minimum value, and rightly so, even if he has not clearly verbalized his choice.

The correct explanation is as follows. Even though subject a is χ(a), we have no given that χ(a) is enough to give rise to μ(a). All we are given is that χ(b) is enough to give rise to μ(b), and that χ(a) implies χ(b). Whence, we can only deduce that a is μ(b). This is what Guggenheimer means by “the only possible solution” – that we do not have enough information to deduce more than this minimum, and cannot conclude that ‘a is μ(a)’.

His reference to “μ(x) = χ(x)” is not correct. It again only reflects his personal confusion as to the precise role played by these two (and indeed all) terms. The subsidiary term obviously does not equal the corresponding middle term, which may be something quite different and incommensurable. What he was trying to say is that the value of μ in the conclusion cannot exceed the value of μ in the minor premise (which is true, so long as no additional premise is brought into play to justify proportionality).

From this reading, we can comprehend Guggenheimer’s “axiom of definiteness” as corresponding (in the present context, at least) to my “principle of deduction.” There is an “axiom of definiteness” in some versions of set theory (though others eschew it), having to do with the equation of certain items. But I am not sure that Guggenheimer’s “axiom of definiteness” strictly corresponds to the latter in the present specific context; I think he just put forward this impressively named idea to buttress his conclusion, i.e. to make it seem “scientifically” justified.

Guggenheimer had to appeal to a vague “axiom” – a seemingly unassailable first principle, however obscure – in order to rationalize his proposed conclusion, because he was not clear as to its precise justification. And he was not clear about its precise justification because he was not clear about its precise content. Knowing the latter, we can more clearly see that the minimalist conclusion μ(b) is all we can draw, because we do not have sufficient information to draw a more weighty conclusion, viz. μ(a).

Needless to say, all the above effort refers only at best to positive subjectal a fortiori argument. As with many other researchers, he does not mention negative subjectal, or predicatal, or implicational, moods of the argument. Also note that, though he rightly upholds the minimal non-proportional conclusion, he does not foresee the possibility of a proportional conclusion given additional information.

3. A faulty approach

Let us now review our examination of Guggenheimer’s thesis. Although we are able to guess at what his intentions probably were, we can see after detailed analysis that he did not succeed in putting them down on paper convincingly. He perceived some relevant aspects of a fortiori argument, but not all. He understood some relevant aspects of a fortiori argument, but not all. He spotted the terms involved, but not their exact roles in the discourse. He roughly intuited the right conclusion, but due to his confusion was not able to formulate it correctly. On the whole, then, his analysis was incoherent and of little value.

We should also take note of the following faults in his approach. The exact forms of the premises and conclusions, and positions of the terms in them, are not exposed. Vague statements are considered sufficient. No logical explanation is given as to why the putative conclusion proceeds from the given premise. The inference is merely rationalized with reference to an arbitrarily declared “rule,” instead of being justified by rational insight. Additionally, an obscure “axiom” is brought into play as convenient.

And lo and behold, the “axiom” proposed here just happens to correspond, at least in some circumstances, to the rabbinical dayo principle that Guggenheimer is trying to explain or justify: “This particular application of the principle of definiteness is known as dayo, Baba Kama, 24b. Its Biblical root is Numbers XII, 14 concerning the punishment of Miriam for her slander of Moses. The comparison is between punishment by the Deity and punishment by the father. Even though the first is infinitely stronger than the second, the punishment is quantitatively the same.”

(I should add that Guggenheimer’s consideration of the dayo principle here only covers one version of it, the objection of the Sages in reaction to R. Tarfon’s first argument in the Mishna Baba Qama 2:5. The second objection of the Sages, in reaction to R. Tarfon’s second argument, is significantly different, and remains unexplained by Guggenheimer’s interpretation.)

What we have here is “custom-built” logic. From the point of view of formal logic, Guggenheimer’s thesis is non-existent, a fantasy. He has taught us nothing. All he has said is that he believes there is, somehow, such a thing as a fortiori argument, and it does, somehow, as the rabbis seem to have advocated[9], yield a non-proportional conclusion. His ‘logic’ is one of vague and unsubstantiated claims.

Actually, Guggenheimer does not make these claims in his own name, but regards them as implied in the Talmudic system of logic. According to Wiseman (note that by “QC” he means qal vachomer):

“Guggenheimer says that the QC as a rule ‘transcends elementary logics’ and derives from the Bible. ‘Accordingly, the statement of the QC rule itself is an element in the system ∑ … limited to cases in which the statement χ(b) is true can be proven’, both without the use of the QC and by it alone. In all, although he admits that the QC can fail, it can work in certain cases.”[10]

Thus, Guggenheimer’s claims are not intended by him as absolute, but as relative to Talmudic logic specifically. They are not universal logical principles, but descriptive of a particular world of discourse. This new claim is, needless to say, quite untrue! A fortiori is a definable and demonstrable form of argument, of universal validity and widely used. Guggenheimer thought otherwise simply because he was unable to actually formalize and validate the argument.

Guggenheimer’s essay has some prestige in certain quarters due to his appeal to ‘modern logic’. But from our closer scrutiny of his work, it is evident that his recourse to symbolic logic did not aid him, and even on the contrary confounded him. As I keep repeating, there is no magic in symbols. The thought that they can somehow make an analysis clearer and more scientific is a lure. Symbols are just shorter words; they only serve to abbreviate longer discourses. They say nothing more than ordinary words do – indeed, they say less, and thus lose much of their semantic potential.

A symbolic treatment can never be more accurate than the underlying insight driving it. If a researcher has not understood the subject matter, his recourse to symbols won’t help him – it will make things worse for him. There is no harm in symbolizing when one has attained understanding; whereas to symbolize before that is sure to lead to error. People who symbolize early generally do so because it allows them to conceal the triviality, if not the confusion and error, of their thinking. People who are impressed and attracted by symbolic treatment condemn themselves to being misled. There is no substitute for understanding and insight. It is interesting to note that (so far as I know) none of the advocates of modern logic who succeeded Guggenheimer have noticed the errors in his treatment.



[1] In: The Great Society: Confrontations with Judaism. A Symposium. Ed. Philip Longworth. London: Blond, 1966.

[2] London: Routledge Curzon, 2002. See online excerpts at: www.mucjs.org/qalvahomer.htm.

[3] As regards the deficiencies in the Gemara mentioned by Maccoby, the latter as we have seen was not really in a position to judge the matter since he had not himself fully correctly understood it.

[4] A Contemporary Examination of the A Fortiori Argument Involving Jewish Traditions (Waterloo, Ont.: University of Waterloo, 2010), pp. 89-91.

[5] As in Numbers 12:14-15, where the penalty prescribed for offending God would rationally seem to have to be more severe than that for offending one’s father – although, in truth, the Bible prescribes the same penalty in both cases.

[6] I have left out some of Guggenheimer’s discourse, considering some ideas as too irrelevant to even mention. For instance, his “predicate of second type,” which tells us that one predicate is stronger than another, is a useless complication. Again, his claim that transitive relations may be circular is absurd, the relative severities he gives as examples being mere figments of his imagination that are nowhere upheld in the Talmud and do not make sense anyway.

[7] I do not know if the word “to” is found in Guggenheimer, or is an error by Wiseman.

[8] To my mind, anyone who speaks of “rules” of inference in that way does not really know logic. Logic is not about abstract “rules” imposed on mankind by modern logicians; it rather has to do with understanding.

[9] As a matter of fact, the rabbis did not consistently advocate or apply the dayo principle. Guggenheimer apparently did not study the Gemara he cites very closely. As for the claimed axiom or principle of definiteness, Wiseman is quick to reject it, rightly pointing out that proportional a fortiori argument is a common occurrence in both general and rabbinic discourse (p. 94).

[10] Guggenhiemer, p. 183; Wiseman, p. 90.

2016-06-14T04:54:23+00:00