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© Avi Sion, 2014.


Addendum to A Fortiori Logic, chapter 25. This addendum has so far not been published in any book or e-book, although posted in The Logician website. It was written in April 2014.




1.   About “translation”

2.   About deduction

3.   Past attempts

4.   A fortiori argument?

5.   In denial


The present essay is in reply to the general letter (called “A general treatment of Avi Sion’s critique,” signed by Dov, Michael, Uri) and specific comments (authored by Michael, and presumably endorsed by Dov and Uri) that you sent to me on 21.1.2014, in reply to chapter 25 of my book A Fortiori Logic (AFL, henceforth) dealing with your 2009 essay “Analysis of the Talmudic Argumentum A Fortiori Inference Rule (Kal Vachomer) using Matrix Abduction”[1].

I am very glad that you wrote to me because it has revealed to me the alarming extent of your misconceptions concerning the nature and purpose of formal logic, i.e. of the work of the logician, let alone further confirmed my conviction that you do not fully understand a fortiori argument. For convenience, I have numbered the paragraphs of your summary letter L-1 to L-10 and your specific comments C-1 to C-23. I will here try to react to your various statements, though in an order that seems appropriate to me.


1.   About “translation”

The main problem in your understanding of logic is your very odd notion that all attempts at argument description and explanation, i.e. at formalization and validation, constitute “translation” – by which term you apparently mean a “trivial and useless” cosmetic rewording of arguments found in ordinary discourse. According to you, such rewording cannot serve to reveal deductive arguments, but are effectively arbitrary. I quote you:


Explanation and validation of the basic KVH itself is trivial, and make almost no difference to non-deductive arguments. To my opinion it is just a matter of translation. (C-6)

Every non-deductive argument can be translated to a deductive one, if you add premises. Take simple analogy between A and B (if A is X than B is also X). If you add a premise that their common attribute implies X you make it deductive. This is trivial and useless. That is exactly what I mean by translation in comparison to logical modeling. (C-7)

Because we are not translating, as you do. We deal with the logical dimensions of the SUGIA. (C-22)


I would say that so long as you do not discard such attitudes, so long as you do not open your mind to more perspicacious and profound work by others, your logical comprehension and skills are bound to remain at a low level. “Translation” from what to what exactly? You do not say. You do not detail even one example of this alleged rewording process nor demonstrate why the results are, and indeed are bound to be (according to you), “trivial and useless.” It is all just a vague notion in your mind, through which you can conveniently summarily dismiss any work you evidently do not understand or take the trouble to study closely.

One could perhaps regard the rewording by A. Schwarz of a fortiori argument in syllogistic form as a mere “translation” – but that would certainly not do it justice. The reason why Schwarz tried to formulate a fortiori argument as syllogism was in order to justify it, syllogism being a more easily and widely understood form of argument than a fortiori argument. As it turns out, Schwarz’s hypothesis was wrong – but not for the reason you assume, i.e. not because it constituted “translation.” The reason it was wrong, as I clearly show in AFL chapters 5 and 14 (read them!), is that, even though they overlap somewhat, the information contained in a fortiori argument and the information contained in syllogism are not coextensive. Hence, Schwarz’s syllogism cannot fully justify a fortiori argument. In other words, the mere fact of “translation” has, contrary to your supposition, nothing to do with the rejection of Schwarz’s theory. His attempt to justify a fortiori argument by reducing it to syllogism is perfectly legitimate and potentially informative, but it is just not supported by the facts.

Moreover, while one might represent Schwarz’s attempt as a “translation,” as just mentioned, one could in no wise reasonably represent L. Jacobs’ simple and complex a fortiori types of a fortiori argument as “translations.” Again, from what to what? He is not changing the wording of arguments found in discourse, but merely attempting to find their common features by putting letters (A, B, x, y) in place of terms commonly found in discourse. He is by this means just trying to summarize information. There is no “translation” whatsoever involved, contrary to your imagination. The reason why his theory also fails (at least in part) is not due to its involving “translation,” but due to its not being applicable to all cases (i.e. it is overhasty generalization from some instances) and to its being incomplete for the cases it does apply to (i.e. his formalization does not bring out all relevant features and therefore cannot be formally validated). This is thoroughly demonstrated in AFL chapter 16 (do read it!).

All the more, the theory of a fortiori argument presented in my 1995 book Judaic Logic, chapter 3, which is a more fully conscious effort at formalization and validation of a fortiori argument, cannot be characterized as you claim so offhandedly as mere “translation.” To help you understand ‘where I’m coming from’ when I say all this, let us go back to the beginning of formal logic, i.e. to the work of its founder, Aristotle. What exactly did Aristotle do when he developed the syllogism? He must have noticed himself and others thinking or saying or writing arguments like:

Socrates is a man, therefore he is mortal.

Or like:

All men are mortal, therefore Socrates is mortal.

He must have asked himself: what makes such arguments cogent, what makes them convincing to our rational faculty? He must have carefully examined and thought about many such statements till he saw, perhaps in a flash of insight, what they had in common and why they were intuitively credible. He proposed the theory that the full argument in the above two examples was:

All men are mortal, (major premise)
and Socrates is a man, (minor premise)
therefore Socrates is mortal. (conclusion)

He was not really thereby “adding” a statement to either of the preceding arguments, but merely bringing out into the open tacit but intended information in each case (the major premise in the first case, the minor premise in the second case). He realized that both these premises were needed to justify the said conclusion, and that neither could do the job by itself.

Moreover, Aristotle could not really have done this without realizing just why these two premises led to this conclusion. He could not have properly described things if he had not properly explained them. Here, he developed the idea of individuals (like Socrates) belonging (that’s the word he often uses) in a group of similar individuals (like mankind), and through this intermediary also belonging in a larger such group (like mortals). That is, he discovered the middle term which made the transition from the minor term to the major term possible. Once he grasped this reasoning process, he was able to formalize the syllogism in question to:

All B are C,
and this A is B,
therefore this A is C.

This was not the end of his task, but in fact just the beginning. He had to now test his theory with reference to many cases which did not seem to fit in to this pattern. For example, he had to realize that the argument:

All men are mortal,
and my cat is not a man,
therefore my cat is not mortal.

… is not valid and why it is not so. Moreover, he tried out every permutation and combination of terms in premises (the four figures, and the 64 moods in them), and thus developed a full theory of syllogism, listing both the valid and invalid moods (for more details, see my book Future Logic, chapters 8-10).

Now, Michael, Dov and Uri, everyone who has in his youth learned formal logic knows this story, and most students supposedly have understood it. But apparently it has escaped your notice or comprehension. Aristotle’s development of the syllogism has obviously nothing to do with mere “translation.” There is no “item translated” and “item emerging from translation” about it – it is work of formalization, and validation or invalidation.

Note well that the process of validation (or invalidation, as the case may be) consists in determining precisely under what conditions (if any) a given premise or set of premises implies a putative conclusion, or precisely through which premise(s) a desired conclusion can (if at all) be obtained. Contrary to what you imagine, it is not arbitrary and futile manipulation of data; it is a very demanding process with very significant results.

The utility of this logical work is that it allows us to understand how and why we reason in certain particular ways all the time, i.e. why we naturally find this or that argument convincing and informative. It is of great importance to epistemology (and to our peace of mind, mental health and survival as living organisms) to know exactly how we reason and to justify, where possible, such reasoning.

The starting point of all logic theorizing has to be ordinary human discourse, which we become aware of by personal introspection and by observing other people’s speech and writings. Once the logician has isolated an argument type as an object of study, he tries first to describe it, then to explain its workings, and thence to prescribe how to correctly use it. The logician is not a superior being, above the crowd; he is an informed and skilful layman. He can and does improve his own and other people’s thought processes because he has studied the matter more than most people. It is clear from the following comment that you do not understand the development of logic as research primarily aimed at perfecting human thought:


That is exactly the difference between a translation and logical model (as ours). A logical model need not to follow literally a layman way of thinking. To my opinion, no layman needs help in order to deduce a conclusion by KVH. It is quite simple. That is exactly the reason why such a work is at most a translation. Can you give me an example in which your validation prevents getting a mistaken conclusion by a layman? I can give you many such cases in our model. (C-11)


If you want many examples of mistaken conclusions by laymen, read the assessments in part 3 of AFL. Many people do make errors of reasoning – which they would not make if they had studied logic more assiduously. You too, as I show here, are far from skillful in reasoning. Study of formal logic improves thinking. As for your claim here that your “model” can prevent mistaken conclusions by laymen – I very much doubt it. It is so confused it can only cause confusion.

The development of a formal theory of a fortiori argument follows the very same process of search and discovery, and justification, as you can see if you read my descriptions of it. There is nothing fanciful or trivial about it; it is a scientific and revealing process. You should take the time to study my book AFL, at least chapters 1, 2 and 3; and if you wish to be real scholars the whole 700 page volume, before making any more silly statements about “translation” and “non-deduction.” There is nothing wrong with not-knowing something; but there is much wrong in entrenched refusal to learn and evolve.


2.   About deduction

According to your statements above quoted, all the deductions thus decorticated and authenticated by formal logic are arbitrary and trivial. You repeat this outlandish claim in your summary:


You repeatedly assert that according to your approach an AFA is a deductive method. You base this on an ad hoc addition of basic assumptions. If that is done, every non-deductive assertion can easily become a deductive assertion. Hence, the discussion whether an assertion is deductive is moot, it relates mainly to the translation of the inference but not to its logical meaning. (L-7)


You imagine that we can justify any claim that, say, conclusion Z can be “deduced” from premises X and Y, by merely adding a premise that “X and Y together imply Z”. In that case the argument becomes, not:

“Given premises X and Y, one can infer conclusion Z”

– but:

“Given that ‘X and Y together imply Z’ and that ‘X and Y are true’, it follows that ‘Z is true’.”

But these are very different arguments! The latter argument is a mere apodosis (modus ponens, affirming the antecedent) – whereas the former is a more demanding construction.

How do we know that “X and Y together imply Z”? That is the question you do not ask. Is this premise true or false? You do not ask. Can one just affirm this connection of any triad X, Y, Z? You obviously think so. You base that belief on the example of analogy (in C-7) – but such simple analogy is not deductive argument, and therefore cannot exemplify your claim about the arbitrariness of deduction! The truth is, no non-deductive can be made out to be deductive by artificial means.

You, however, say (in C-7): “Every non-deductive argument can be translated to a deductive one, if you add premises,” and again (in L-7): “every non-deductive assertion can easily become a deductive assertion” – but you do not say how you know in the first place that the assertion in question, whatever it is, is “non-deductive”! You obviously think all assertions are in fact non-deductive.

Where have you first logically demonstrated that the assertions by Schwarz or Jacobs, let alone mine, are “non-deductive”? We can only label something as non-deductive if we are sure it is not deductive. Since you obviously do not understand the nature and basis of deduction, you are hardly in a position to judge what is or is not “deductive.” All you do is throw opprobrium on others’ theories by calling them “translations,” as if that cliché clinches the matter. Tell me, where in general logic theory – in metalogic – is the concept and theory of “translation” developed and validated? Nowhere – it is just a figment of your imagination.

According to you, then (since your statements are general, i.e. you say and repeat “every”), Aristotle’s syllogism is also just “translation” – based on nothing more than the arbitrary apodosis “Given that ‘All B are C and this A is B’ together imply ‘this A is C’, it follows that if All B are C and this A is B, then this A is C.”

Have you not noticed the self-inconsistency of your sweeping claim? You reject all of formal deductive logic except apodosis. But if all deductive logic can only be justified by an arbitrary apodosis, then even that arbitrary apodosis can only be justified by an arbitrary apodosis, and likewise the latter in turn, and so on ad infinitum. Do you understand this, which shows the naivety and absurdity of your anti-deduction belief? I repeat for your benefit: if the only way to prove that “premises (X and Y) imply conclusion (Z)” is by saying: “if (X and Y imply Z), then if (X and Y) then (Z),” then the latter argument must in turn be justified by saying: “if (X and Y imply Z implies that if X and Y then Z), then given (X and Y imply Z), it follows that (if X and Y then Z); and so on without end.

In other words, your vision of deductive argument is that it is nothing more than tautology, in which case there is no such thing as deductive logic. There is, in your view, only inductive logic. You think it is useless and trivial to look for any necessary connections between things or ideas, and exclusively acknowledge your “logical modeling” method as capable of inferring more information from given information. If anything deserves to be characterized as “trivial and useless” it is surely this belief of yours, which is based on shocking ignorance of the genesis of formal logic and on faulty reasoning.

It is no surprise, in view of such attitudes, that the starting premise of your 2009 paper on a fortiori argument is that such argument is necessarily non-deductive. In response to a remark by me that this basic premise is mistaken, you reply:


Nonsense. See below. (C-8)

You have to take into account that due to most of the commentators, the Talmud itself looks at KVH as a doubtable conclusion. That is the reason why you get no punishment if you committed an offence which is deduced by KVH. (C-9)


The fact that the Talmud views a fortiori argument as open to rebuttal does not justify your view, for any deductive argument can indeed, even if formally valid, always be rebutted by attacking its premises. As I point out in my original essay (i.e. in AFL), your English paper contains no analysis of the rabbinic doctrine of rebuttal, so you have not demonstrated your understanding of it. This doctrine does not constitute a denial of the deductive power of a fortiori argument, contrary to what you evidently imagine. This comment of yours shows that you have not yet grasped the difference between formal validity and concrete truth of premises. It is amusing that someone who does not know even such elementary aspects logic says “Nonsense” with such superb aplomb.

More than this, as your following comment makes clear, your opinion of deductive logic is so negative (due to your ignorance of it) that you regard any claim of deduction as ridiculous:


Constructed deductively of course!! Let us remember that due to Sion KVH is deductive inference. (C-20)


You criticize me for being angry (in L-2 and C-15). Well I might be outraged when I see so much nonsense being peddled as logic. But I assure you my judgment is not clouded by my just indignation. I am not in any way hostile towards you as persons, or in any way jealous of your or anyone else’s discoveries (if any). My only concern is with truth and falsehood, and this concern certainly should be passionate. It is in truth out of kindness that I have taken the time to criticize your wrong attitudes, and to advise you not to cling to them if you want to progress as logicians. If you are truly interested in logic, and in particular in a fortiori logic, you will follow my advice and study the matter further. Nobody forces you to do so; the choice is yours.

The truth is, formal deductive logic is the ultimate in “logical modeling.” When Aristotle provided valid syllogistic forms, or I provided valid a fortiori forms, this was giving people 100% reliable models of logical discourse. Given certain information, we might well be able to infer from it some other information, which is implicit in it but not immediately apparent. This is certainly not always possible, but it is certainly sometimes possible. To say that it is never possible (as you effectively do) is a silly as to say that it is always possible. The primary work of the logician is to try and find ways to demonstrate the connection between various premises and conclusion. Only if such demonstration is not found possible can one declare that there is no necessary connection, i.e. no deductive argument. In that event, one might try and find less than necessary connections, i.e. inductive arguments.

As regards the difference between deduction and induction, and in particular the difference between deductive and inductive a fortiori argument, I advise you to carefully read AFL chapter 3, section 3. A fortiori argument has in principle deductive force – when it is perfectly formulated. Nevertheless, when an attempted a fortiori argument is incomplete, i.e. when some needed elements of it are not clearly put forward, it can still be looked upon and treated as an inductive a fortiori argument, i.e. as one with less than deductive force, at least until evidence or proof to the contrary is provided.

The funny thing is that your denigration, and even denial, of deductive logic is quite unnecessary. You think you need to take an antagonistic stance in order to give value and importance to your proposal of “matrix abduction” of a fortiori argument, but in fact you don’t. There is room for your proposal even granting the existence and validity of deductive a fortiori argument, because your method (as I understand it, see below) can still in principle be applied to determination of quantitative variations within deductive a fortiori arguments, and all the more so to determination of quantitative variations within inductive a fortiori arguments.

Note that neither here nor in my original critique (i.e. in AFL) do I try to deny the possible utility and novelty of your work as such, as you suggest in your letter (quoted below). Why would I?


Further on you write that our concept of abduction is not new. It seems you did not understand that we only use a certain type of abduction (which is indeed novel) as a logical tool for our purposes. Contrary to what you write, abduction is not the aim of our research, and in principle there is not supposed to be anything innovative in what we have to say about it. We here bring a new abductive algorithm for our logical purposes and not anything new about the concept itself. (L-4)


I have no desire to put you down, no wish to deny or diminish your achievements. I rejoice at all advances in logic theory, or any scientific endeavor, whoever is their author. My only concern throughout is with the accuracy and relevance of your proposals in relation to a fortiori argument (as against other forms of proportional reasoning); I do not think them altogether worthless as you suppose.


3.   Past attempts

I think I know how you have got your idea that all attempts are formalization constitute mere “translation,” and are thus invalid, being non-deduction posing as deduction. Back in 1992, Michael Avraham published a paper called “The ‘Kal Vachomer’ as a Syllogism – Arithmetic Model” in Higayon (vol. 2.  Pp. 29-46). In this essay, he made a first and only attempt at formal description of a fortiori argument, as follows (my translation from the Hebrew):


If A is light in ‘a’ and heavy in ‘b’, then B which is heavy in ‘a’ will obviously be heavy in ‘b’.


I analyze and criticize this proposed formulation in AFL chapter 20, and will not repeat myself here. Suffice to say here that this attempt, though valiant, was unsuccessful. It is incidentally interesting that you do not mention this essay and my review of it in your letter or comments. It suggests to me that you did not read chapter 20, which suggests to me that you did not even take the trouble to look through the table of contents of AFL, let alone read any chapter of the book other than chapter 25 (and possibly not all of that – see further on). This tells me something about your study methods.

Anyway, it is clear from this episode that Michael must have, if only subconsciously, realized the failure of his formula for a fortiori argument, and thus lost faith in formalization (which he thereafter called “translation”). It is significant that he made no attempt at its formal validation or invalidation. But it is clear from the fact that he thereafter, in the same essay, went on to try out a more inductive approach – albeit one, in my assessment, leading nowhere – that he came to regard a fortiori argument (as he saw it through his attempted formalization) as “non-deductive.” Apparently, he passed on his credo of “non-deductive translation” to his colleagues (Dov and Uri) in 2009.

But of course, these conclusions were just hasty generalizations on Michael’s part. Just because his own single attempt at formalization was not very convincing, it did not follow that other attempts might not be more successful. As I show in AFL, many people have over time tried their hand at describing and explaining a fortiori argument. Michael was certainly not the first or the last. Yet he made no effort to find out whether other people did so better (or worse) than him. In the passage of your letter quoted below, you point to your lifetime experience in logic research; I have not read your other works, but judging from the two articles of yours that I have read you ought to humbly revise this favorable self-assessment of yours:


Reading your words one gets the impression that there is nothing correct in our work, and that we actually do not understand what an AFA is all about, and what makes it different from analogy. That would be quite strange coming from people involved in all branches of logic research for several decades. (L-3)


Returning to your 2009 paper, you do there (as we have already seen) mention the attempts by Schwarz and Jacobs, not to mention my own; but my complaint is that you do not make any effort to analyze them in detail. You merely list them and more or less reject them, but you do not say why exactly they deserve rejection. This is inadequate methodologically – you cannot claim knowledge that you have not publicly demonstrated.

Incidentally, in your comments, you complain that I criticize your brief exposition of Schwarz’s syllogistic formula for a fortiori argument, saying:


As I remember, the identification with Aristotelian Syllogism is Schwartz’s main statement. We think and wrote exactly the opposite. (C-2)

That is not ours but Schwartz’s analysis of KAL VAHOMER. (C-3)


Yet I clearly wrote: “The authors do not, however, intend thereby to subscribe to Schwarz’s theory.” So you misunderstood my criticism. My point here is that it is not enough to declare that you disown Schwarz’s approach – you need to truly justify your distance from it.

I was also perplexed by your comments concerning Jacob’s simple and complex type of a fortiori argument:


There is no such KAL VAHOMER as the first one. The second is our’s Talmudic one (sic). The biblical is a third kind that Sion totally ignores here. (C-4)

This is all a misunderstanding. Look at my former note. (C-5)


Here again, without any proof you claim that there is no qal vachomer of the first type (contrary to Jacobs, who claims all Biblical qal vachomer are simple); then you claim (like Jacobs) that Talmudic qal vachomer arguments are of the second type; then you claim that Biblical qal vachomer are of a third type that I (and presumably Jacobs too) totally ignore, a third type that you never go on to describe! Now, I ask you – is this serious scholarship? Where is your formal study of Jacobs’ two types; and what is your mysterious third type? Where is your empirical study of all Biblical and all Talmudic qal vachomer that allows you to make such sweeping statements? You here speak with a tone of authority, but you are just expressing your vague prejudices again.

I can speak knowledgeably about Jacobs’ theory of a fortiori argument because I have read all of his writings on the subject and written a thoughtful 25-page essay on it (see AFL, chapter 16). And I can speak about Biblical a fortiori argument because I have done empirical research in that field, listing and analyzing 46 arguments, many of them previously undiscovered (see AFL, appendix 1). As regards the Talmud, I have listed and analyzed (following Samely in most cases) 46 Mishnaic a fortiori argument (see AFL, appendix 2), and have made efforts to examine the use of many such arguments in the Gemara (see AFL, appendix 3). Your statements, on the other hand, are unsubstantiated assertions, not based on systematic research.

I show in my book that Jacob’s first type is, from a formal point of view, merely an abridged version of his second type. In the former, the middle term is not explicitly mentioned, whereas in the latter it is. Moreover, I show that there are arguments of both types in both the Tanakh and the Talmud (thus rebutting Jacobs’ claim that the first type comes from the Bible and the second from the Rabbis) – and also that there are arguments of other types (i.e. that Jacobs’ formulation is far from exhaustive). You just make wild claims.

As far as I am concerned, as already explicated earlier, both Schwarz and Jacobs were way ahead of you in their understanding of the task, even if they did not find the correct or complete solution to the problem of a fortiori argument. But in any case, your mention in passing in your paper of these two authors, and of my Judaic Logic work, does not constitute scholarship. As far as I am concerned, this is just name-dropping, to give an illusion of having looked at the literature. Your passing mention of the Islamic qiyas and the Indian kaimutika fall under the same category, as far as I am concerned. If you wished to scientifically claim that all theories of a fortiori argument preceding yours were worthless (as you have the chutzpah to do), you were duty-bound to list all the theories you had in mind and to closely analyze them all. Don’t take people for fools – make a real effort.

In my book I analyze the work of some thirty modern authors, not to mention many less recent ones. So I can tell you who said what, and why what they said is wholly or partly right or wrong. There is no guesswork involved, but patient (ad nauseam) sifting through actual data and careful reflection.

One of the most interesting findings of this research was the discovery of the contribution of Moshe Chaim Luzzatto (the Ramchal). His work is distinguished by its clearly listing the four main forms of a fortiori argument (which I have called the positive and negative subjectal and the positive and negative predicatal). Even so, his work is not the definitive solution, for two related main reasons. He failed in his formalization to realize and mention the crucial factor of “sufficiency or insufficiency of the middle term,” and he made no effort at validation (which would in any event have only been possible through the sufficiency or insufficiency of middle term).

Your approach to a fortiori argument, lacking such subtleties, is incapable of analyzing and judging past contributions.


4.   A fortiori argument?

This brings us to your claim to know a fortiori argument and to have devised a way to represent it. I have no doubt that you can, in most cases, tell at a glance that a certain argument is intended to be a fortiori. After all, whoever wrote the text (or spoke the words) intended his argument to be so received. But to recognize an a fortiori argument’s grammatical form (or the keywords indicative of such argument) is not the same as to understand its logical structure and workings. You have not shown yourself to have grasped the latter.

As I have shown in AFL, chapter 25, the techniques you use to represent a fortiori argument, such as tabulation, are not capable of distinguishing between the different forms of the argument. A fortiori argument may be copulative or implicational; if copulative, it is subjectal if the major and minor terms given in the major premise are the subjects of the minor premise and conclusion, and it is predicatal if the major and minor terms given in the major premise are the predicates of the minor premise and conclusion (similarly, if implicational, it may be antecedental or consequental, according to where the major and minor theses appear); these arguments may be positive or negative; there is also a distinction between primary and secondary arguments; there is also a distinction between purely a fortiori argument and a crescendo argument, the former being non-proportional and the latter proportional.

Your method is too rough to discern and take into account these various fine distinctions. Yet these distinctions are all very significant in determining the conclusion from given premises. For instances: positive subjectal and negative predicatal arguments go from minor to major, whereas negative subjectal and positive predicatal arguments go from major to minor. In your approach, you can go equally well from minor to major or from major to minor in all cases. In a fortiori argument, this is impossible (except in the special case of a pari argument, which is bidirectional). This goes to show that your method is not essentially concerned with a fortiori argument, but with quantitative analogy or pro rata argument. It is simply about predicting missing quantities from given quantities, given that some proportionality is present.

This is why you can claim that your method applies equally well to a fortiori argument and to simple analogy and to more complex analogy (binyan av). You evidently think that this is a bargain – two or more articles for the price of one! Thus, you write the following comments:


Our main statement is that analogy, a-fortiori and many other logical inferences can be put together on the same basis. This is another advantage when one deals with logical model rather than a translation. It appears that Sion didn’t understand this basic (and novel) issue in our work, so it is not surprising that he sees no new concept here. (C-19)

Here again you miss our main point. A-fortiori and analogy can be put on the same general logical basis. (C-23)


The truth is that your method only addresses the lowest common denominator of these three forms of argument – namely, proportionality. Given certain specific ratios are true, or given a certain general formula regarding concomitant variation, one can fill in blanks in information. But, though this element of proportionality is found in some a fortiori arguments (specifically, in a crescendo arguments), it is not really found in all of them. Moreover, even where it is found, when dealing with a fortiori argument we cannot normally (i.e. except in egalitarian moods) reason in any direction we please. We can only reason from minor to major or from major to minor. If we reason in the wrong direction, given a certain structure, we are making an illicit inference – it is fallacious reasoning. Your method cannot integrate this crucial distinction; it is too indiscriminate.

You, of course, repeatedly claim to be able to distinguish these various forms of argument in your system – but you provide no evidence or proof of such ability. For instances, in your letter, you write:


You repeatedly assert that that we do not understand the difference between an AFA and analogy, and mix the concepts up. But we are well aware of the differences between these concepts, and do not compare them anywhere. (L-5)

You do not seem to have understood our principal innovation in this and other papers, which is the following: Consider inferences of the following types: AFA, analogy (binyan av based on one or more verses), their rejections and also expanding structures built from such inferences. All these inferences can be united within one conceptual and logical structure, and can be analysed in a similar manner. It does not mean that they are all identical (as you think we are saying), rather that their differences may be exposed using the same box of mathematical and logical tools. (L-6)


Note your admission that you “do not compare them anywhere,” even as you assert that “their differences may be exposed using the same box of mathematical and logical tools.” You perhaps do not equate them – but you certainly do not clarify their differences. In truth, to repeat, your tools can only deal with proportionality, which is the common feature of simple and complex quantitative analogy and of a crescendo argument. Your tools refer to the commonality of these processes, but are not able to differentiate them.

Regarding the figures and moods of a fortiori argument I invite you to study AFL chapter 1. Regarding the distinction between pure (non-proportional) and a crescendo (proportional) a fortiori argument, I invite you to study AFL chapter 2. Regarding the formal distinctions between a fortiori argument and simple and complex analogy, I invite you to study AFL chapter 5, section 1. If you have not studied these chapters, you cannot claim to have taken these issues into consideration in your work. You cannot claim knowledge or ability that you have not explicitly displayed for all to see. You have to study – it is not a substitute to bluntly “deny” the work or criticisms of other people without taking the time to study. If the subject does not interest you, that’s fine. But to claim that it does interest you and yet not to do the necessary work, that’s wasting your time and others’. Study is not a hardship – it is enriching.

In one of your comments, you ask:


Does Sion’s attitude explain more complex structures than ours? (C-6)


The answer to your question is surely yes. If you had taken the trouble to study the work before you, you would not have asked such a question.

I have taken into consideration a great many subtleties in my analysis of a fortiori argument that you do not even dream about, let alone mention or deal with. Your approach is naïve and simplistic. Look at the table of contents of my book AFL. Have you even looked into any of the issues there raised? E.g. Have you considered quantification (ch. 3.2)? Or antithetical items (ch. 3.4)? Or the differences between ontical, logical-epistemic and ethical-legal a fortiori arguments (ch. 4)? It is evident from your work that you are only (if vaguely) aware of the simplest form of a fortiori argument, namely the positive subjectal mood. You do not even treat the corresponding negative mood, let alone the positive and negative predicatal moods.

Even though in your letter (L-6) you claim to have dealt with “rejection” of arguments, you have in fact not clarified how your approach effectively differentiates between positive and negative arguments. Moreover, you even confuse the positive subjectal mood of a fortiori argument with the positive subjectal mood of quantitative analogy, since as already pointed out your arguments are reversible (i.e. do not only proceed from minor to major). You think I am wrong in levelling this criticism against your work because you have not yet, even now, understood this criticism.

In the following comment you express indignation at my referring to your table as mere analogy, and not a fortiori argument:


1. Please just read what it says! Fig. 46 deals with the second step, and not with step 1. It is stated clearly. Step 2 is exactly similar to this table, with no translation or hidden premises. 2. According to your approach step 2 is analogy and not a- fortiori. This is false of course’ as one can see in the Talmud itself. (C-21)


This comment once again shows you do not understand that such a table cannot represent an a fortiori argument per se, but only the analogical subtext of a fortiori argument, for the simple reason that the issue of the sufficiency of the middle term, which is a distinctive feature of and essential to a fortiori inference, is simply not reproduced in this sort of tabular representation. Just because you have in the back of your mind the thought that this table is intended to represent an a fortiori argument does not mean that the table actually performs this desired function – all it does is illustrate the underlying proportionality. It is useless for you appeal to the Talmud’s labeling of its argument as a fortiori – this does not prove that your representation of it is entirely accurate. You need to become more conscious if you wish to progress in logic; you are still very much a novice.

You refuse instruction offhand. In my original critique (i.e. in AFL), I try to draw your attention to the difficulties inherent in quantification within a tabular representation like yours, and here is what you remark:


1. If you take an example (like a simple KVH 2×2 table) you can see that we’re quite right. Is half payment of KEREN in RESHUT HARABIM a general or particular statement? Is it right in most cases or at all of them? 2. According to your suggestion I would rather take fuzzy logic with infinitely many values. (C-10)


This reaction is indeed the product of a fuzzy mind. Your thinking is far too rough; you are satisfied with mere approximations; you do not show the patience to deal with fine details. One cannot opt for fuzzy logic when precise logic is available – it can only be a last resort.


5.   In denial

Your letter and the related comments display one common feature throughout – you are in denial, refusing any and all criticisms of your 2009 article. I do not doubt that in your own assessment, you have committed no errors or lacunae; but I suggest that this is because since writing that essay your thought has not evolved, because you have made no effort to study the matter further. Although you deserve high commendation for having the scientific spirit to reply to my criticism (for many others whom I have criticized have not displayed such spirit), you do not show the same spirit with regard to acknowledgment of any errors or lacunae in your approach. In your letter, you state:


“You express your appreciation of Dov Gabbay’s “scientific sportive spirit”, i.e., his readiness to examine criticism of his work in an open and honest manner.” (L-9)


This is a good definition of scientific spirit (I never said “sportive,” by the way, but that’s unimportant). But I would add that this spirit includes willingness to admit errors or lacunae. I do not see such willingness even in your reaction to the most obvious errors which I pointed out to you.

I refer here to AFL chapter 25, section 3, where I point out to you that:

  • given that only three quantities are possible, namely 0, ½, or 1, then:
  • if we have four variables A, B, C, D, and we are given that C:D as A:B, it logically implies that whenever B>A, it must follow that D>C: so that:
  • (a) in the specific case where A=0 and B=1 and C=½, it follows that D=1 (and cannot be ½);
  • similarly, (b) in the specific case where A=0 and B=½ and C=1, it follows that D=1 (and cannot be ½).



This is a simple ratio calculation and not open to debate – ask anyone in your university’s mathematics department. Yet you refuse the criticism and continue to blithely claim that D may = 1 or ½ – and even, here, therefore just ½ (!) – as is evident in your comment:


“When the conclusion is X>=1/2, this means that practically you cannot take more than 1/2. This needs no further explanation in the logical context, and there is no discrepancy. (C-14)”


No, sir. The only conclusion you can draw either way in your tabular approach is 1 (like R. Tarfon), and there is no place in it for the conclusion ½ (and therefore also no place for the vaguer conclusion “1 or ½”). Yet you insist on the latter conclusion, in order to make your approach seem capable of addressing the Sages’ disagreement with R. Tarfon. Whether this is lack of attention or lack of intelligence or dishonesty on your part, I cannot say – but I can say that it is error, and you must admit the error. If you admit the error, it is a minor issue; but if you refuse to admit it, it becomes a major issue.

The scientific spirit, ideally, is to be only concerned with truth and falsehood, and not with one’s personal standing or any ideological prejudice. In your following comments, you protest that it was not your intent in your original English paper to explain the debate between R. Tarfon and the Sages (which is presented in the Mishna Baba Qama 2:5), and you accuse me of miscomprehension and my critique of irrelevancy if not dishonesty:


You missed the whole issue. We use this KVH just to demonstrate a simple KVH, and we did not intend to explain the SUGIA itself. You decided that we have to explain the SUGIA, then criticize us about the missing exact page, and about not explaining the TANAIC concepts and debate, and especially DAYO. This is an unfortunate case of reading mis-comprehension. It seems to be due to your evident anger. You must put that aside in order to keep your ability to understand and criticize honestly. You can delete about 2 irrelevant pages of discussion here. It is all based on a mistake. (C-15)

The AFA that we cited from Babba Kamma (R. Tarfon’s dayo) was meant as an illustration only, to show the reader an example of a Talmudic AFA. You spend a lot of effort (over several pages of the chapter) to explain the issue (sugiya). You thus attempt to show that we did not explain the opinions of the tannaim and their argumentation (and even did not include a reference to it). But it never was our intention to explain all this or relate to it, but only to show an example of a Talmudic AFA. (L-8)


This is evading the issue, which is that your method of a fortiori argument analysis is, contrary to your claim, incapable of representing even this one fundamental Mishnaic debate which you take as an example, let alone all possible debates. You perhaps imagine that you are successfully representing the debate, because you are unaware of your error of calculation. But I show you that your method in fact can only explain R. Tarfon’s position (viz. full compensation either way), and cannot explain the Sages’ counter posture. Surely that is a very relevant objection; and since I prove my case it is neither miscomprehension nor dishonesty on my part. Moreover, the fact that your method can only duplicate R. Tarfon’s position and has no place for the Sages’ is indicative of its simplicity – it just deals with proportions, i.e. it is technically only about analogical reasoning and not about a fortiori reasoning.

You cannot use a text (the said sugya) as an example and misrepresent it, and then complain when someone objects to such misrepresentation that it was not your intention to fully analyze the issues involved. Your intellectual duty here was to lay out the relevant elements of this sugya – including the fact that R. Tarfon twice concludes 1, whereas the Sages opposing him twice conclude ½ (motivated by a dayo statement, which therefore also needs some mention and explanation). Your method, objectively, could only come up with R. Tarfon’s conclusions (1) and could not reflect the Sages’ position (½), let alone its motive (dayo). This is not “missing the whole issue” – it is very much the issue.

It is not enough in this context to refer me (or any reader) to your Hebrew paper, as you do in these comments:


In order to understand R. Tarfon and the others you can read the Hebrew paper. Here we don’t explain it. (C-16)

If you will read our Hebrew paper you will get the exact explanation to the DAYO also. But this is not our aim at this stage. (C-17)

We had no intention of explaining it here. (C-18)


No one expects you to reenact the whole sugya in the English paper, if that is not your intent. But you must state the relevant elements of your interpretation in a few words – not just to show you understand the discussion, but to enlighten the reader with your lights. I have not read your Hebrew paper; but I am very skeptical that you have in it accurately analyzed the rabbinic discussion, for the simple reason that one cannot truly understand it if one does not have a clear idea of a fortiori reasoning, and you do not. If you have not grasped the different forms of such argument, and the way their premises might be arrived at, you cannot grasp the nature and variety of the Sages’ dayo objections.

You may think you have “the exact explanation of the dayo;” but I doubt it. I challenge you to produce and send me a faithful English translation of your Hebrew paper (without any ex post facto revisions, please), and I will tell you exactly where you went wrong. Alternatively, read my account of Baba Qama 2:5 and 25a-b (in AFL, chapters 7-8), and see for yourself where we agree or differ. You might well learn something new!

If you discern anger in my rebuttal, it is because I do not like being taken for a ride. It is obviously comforting to you to imagine that, as you put it in your letter, I display “a basic lack of understanding” of your work, but I think I understand it only too well – well enough, surely, to find fault with it. You use this accusation to close the subject, saying:


After realising that your words show a basic lack of understanding of our work, we did not proceed any further. (L-1)


And indeed, your comments stop well before the end of my critique (a bit past the halfway mark, to be more precise)! Perhaps you think that everything I said after that point was just (more) hot air – or maybe you could not face the criticism or understand it, and preferred to walk away from the discussion under the pretext that I do not understand your work. Did you read the remaining pages? If so, was there nothing in them that you found relevant or significant?

Interestingly, just after your last comment (C-23) I propose to you a “modified table” in which the issue of sufficiency of the middle term is taken into consideration. I suspect you have no comment on that suggestion because you do not understand the need to take this issue into account if your tabular representation is to refer to a fortiori argument instead of just to analogy. Am I here misunderstanding your work, or are you failing to grasp my criticism?

Thereafter, I show how your analysis of the sugya in question (Kiddushin 5a-b) “mixes apples and oranges,” i.e. indiscriminately lumps together arguments that are a fortiori and others which are not, in a complex tabular unfolding. I explain clearly that such compound tables must be avoided because they blur the boundaries between individual arguments in the chain of reasoning (rather than “unite” them as you claim in L-6). Furthermore, the fact that these “expanding structures” (as you call them in L-6) are not mechanically generated but require human intervention to build up is cause for suspicion. Perhaps you can understand these concerns more readily if I refer you to my critique of G. Abitbol in AFL, chapter 21, section 4. He, like you, dealt with refutations by expanding the initial table of four cells, i.e. by adding cells (vertically or horizontally) – whereas it would be more accurate and safer to construct antagonistic tables that respect the number of terms operative in each individual argument.

In the next section of chapter 25, I criticize your overall methodology by pointing out that an inductive method such as yours cannot be “proved” by testing it once or twice, particularly if the material used for the test is itself open to doubt or at least still unproven (as is the case with examples drawn from the Talmud). If you had tested your method say twenty or fifty times, on all sorts of reasonably trustworthy material, and found it fitting every time, its reliability would certainly be highly confirmed. But as your paper stands, that was not the case. Perhaps you can understand these concerns more readily if I refer you to my critique of Y. Ury in AFL, chapter 29, sections 2-3. He devised a way to represent a fortiori arguments and, like you, tested it on a few Talmudic arguments; after which he too optimistically predicted it applicable to all arguments.

Surely, if you disagreed with these criticisms of your method, you should have defended yourself; and if you agreed, you should have said so. Yet you chose to ignore them all, under the pretext that I misunderstand your work. I have written a haiku (that’s a Japanese style of poetry consisting of 5-7-5 syllables), which sums up the present essay for you:


Pearls on a platter –

This food is too rich for him.

Prefers his own ‘ful’!


With thanks for your kind attention, and friendly regards,

Avi Sion



[1]              Studia Logica 92(3): 281-364. For information, your original essay was 84 pages long; my chapter 25 was 15 pages (A4) long; your general letter was 2 pages long, and the 23 additional comments amount to about 1.5 more pages. The present retort is 13 pages long.