A FORTIORI LOGIC
CHAPTER 26 – Stefan Goltzberg
Stefan Goltzberg apparently started writing quite recently, showing from the beginning considerable interest in a fortiori argument. We shall here review his essay in English “The A Fortiori Argument In The Talmud” (14p.), published in 2010. He has written at least two more papers, in French, which also treat of a fortiori argument, though not exclusively. However, they do not bring much more to the subject than the English one, so most of our attention here will be on the English one. All quotations come from the latter unless otherwise specified.
Goltzberg proposes the following, if only “provisional,” definition of the argument:
“The a fortiori argument… structure is that if p applies in case A, and since B is more x than A, then p applies at least as much in B. p is any category; A and B are situations; and x is the scalar feature of a situation by which a category applies.”
Notice that he does not limit the argument to legal contexts. Notice, too, that he places in his discourse the minor premise before the major premise and conclusion; this is intentional on his part, because he regards these as three “stages” of argumentation. This definition seems essentially correct, though as we shall see it is not all-embracing. Let us to begin with put it in standard format, to better understand it:
B (P) is more x (R) than A (Q),
and, A (Q) is x (R) enough to be p (S);
therefore, B (P) is x (R) enough to be p (S).
We see that what Goltzberg has in mind is positive subjectal argument. His vague description of the subsidiary term p as “any category” that “applies” in “situations” A and B (the minor and major terms, respectively), and of the middle term x as “the scalar feature of a situation [i.e. A or B] by which a category [i.e. p] applies,” may be viewed (generously) as perhaps an attempt on his part to expand his definition to include positive antecedental argument. But in any case, formally speaking, Goltzberg has here put his finger only on a minori ad majus argument, with no reference to a majori ad minus argument, although the latter is commonly known. More specifically, he mentions neither negative subjectal argument, nor positive or negative predicatal arguments, nor their implicational equivalents.
Additionally, though he does mention a middle term (x), his exposition lacks mention of the indicator of sufficiency (i.e. A or B is x enough to be p); yet, this indicator is necessary in order to signify that there is a threshold value of x as of which p is applicable. Like many other commentators, Goltzberg has not understood that without this crucial factor in the minor premise, the desired conclusion cannot be derived from it. Given that “B is more x than A,” and merely that “p applies in case A,” it does not logically follow that “p applies in B.” In fact, Goltzberg’s given premises are equally compatible with the contradictory of the desired conclusion, i.e. with “p does not apply in B.” It is only with the minor premise “A is x enough to be p” that we can logically guarantee the conclusion “B is (x enough to be) p.”
So, as regards the structure of a fortiori argument, Goltzberg’s viewpoint is inappropriately narrow and sorely deficient. He could have remedied some of these imperfections by referring to the definition proposed (perhaps with other authors) by P. A. Lalande, in his Vocabulaire technique et critique de la philosophie (1926), which I cited in my Judaic Logic:
“Inference from one quantity to another quantity of similar nature, larger or smaller, and such that the first cannot be reached or passed without the second being [reached or passed] also” (my translation).
Here, we see that both directions of inference are mentioned, and moreover the idea of a threshold is emphasized. However, this definition too is imperfect because, as I remarked there, it “fails to specify that the positive movement from large to small is predicatal, while that from small to large is subjectal; and it ignores negative moods altogether, as well as differences between copulative and implicational forms.”
Nevertheless, Goltzberg’s definition is technically passable as a first attempt. But the question arises, where did he get it? Let us engage in a bit of ‘higher criticism’ and find out. Is he claiming it as his own discovery? One might think so, seeing how breezily he put it forward it, almost in passing, as if it is obvious. Or perhaps he regarded it as so commonly known that it does not require a reference.
At first glance, this definition reminded me of the one given by the Encyclopedie Philosophique Universelle, which I also cited in my Judaic Logic:
“A fortiori argument rests on the following schema: x is y, whereas relatively to the issue at hand z is more than x, therefore a fortiori z is y” (my translation).
But of course, this definition was incomplete in that it did not specify a symbol for the “issue at hand” (i.e. the middle term). Perhaps Goltzberg read that and substituted the symbols A, B and p for x, z and y, respectively, and thought to add a fourth symbol (his x). Alternatively, he might have based his definition on the work of Moshe Chaim Luzzatto called Sepher haHigayon (1741), which was published in English translation under the title The Book of Logic in 1995. Four forms of a fortiori argument are there defined, including the positive subjectal: “Quantified commensurates [are terms that] share a common quality, but not in the same degree. One exhibits a Greater degree and the other a Lesser degree of the same quality.… Whatever is affirmed about the lesser will surely be affirmed about the greater”. But there is no evidence he has read that book, either.
In any case, he does not refer to these possible sources of his definition, or any other. He does however say in a footnote to his definition: “Avi Sion puts forward that ‘Aristotelian syllogism deals with attributes of various kinds, without effective reference to their measures or degrees’ (Sion, 48).” So it is reasonable to suppose he got his definition from the presentation in my Judaic Logic, which (certainly some 15 years before him) clearly formulates a fortiori argument as comprising four terms (or theses) in three propositions (two premises and a conclusion). The questions arise: Why does he not say it? Why does he so carefully change the symbols used? Why did he only mention the positive subjectal form and no other? Why did he not mention the threshold involved? I suspect there was a conscious or subconscious desire to conceal my contribution, so as not to be overshadowed by it.
I resent such treatment. Such behavior is contrary to the ethics of academic work (and, for Jews, contrary to rabbinic teaching, see Pirqe Avot, 6:3,6). One should always try to acknowledge the sources of one’s ideas. Even if one thinks one arrived at an idea independently: one may say so, but one must still admit that some other(s) arrived at it before – for the simple reason that one can never be sure not to have been indirectly and unconsciously influenced. One is duty bound, too, to pass on information correctly, and as fully as appropriate to the context. All the more so, one must not misquote someone, or quote him out of context in such a way as to make him appear to say something he did not say, or (worse still) the opposite of what he said. How can logical and other studies progress if people studiously avoid relaying new findings? 
Goltzberg also needs to be reproved for his failure to mention, not only my formalization of the various moods of a fortiori argument, but also my original formal validation of those moods. How can something so important to the subject of study have been simply ignored? Obviously, he did not grasp its significance. He does, nevertheless, to his credit, on various occasions, acknowledge the logical force of the argument, saying, for instance:
“I think that … a fortiori … is … unrefutable (sic) … iff one accepts the premisses and the hierarchy between the terms.”
That is to say, as I understand him, the argument is formally valid in itself, although of course (as in all argumentation) the truth of the conclusion depends on that of the two premises. It is not mere rhetoric, but logic. So well and good; we seem to be in agreement on this point. But this is not, as he seems to “think,” a mere matter of opinion or authority – it can be formally proved! And Goltzberg fails to mention the important fact that I have indeed demonstrated its validity. For this reason, I get the impression that he merely glanced through my Judaic Logic, gleaning from it snippets of information without bothering to actually study the text as a whole. This impression is reinforced by the following passage:
“Avi Sion has devoted many pages to a fortiori arguments within the Bible and the Talmud. At the end of a chapter on formalities of a fortiori arguments, Sion writes: “I did not prove the various irregular a fortiori to be invalid, but rather did not find any proof that they are valid” (Sion, 1997: 46). Sion claims that an a fortiori argument’s validity, if not rebutted, is not yet demonstrated either. We do not claim to provide the reader with such a logical proof – Sion is right. The a fortiori argument is not only a logical but also linguistic device. This is why a logical approach to the a fortiori argument is insufficient to grasp its linguistic specificity.”
Notice how, in his response to the statement of mine that he cites, Goltzberg drops out the word “irregular.” I find this remark of his very irritating, both because of its unsubstantiated, pretentious claim that logic is insufficient in this matter, and because of its condescending, narcissistic tone. When I chanced upon it (in Aug. 2010), I wrote to its author the following complaint:
“You have here taken what is effectively an endnote (to chapter 3.3 of my Judaic Logic), concerning ‘additional details’ within a fortiori logic – namely the status of certain arguments that resemble the a fortiori, but are doubtful (labeled ‘irregular’) – and you have made it seem as if I subscribe to your idea that the (regular) forms of a fortiori argument cannot be formally validated! You are careful not to mention the four (or eight) valid moods of a fortiori that I have originally identified (in chapter 3.1) and the validation procedures that I describe in detail (in chapter 3.2), though these are the crux of the matter. …”
It is bad enough to conceal relevant information. But in that statement Goltzberg depicts me as saying the exact opposite to what I intended in my book, apparently to make it look as if I effectively endorse his allegedly more discerning view. Forgive my harsh words, but this is clearly intellectual dishonesty on his part. Let me emphasize: I firmly believe that the principal moods of a fortiori argument are demonstrable; there are no ifs and buts about that. This is not a mere opinion, but is the product of long and hard research work. Of course, as with all argumentation, one has to verify the truth of the premises and ensure the process is correctly carried out; but granting these precautions, if the mood concerned is formally validated, the conclusion is not subject to discussion. As for the irregular moods I mention in the said book, I have more recently developed the technical means to deal with them (i.e. formally validate or invalidate them) – namely, matricial analysis – so the uncertainty Goltzberg relied on in his remark is in principle no longer present.
Let us move on. Goltzberg also mentions the work of Aristotle, saying:
“As far as we can see, Aristotle does not explicitly look into the so-called a fortiori argument. He mentions the topos, ‘He who can do more can do less’ in the books of the Topics (II, 10). …He presents the a fortiori argument as a topos among others, in other words, as a defeatable argument.”
Actually, I have not found the maxim “He who can do more can do less” anywhere in Aristotle’s Organon, but I do recall seeing it mentioned by Ventura in the name of Lalande. Maybe it comes from some other Greek or Roman writer. This said in passing – it is not very important. A more significant question is: is it accurate to say that Aristotle “does not explicitly look into a fortiori argument”? No – he describes the argument, or at least some aspects of it, pretty well albeit briefly in his Rhetoric (II, 23) and Topics (II, 10 and III, 6). It would, however, be true to say that, unfortunately, Aristotle does not analyze this form of argument in a systematic manner comparable to his study of syllogism: he does not formalize it, nor identify all its moods and figures, nor make any attempt at its validation.
A second important question is: does Aristotle consider a fortiori argument as “defeatable”? I would answer, emphatically: no. I quote him: “Another line of proof is the a fortiori. Thus it may be argued that if even the gods are not omniscient, certainly human beings are not.” It is clear from this excerpt and the rest of his discussion that he (intuitively) regards the argument as deductively strong and not open to rebuttal. Note that he calls it a “line of proof” and says “certainly” in the conclusion. If he had considered the argument as inherently weak, he would surely have said so, and given at least one example of counterargument. Goltzberg’s error here is to generalize too readily, imagining that just because many of the arguments described in Aristotle’s Topics and Rhetoric are rhetorical in character it follows that none of them are logically forceful.
It should be said here that Goltzberg is a fan or disciple of Chaim Perelman. I have not personally studied Perelman’s work, but I gather from third party summaries that it is very interesting and influential. It is possible that Goltzberg’s reading of Aristotle was influenced by Perelman. According to Goltzberg, the latter’s position on a fortiori is as follows:
“Among modern types of topical theories of argumentation, the new rhetoric of Chaïm Perelman deals with the a fortiori argument and considers it as a sort of analogy argument (Perelman 1977: 155) and stresses the fact that the a fortiori argument is not part of formal logic since there are laws that limit the use of a fortiori arguments (Perelman 1976: par. 33). Perelman, in this manner, renews the topical theory of argumentation. According to him, there is no undefeatable argument, not even the a fortiori argument. The a fortiori is then not set apart from the other types of arguments.”
Needless to say, I firmly disagree with Perelman’s apparent exclusion of a fortiori argument from formal logic and view of such argument as “defeatable.” That it is a sort of analogy is obvious – but it is a very sophisticated sort, which is particularly reliable because it is based on precise quantitative and implicational relations. That the fact that “there are laws that limit the use of” such argument should make it “not part of formal logic” is a very mysterious notion; all of formal logic can be viewed as a study of the laws limiting the use of all types of arguments! It is only through such study that we can determine with certainty which arguments are valid logic and which are mere rhetoric. It would seem that Perelman did not understand the nature of logic. But anyhow, my sense is that, though Goltzberg admires Perelman, he does not entirely agree with him as regards a fortiori argument.
Incidentally, Gabriel Abitbol also mentions Perelman – with disapproval. Whereas Perelman’s thesis is that legal dialectics “constitutes an argumentation or a set of proofs tending to persuade, if not convince, as to the legitimacy of the premises,” Abitbol’s view is that “Talmudic dialectic” is never about “the basis of premises,” since in its case these “are axioms written in the Torah” (pp. 336-7). In other words, according to Abitbol, Perelman’s analysis of legal debate, though it may reflect somewhat the discourse involved in practice in secular contexts, does not apply to Talmudic discourse. I would add that logic (inductive as well as deductive) is certainly preferred to mere rhetoric in all legal systems, including the secular and the Talmudic; the use of mere rhetoric is always only a last resort, when logical means, which more persuasive, are not found.
One feature of Goltzberg’s above-cited definition that I have not mentioned so far is his inclusion of the qualification “at least as much” in the conclusion. As we shall see, this interpolation is crucial to his “theory of a fortiori,” so it is no accident.
However, even though this qualification is in many cases tacitly intended, so that it is legitimate to make it explicit, it is an error to include it in the definition, for the simple reason that it is very often not applicable. Very often, the subsidiary term allows of no measures or degrees (if only because it is already the highest possible measure or degree, so that to say “at least” is a redundancy), so that what is said of it in the minor premise is bound to remain the same in the conclusion. For examples, something ‘imperative’ or ‘black’ cannot be more or less so – either it is so or it is not. In such cases, to repeat, it is inappropriate to use the qualification “at least.”
Goltzberg’s emphasis on the expression “at least” is, of course, motivated by the desire to integrate the rabbinical ‘dayo’ (sufficiency) principle into the definition of a fortiori argument once and for all. This is commendable, but as already explained inaccurate and misleading. Prima facie, the dayo principle corresponds to the logical rule that we cannot bring more information into the conclusion than was given in the premises – which I have lately labeled ‘the principle of deduction’. This rule is universal: it is not peculiar to a fortiori argument, but equally applicable to syllogism and all other forms of deduction. Of course, inductive reasoning is not subject to the same restriction. Induction is precisely the effort to extrapolate from given information and predict things not deductively implied in it.
Goltzberg apparently interprets the dayo principle in this manner when he says (all italics his, square brackets mine):
“…this instruction aims at insisting on the fact that the second situation [i.e. the conclusion] deserves the judgment applied to the first situation [i.e. the minor premise], in a degree that is at least as great but not greater. …Dayo, as a claim that the second situation be treated precisely as the former, is not added to the a fortiori argument. It is simply inherent in it. The merit of the Talmud is not to have added this device but to have made it clear that one should respect the principle of the dayo.”
Note well: according to him, dayo is not an artificially “added” limitation, but one naturally “inherent” in a fortiori argument. He denies the alternative interpretation, according to which:
“The function of the dayo clause is [i.e. would, according to such alternative interpretation, be] the following: it prevents someone from applying a higher rate/price/praise/blame to a situation that obviously deserves it at least as much as the former and probably more, as one would want to continue the proposition. The Talmud would have added the dayo device and transformed thereby the very structure and use of a fortiori arguments.”
Thus, Goltzberg is well aware that some people view the natural conclusion from an a fortiori argument to be ‘proportional’ and view the dayo principle as an artificial prohibition. But he disagrees in principle with them, though he adds in a footnote:
“Several persons to whom I said this brought examples from the Talmud in which, according to them, some opinion stressed the fact that there was an a fortiori argument but the dayo was refused. In fact, this issue deserves a closer scrutiny. It is possible to make it clearer by the distinction between de re (‘in fact’) and de dicto (‘supposedly’): someone may be said to claim (de dicto) that there is an a fortiori without dayo, but no one could possibly think that there is de re an a fortiori without dayo. This point merits wider examination.”
Here we see that Goltzberg is not personally acquainted with the discussion in Baba Qama, 25a, but has relied on hearsay concerning it. It is correct to say that there are instances in the Talmud where a fortiori argument is not subjected to dayo. But Goltzberg’s attempt to explain this away by saying that perhaps those who think so confuse supposition with fact is not correct. I am afraid that I may be partly responsible for Goltzberg’s misapprehensions in this matter, because of my emphasis in my Judaic Logic on the dayo principle as a formal limitation on a fortiori inference. In truth, I did there admit that the strict conclusion (in accord with dayo) could be surpassed by means of additional deductive or inductive argument. But I had not at the time yet developed precise formal conditions for this; I had not yet developed the concept of a crescendo argument.
I know now, after “closer scrutiny” of Talmudic a fortiori argument earlier in the present volume, that the dayo principle functions differently. Looking at the Mishna of Baba Qama 25a, whereas the prima facie idea that the dayo principle refers to the principle of deduction is consistent with the first argument of R. Tarfon, it is clearly not consistent with his second. This implies that the dayo principle is something else entirely, which, though in some cases it intersects with the principle of deduction, in other cases (as Tosafot remarked) it operates at a different level.
We could interpret both of R. Tarfon’s arguments as mere analogical ones (pro rata, to be exact), and the two dayo objections by the Sages as ad hoc interdictions of proportionality. But this has nothing to do with a fortiori logic, with which these arguments are traditionally associated; so let us ignore it here. If we interpret the first argument by R. Tarfon as a crescendo, then the Sages’ first dayo objection may be conceived as aimed at interdicting the additional premise in it that justifies ‘proportionality’, and therefore as advocating a purely a fortiori argument.
This scenario could be interpreted as an application of what I have lately called the principle of deduction, i.e. the rule of deductive logic that the conclusion cannot make claims that the premises cannot sustain. However, upon reflection it is perhaps not wise to here effectively identify the dayo principle with the principle of deduction, for the latter principle is applicable not only to purely a fortiori argument but also to a crescendo argument! If in the latter form of argument we advocate a conclusion that contains information not found (explicitly or implicitly) in the premises, then here too we would reject the putative conclusion as illegal (i.e. contrary to the principle of deduction).
In any case, the dayo principle cannot be limited to its role in the first Mishna debate, of ensuring that the conclusion be quantitatively as well as qualitatively identical to the minor premise. It cannot, because in the second argument of R. Tarfon the conclusion is the same whether that argument is construed as a crescendo or purely a fortiori. In other words, here the first dayo objection by the Sages is already obeyed. So their second dayo objection must mean something else: it must (since there is no other explanation for it) be aimed at the generalization preceding the formation of the major premise of R. Tarfon’s a fortiori argument. So, there are at least two types of dayo objection.
What is clear from this analysis of the dayo principle is that it is not a logical principle. It is not to be confused with the principle of deduction. It is a rabbinical decree, apparently based by the Gemara on a particular reading of a Torah passage (namely Num. 12:14-15). It is a legal posture, apparently aimed at preventing punishments based on inference to be in excess of those found in the source-text. It is a morally motivated principle, to temper justice with mercy. Maybe also an epistemological principle, to avoid errors of judgment based on errors of inference. This being so, the idea found in the Gemara that the dayo principle may in some circumstances be bypassed is not all that surprising.
All this is said to answer Goltzberg’s uncertainties and doubts. We could relate all this to his definition of a fortiori argument as follows. We could say that the qualification “at least” found in it serves to include both purely a fortiori argument and a crescendo argument in a broad definition. When the minimum quantity is not surpassed, we have pure a fortiori argument; when it is surpassed, we have a crescendo argument. This would be a neat way to merge the two forms of a fortiori argument into a single, generic form. However, we would still need in such a broad definition to mention the additional premise that makes possible a pro rata conclusion in certain cases.
Goltzberg’s definition as it stands does not specify under what conditions the minimum quantity may logically be surpassed; it would need to be modified appropriately to do so. This would result in a statement like: ‘Given B is more x than A: if p applies in case A, then p applies at least as much in B; if it is additionally known that p is proportional to x, then p applies pro rata more to B than to A; otherwise, if such proportionality is not known to apply – or if it is known not to apply (e.g. through a dayo objection) – then p applies exactly as much to B as to A’.
All that we have presented and discussed so far seems to be, in Goltzberg’s essays on the subject, merely introductory or incidental to his central thesis, which he calls grandiloquently the “two-dimensional theory of a fortiori.” According to this insight:
“Arguments are structured by two main parameters: orientation and strength. The four types of arguments may be analyzed through the following transitional keywords examples. Keywords may be co-oriented or counter-oriented and stronger or weaker.”
Goltzberg observes that in statements like “p or at least q” or “p or even q,” thesis q is “co-oriented” with p, because it points in the same direction, differing perhaps only in measure or degree. Whereas in statements like “p even if q” or “p unless q,” thesis q is “counter-oriented” to p, because it points in the opposite direction. That is to say, in the former cases, q is “an agreeing argument” with p; whereas in the latter q is a “counterargument” to p. Moreover, as regards “the strength parameter”: in “p or at least q” or “p even if q,” q being weaker strengthens p; in “p or even q” or “p unless q,” q being stronger weakens p. He gives simple illustrations, like “He can run 10 miles or at least 5 miles.”
The fact that Goltzberg repeats this theory in all his essays on the subject makes me call it his pet theory, and assume that it is his own for that very reason. He seems fascinated with its symmetry and versatility, and with the facts that theses (or clauses of theses or terms) may point in opposite directions as regards their meaning and that one may reinforce or undermine another. He calls this “two-dimensionality” to imply that both these sets of factors must be taken into consideration if we are to fully comprehend the relationship between the theses concerned. However, he does not precisely define or explain these various concepts, but only uses them intuitively.
Before further ado, let me say some things about the concept of “at least,” which for his part Goltzberg hardly says anything about, though he uses and relies on the expression quite a bit. Underlying a fortiori argument is the known mathematical field of comparisons of quantity. As I have shown in the context of validation procedures, the concepts of ‘more than’, ‘less than’, and ‘as much as’ are involved not only in the (commensurative) major premise but also in the (suffective) minor premise and conclusion. The expression ‘at least’ means ‘not less than’; similarly, ‘at most’ means ‘not more than’ and ‘exactly’ means ‘neither more nor less than’. So these expressions fall squarely in the said larger context, as negations of positive comparisons. Propositions involving them are members of one big family.
Furthermore, it is important to realize that these expressions are tacitly modal. When we say of something that it is ‘at least’ X, we intend X as a minimum quantity; we impose the impossibility of a lesser quantity and leave open the possibility of a greater quantity (than X). Likewise, ‘at most Y’ intends Y as a maximum, with lesser amounts possible and greater amounts impossible; and again, ‘exactly Z’ implies that Z is both minimum and maximum. These modalities, possibility and impossibility, may be intended in any of the various modes (or types) of modality. When ‘possibility’ is intended in the logical mode, we mean that the event concerned is conceivable in the given context of knowledge; in the extensional mode, we mean that there are known cases of it; in the natural mode, that it occurs in some circumstances; in the temporal mode, at some times; in the spatial mode, in some places; and in the ethical mode, we mean that it is permissible, i.e. a means compatible with our ends.
As regards the “keywords” Goltzberg uses in the above table, let me add the following. The expression “p or at least q” means “the greater quantity (p) is possible, the lesser one (q) is sure;” whereas “p or even q” (i.e. “p or as much as q”) means “the lesser quantity (p) is sure, the greater one (q) is possible.” These two statements are very similar in form, though with different emphases; however, in view of the switching of symbols for greater and lesser, they are contraries. Likewise, the expression “p unless q” means “if q, then not p;” whereas “p even if q” means “if q, not-then not p;” so these two statements are contradictories. These are four representative examples; but evidently they are not a symmetrical and exhaustive list of possibilities. Goltzberg does give a few other keywords in other essays, but he does not treat the matter systematically.
Now, as already said, Goltzberg does not seem to realize the exact meanings and sources of his concepts of orientation and strength. The idea of orientation is an old deductive concept – it is the idea that ‘every thesis has a counter-thesis’. The idea of strength is an old inductive concept – it is the idea that ‘a thesis may be more or less confirmed’, ranging in probability from 0% (definitely denied, or impossible) to 100% (definitely affirmed, or necessary). And probability is a modal concept, which may be intended in any of the various modes of modality already listed above. As regards the concept of confirmation and its many relatives, I refer you for instance to my essay Principles of Adduction, where the seesaw between theses and counter-theses is well explained. So these concepts are nothing new, and Goltzberg’s use of them is neither innovative nor very profound.
But the bottom line is pragmatic. What is the practical value of Goltzberg’s insight or observation regarding the “at least” factor in a fortiori discourse specifically? Its main function seems to be to institutionalize the dayo principle, which he understands as equivalent to what I have called the principle of deduction, but also apparently to clarify the role of the conflicting marker “all the more.” The latter expression creates an expectation for a heavier claim, while the former principle signifies that we will nevertheless rest content in the conclusion with the lighter claim made in the minor premise. The conclusion neither surpasses nor annuls the premise it is based on, but modestly adapts to it, as it were. This approach relates to rhetoric rather than to logic, in accord with Goltzberg’s underlying interests. From the formal point of view, it is neither useless nor harmful, but not of great moment.
Goltzberg, in a more recent paper,
apparently generalizes his above-described idea that “arguments are
structured by two main parameters: orientation and strength” to all forms of
argument, and calls it “the two-dimensional theory of argument.” Judging by
this title, one would expect the idea to be as revolutionary as, say, “the
special theory of relativity”! But as we have seen, there is not much to it.
It is just old stuff repackaged with a bit of grandstanding. This remark is
not, of course, intended to put down or discourage the author, but only to
recommend more modesty.
 In: Judaic Logic, ed. Andrew Schumann. Piscataway, N.J.: Gorgias, 2010. This can be read online at: stefangoltzberg.files.wordpress.com/2010/11/stefan-goltzberg-2010-the-a-fortiori-argument-in-the-talmud.pdf.
 These are: “Esquisse de typologie de l’argumentation juridique” (19p.), published in the International Journal of the Semiotics of Law, vol.21 (2008), pp.363-375, and posted online at: stefangoltzberg.files.wordpress.com/2009/04/stefan-goltzberg-2008-esquisse-de-typologie-de-largumentation-juridique-version-site.pdf; and “Logique et argumentation” (65p.) seemingly not yet published (Syllabus, U. of Mons, Nov. 2010), posted online at: stefangoltzberg.files.wordpress.com/2010/11/logique-et-argumentation-syllabus-stefan-goltzberg-novembre-20101.pdf. The latter paper is his most polished.
 As he explains in his 2010 French essay. This is of course commonly done and acceptable, if we think in psychological or rhetorical terms. However, the major premise is conventionally put forward first in formal logic discourse, because it is what justifies the inference from the minor premise to the conclusion.
 Paris: PUF, 1972.
 Paris: PUF, 1990.
 Pp. 89-90. Note that the two premises and conclusion, and the four terms in them, are all present and accounted for, although not symbolized.
 In the same spirit, I believe, Goltzberg asserts: “Ten a fortiori arguments are to be found in the Tanakh” – without mentioning that I uncovered and listed an additional twenty or so cases! Though he does mention “the forgotten a fortiori arguments,” with a reference to Moshe Koppel’s Meta-Halakha. Logic, Intuition And The Unfolding Of Jewish Law (Northvale, NJ, Jason Aronson, 1987). Perhaps, then, someone preceded me in this matter. I have not seen what is actually said there, and so cannot judge. People should make the effort to more precisely relay information they refer to.
 This said in a private e-mail to me dated 20/9/07, when he first made contact with me after discovering my book, Judaic Logic. He says about the same in French in his 2008 paper: “Les arguments indéfaisables [lit. which cannot be undone] incluent également l’argument a fortiori;” adding in a footnote: “Il va de soi que les arguments a fortiori ne sont contraignants que sous la clause ceteris paribus [i.e. all other things being equal].” And further on: “Pour être exact, un argument a fortiori est certes toujours défaisable, mais contrairement aux arguments a pari et a contrario qui peuvent être évincés sans autre forme de procès, la réfutation de l’argument a fortiori suppose que l’on rejette non seulement la conclusion mais toute l’échelle argumentative qui le sous-tend.” I have not found a similar statement in his 2010 English essay, however.
 When I said as much to him, Goltzberg replied: “I am not dishonest. I surely am not. It is always possible to misunderstand a text, but it is not necessarily the case when one quotes you. I did not want to distort your claim, which I’ve quoted. I did not even criticize your position and I do not behave ‘as if’ you said something else. I am not responsible for the hypothetical interpretations beyond what is said in my text.” Readers may judge for themselves.
 See my work The Logic of Causation, Phase III.
 It was on p. 76 of Terminologie Logique, to be exact. (Paris: Librairie Philosophique, 1935.)
 It has a Latin equivalent: Qui magis potest minus potest. This may be of Scholastic origin. But the maxim also seems proverbial in many other languages.
 Poland, 1912 – Belgium, 1984.
 The references are to: Logique juridique. Nouvelle rhétorique (Paris: Dalloz, 1976) and L’empire rhétorique: Rhétorique et Argumentation (Paris: Vrin, 1977).
 Presumably this refers to the dayo principle (see next section).
 Moreover, I would criticize Abitbol’s position and say that even if Talmudists do not dispute the basic premises given by the Torah, treating them as axiomatic, it remains true that they are objectively open to dispute – and indeed have become hotly disputed in modern times. Moreover, even within the framework of the Torah as an uncontestable given, Talmudists do not always argue logically, but sometimes resort to mere rhetoric.
 Goltzberg ignores this when he contrasts, for example, “forbidden” and “even more forbidden”!
 As I did when I wrote my Judaic Logic, I must admit.
 See my work Future Logic, chapter 11, on this topic.
 p refers to the greater in one and the smaller in the other, while q refers to the smaller in former and the greater in the latter. To be more exhaustive, we would have to also consider: ‘q or at least p’ and ‘q or even p’. Also, as above mentioned, ‘at most’ and ‘exactly’. And negations of all these.
 See also my 1990 work Future logic, chapters 46-49. This essay was written soon after that book (the same year) but only published in 2003 in my Phenomenology (as chapter 7.1). I do not of course claim being the originator of these concepts.
 Théorie bidimensionnelle de l’argumentation : Définition, Présomption, Argument A Fortiori. I have not read this document (dated 2010-11), though it devotes a chapter to a fortiori argument, because it is not yet published. But its table of contents is posted online at: crfj.academia.edu/StefanGoltzberg/Papers/1441244/Theorie_bidimensionnelle_de_largumentation_definition_presomption_et_argument_a_fortiori.