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A FORTIORI LOGIC

© Avi Sion, 2013 All rights reserved.

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A FORTIORI LOGIC

CHAPTER 24 – Lenartowicz and Koszteyn

1. The form of the argument

2. The dayo principle

3. Epistemic substitution

1. The form of the argument

Piotr Lenartowicz SJ and Jolanta Koszteyn, in their paper “On Paley, Epagogé, Technical Mind and A Fortiori Argumentation” (Cracow, 2002)[1] refer to and comment on my theory of a fortiori argument in Judaic Logic. I would like here to respond to their comments. They begin promisingly, as follows:

“We have decided to follow Avi Sion’s version of the a fortiori argumentation (Hebr. qal vachomer). The formal structure of his version is complex. According to Avi Sion it involves two elements, the scheme of a fortiori “syllogism” and the Dayo (Sufficiency) Principle. The a fortiori “syllogism” is this:

P is more R than Q is R, and

Q is R enough to be S; all the more (a fortiori)

P is R enough to be S.



It is not a typical syllogism. A typical syllogism has just three terms: P, Q (or a middle term) and S. Here we are dealing with four different concepts. The concept of R refers to a trait that decides whether something is S, or not. In the first ‘premise’ P is asserted to possess the trait R in a higher degree than Q. In the second ‘premise’ Q is asserted to possess the R trait in the sufficient degree to be S. In the conclusion P is asserted to possess the trait R in the sufficient degree to be S.”

The authors here show they have clearly grasped the difference between Aristotelian syllogism (“typical syllogism”) and a fortiori argument (“a fortiori syllogism”) – one having three terms and the other four terms, whose functions in the inference are very different. Moreover, they have well understood that the fourth term R, found in all three propositions of a fortiori argument, “refers to a trait that decides whether something is S or not.”

However, the form of a fortiori argument they list here is the positive subjectal, which is only one of many forms, albeit the most typical. It is therefore misleading to say as they did that this is “the” form of it. Moreover, it would be best, in order to avoid confusion, not to use the word “syllogism” at all with reference to a fortiori argument, although this is strictly speaking of course not erroneous, since the etymology of the word is syn + logism, meaning ‘merged discourse’.

2. The dayo principle

Next, the authors comment on the rabbinic dayo (i.e. sufficiency) principle:

The Dayo (Sufficiency) Principle requires that the conclusion of an a fortiori argumentation be kept within the limits of the minor premise. Avi Sion’s interpretation seems to reduce the classical a fortiori argument to the a pari argument. It changes the [form presented in the previous quotation] into

P is as R as Q, and

Q is R enough to be S. Therefore

P is R enough to be S.



So, in our opinion, the Dayo (sufficiency) Principle might be called an Overcaution Principle. It gives a firm assurance of the right conclusion, although cognitively it is a wasteful, costly instrument.”

Here unfortunately the authors display some confusion. Of course, I know where their criticism is coming from: they are thinking of ‘proportional’ a fortiori argument, i.e. what I have lately called a crescendo argument. In that specific context, the conclusion’s subsidiary term (S2) is indeed greater than the minor premise’s subsidiary term (S1), in proportion to the values of the middle term (R) for the major and minor terms (P and Q), respectively. However, what they do not realize is that such argument is not purely a fortiori, but a compound of a fortiori and pro rata arguments.

The argument they present in the above quotation, drawn from my book Judaic Logic, is clearly intended as purely a fortiori. Given only the two premises shown, one cannot but deduce the conclusion shown. There are no ifs and buts about it, no way to arrive at a ‘proportional’ conclusion. There is no “overcaution” in the non-proportional conclusion – it is an undeniable fact of logic. If we want to obtain a ‘proportional’ conclusion, we must provide an additional premise, which affirms that the subsidiary term (S) is ‘proportional’ (in some way: strictly or roughly, directly or inversely, or however) to the middle term (R). Given this third premise, which makes possible a pro rata argument, we can with the full blessing of logic deduce the putative ‘proportional’ conclusion. But in the absence of such added information, we are logically forced to remain content with the said non-proportional conclusion.

The definition of the dayo principle by Lenartowicz and Koszteyn, viz. that “the conclusion [must be] kept within the limits of the minor premise” is accurate enough in the context. This is the way I understood the Talmudic principle at the time I wrote Judaic Logic, although I have more recently come to the realization that the original rabbinical understanding of it was different. But for the moment here let us read it as the two authors do. In the context of purely a fortiori argument, the dayo principle may be equated to the principle of deduction, as I now label the logical rule that we cannot end up with more information in the conclusion than we are given in the premises. Granting this equation, as these authors do in their definition, they have no leg to stand on when they deny the principle, calling it “overcautious” and “cognitively wasteful,” implying that it causes loss of information. It does not hide or jettison information, but merely prevents the unwarranted addition of information.

The principle of deduction is a truth of logic applicable not only to a fortiori argument but to all deductive arguments. As regards purely a fortiori argument, if we claimed that the difference in the magnitudes of R relative to P and Q implies that the S in the conclusion should be greater than the S in the minor premise (typically, though not necessarily or exactly, in the same proportion as P is more R than Q) – we would effectively have two subsidiary terms instead of S, viz. S1 (predicated of Q in the minor premise) and S2 (predicated of P in the conclusion). That is, we would have a total of five terms instead of four! Such ‘proportional’ reasoning is clearly fallacious, as can be shown by reviewing how such purely a fortiori argument (positive subjectal) is actually validated[2].

Many people find it difficult to be reconciled with this strict limitation set by deductive logic. Lenartowicz and Koszteyn are here evidently also recalcitrant, since they characterize it as overly cautious and wasteful. What of course they and others like them fail to realize is that the principle of deduction only insists that the value of the subsidiary term S in the conclusion of such an a fortiori argument (from only two premises) should be at least equal to its value in the minor premise; it does not forbid an increase in value based on other reasoning processes or on the basis of further observations. That is, there is no logical objection to discovering a change in the value of S by other means. All logic insists on is that the a fortiori argument by itself cannot effect such a change.

But such a change may well be deduced from other considerations, or be additionally proposed inductively. In the latter case, a hypothesis is formulated that the value of S in relation to P is greater than that in relation to Q, and perhaps (as is often the case) that these two values of S are in the same proportion as the values of R in relation to P and Q respectively. The a fortiori argument may even be viewed as one confirmation (among others, hopefully) of the said hypothesis, provided such confirmation is not confused with definitive (i.e. deductive) proof, since it takes us only part of the way towards the desired ‘proportional’ conclusion. Under such conditions, evidence might well still be adduced that refutes the claimed change in value of S, without invalidating the a fortiori inference of S at its initially given value.

It should be added that the principle of deduction applies equally well to a crescendo argument, though in a different way. Since there are three appropriate premises in a crescendo argument, the ‘proportional’ conclusion can be drawn in full accord with the principle of deduction. What the latter principle would here interdict, on the other hand, would be some fanciful conclusion other than the ‘proportional’ one that strict logic upholds. For instance, a conclusion that is manifestly out of proportion or that concerns a completely different predicate. Any inference beyond the bounds of the given data would be judged illicit.

One reason these authors, and many others like them, are confused in these matters is simply that they are misled by the traditional expression “all the more so” used to signal a “therefore” in a fortiori argument. This expression gives the impression that the conclusion may be quantitatively superior to the minor premise. But it is just hyperbolic. All it really means is that the conclusion is as sure as the minor premise, as is true of any deductive argument. We could equally well rhetorically say that the conclusion of an Aristotelian syllogism is “all the more” true given the premises. But we never do, reserving this phrase and others like it to flag a fortiori argument.

Now, as regards the dayo principle. No doubt Lenartowicz and Koszteyn express skepticism towards it because they perceive it as of Talmudic origin, and being Christians they do not feel obliged to respect this source. And they are right to do so. This principle was of course not first formulated by me, but was put forward long before by Talmudic rabbis. At first I, and others like me (such as Maccoby), interpreted this principle as equivalent to the purely logical and quite universal principle of deduction, i.e. the idea that there cannot be more information in a deductive conclusion than was tacitly implied or explicitly mentioned in the premises. However, I have in the course of the present study found this interpretation inadequate and proposed a more accurate one.

In truth, the Talmudic dayo principle is quite different from the principle of deduction, although the two happen to intersect in some examples. The Talmudic dayo principle has a moral and not a logical significance, and is not applicable universally but in the particular context of Jewish law and even then under special conditions. Moreover, though in some cases its application consists in limiting the concluding predicate to the minor premise’s predicate, there are other applications of it, notably the prevention of certain generalizations prior to a fortiori argumentation. It would be quite legitimate to claim that this dayo principle does not need to be adhered to outside Judaism.

We can apply the dayo principle to purely a fortiori argument, by limiting the conclusion to a mirror image of the minor premise (as the principle of deduction teaches anyway); we could also conceivably do so by (for some external reason) interdicting the formation of the major premise by generalization. However, the proper field of action of the dayo principle is a crescendo argument. For here it means, on the basis of its presentation in the Mishna (Baba Qama 2:5), either that the pro rata underpinning of the a crescendo argument is interdicted, or that the generalization through which the major premise is constituted is prevented. In either of those cases, the putative ‘proportional’ conclusion is stopped.

Thus, the resistance to the dayo principle can be justified, but not as attempted by Lenartowicz and Koszteyn. Their rejection is wrongful, because it is actually directed at the principle of deduction, i.e. it is an attempt to justify a ‘proportional’ conclusion from purely a fortiori premises, which is contrary to formal logic. But had they argued that the dayo principle in its true sense (as distinct from the principle of deduction) is a specifically Judaic legal principle – that would have been okay.

Additionally, these authors are completely wrong in their reading the dayo principle (i.e. the principle of deduction, in the present context) as implying that the initial a fortiori argument has been turned into an a pari (i.e. egalitarian) one. Certainly, I have never suggested such a thing, and they are wrong to think it. The said principle certainly does not require that, in purely a fortiori argument, P and Q be equally R (though in some cases this may happen). It only states that the subsidiary term S must be the same in both minor premise and conclusion (i.e. S is not affected by the difference between P and Q).

3. Epistemic substitution

Finally, Lenartowicz and Koszteyn propose to substitute an epistemic version of a fortiori argument for my more ‘ontical’ version, stating:

“The true a fortiori argument therefore might be represented by the following scheme:

P is more evidently R [than] Q is R, and

Q is R evidently enough to be S. All the more (a fortiori)

P is R evidently enough to be S.



In other words, if it is irrational and non-empirical to doubt that Q is S, then it is even more irrational and non-empirical to doubt that P is S.”

Now, though this proposed epistemic a fortiori argument may seem reasonable at first sight, it can be severely criticized as follows.

Here, note firstly, the major premise no longer informs us that the quantity of R predicated of P is greater than the quantity of R predicated of Q, but instead that the amount of evidence available for the hypothesis that ‘P is R’ is greater than the amount of evidence available for the hypothesis that ‘Q is R’. It may well be, in this perspective, that the quantity of R predicated of P is equal to or even less than the quantity of R predicated of Q, and it may even turn out that P is not at all R and/or that Q is not at all R. In other words, though it looks similar to the original, the major premise is now an entirely different proposition. As a result, not only has given information been irretrievably lost, but also new information is being claimed without justification. This is a very serious objection, but this is not the main objection.

The main objection is that the minor premise and conclusion are in fact meaningless statements. As I have explained in more detail earlier in the present work[3], it never happens in human knowledge that we say “X is R evidently enough to be S” (where X here stands for Q or P). What would such a statement possibly mean? Presumably, that if the probability that ‘X is R’ is true is great enough, then ‘X is S’ becomes true. Note well, ‘becomes true’, and not merely ‘becomes as good as true for us, i.e. can be reasonably assumed to be true’. Yet we never in formal logic infer a definite truth from a probable truth. Such a thing is unheard of and not to be confused with the following sort of apodosis, where given that one thesis implies another we can pass on the probability of the first onto the second:

If ‘X is R’, then ‘X is S’, and

‘X is R’ is probably (to some degree) true,

therefore: ‘X is S’ is probably (to the same degree) true.



The conclusion here is, note well, only probable to the same degree. If it so happens that the given implication in the major premise (of this apodosis) is itself only probable then the probability of the conclusion is proportionately less than that of the minor premise. But whatever the case, we never have a probability implying a settled truth. Therefore, the minor premise and conclusion of the epistemic a fortiori argument proposed by these authors are not meaningful propositions. The argument may look like something, but it is not. Really, what the authors were trying to say, which is meaningful, is the following:

a) There is more evidence that P is R than that Q is R.

b) Given that ‘Q is R’ implies ‘Q is S’; if there is some evidence that Q is R, then there is some evidence that Q is S.

c) Given that ‘P is R’ implies ‘P is S’, if there is some evidence that P is R, then there is some evidence that P is S.



But this triad is not a valid a fortiori argument, indeed not even a valid argument! The proposition labeled (a) corresponds to their major premise. The proposition labeled (b) is my interpretation of their minor premise, “Q is R evidently enough to be S;” notice that this is an argument (apodosis) and not merely a proposition. The proposition labeled (c) is again my interpretation of the conclusion postulated by the authors “P is R evidently enough to be S;” this just mirrors my interpretation of the minor premise, note well. To validate the epistemological argument proposed by Lenartowicz and Koszteyn, we must show that the putative conclusion can be inferred from the given premises. My point is that such validation is not possible. Why?

Let us first look at proposition (a): all it tells us is that the amount of evidence in support of ‘P is R’ is numerically greater than that the amount of evidence in support of ‘Q is R’. This does not imply, note well, that these two packets of evidence overlap at all; so the evidence in support of ‘Q is R’ is useless as evidence in support of ‘P is R’ and vice versa. Nevertheless, we can infer from this comparative proposition the two clauses: “there is some evidence in support of ‘P is R’” and “there is some evidence in support of ‘Q is R’,” which clauses we do need in the next two segments. Segment (b) is an argument wherein the proposition “‘Q is R’ implies ‘Q is S’” is the major premise; if this is granted, then using the clause “there is some evidence in support of ‘Q is R’” found in proposition (a), we can infer that “there is some evidence in support of ‘Q is S’.” Segment (c) is likewise an argument wherein the proposition “‘P is R’ implies ‘P is S’” is the major premise; if this is granted, then using the clause “there is some evidence in support of ‘P is R’” found in proposition (a), we can infer that “there is some evidence in support of ‘P is S’.”

We could at most add the final conclusion that: “the amount of evidence in support of ‘P is S’ is numerically greater than the amount of evidence in support of ‘Q is S’.” This would serve to justify their statement that “if it is irrational and non-empirical to doubt that Q is S, then it is even more irrational and non-empirical to doubt that P is S.” But in any event, the premises originally given by the authors do not suffice to draw their putative conclusion that “P is indeed S;” with those premises alone we do not have a valid argument.

Their proposed argument can be constructed, provided we grant two additional premises (namely, the major premises of segments (b) and (c)). However, if these additional premises are granted, the argument emerging is not a fortiori in form. For, though we do rely on the two clauses implicit in proposition (a), we do not at all make use of the comparison it effects between the respective amounts of evidence for these clauses; the “more evidence” for this element than for that element is redundant, and it plays no role in the inference actually made. Furthermore, segment (b) is totally useless in the process of drawing the conclusion of segment (c), i.e. the desired conclusion. All we need to draw that conclusion is the clause “there is some evidence in support of ‘P is R’” implied by (a), plus the additional premise “‘P is R’ implies ‘P is S’” separately granted in (c); clearly, (b) is redundant and plays no role in this inference.

Therefore, the scheme vaunted by Lenartowicz and Koszteyn as “the true a fortiori argument” is not an a fortiori argument and not an argument at all. It looks superficially meaningful and valid, but it is neither. In short, this proposal is mistaken and cannot be rationally upheld. Of course, its authors meant well; but they did not carefully think their proposal through. The lesson to draw from this episode is the importance of linguistic precision. It is very easy, when dealing with symbols, to get carried away by appearances and fail to connect to reality. And this is particularly true when logical or epistemological concepts are involved. Even so, there is a valid way to express what the authors were trying to say. They came close to it when they tried to explain their “true scheme,” saying:

“If it is irrational and non-empirical to doubt that Q is S, then

it is even more irrational and non-empirical to doubt that P is S.”



This statement of theirs can indeed be construed as a valid a fortiori argument, although I would have put it more positively as follows:

Since thesis P is more evident (R) than thesis Q:

if thesis Q is evident (R) enough to be believed (S),

then thesis P is evident (R) enough to be believed (S).



Let us now analyze these two proposals, mine first and then theirs. My version is a hybrid-seeming a fortiori argument, involving two theses P and Q, respectively the major and the minor, compared by a logical middle term (R), referring to amounts of evidence, and an epistemic subsidiary term (S), referring to belief; this fits in the standard model and is valid. Now look at their version. What is its major premise? Presumably, it is that given in their “true scheme,” viz. “‘P is R’ is more evident than ‘Q is R’.” But their minor premise and conclusion (rightly) only mention ‘Q is S’ and ‘P is S’; they say nothing about a term R. So their major premise should really be: “’P is S’ is more evident than ‘Q is S’.” Or, to use the same language as they use: “’P is S’ is more ‘irrational and non-empirical to doubt’ than ‘Q is S’.” With this modified major premise, their conclusion can indeed be drawn from their minor premise.

In case the reader has been confused by all the Ps and Qs – let me now compare these two versions:

Since ‘P is S’ (my ‘thesis P’) is more ‘irrational and non-empirical to doubt’ (term R, my ‘evident’) than ‘Q is S’ (my ‘thesis Q’):

if it is ‘irrational and non-empirical to doubt’ enough that ‘Q is S’ is true (term S, my ‘believable’),

then it is even more ‘irrational and non-empirical to doubt’ that ‘P is S’ is true (term S, my ‘believable’).



As you can see, the items I called ‘thesis P’ and ‘thesis Q’ correspond to the propositions ‘P is S’ and ‘Q is S’ (wherein P and Q are terms related to a third term S) of Lenartowicz and Koszteyn. Their middle term is really the epistemic qualification ‘irrational and non-empirical to doubt’ (corresponding to my ‘evident’) – this is what they should label R (though they do not label it at all). Their subsidiary term is, I suggest, the logical qualification ‘true’ (corresponding to my ‘believable’) – and this is what they should label S (though they have not mentioned it at all). Thus, we see that the authors got confused by the symbols involved, and for this reason produced a fallacious “true scheme.”

Finally, let me again address their claim that the resulting a fortiori argument (after applying the various clarifications and corrections just seen) is the “true scheme.” What they are saying is that the original ontical a fortiori format is not as interesting or valuable as the more epistemic format they propose. This claim is wrong, because (as already mentioned) in passing over from the ontical to the epistemic format much information has been lost on the way, and other information introduced ex nihilo in its stead! And furthermore, far from being the essence of a fortiori reasoning, the epistemic version is not even formally implicit in the ontical one. Consider the following two arguments:

Ontical version

Epistemic version

P is more R than Q is R, and

‘P is S’ is more evident than ‘Q is S’, and

Q is R enough to be S;

‘Q is S’ is evident enough to be believed;

so, P is R enough to be S.

so, ‘P is S’ is evident enough to be believed.



We see here that, although both these a fortiori arguments result in the same effective conclusion, viz. that ‘P is S’ – the ontical version has this conclusion as an established fact, whereas the epistemic version is only capable of yielding a probable conclusion, i.e. a less certain one. Moreover, the original (ontical) major premise, with all its information about P being R and Q being R, and the R of P being greater than the R of Q – all that data is absent in the revised (epistemic) version. There is therefore no wisdom in preferring the epistemic version to the ontical one.

Furthermore, given the ontical version does the epistemic version logically follow as these authors seem to think and suggest? Obviously not: the information given in the original version does not justify the revised version’s claim that ‘P is S’ is more evident than ‘Q is S’; nor for that matter that ‘P is R’ is more evident than ‘Q is R’. We could say that the premises of the ontical version constitute evidence for its conclusion – but that does not mean that the conclusion is more evident than its premises. It is not with reference to evidence that the conclusion proceeds from the premises in the ontical version, but with reference to the given middle term R. So the proposed substitution of an epistemic format for the ontical a fortiori argument is an illicit process – i.e. it is fallacious.

This concludes my appraisal/criticism of the ideas of Lenartowicz and Koszteyn concerning a fortiori logic. They began wisely by understanding and adopting my standard model (or more precisely, one form of it) – but then they unwisely swerved off into denial of the principle of deduction, as well as misconception of epistemic a fortiori argument and preference for it over the ontical form. My purpose here, as throughout this volume, was of course not to publicize errors but to make sure they are not perpetuated.



[1] In: Forum Philosophicum Facultas Philosophica Ignatianum Cracovia – Cracow, 7: 2002, pp. 49-83. This paper is available in pdf format online at: www.jezuici.pl/lenartowicz/articles/POZ/fpaley.pdf (see pp. 12-14).

[2] See in my Judaic Logic, chapter 3.2.

[3] See the discussion of ‘Certainty from mere probability’ in the section ‘Probable inferences’ in the chapter ‘Apparently Variant Forms’ (4.5).

2016-06-14T04:59:49+00:00