A FORTIORI LOGIC

The most recent attempt to study a fortiori argument may be that of Hubert Marraud, whose “From a Stronger Reason,” a draft paper of 14p, was posted online in 2012[1]. From the outset, it is evident that this author’s orientation is ‘rhetorical’ rather than logical. This rhetorical approach is, of course, not without value, and rather commonly used; but it does not go to the core of the issues, and for that reason – as we shall see – makes possible someserious errors of logic. I shall here only comment on Marraud’s ideas directly concerned with a fortiori argument, and not deal with some of his views relating to argument in general.

1.Warrants and premises

Marraud begins his presentation by stating: “A fortiori argument (henceforth AF argument) is a traditional kind of argument that has been largely ignored by contemporary argumentation theorists.” In fact, as we have seen in the present survey, quite a number of contemporary writers have ventured into the a fortiori field, although most of those who have done so have proved to be rather amateurish. What is most striking is the silence on this subject of big name modern logicians. In the course of his essay, Marraud cites some authors of papers concerned with a fortiori argument that I was unaware of. Not having found copies of their papers, I cannot assess them in the present volume. Notably absent from Marraud’s references, however, is my own work on the subject, in myJudaic Logic(1995). It is a pity for him that he did not discover it, even though it has been freely available on the Internet since 2001; it would no doubt have improved his work and saved him much embarrassment.

He traces “the concept of” a fortiori “back to Aristotle’s topic from more and less (Rhetoric II, 23).” I take it that Marraud here means that Aristotle was the first to draw our attention to the use of a fortiori as a distinct form of reasoning. But he (Marraud) does not explicitly mention that the argument can be found used in earlier literature. From his remark that “In fact most translators usea fortioriinstead offrom more and less,” it is evident that Marraud looked for the literal translation of Aristotle’s Greek words for this sort of argument. He further informs us that “Aristotle’s examples include both theoretical arguments (If not even the gods know everything, human beings can hardly be expected to do so) and practical arguments (If Hector did well to slay Patroclus, Paris did well to slay Achilles).” But this theoretical-practical distinction is certainly not clear; and Marraud does not clarify it for us.

Be that as it may, let us examine Marraud’s views concerning a fortiori argument and look at and evaluate his treatment of specific examples. Marraud tells us that he follows a “Toulmin model” for the description of different a fortiori arguments (p. 5). This model consists of premises and a conclusion—and a “warrant.” This constitutes an “argument scheme.” The framework used is: “premises so conclusion since warrant.” The warrant is what explains how the premises “lead to” the conclusion. The “strength” of the argument depends, we are told, on the kind of warrant involved; but it is not made clear (as far as I could tell) just what “strength” is and just how the said dependence is to be determined[2]. Marraud presents his arguments in diagrams like the following (the diagram below is my own generalization from his various applications):

Diagram 30.1

The symbol he uses (an arrow on a stand[3]) apparently means that the premises (p) imply the conclusion (q) on the basis of the warrant (w) – the letters p, q, and w, for the three items involved, being added by me. Logically, of course, this diagram signifies: If the warrant is true, then the premises do together imply the conclusion; or, in other words: If w, then if p then q. But this nested hypothetical proposition in turn formally implies: If w + p, then q – which suggests that w is just another premise. However, Marraud is careful to advise us that the warrant, whether explicit or implicit, is not conceived as an additional premise. It is rather what Aristotle called the “principle” underlying the argument, as in the example:

“The argument that a man who strikes his father also strikes his neighbours follows from the principle that, if the less likely thing is true, the more likely is true also.” (RhetoricII, 23[4].)

I must first comment on this proposed model and the ideas it enshrines. It is true, as Marraud and presumably before him Toulmin observes, that when people reason, they sometimes – though not always – explain or justify their reasoning. So, there is empirical basis for the concept of a “warrant.” However, what is the difference between a warrant and a premise? Marraud does not look into this issue very deeply. As a result, as we shall see when we examine his specific examples, all of his alleged warrants are in fact premises! To avoid this confusion, we must clarify the difference.

A warrant differs from a premise in that the former is more abstract than the latter. A warrant concerns the form of an argument, whereas a premise tells us something about the content. In the symbolic terms, the content of a warrant w for an argument with specific premises p and specific conclusion q would be: “premises of form p do imply a conclusion of form q.” In the example from Aristotle above given, “if the less likely thing is true, the more likely is true also” is a warrant, while “a certain man occasionally strikes his father” is a premise. The former principle is indeed a warrant, because it is a proven general statement about a certain form of argument; whereas the latter information is a mere premise, because it is a statement involving particular terms (namely: man, strikes his father), whose implication of the conclusion (that the man may strike his neighbors) needs to be established.

Actually, this definition of a warrant is not entirely correct. A more precise statement would be the following. It could be said that the whole science of logic is a search for argument forms and their warrants. Identifying an argument form is formalization, and validating it is finding out how to warrant it. The primary warrant of any argument form isthe processthrough which it is validated. But once this initial validation is effected by some logician, the argumentformbecomes a warrant in its own right. For example, once Aristotle validated the moods of 2^ndfigure syllogism by reductio ad absurdum to moods of 1^stfigure syllogism (which were already validated by exposition), then anyone reasoning by way of 2^ndfigure syllogism could henceforth warrant hisparticularargument either by engaging in aparticularreductio ad absurdum process, or by appealing to the already establishedgeneralform of that syllogism without the need to repeatedly validate it.

Thus, particular arguments can in practice be warranted – i.e. explained and justified – in two ways. We may either use an appropriate logical validation process for that particular argument, which may be complicated and long-winded. Or we may simply evoke the form of the argument – in wordless intention, in words or using symbols – implying that this form of argument has already been validated in theoretical terms by someone for all time and there is no need to repeat the validation process in this concrete case. However, it should be kept in mind that the latter sort of warrant is an intellectual construct; it is something relatively artificial and static, although full of potential. The true warrant is always the underlying validation process, which calls for actual rational insights. If we lack these actual insights, and therefore cannot really understand and validate a certain principle, the principle is for us an empty shell and appeal to it does not in fact constitute a warrant.

Contrary to common misconception, the validation process does not consist in applying the ‘laws of thought’ (conceived as axioms, i.e. irreducible primaries) to particular arguments, but in the ad hocuse of rational insight. This ad hoc view of validation has to be upheld, since otherwise we could not explain or justify the supposedlyfirstapplication of these laws. That is, we cannot appeal to syllogistic inference from axioms before we have validated such inference. The axioms are thus ex post facto summaries of what we effectivelydo– they are not a priori or arbitrary principles that weapply. Rational insight is an empirical act, whereas application of principles is conceptual reflection. Therefore, though it is true that we often think by appeal to established logical principles, we cannot be said to truly understand these principles if we are unable to validate themwithoutresort to a prior warrant.

We cannot logically say that “p implies q” really means “p implies q, if and only if w;” i.e. that the former statement is always a truncated expression of the latter. This would mean that p is not sufficient by itself to imply q; i.e. that only in conjunction with w (or some other such warrant) can p imply q. And since this in turn means “(w + p) implies q,” it calls for a further (anterior) warrant; and so on, ad infinitum. Therefore, wemustadmit the intervention, andlogical autonomy, of primary rational insights at some stage. If we admit it immediately for “(w + p) implies q,” we might as well admit it immediately for “p implies q.” Thus, the Toulmin model is logically untenable if taken to extremes. The buck has to stop somewhere. We have to admit that not all reasoning calls for a warrant, let alone for further “backing” for the warrant. Otherwise, there would be no end to it – no possibility for rational knowledge at all.

2.The main form of a fortiori

Let us move on and analyze what Marraud refers to as “the positive form of arguments from more and less,” which he presents as follows (p. 6):

Taking him at his word. The first thing to note is that Marraud’s proposed warrant (the bottom line in his diagram) is not a warrant but a premise. It is not an abstract principle of logic which explains and justifies the proposed argument (the top line); it is a concrete proposition with specific terms, and is therefore part and parcel of the argument. Thus, the argument presented here does not involve an explicit warrant – it is only superficially in accord with the Toulmin model. This means that the argument should be rewritten as follows. Note that I have changed the symbols – putting, in place of Marraud’s O, O’, R, P, my by now standard symbols P, Q, R, S, for the major, minor, middle and subsidiary terms, respectively[5].

The first question to ask is: do the given premises (a), (b) and (c), indeed imply the conclusion (d) – that is, is the argument valid? The answer is: yes; and this can be proved as follows: Premise (a) implies that P is R and that Q is R. Premise (b) implies that whatever is R is S. Whence, we can infer that P is S and that Q is S. We have thus established that the putative conclusion (c) does indeed logically follow from the given premises, and they alone. This validation process constitutes the true warrant of this argument, and not the proposition (here labeled (b)) that Marraud claimed as the warrant.

Our next question is: was all the information given in the premises needed to obtain the conclusion, or was some of it redundant? Well, for a start premise (c) was redundant, since it can be inferred from premises (a) and (b), and anyway was not used to get to the conclusion. But moreover, the comparison inherent in premise (a), i.e. the fact that Rp > or < Rq, was redundant information, since we only used the incidental implications that P and Q are R. Similarly, the concomitant information inherent in premise (b), i.e. the fact that S varies in proportion to R, was redundant information, since we only used the incidental implication that All R are S.

Thus, the argument could have been formulated much more succinctly than it was, as: (a) Q is R, and (b) All R are S; therefore, (d) Q is S. This is just positive first figure syllogism. Thus, although Marraud’s argument looks at first sight like a complex a fortiori argument, it turns out to be a very simple syllogism. This does not mean that a fortiori argument is syllogism, but means that what Marraud has presented as a fortiori argument is in fact not a fortiori argument. For one cannot claim to have formalized a fortiori argument if one has not, among other things, taken into account the quantitative comparison, viz. Rp > Rq, which such argument is generally acknowledged to involve. Merely to have this information in a premise is not enough – it has to be actually used to infer the conclusion. That quantitative comparison is a defining aspect of a fortiori argument is acknowledged by Marraud through the very fact that he includes it in his schema. But his inclusion of it is very superficial – it in fact plays no active role in his argument.

Ignoring the so-called warrant. We could end our analysis of Marraud’s thesis here, for what we have found so far is bad enough. But we shall explore his work further and show up its other faults. Let us for a while, for the sake of argument, ignore Marraud’s “warrant” (i.e. premise (b)), and examine what he obviously regards as the core argument (namely, premises (c) and (a) and conclusion (d)). This is worth doing, since in his estimation the warrant stands apart from the argument, as evidenced by his diagram which has the former in the bottom line and the latter in the top one. Thus, we could say that for Marraud the essence of a fortiori argument is:

Now, this looks more like what is generally considered as the form of a fortiori argument. However, it is not. Although it somewhat resembles the positive subjectal mood of such argument, there are two serious errors in this representation, one being an error of commission and the other being one of omission. The first serious error is that the major premise says “P is ± [i.e. more orless] R than Q,” implying that positive subjectal a fortiori argument can proceed in either direction, which is wrong. I do not think this is an error of inattention, because he repeats it many times. He might have been unconsciously influenced into making this error by Aristotle’s label for the argument as “from more and less.” But unlike Aristotle, he displays no awareness of the difference between argument “from major to minor” and that “from minor to major”[6]. So I believe Marraud’s use of “± R” is indicative of his uncertainty in this matter, and is designed to fudge the issue.

This is further confirmed by the following observations. I have applied the labels P and Q in the order of appearance of Marraud’s objects O and O’, and everywhere in my work these labels stand for the major and minor terms, respectively. It would appear, judging by Marraud’s description, further on, of the corresponding “meta-argument with scalar warrant,” where he has the premise “O ismoreR than O’,” that he thinks of O as the major term and O’ as the minor term; so, my choice of substitute labels is confirmed. However, this would imply, in the present context, where ‘P is S’ is the minor premise and ‘Q is S’ is the conclusion, that Marraud erroneously believes that positive subjectal a fortiori argument proceeds from major to minor! However, this supposition is belied by Marraud’s description of “meta-argument from strength comparison,” in which the movement is clearly (as it should be) from minor to major. So it looks like Marraud is not sure what to think or is confused, sometimes ignoring the issue, sometimes opting for major-to-minor, and sometimes opting for minor-to-major.

But even if we generously take the above core argument to read: “P islessR than Q, and P is S; therefore, Q is S,” it is faulty. For, even with the correct major premise, the putative conclusion (Q is S) cannot logically be drawn from the given minor premise (P is S)[7]. The said conclusion can only be drawn from a minor premise which reads: P isR enough to beS. This is a very different proposition from the simpler ‘P is S’ (which it implies, however). Marraud is evidently unaware of this essential condition for validation, since he does not mention this missing factor in his formula. This lacuna again shows he does not fully understand a fortiori argument. We cannot assume that his “warrant” is indicative of awareness of this issue. For this proposition of his only affirms a concomitant variation between the middle term (R) and the subsidiary term (S). It clearly does not state that there is athreshold valueof R for acceding to predicate S.

Reinserting the so-called warrant. Now, let us consider Marraud’s “warrant,” i.e. the proposition “± an object has property R, ± it has property S,” which we chose to ignore momentarily. We have seen that, once that is treated as a premise (b), in conjunction with premise (a), i.e. ‘P is ± R than Q’, we canwithout need of any further informationinfer both that ‘P is S’ and that ‘Q is S’. Thus, given the premise (b), it does not matter whether P is more or less than Q, i.e. premise (a) may well read ‘P is ± R than Q’, for the argument works as well both ways, yieldingbothconclusionseitherway. Of course, that is not magic, but simply due to the fact – as already explained – that the comparative function of the major premise (viz. that Rp > or < than Rq) is in fact totally unused.

Another impact that consideration of the “warrant” would have is quantitative differentiation between the two conclusions. One would read ‘P is Sp’ and the other would read ‘Q is Sq’, since the value of R for P would be different from the value of R for Q, and S being proportional[8]to R the value of S for P would accordingly be different from the value of S for Q. But Marraud is unaware of this issue of quantitative variation in the predicate S, or at least he does not display awareness by using different symbols for the S of P and the S of Q. Even though he makes a big thing of the proposition “± an object has property R, ± it has property S,” characterizing it as the warrant for the core argument, he in fact does not apply the proportionality inherent in it in this formula. This is surprising, considering his following statement:

“The asymmetry underlying AF arguments is explained through the scalar principle ortoposserving as warrant. Anscombre and Ducrot (1983) define atopos(pluraltopoï) as a general principle authorizing the step from premises to conclusion and consisting in a correspondence between two scales… This implies that the predicates in thetoposadmit degrees.”

After saying that, it occurs to him that in one of Aristotle’s examples, viz. “If not even the gods know everything, human beings can hardly be expected to do so,” the predicate (i.e. the subsidiary term S, “omniscient”) is not “gradual” (i.e. either one knows literally everything or one does not) – so, no proportionality is appealed to. His response to this “complication” is to ignore it “for the moment,” although he does not return to it later as far as I can see. This suggests that he views a fortiori argument as essentially proportional, even though the conclusion of his above argument, as already pointed out, has the same predicate S, and not a proportional predicate, S+ or S–.

It could be supposed that Marraud’s proposition “± an object has property R, ± it has property S” corresponds to what I have identified as the additional premise of a crescendo argument, which I formulated as “for whatever is R and S, S varies in proportion to R.” But, whereas I make it clear that the variation of S is tied to that of Ras ofthe threshold value of R which gives access to S (as the clause ‘R enough to be’ of my minor premise emphasizes), Marraud’s proposition is quite general. Thus, while my premise about proportionality only applies to a limited range of R (namely, where an R is indeed S), and thus implies thatnot allR are S, Marraud’s proposition is indiscriminate, i.e. it implies thatallR are S. This distinction explains why my arguments are indeed a fortiori whereas Marraud’s argument is merely syllogistic.

Moreover, whereas I use this premise about proportionality to distinguish a crescendo argument from purely a fortiori argument, Marraud apparently considers it as necessary (if only in the way of a “warrant”) even for purely a fortiori argument, and he remains oddly silent concerning a crescendo argument. In truth, such a proposition about proportionality isredundantin purely a fortiori argument, and in applicable casesnecessitatesa crescendo argument. So while Marraud can be commended for having, like many before him, mentioned proportionality in the context of a fortiori argument, it cannot be fairly said that he grasped its full significance.

In any case, as already explained, the proposition about proportionality is not a true warrant but a mere premise, since it contains useful information with specific terms. And of course, once this proposition is taken as a mere premise, in conjunction with Marraud’s other premises, the argument as a whole ceases to be a fortiori in type, and becomes syllogistic, so that the proportionality implied by it as well as the comparison implied by the major premise become completely irrelevant and unused. Thus, not only is it not a warrant, but much of the information it contains is not used as a premise. The true warrant of any material a fortiori argument is the standard form that corresponds to it, and the process through which that form (and thus any of its material instances) can be validated.

In any case, Marraud’s treatment does not offer us a logical validation of a fortiori argument. Presumably, he regards the stated “warrant” as having a validating effect – but as it turns out, his “warrant” is inappropriate – it does not actually explain or justify the reasoning process in the way he means it to. This is true, whether we consider the implications involved in his schema – namely, the arrow symbol and the “since” – as deductive or inductive. I took them in my above analysis to refer to deductive implications, but the same critique holds if they are taken as more inductive. For, if the schema is not successful assuming the stronger deductive implications, it is all the more unsuccessful assuming weaker inductive ones. An inductive argument is only credible if it is formally based on a validated deductive argument.

The issue of origin. Even if some of the errors in Marraud’s treatment above pointed out are inexcusable, he still deserves some praise for glimpsing some aspects of a fortiori argument as such, namely: that it involves a comparative major premise (his “P is ± R than Q”), a predication as minor premise (“P is S”), and a somewhat similar one as conclusion (“Q is S”). The details of his presentation are inaccurate, as already explained; but these rough ideas are still laudable (though far from novel in 2012, of course). The question is: are they his own ideas, or did he get them from elsewhere?

Marraud’s above definition, leaving out the “warrant” part, looks very much like the deficient one proposed by Allen Wiseman inA Contemporary Examination of theA FortioriArgument Involving Jewish Traditions(2010). But there is no evidence that he was aware of that dissertation, even though it was immediately published on the Internet. I suspect Marraud was probably influenced by Stefan Goltzberg’s equally deficient definition: “Thea fortioriargument… structure is that if p applies in case A, and since B is more x than A, then p appliesat least as muchin B,” since he refers to the latter’s essay, called “The A fortiori Argument in the Talmud,” in his paper, even though he does not specifically mention Goltzberg’s definition[9]. Needless to say, Goltzberg’s definition was, as I show in the chapter devoted to him the present volume (26), itself influenced by earlier definitions.

Since Marraud in his reference to Goltzberg’s essay mentions the collection calledJudaic Logicedited by Andrew Schumann (2010), it is possible that he also read my essay in it, viz. “A Fortiori Reasoning in Judaic Logic” (pp. 145-175). This essay includes my clear and definitive definition of a fortiori argument in all its figures and moods, already presented in my earlier bookJudaic Logic(1995). However, I doubt he did read that essay, or the earlier book, because I cannot imagine if he had how on earth he could have made the mistakes he made (detailed above)[10]. Moreover, it is noteworthy that Marraud’s treatment of a fortiori argument, such as it is, focuses exclusively on the positive subjectal mood of it. He is apparently aware that this argument is also possible in negative form, since he says: “the positive form of arguments from more and less can be represented as follows” – but he nowhere goes on to specify what the negative subjectal mood might look like. In any case, he shows no awareness of the predicatal form of the argument, whether positive or negative. He is also unaware of the difference between copulative and implicational arguments.

3.So-called meta-arguments

Marraud next tries to add depth to his analysis, by distinguishing three “different though interrelated forms of a fortiori argument,” three “senses” of the “argument scheme” so called. In addition to the above described basic or main sense of “an argument with a scalar warrant” (also called argument “from more and less”), he discerns two additional senses: “a meta-argument concluding that an argument is even stronger than a previous one” (also called meta-argument “with a scalar warrant” or “from more and less”) and “a meta-argument establishing the sufficiency of some argument on the grounds that a weaker argument is sufficient” (also called meta-argument “from strength comparison”). He defines a “meta-argument” as “an argument about arguments.” Let us now look into these two “more complex” forms of a fortiori argument.

Meta-argument from more and less. The argument he refers to as “sense 2” proceeds as follows (p. 6):

The two “arguments” this meta-argument is about are A1: “O is R, so O is P” and A2: “O’ is R, so O’ is P”. We may call these the “sub-arguments,” as the author himself occasionally does. The conclusion is to be read as “argument A1 is stronger than argument A2.”

The first thing to note, here again, is that Marraud’s proposed warrant (the bottom line in his diagram) is not a warrant but a premise. This means that the argument should be rewritten as follows. Note that I have here again changed the symbols – putting, in place of Marraud’s symbols O, O’, R, P, my by now standard symbols P, Q, R, S, for the major, minor, middle and subsidiary terms, respectively.

Comparing this argument form to the preceding one, we notice a number of details. The major premise (a) here has it that P is “more R” than Q is (i.e. the major term P has more of the middle term R than the minor term Q does), whereas previously P was said more vaguely “± R” than Q. Marraud had to be more specific here, because his thesis is precisely that the argument involving the greater value of R (viz. P) is “stronger” than the one with the lesser value of R (viz. Q). The premise (b) remains the same, while the minor premise “P is S” (which was redundant, anyway) has been dropped. The conclusion is very different – before it was simply “Q is S,” whereas now it is a complex comparison of two sub-arguments. Even so, the present meta-argument consists essentially of two simpler arguments.

Here again, from premise (a) we can educe that P is R and Q is R; and from premise (b) we can educe that All R are S. Thence we can deduce syllogistically that P is S and Q is S. Thus, a full statement of the two sub-arguments, A1 and A2, would be “All R are S, and P is R; therefore, P is S” and “All R are S, and Q is R; therefore, Q is S.” Note that Marraud’s definition of the two sub-arguments as “P is R, so P is S” and “Q is R, so Q is S” is deficient, since it does not specify the premise “All R are S” of the two syllogisms. The reason he does not make this premise explicit is probably, I submit, that he does not discern it; I suspect that he imagines that premise (a) suffices – without the active participation of premise (b), other than as “warrant” somehow buttressing the argument – to infer the two sub-arguments and their comparative strengths.

Thus, we can prove for Marraud that the two “sub-arguments” he refers to as A1 and A2 are indeed implied by his premise (a) and his “warrant” (i.e. his premise (b)). But this is not the purpose of his “meta-argument,” which a claim that A1 is “stronger” than A2. The obvious next question to ask is: do the given premises (a) and (b) indeed imply the conclusion (c) – that is, is the argument valid? The answer is, resoundingly: no! For a start, Marraud does not prove his claim, but takes it to be obvious. In his mind’s eye, evidently, the mere fact that argument A1 concerns an object (i.e. a logical subject) “more R” than argument A2, logically implies that A1 is “stronger” than A2. Yet, a moment’s thought should have shown him the absurdity of this assumption, in view of the reversibility of comparative propositions.

For every term R1, we can posit a relative term R2, such that if P is more R1 than Q, then Q is more R2 than P, and vice versa. To give an example, if Jack is taller (moretall) than Jill, then Jill is shorter (moreshort) than Jack, and vice versa. Thus, the same “meta-argument” would yield conflicting conclusions, according as we read its major premise (a) in the form “P is more R1 than Q” or in the reverse form “Q is more R2 than P.” In the former case Marraud would conclude “A1 > A2” and in the latter case he would conclude “A2 > A1.” He would thus draw contrary conclusions from logically identical premises! This is manifest nonsense, and it formally proves that his conclusion about the relative “strength” of A1 and A2 is invalid. Anyway, why should a greater quantity of something (“more R”) give rise to more certainty (“strength”) than a lesser quantity of it? It is a silly notion[11].

Another proof of the absurdity of Marraud’s claim is that the quantitative comparison inherent in the major premise “P is more R than Q,” i.e. the implication that Rp > Rq (i.e. that the quantity of R of P is greater than that of Q) is in factunusedin the formation of the two sub-arguments, A1 and A2. If it is unused, it cannot possibly affect the result; i.e. it makes no difference to these sub-arguments whether Rp > Rq or Rp < Rq, or even Rp = Rq. The two sub-arguments only make use of the implications “P is R” and “Q is R” of the major premise; and these implications have no quantitative differentia. And indeed, Marraud betrays his uncertainty or confusion further on, when he refers to “greater or lesser presence of R” and “more/less R” even though in the diagram he only has “more R”[12].

Still further proof of the absurdity of Marraud’s claim is that the two sub-arguments areidenticalin form (they are both syllogisms of form 1/AAA), which share the exactsamemajor premise (All R are S, implied by “± an object has property R, ± it has property S”) and whose minor premises (P is R and Q is R) were both obtained by thesameprocess (immediate inference) from the verysamesource (namely, “P is more R than Q” – or even if we wished, “P is ± R than Q”). How can two processes, identical in all respects, be characterized as having different “strengths”? Logic is formal – it is ruled by law. Processes that are formally indistinguishable must be admitted to have the same “strength.” Clearly, Marraud did not reflect on the untenable implications of his claim[13].

It should be added that all the above critique would be applicable equally well if Marraud had used the proportionality implied by premise (b) to infer different values of S for P and Q – i.e. to infer “P is S1” and “Q is S2.” Had he done so, the contents of arguments A1 and A2 would be a bit more different – but their form and therefore their logical status would remain the same, i.e. it would still be formally impossible to conclude that A1 is “stronger” than A2. The only way Marraud could defend his thesis would be by introducing additional information that would somehow alter the logical status of one or the other of the arguments. But then the “meta-argument” would have a different form than the one he here proposes.

The term “stronger” used here supposedly refers to logical strength – i.e. to a probability rating. Maybe Marraud thought the sub-arguments are inductive rather than deductive, and therefore likely to differ in probability. But as we showed above, they are in fact both deductive, and even simply syllogistic, and based on the same premises. And it can be said for any deductive argument: if the premises are sure, so is the conclusion; the conclusion is exactly as sure as the two premises that give rise to it – neither more nor less. If the argument is well-formed, but one or both premises is/are less than sure, so that the argument is effectively inductive, the probability rating of the conclusion is the product of the probability ratings of the two premises. Probability ratings do not emerge haphazardly, but in accord with reasonable rules.

It could be that Marraud, like many commentators before him, was misled by the name “a fortiori” or by the traditional marker “all the more” into the belief that the argument he describes involves a change of strength. But it is hard to see how it reasonably could, since both these expressions, and others like them, suggest that the putative conclusion is ‘stronger’ than the minor premise it was based on. The Latin phrasea fortiori ratione, remember, means ‘withstronger reason’[14]. But this is just hyperbole, for as above explained, it cannot be literally correct. For sure, the phrase does not mean, as Marraud translates it in the title of his essay, ‘froma stronger reason’. That is, it does not mean that the minor premise is stronger than the conclusion; for in truth, the minor premise ofanydeductive or inductive argument, whatever its form and content, is always either stronger or equal in strength to the conclusion, and never of lesser strength, so to say ‘from a stronger reason’ would be a redundancy.

In any case, Marraud’s thesis here is not that the conclusion of a fortiori argument is stronger than its minor premise, or vice versa, but that one sub-argumentas a wholeis “stronger” than the other sub-argumentas a whole. There is clearly zero logical justification for that claim; it is simply a personal fantasy, which reveals the extent of its author’s misunderstanding of logic in general and a fortiori argument in particular. In sum, there isabsolutely no logical basisfor Marraud’s claim to have here identified a distinct form of a fortiori argument deserving to be characterized as a “meta-argument.” His “meta-argument” cannot be claimed to be anything more than a conjunction of two quite ornery “arguments” – there is in fact no “argument about arguments” in it. To speak of a “meta-argument” in this context is, frankly, pretentious nonsense.

Meta-argument from strength comparison. Marraud next very briefly presents the argument he refers to as “sense 3,” saying: “Once it has been established, A₂< A₁may serve as a principle for transferring sufficiency from A₂to A₁,” and depicting the argument as having the following form (p. 7):

Marraud then explains: “we are bound to accepta fortiorian argument because of our prior acceptance of a weaker argument.” And he calls this “a variant of arguments from strength comparison.” It is not difficult to back-engineer the image at the back of Marraud’s mind, which caused him to formulate this “meta-argument.” I would say his subconscious thought was the following a fortiori argument, in which “sufficient” is taken to mean “sufficiently strong to be relied on”:

This is a valid positive subjectal mood of a fortiori argument, assuming we manage to establish the premises somehow[15]. To my mind, it is an argument like any other argument – not a transcendent “meta-argument.” It just so happens that its subject-matter – i.e. its major and minor terms A1 and A2 – consists of ‘arguments’. Its middle term (R) is ‘logical strength’ and its subsidiary term (S) is ‘reliability’ (or some such indicator of credibility or persuasiveness, which causes our “acceptance”). The expression ‘sufficiently strong’ or ‘strong enough’ indicates that the threshold at which logical strength has a magnitude or degree capable of generating ‘reliability’ has beenreached or surpassed. To repeat, this is a normal, truly a fortiori argument – nothing special, nothing calling for a new name. Its content does not affect its form. What can be said to characterize it is that it is a logical-epistemic argument, rather than an ontical one.

If we now look at Marraud’s rendering of this argument, we can discern various problems with it. First, as usual, what he claims to be the “warrant” of the argument (viz. “A₂< A₁”) is not an abstract, external warrant, but a concrete major premise without which the conclusion (“A₁is sufficient”) could not be drawn from the minor premise (“A₂is sufficient”). It does not matter that this “warrant” is different from the preceding “warrants” (which affirmed proportionality); what matters is the role it plays in the inference. The true warrant of this inference is the standard form of positive subjectal a fortiori argument, or the validation process for that form.

Second, the minor premise as he has it is logically incapable of yielding the conclusion. In the proposition “A₂is sufficient,” Marraud does not say what he means by “sufficient.” Sufficiencyof whatandfor what? And how exactly is the degree of sufficiency to be determined? And what is the relation of this notion to that of strength? He does not say. In truth, this premise must be formulated as “A2 is sufficiently strong to be relied on,” in accord with the standard formula “Q is R enough to be S” (here, the minor term P is A2, the middle term R is ‘strength’, and the subsidiary term S is ‘reliable’). We can then draw the conclusion “A1 is sufficiently strong to be relied on” (which implies Marraud’s “A₁is sufficient”).

Notice that the reasoning here is, rightly, ‘from minor to major’ (i.e. from the minor term in the minor premise, to the major term in the conclusion). That is, from the weaker argument (A2) to the stronger one (A1). In the previous type of “meta-argument,” however, Marraud conceived the conclusion as establishing one argument (A1) as “stronger” than the other (A2) – i.e. it seemed to go ‘from major to minor’, or at least to favor the stronger argument over the weaker. It is surprising that Marraud does not stop and wonder about this reversal of fortune. In any case, his adoption of the present argument, viz. “A₂< A₁and A₂is sufficient; therefore, A₁is sufficient,” is quite intuitive – he does not attempt to validate it.

As we have seen, Marraud does not have the crucial factor of “R enough to be” in his primary scheme; i.e. he does not realize that his minor premise imperatively needed to be “P isR enough to beS” (rather than merely “P is S”) to make his conclusion “Q is S” formally possible. Even so, to his credit, he shows an intuitive sense of the need for this factor somehow, when he says “the presence of (a certain amount of) R is a sign of S” and when he here mentions “transferring sufficiency.” But this is not a scientific theory, with substance and consistency. It does not qualify as knowledge. Confusion is bound to ensue. This man needs to study logica lot, before attempting any further ado in this demanding field.

4.Paulo minor argument

Apparently determined to innovate somehow, Marraud tries to introduce a fourth form of a fortiori argument (pp. 8-9), which he dubs aspaulo minor, which is Latin for ‘a little less’. Although that name hardly matches the three examples he gives[16], we shall see that it is appropriate.

He gives the following as its prime example: “If demigods are little more than humans, they are also slaves to their passions.” Here, presumably, the words “little more” make him think this constitutes some sort of a fortiori argument. But in fact this argument is merely rhetorical, for the inference it proposes is fallacious. “Demigods are little more than humans” implies that “Demigods are more than humans,” even if the difference is only a “little” bit. Therefore, onecannotlogically infer from the fact that humans are “slaves to their passions” that demigods are also “slaves to their passions.” This can be clearly seen if we cast the argument in standard form (using “godlike” as the middle term):

As can be seen, the argument is actually a negative subjectal one, and yet it proceeds ‘from minor to major’ – therefore, it is invalid. In other words, even if the demigods are little more than human, they might still have more power over their passions than humans do; only creatures that are less than human may logically be expected to be as much or more enslaved to their passions. That Marraud finds the said discourse convincing shows his lack of logical acumen.

The second example looks nothing like the first, and moreover contains glaring inconsistencies due only in part to the author’s bad English. He starts with the sentence: “Anne is almost taller as Betty” – I am not sure whether that means “Anne is almost as tall as Betty” (which implies she is not as tall) or “Anne is almost taller than Betty” (which implies she is perhaps as tall, though not taller). Then he states that “this sentence implies ‘Anne is tall’ and ‘presupposes Betty is tall’” – showing he does not understand that ‘presupposes’ is effectively equivalent to ‘implies’. Then he says that “the underlying reasoning can be reconstructed as an argument from more and less,” and forces the argument into his usual diagram, with the premises “Betty is tall; Anne is almost taller as Betty;” and the conclusion “Anne is tall,” with the alleged warrant: “x is taller than y, so x is tall”[17]. Notice how the “warrant” differs from the example: “y is tall” is missing from it and the wording “almost taller as” is replaced by “taller than.” This is quite a mess.

In truth, “Anne is taller than Betty” formally implies both “Anne is tall” and “Betty is tall” by mere eduction (i.e. immediate inference), where “tall” may have any value. “Betty is tall” is not a premise for “Anne is tall;” and there is no need for a fortiori argument to draw the desired conclusion, and anyway the argument Marraud presents as a fortiori is not one. Additionally, Marraud tries to spin the argument as one “by strength comparison,” saying: “The warrant correlates tallerness to tallness, so that an accrual of tallerness appears asan accrual of reasons fortallness … there are no degrees of tallness, and degrees of tallerness are to be correlated todegrees of justified belief.” (The italics are mine.) Here again, as with his earlier “meta-argument from more and less,” he is trying to tie epistemic credibility to ontical size – which is balderdash.

The third example Marraud gives us looks nothing like the preceding two. It reads: “If you’re planning to become pregnant, taking certain steps can help reduce risks for both you and your baby. Proper health before deciding to become pregnant is almost as important as maintaining a healthy body during pregnancy.” He depicts this pictorially as follows:

Where, “A₂= You’re planning to become pregnant, so eat a balanced diet.” and “A₁= You’re pregnant, so eat a balanced diet.” This is just more spin on Marraud’s part. All the given discourse says is that proper health care before pregnancy is almost as important as proper health care during pregnancy. Presumably “importance” here refers to “importance for the future baby’s health,” i.e. to the power to cause health in the future baby[18]. In other words, if she wants to have a healthy baby, a woman should take care of her health before she actually gets pregnant as well as during her pregnancy. Health care during pregnancy plays a major role in ensuring the baby’s health, but health care before pregnancy also plays a role even if a slightly lesser one.

There is no inference involved; it is just a statement of fact. The words “almost as important” used here are not indicative of a fortiori argument, let alone a “meta-argument from more or less” or an “argument by strength comparison” or a “paula minorargument.” Indeed, if we try to formulate a standard a fortiori argument, all we get for our troubles is yet another non-sequitur, since the argument is positive subjectal and yet goes from major to minor: “Health during pregnancy (P) is a bit more important (R) than health before pregnancy (Q) is, and health during pregnancy (P) is important (R) enough to make it recommended (S); therefore, health before pregnancy (Q) is important (R) enough to make it recommended (S).” This is, to repeat, invalid reasoning.

To conclude: Marraud’s idea ofpaula minora fortiori argument is clear enough, even if he does not manage to express it clearly – but it is wrong if deductive argument is intended. Given, P is more R than Q, one cannot deduce, from “Q is ‘almost but not quite’ R enough to be S”, that “P is likewise ‘almost though not quite’ R enough to be S.” Similarly, one cannot deduce, from “P is ‘almost but not quite’ not R enough to be S”, that “Q is likewise ‘almost though not quite’ not R enough to be S.” A little less than enough is just not good enough – if the threshold for something is stated as Rx, than nothing less than the value Rx will do the trick. Deductive logic does not allow approximations or compromises.

One might, however, argue inductively that, given that P is R enough to be S, and Q is only a little less R than P is, there is a reasonable chance that Q is R enough to be S. This would be based on the thought that the threshold value of R for S (i.e. the minimum Rx)might bebelow both the values Rp and Rq, since that is often de facto the case. Similarly, given that Q is not R enough to be S, and Q is only a little less R than P is, there is a reasonable chance that P is not R enough to be S. This would be based on the thought that the threshold value of R for S (i.e. the minimum Rs)might beabove both the values Rq and Rp, since that is often de facto the case. Of course, such thoughts are speculative; but they do suggest there is some probability in the proposed conclusion. The closer the values of Rp and Rq, the greater the probability; the wider the spread between them, the lesser the probability.

The above concerns subjectal a fortiori argument, but obviously similar reasoning can be applied to predicatal argument. As regards the positive mood: given that more R is required to be P than to be Q, one cannot deduce, from “S is ‘almost but not quite’ R enough to be P”, that “S is likewise ‘almost though not quite’ R enough to be Q.” Similarly, as regards the negative mood: one cannot deduce, from “S is ‘almost but not quite’ not R enough to be Q”, that “S is likewise ‘almost though not quite’ not R enough to be P.” These two arguments are of course also invalid, since ‘almost though not quite’ implies ‘not’. But they could be taken as inductive. We could also formulate implicational equivalents of the four copulative forms mentioned above.

Thus, from aninductiveperspective, positive subjectal (and similarly, negative predicatal) argument could proceed from major to minor, and negative subjectal (and similarly, positive predicatal) could proceed from minor to major. While this sounds reasonable, I wonder if such discourse can truly be referred to as inference, even if only as inductive inference. For ultimately, it seems to me, since we lack precise information, the probability is really always fifty-fifty. Strictly speaking, then,paula minorarguments are invalid. They might however retain some power of conviction as inductive inferences, although that is open to debate. Thus, if as Marraud suggests they have some rhetorical weight in ordinary discourse, it is at best merely inductively and at worst through sophistry.

5.Legal a fortiori argument

Marraud considers some examples of legal reasoning, drawn from actual law cases, and tries to show how they fit into his conception of a fortiori argument (or meta-argument) as comparison of the strengths of two arguments (or sub-arguments).

The first example of legal argumentation he presents (pp. 2-3) is put forward by an American judge in an actual case:

“Possession by an accused of recently stolen property is sufficient to sustain a conviction of theft where a satisfactory explanation is not given, particularly where the nature of the items and their condition support an inference that they have been stolen.”

This is a straightforward a fortiori argument, which can be put in standard (positive subjectal) form as follows. The tacit major premise is: Possession by an accused of recently stolen property where a satisfactory explanation is not given and where the nature of the items and their condition support an inference that they have been stolen (P) is more damning evidence (R) than Possession by an accused of recently stolen property where a satisfactory explanation is not given and where the nature of the items and their conditiondo notsupport an inference that they have been stolen (Q). The minor premise is explicitly: Q is sufficient (i.e. sufficiently damning evidence, R) to sustain a conviction of theft (S); and the conclusion is explicitly: P is sufficient (i.e. sufficiently damning evidence, R) to sustain a conviction of theft (S). Note that the word “particularly” used here serves merely to signal a fortiori argument, in the same way as expressions like “a fortiori” or “all the more” would do.

Marraud, however, conceives the judge’s argument as consisting of two “arguments,” the first being effectively that ‘Q implies S’ and the second that ‘P implies S’. He considers the first argument as “cogent,” but the second one as “a stronger argument, as the word ‘specially’ indicates.” Then he adds: “So far as the first argument provides sufficient evidence, the second seems unnecessary. This leaves open the question of why an arguer would use such a redundant way or arguing.” He speculates that the judge may by this means be “anticipating the rebuttal of someone rejecting the sufficiency of the first argument.” That is why, he tells us, “it is often said that an AF argument reinforces a claim already established.” In Marraud’s view, the judge “thinks that the premise ‘the nature and condition of the items support an inference that they have been stolen’ can be suspended without ruining the argument.”

In other words, for Marraud – who conceives of the overall argument as a comparison in “strength” of two implications, or if–then propositions – the first antecedent (Q) is strong enough to prove the consequent (S) and the second (P), though stronger (due to providing an additional reason), and therefore also able to prove S, is redundant. Marraud says this, about redundancy, because he considers the two antecedents, which I have labeled Q and P, to be two related theses: Q being “A” and P being “A + B” – both able to imply the conclusion “C” (my S). Clearly, his thought is that if A alone suffices to imply C, then the B in A + B is redundant. But these are not the true antecedents involved. The two these are in fact ‘A + notB’ and ‘A + B’. ‘A alone’ here means ‘A without B’, and not as Marraud thinks ‘A, whether or not B’. Note well the difference[19]. Once this is realized, it becomes clear why there is no redundancy, but new information has been uncovered.

Thus, though Marraud’s explication sounds reasonable on the surface, it is not at all a correct description of the overall argument. The two propositions ‘Q implies S’ and ‘P implies S’ are quite distinct and, from a logical point of view, equally cogent—equally “strong.” Neither is more reliable than the other (at any rate, the second is not more reliable than the first, since it depends on both the minor and premises). The a fortiori argument, we might say, simply consists in logically deriving the second implication from the first. That is to say, the first helps us to discover and justify the second. But once that discovery and justification effected, the net credibility of the conclusion is neither greater nor smaller than the joint credibility of the given major and minor premises. There is no comparison of “strength” involved, and no part of the argument is “redundant.”

Marraud presents his next example of legal argumentation (pp. 9-10) as follows, in his own words:

“Justice Souter uses first an argument from precedent: the Court of King’s Bench held that a private person needed no warrant to arrest a common cheater whom he discovered cozening with false dice; so in this case no arrest warrant was needed. Then he goes on to reinforce his argument with another: by a stronger reason a police officer can execute a warrantless arrest because a police officer has even more of a right to arrest than another person.”

Before analyzing Marraud’s take on this reasoning, let me say how I see it. In my view, the first argument is an argument from precedent, meaning that a past judgment of a court is held up as a case in point. The latter is implicitly generalized, meaning that all subsequent cases that are reasonably similar may be subjected to a like judgment. Then this generality is syllogistically applied, meaning that the present case is judged in accordance with the said rule. This is not a fortiori argument, but more akin to the rabbinical technique ofbinyan av– i.e. it is complex analogical argument. The second argument can be construed as a fortiori, positive subjectal in form, as follows:

Marraud, for his part, analyzes the judge’s reasoning in much more convoluted terms. He perceives the reasoning as a comparison between arguments of different strengths, and thus reconstructs the argumentation “as a meta-argument from more and less.” This is presented diagrammatically as follows:

In this schema, argument A1 is that “that there are reasons X to allow a private person to execute a warrantless arrest,” and argument A2 is that for the same reasons X, “a police officer can execute a warrantless arrest.” And Marraud is here claiming to have proved, given the stated premise and warrant, that A2 is “stronger” than A1. But as we have seen before, this schema is logically incapable of proving any such thing. It should additionally be noted that Marraud is here, as usual, unclear about the difference between the “warrant” of an argument and a premise. The proposition “a police officer has more of a right to arrest than a private person” was previously described as a “warrant” and here placed in the role of premise. As regards the “warrant” in the diagram, simply stated as “scale of arrestors,” it was previously presented with the words: “The implicit warrant will be some scale of arrestors (taking into account their status, the circumstances of the arrest, the nature of the offence, etc.) and placing police officers at the top.” Clearly, these two say much the same thing, even if he labels them differently to make two propositions out of them, and project the first as a concrete premise and the second as a “principle.”

After presenting this “meta-argument from more and less,” Marraud suggests that “An associate argument from strength comparison leads to the main conclusion: police officers can arrest someone for a minor criminal offense without warrant.” Though he does not detail this new argument, we can assume it is that since the (allegedly) weaker argument A1 is “sufficient,” then the (allegedly) stronger argument A2 must be “sufficient.” This argument is indeed a fortiori in intent, even though incompletely verbalized (the meaning of “sufficient” having not been clarified by him, as earlier explained). However, the argument is wrong anyway, since its major premise, viz. that “A2 is stronger than A1” has, as we just pointed out, not been proved by the preceding argument! In conclusion, even though Marraud tries to use the said example of legal reasoning as evidence for the utility of his two “meta-arguments” – it turns out that this is mere spin on his part. In fact, that example cannot be explained by these means, but only as I previously explained it.

Another example of legal reasoning proposed by Marraud (pp. 10-12) is the following:

“Suppose a thief steals a wallet and the £20 note therein. His victim will undoubtedly have a claim against him in wrongs, more specifically, in the tort of conversion. […] What is a matter of debate is whether the victim can maintain a common law strict liability claim in unjust enrichment. The existence of such a claim is said to flow as a matter of deductive logic from the availability of strict liability common law claims in unjust enrichment for mistaken transfers. Mistaken transferors recover because their consent to the transfer was impaired. In the posited case, the victim of the theft gave no consent whatever to the ‘transfer’ of the wallet and note to the thief. He was ‘ignorant’ of it. His ability to claim in unjust enrichment is, it is said, a fortiori from mistake.”[20]

As before, let me analyze it prior to considering how Marraud perceives it. In my view, the argument is clearly as follows: Since the transfer of the stolen property to the thief involves no consent whatever by the victim of the theft, it isanalogous to, or classifiable under, mistaken transfer, i.e. transfer in which consent was impaired. Therefore, since mistaken transferors recover their property under a common law strict liability claim in unjust enrichment, it follows that victims of theft can recover their property under the same law. This reading implies two possible interpretations.

The first interpretation is syllogistic: it considers that “transfer of property with no consent” (i.e. through theft) is simplya case of(even if only a limiting case of) “transfer of property with incomplete consent” (i.e. mistaken transfer). That is, here, “incomplete consent” is understood to mean “less than complete consent,” and thus include “zero consent”[21]. Thus, the argument takes the form of 1/AAA syllogism, with minor term P, middle term Q and major term S, as follows:

Alternatively, we can place “transfer of property with no consent” and “transfer of property with incomplete consent” on a common continuum, R (say, degrees of dishonesty). Here, “incomplete consent” implies “some consent” and thus excludes “no consent.” We can then interpret the argument as a fortiori, of positive subjectal form, with major term P, minor term Q, middle term R, and subsidiary term S, as follows:

Note the simplicity of both these interpretations. Marraud, for his part, makes something rather more complicated of it. He sees it as “a comparison of the strength of two arguments.” The first argument is that since “a mistake is recognized by law as an unjust factor,” then “suppose the transferor’s consent to a transfer were vitiated by a mistake,” it would follow that “the victim could claim in unjust enrichment.” The second argument is stated as: “suppose a thief steals a wallet and the £20 note therein,” then “the victim can claim in unjust enrichment.” Both these sub-arguments are presented in the usual diagrammatic form. The “warrant” given for the first argument is, of course, as usual, not in fact a warrant but a premise; the argument being essentially syllogism (or apodosis), as Marraud admits when he characterizes it as “hypothetical.” The second argument, on the other hand, contains a question mark in lieu of a “warrant.” The first argument is “invoked to gain acceptance for” the second one, which is called “the main argument.”

Marraud now searches for an appropriate warrant for the second argument. He needs to do this because the relative strengths of the arguments will depend on their respective warrants. The warrant of the first argument cannot, however, be carried over to the second argument, since the subject-matter is different. If “the missing warrant” were taken to be “theft is recognized by law as an unjust factor,” this would amount to “begging the question.” If the needed warrant were assumed to be “ignorance is recognized by law as an unjust factor,” a debate would ensue as to the truth of this under English law, and anyway this would (for some unstated reason) “lend less force” to the conclusion of the 2^ndargument than “mistake is a recognized by law as an unjust factor” does to the conclusion of the 1^stargument. Therefore, Marraud considers that “some principle like ‘mistake is less than ignorance’… is needed to transfer acceptability from one argument to the other.”

After that, he thickens the plot further by introducing the notion of “backings” to “warrants,” explaining that “backings may confer different degrees of force to the warrants they justify.” To my mind, all this discussion – Marraud’s lame search for a warrant, and then for a backing for it – is close to useless chatter, being based on a wrong perception of the reasoning involved in the said example. The reasoning has nothing to do with comparison of the “strengths” of different “arguments.” The reasoning is, as above shown, simply syllogistic or a fortiori with reference to specific terms (given or implied). Marraud gets entangled in a complicated discussion, because he is unable to see the straightforward solution to the problem. And the reason for that is that he has not sufficiently studied formal logic, but evidently thus far only studied the ‘rhetorical’ approach to analysis of arguments.

The rhetorical description of human thought processes is interesting, but essentially superficial being incapable of judging validity. Only formal logic can provide a credible and deep understanding of reasoning in general, and legal argumentation in particular.