A FORTIORI LOGIC
CHAPTER 19 –Jonathan Cohen
Alexander Samely tells us, in hisRabbinic Interpretation of Scripture in the Mishnah:
“In an informal conversation with me in July 1984 the Oxford logician Jonathan Cohen once worked out the following representation of thea fortiori, according to my notes:(n)[(x)(y)(F(x,y) = n → G(x,y)=m) → (a)(b)(F(a,b) > n → G(a,b)>m)].”
Samely does not further clarify or justify this symbolic formula, as if its meaning and truth are obvious. We shall assume here that he reported it fully and correctly.
Allen Wiseman proposesthe following interpretation of Cohen’s formula in plain English: “For all n, and for any x and y, if they have a feature of equality in n, then another factor is equal to it as m, so that for any actual cases a and b, with the feature between them greater than n, then the other factor is greater as m.” However, although he is a commentator well versed in issues ofdayo, Wiseman does not discuss the validity of Cohen’s formula.
My own reading in ordinary language of Cohen’s formula would be as follows:
Given that:a certain thesis F, involving terms x and y, containing some quantitative factor n, implies that thesis G, involving the same terms x and y, contains some quantitative factor m,
it follows that:a like thesis F, involving terms a and b, containing some quantitative factorgreater thann, implies that thesis G, involving the same terms a and b, contains some quantitative factorgreater thanm.
Clearly, what we have here is a premise and a conclusion, each of which has hypothetical form (i.e. consists of an if–then statement). Cohen is saying: Given the truth of the first if–then statement, it follows that the second if–then statement is true too. No proof of this is provided, note well; it is just an assertion. We are not toldwhythe first statement should be taken to formally imply the second. More on this issue in a moment.
According to Wiseman, Cohen intends a and b as particular instances of abstractions x and y (respectively). This may well be the case. But if so, note well that the premise (involving x and y) is not identical in form with the conclusion (involving a and b) – so we do not here have a simple case of application of a general principle to a particular instance. It follows that nothing is gained, no progress is made toward validation, by having a and b as instances of x and y – if anything, we would be artificially limiting the applicability of the argument presented by Cohen if we in fact managed to validate it. In any case, as my analysis of a fortiori argument well shows, there is no rule concerning it, such as the one Cohen apparently assumes, that the items involved must be generalities and particulars.
In my opinion, therefore, mention of the terms x, y and a, b is superfluous, and only obscures the essence of the argument proposed. It is obvious that Cohen inserts these extra details in order to be in servile compliance with the pretentious conventions of modern symbolic logic. But they are encumbrances, useless complications, which do not in fact affect the process of inference of concern to us. We could considerably simplify the statement by referring to F(x,y) and G(x,y) as F1 and G1, and to F(a,b) and G(a,b) as F2 and G2. The suffixes 1 and 2 suffice to indicate that F and G with the same suffix have certain terms in common, and that the F and G in the premise are different in some way from the F and G in the conclusion. That is all that is needed to express the situation. Granting this clarification, Cohen’s statement is effectively that:
Given that:a certain thesis F1 involving a quantitative factor n, implies that thesis G1 involves a quantitative factor m,
it follows that:a like thesis F2 involving a quantitative factorgreater thann, implies that thesis G2 involves a quantitative factorgreater thanm.
Looking at this simplified statement, we could well suppose that what Cohen had in mind was not a sophisticated a crescendo argument, but the simpler pro rata form of argument. But if we grant him the benefit of the doubt and suppose that he indeed did have a fortiori argument in mind, the following would be the standard form (using our usual P, Q, R, S terminology) that would most adequately reflect such underlying thought:
F2 (P) involves more n (R) than F1 (Q) does;
and F1 (Q) involves n (R) enough to imply G1 to involve some m (S);
and m (S) is proportional to n (R);
therefore, F2 (P) involves n (R) enough to imply G2 to involve more m (>S).
Cohen’s formula evidently refers to a positive antecedental a crescendo argument, going from minor to major. The sub-theses F1 and F2 (the antecedents in his premise and conclusion) are respectively the minor and major items (P and Q). The quantitative factor ‘n’ in common to them is the effective middle item (R), and the quantitative factor ‘m’ in common to the consequents in Cohen’s premise and conclusion is the effective subsidiary item (S).
Cohen’s formula is in fact not as sophisticated as I here present it on his behalf. He does not explicitly mention the major premise, though we can readily see that it was part of his thought process. Nor does he anywhere hint at the additional premise about proportionality between m and n; more on this serious lacuna in a moment. Nor does his formula contain the crucial notion of sufficiency, which gives the sense that there is a threshold of the middle item (n) to cross before the subsidiary item (m) can be applied. I have kindly filled in the gaps in Cohen’s argument, to show what he was probably aiming at; but his formula as it stands did not hit the target, to say the least.
As regards the subsidiary item, more precisely put, it is not just ‘m’ but the whole implied proposition that ‘G involves m’; this means that mention of sub-theses G1 and G2 was really an unnecessary complication. That is to say, substituting symbol M for ‘G implies m’, it would have sufficed for Cohen to say:Given that:‘F1 implies n’ implies M1,it follows that:‘F2 implies more n’ implies M2 (where M2 > M1). Cohen’s mention of G and m is a fault, because it is a redundancy that conceals the essence of the argument; it introduces inessential elements, which make his formula more specific than it needs to be. The job of logicians is to reduce arguments to their bare essentials, avoiding all irrelevant embellishments.
The argument must be classed as ‘antecedental’ because it involves implications: ‘Fimpliesn’ and this proposition in turnimpliesM. Cohen’s formula is too narrow in that it includes only positive antecedental argument, and makes no mention of the negative antecedental form, or for that matter of the positive and negative consequental forms, of the arguments. Furthermore, Cohen’s formula is too narrow in that it fails to mention the much simpler forms of copulative argument, involving the relation ‘is’ between terms instead of the relation ‘implies’ between theses. That is to say, he misses out on the positive and negative subjectal and positive and negative predicatal moods of a fortiori argument. To be general, his symbolic formula should have clearly included all possibilities.
But the most important issue to raise in relation to Cohen’s formula is that it is in fact fallacious – it posits anon sequitur. That is, his conclusion does not deductively follow from his actual premise. He typically (as many other commentators unthinkingly do) infers a ‘proportional’ conclusion without justification. The a fortiori argument proposed by Cohen is actually not the one I have kindly proposed for him above but the following:
F2 (P) involves more n (R) than F1 (Q) does;
and F1 (Q) involves n (R) enough to imply M1 (S1);
therefore, F2 (P) involves n (R) enough to imply M2 (S2),
where M2 > M1 (presumably in proportion to n2 > n1).
We might call this ‘the fallacy of diverse weights’. No reason is given here why M2 should be greater than M1 (presumably in proportion to the greater magnitude or degree of n2 over n1). The proportionality is taken for granted by Cohen, without provision of an explicit premise that would make it logically possible. In formal logic, the proportionality has to first be legitimatized by means of an additional premise, before a ‘proportional’ conclusion can be drawn; without such additional premise, only a ‘non-proportional’ conclusion is logically legitimate. Cohen’s formula shows him to be unaware of this indisputable logical rule. Had he been aware of it, his premise would have included a clause to the effect that ‘m is proportional to n’. Without this addition, the only possible conclusion from Cohen’s premise is that “F2 (P) involves n (R) enough to imply M1 (S1)” (note the suffix 1 in the subsidiary term).
Therefore, we must judge Cohen’s formula to be invalid. If we take Cohen at his word, since his formula does not explicitly justify proportionality, we can say that he is advocating invalid ‘proportional’ a fortiori argument; that is to say, he erroneously imagines that the two premises of purely a fortiori argument suffice to yield a proportional conclusion, which is wrong. If we put words into Cohen’s mouth, and say that he tacitly intends a crescendo argument (i.e. a fortiori cum pro rata), we can ask why as a professional logician he failed to mention the crucial tacit assumption that ‘m varies concomitantly with variation of n’. Thus, his formula is either fallacious or formally deficient. In any case, this means that Cohen has not understood that the essence of a fortiori argument isnon-proportional; he clearly thinks the opposite.
Moreover, although we have here done our best to form an a fortiori argument of some sort from his formula, his actual formulation is not in accord with the way people ordinarily think a fortiori thoughts; his statement is an artificial straitjacket, which misleads him as well as others. The job of a logician is not merely to propose a theoretical way to obtain some conclusion, but also to relate his proposal to the way people think in practice. Before logicians can prescribe, they must first be able to accurately describe.
Finally, not only is Cohen’s formula incomplete and inaccurate in the stated ways, but he apparently made no attempt to validate it. He just takes it for granted, as obvious on sight, without any attempt at explanation or justification. Perhaps he thought of it as an inductive leap, rather than as a deductive step. But if so, how come he uses the same symbol (an arrow) for inductive (merely probable) implication as he does for deductive (necessary) implication? It is of course possible that Samely recorded Cohen’s formula incompletely or inaccurately, and without taking note of its validation. But I would confidently say that had Cohen in fact made an effort of validation, his formula would have turned out very differently. Its defects and errors, which we have highlighted above, are evidence of a lack of effort of validation. Modern logicians often function that way, thinking that it suffices to hand down a formula that seems ‘intuitively’ logical to them. No, my friends, everything has to be proved; and in the process of seeking proof one improves one’s statement.
I should stress that the above comments are based on just one brief remark by Samely. It is conceivable that somewhere in Cohen’s works there is a more complete and accurate theory of a fortiori argument. I have not read his works, which judging by their titles could be quite interesting. Needless to say, I would not presume to judge them on the basis of that one bit of hearsay evidence.
Oxford: Oxford UP, 2002. See footnote on p. 177. Samely would have been about 24 at the time of his “informal conversation.” Presumably this refers to Laurence Jonathan Cohen (England, 1923-2006).
InA Contemporary Examination of the A Fortiori Argument Involving Jewish Traditions, in a footnote on p. 129.
It is not surprising, in view of this lacuna in Cohen’s treatment, that his student Samely was left with uncertainty as to whether the conclusion of a fortiori argument is proportional or non-proportional. We could conversely infer from Samely’s evident uncertainty that Cohen’s teaching on this issue was misleading, or at least unclear.