A FORTIORI LOGIC
CHAPTER 28 – Allen Wiseman
Allen Conan Wiseman wrote in 2010 an extensive study of a fortiori argument, entitled A Contemporary Examination of the A Fortiori Argument Involving Jewish Traditions. This was not yet published in print form, but was made freely available in pdf format on the Internet. I came across it there, quite fortuitously, in mid-October 2010; and I found it interesting enough to immediately pass the information on to some two dozen academic correspondents.
My present work was directly inspired by Wiseman’s felicitous idea to devote a whole book to this one topic, surveying and discussing contributions to the field, and proposing insights and analyses of his own. Had he not written his treatise, I would not have written mine. I was aware at the time of a few papers on a fortiori argument. But, although I disagreed with their authors in various ways, I felt no call to respond in writing to their work. It was only thanks to Wiseman’s study that I realized that there were many more contributors to the field, and that I was duty bound (as a contributor to it myself) to respond to them all, including Wiseman.
Let us examine and assess Wiseman’s contribution. Having carefully read the whole work, I can safely say that its main goal is to contest the rabbinical “dayo principle” (as he sees it), as an unreasonable and unfair limitation on natural a fortiori argument even in many religious contexts. More than half the book is devoted this topic, and the other half prepares the ground for its discussion. To defend his main thesis, Wiseman introduces and develops two distinctive general theses: a belief in “inductive a fortiori argument” and a belief in “proportional a fortiori argument.” These overriding theses are both based on his peculiar definition of a fortiori argument – or, I should say it outright, his wrong definition of it. We shall first examine his definition of a fortiori argument, and the moods he develops around it, then the two theses he bases on it, and lastly his treatment of the dayo principle by means of them.
Wrong definition of a fortiori. Wiseman’s working definition of a fortiori argument is consistent throughout his work. In the abstract, already, he describes the argument as follows: “Typically, the argument claims this: if a lesser (or greater) case has a feature, a correspondingly greater (or lesser) case has that feature too.” In his introduction, he presents the argument as composed of two premises (P1, P2) and a conclusion (C), as follows:
“P1: A is greater than B.
P2: B has [feature] f.
C: A has f* too.”
Regarding the “superlative relation” of A and B, he tells us that “one is ranked higher while the other lower along a continuum within some common category” (my italics) – which indicates a recognition that there is a middle term, even though this is not explicitly mentioned in the formula. However, his middle term appears (even if mostly implicitly) only in the major premise (P1); there is no reference to its important role in the minor premise (P2) and conclusion (C). Regarding the conclusion he adds: “The [feature] f* here may be the same as or proportional to f” (again, my italics). This formula for a fortiori argument recurs throughout the treatise.
At the end of his first chapter, he proposes the following “general definition” of a fortiori argument: “For congruent items, when one of the ranked items in the comparison has a feature, due to the (likely) heritable property, the other probably or surely has it to some degree too.” His definitive, “reworked definition”, which is presented in his Conclusion, goes like this: “The a fortiori argument compares two ranked items in a continuous, common category, one of which has a key feature, to conclude that the other likely has a form of the feature, which only in heritable cases is deductively valid and sound in that it surely has the same feature or its reasonable ratio.” 
From these many sources, we can formulate the following composite picture he has of a fortiori argument: ‘If A is more x than B, and B is y, then – probably or surely – A is equally y or more than y’. The ‘probably or surely’ part refers to the distinction between inductive and deductive a fortiori argument; while the ‘equally or more than’ part refers to the distinction between non-proportional and proportional a fortiori argument. Thus, his definition is intended to cover all contingencies, as he sees them, relating to such argument.
Additionally, Wiseman describes a fortiori reasoning by means of diagrams, and many examples. His main diagram consists of a horizontal line representing the “scale” (x) along which the major and minor terms (A and B) are compared – i.e. it is the middle term. The feature y being found at one point in this scale (B) can be expected to be found in equal (y) or greater measure (y+) at a later point (A). Thus:
Scale x —– B —– A —– →
Further on, he draws two graphs to show the difference between a “same” conclusion (y) and a “proportional” one (y+). In the former, the x-y curve is a perpendicular line (meaning there is one value of y for all values of x), and in the latter the x-y curve is an inclined line (meaning there are different values of y for the various values of x).
It should be obvious by now why Wiseman’s definition of a fortiori argument is sorely mistaken. Although, to his credit, he implicitly or explicitly does refer to the middle term (x) in the major premise (i.e. as the scale of comparison between the major and minor terms, A and B), he nowhere mentions the crucial role that this middle term is called upon to play in the minor premise and conclusion! His “a fortiori argument” is therefore no more than quantitative analogy.
He nowhere mentions that there is a threshold value of the middle term, which must be crossed before the subsidiary term (the feature y) can be predicated of either subject (B, and then A). Yet this is the basis of the whole a fortiori inference! The fact that ‘A is more x than B’ does not by itself allow us to infer that, if ‘B is y’, then (probably or surely) ‘A is y’ (or more than y). The inference is in truth based on the understanding that if ‘B is x enough to be y’, then ‘A must be x enough to be y’ (from which we can educe that ‘A is y’). Given the major premise, if the “x enough to be” clause is missing from the minor premise, it is logically equally possible that ‘A is y’ or that ‘A is not y’. Since ‘A is y’ is not formally implied by ‘B is y’, it follows that ‘A is not y’ is formally compatible with it.
Wiseman simply has not assimilated this crucial subtext. In one place, he does appear to realize it, when he gives as example the marks for an exam, taking a mark of 50 or less as a failure and any mark over 50 as a pass, and then remarks: “Nothing matters except the mark above or below the critical turnover point”. But in the rest of his work, and in his verbal and symbolic definitions of a fortiori argument, he does not take this lesson to heart. And this is, I must say, very surprising, considering that Wiseman has read and apparently understood my analysis of the argument, which clearly includes the issue of sufficiency (or insufficiency) of the middle term as an essential factor. It would seem therefore that, to his mind, the words “R enough to be” (or “not R enough to be”) were redundant! He never internalized their crucial significance.
This is evident even in his diagrams. Although he states that his diagrams “adapt some of Samely and Sion’s ideas,” it looks like he relied on the former more than on the latter – because the diagram I had in my Judaic Logic, and which I reproduce in the present volume unchanged, included for each mood of a fortiori argument a comparison of three points (Rp, Rq, Rs) on the continuum (R). Even though Wiseman reproduces my three-point diagram for positive subjectal argument when he presents my theory, when he presents his own theory he only compares two points, completely ignoring the crucial third point which serves to correlate these two points. Yet, without an intermediary, there is no way to infer anything.
Even the examples he gives are deficient. For instance, he argues: “If I can lift a kilo with one finger, surely I can lift the same (or more) with my entire hand,” whereas the a fortiori argument involved is really: ‘A whole hand is (obviously) stronger than the fingers constituting it. If one of my fingers is strong enough to lift a kilo, then my whole hand is strong enough to lift a kilo (or more)’. Wiseman often (too often!) refers to examples. He does so not only to illustrate but, I suspect, also to understand. Yet, as far as I can see, it did not occur to him to construct a deviant example, which would show up the absurdity of his definition. For instance: ‘Jack (A) is stronger (more x) than Jill (B). If Jill (B) is beautiful (y), then Jack is surely as beautiful or more so (y)’. Obviously, such argument is nonsense. Why? Because no connection is indicated between the scale of comparison (x) and the predicated feature (y).
The middle term must without fail be specified (or at least clearly tacitly intended) not only in the major premise (as the link between the major and minor terms), but also in the minor premise and conclusion (as the link between the subjects and predicates in them) – otherwise, we have no basis for inference, and the conclusion is arbitrary. We cannot be sure that the middle term mentioned (or intended) in the major premise plays any role whatever in the minor premise and conclusion; the latter may contain another ‘middle term’ or none at all, for all we know.
Judging by his definition of a fortiori argument, Wiseman has not fully understood this crucial point – and, as we shall see, this causes him countless problems throughout his treatise.
The moods of a fortiori. As regards the moods of a fortiori argument, the following should be added. Looking at Wiseman’s above definitions, it is clear that what he has in mind is the paradigm of a fortiori argument, i.e. the positive subjectal mood. A and B are subjects and x and y are predicates. He does not notice, however, that many of his examples are in fact positive predicatal. For instance, the above given example, “If I can lift a kilo with one finger, surely I can lift the same (or more) with my entire hand,” is positive predicatal in form, since the subject (‘I’ in this case) is the same in antecedent and consequent, while the predicates differ (lift a weight with one finger or with whole hand). In positive subjectal argument, remember, the subjects differ and the predicates are essentially one and the same (even if we admit of ‘proportionality’).
The narrowness of Wiseman’s formal definition comes as a surprise, considering that, in his analysis of my theory of a fortiori argument, he reviews my positive predicatal mood in as much detail as he does my positive subjectal mood. But I notice that he there misunderstands the expression ‘from major to minor’, which means from the major term (P, in the minor premise) to the minor term (Q, in the conclusion), as meaning: “from major (greater) to minor premise.” The latter interpretation is of course absurd, and may be merely an error of inattention – but it is still indicative of some confusion on his part. In any case, his definition makes no mention of positive predicatal argument. Nor does it account for the negative moods of either subjectal or predicatal argument. Here again, although Wiseman does duly note in his description of my work that I include negative moods of a fortiori argument, he does not himself explicitly include them in his definition. It is also noteworthy that nowhere in his review of my work does he mention the derivation/validation of the negative moods by reduction ad absurdum from the positive moods. That he failed to notice out loud something so important is significant.
As regards his definition, I guess that Wiseman considers (though he nowhere tells us so) that when he says of some subject that it “has feature f,” this includes cases where it “has feature not-f” – i.e. by the term feature, he tacitly intends negative as well as positive features. This may seem to exonerate him, but in fact it does not, because the positive and negative subjectal moods of a fortiori argument differ radically, in that the former proceeds from minor to major, whereas the latter proceeds from major to minor. This difference of direction is not a trivial detail, but makes all the difference between a valid argument and an invalid one. We could give Wiseman the benefit of the doubt here again, by referring to the definition in his abstract, viz. “Typically, the argument claims this: if a lesser (or greater) case has a feature, a correspondingly greater (or lesser) case has that feature too.” Perhaps what he has in mind here is that in positive cases the inference will go from the lesser to the greater, and in negative cases it will go the other way.
However, this would be a too lenient judgment. First, because his more elaborate definitions (the “general definition” and the “reworked definition” above cited) do not at all mention the issue of direction of inference. They refer two “ranked items” and tell us that if one has a feature then the other one will (probably or surely) have it too, or “a form of” it, “to some degree.” Secondly, because the graphs he uses to represent a fortiori argument, clearly allow motion in either direction (i.e. up or down the curve, even if it has outer limits). Indeed, he says it out loud: “Of course, we could reverse course, to begin higher and end lower down”. And this is the way it must be, logically, if one’s understanding of a fortiori argument does not include the concept of a threshold. Additionally, there are many more places throughout his work where it is obvious that Wiseman considers either direction of inference as okay.
It is true that his above-mentioned symbolic formula, viz. ‘If A is greater than B, and B has f, then (probably or surely) A has f (or more),’ suggests a unidirectional inference, from minor (B) to major (A). But this is clearly only intended as the typical or most common case, as is evident from Wiseman’s subsequent listing of “alternate arrangements of QC terms” (QC refers to qal vachomer, which is the Hebrew term for a fortiori argument). There he presents what are, in his view, effectively, the various valid moods of a fortiori argument.
There, he mentions, as a valid mood: “If the lesser has the feature, surely (or very likely) the greater has the same, equal feature (or more of the feature);” and he also mentions, as an equally valid mood: “If the greater has the feature, surely (or very likely) the lesser has the same feature (or less of the feature).” The first of these is minor to major; but the second is, note well, major to minor. Wiseman also there proposes negative moods, which he refers to as “non-attaining cases.” These are: “If the greater fails to attain a feature, surely (very likely) the lesser fails with the same feature (or fails it even more);” and “If the lesser fails to attain a feature, surely (or very likely) it fails equally (or more) with a greater feature.” The first of these is major to minor; but the second is, note well, minor to major.
Notice that all these moods are subjectal in form. Thus, there is no doubt that, for Wiseman, a fortiori argument, whether positive or negative, may go in either direction. Basically, he accepts these moods, or “variants,” because they seem intuitively reasonable to him. However, he does make a small effort at validation, in that he presents a diagram and an example for each mood. The problem is that these diagrams are, as already explained, logically deficient, in that they only compare two points on the scale, instead of three – i.e. there is no reference to a threshold. That explains why Wiseman is able to regard as valid reasoning the positive-subjectal, major-to-minor mood, and the negative-subjectal, minor-to-major mood. Such reasoning is characteristic of quantitative analogy or pro rata argument, but it is not possible with a fortiori argument.
To make matters worse, Wiseman proposes in the same context another set of positive and negative moods, in which the relations are more complex. One is: “If the greater has the feature, surely (or very likely) the lesser has more of the feature;” in this case, he is presumably thinking of inverse proportionality. Another is: “If the greater has the feature, surely (or very likely) the lesser fails to have that feature;” here, he apparently has in mind a breach in the continuity of the scale. Yet another is: “If the lesser has the feature f (as a defect), surely (or very likely) the greater has less (or fails to have) f;” this is really a more material case, because the concept of “defect” is not formal. Then there are the negative complements of these. These various additional moods are, however, superfluous. The first one, concerning inverse proportionality, is just a special case of direct proportionality. The second, concerning limited continua, can be dealt with by means of conditional a fortiori argument; i.e. within certain limits, this continuum applies, but beyond them, it fails. The third, as already pointed out, is a more material case. So this segment of Wiseman’s attempt at a listing of moods is rather forced. And anyway, Wiseman’s treatment of such special cases is far from exhaustive.
The upshot of all this is that Wiseman effectively considers all moods as valid. He does not identify certain moods as invalid. His view of a fortiori argument is so vague, so open, that anything goes. He would probably not admit it if confronted, but it summarizes what is implied by his doctrine of “alternate arrangements of QC terms.” According to this doctrine, so long as one follows the given x-y curve, one can infer, positively or negatively, from A to B or from B to A. This result is, to repeat, due to Wiseman’s signal failure to grasp the significance of the threshold value of the middle term. In other words, Wiseman has confused a fortiori argument with pro rata argument.
We shall presently go deeper into Wiseman’s view of a fortiori argument, by analyzing his ideas of “inductive a fortiori argument” and “proportional a fortiori argument.” We may view these ideas as derived from his broad, i.e. vague, definition – or, alternatively, we may view his definition as having been tailor-made to fit these ideas, which he needed to support his thesis concerning the rabbinical dayo principle. Probably, the genesis occurred both ways.
Stringent and lenient. But first, let us look more closely at another set of moods of a fortiori argument, which Wiseman develops further on. This doctrine focuses on the words ‘stringent’ and ‘lenient’ commonly found in rabbinic formulations of a fortiori argument, such as that by R. Feigenbaum: “Any stringent ruling with regard to the lenient issue must be true of the stringent issue as well; [and] any lenient ruling regarding the stringent issue must be true with regard to the lenient matter as well”. Wiseman tries to approach this formula more generally and systematically, by considering all conceivable permutations. He accordingly lists the eight “a fortiori possibilities” tabulated by me below:
Minor to major
Contrary to (1)
Major to minor
Contrary to (4)
Wiseman numbered the eight moods (1) to (8), but I have reordered them more logically, finding his list incomprehensibly disorderly (judge for yourself). The two shaded rows (numbered 1 and 4) represent the two rabbinical moods just cited. These two can be validated in the form of standard subjectal a fortiori argument, respectively positive (minor to major) and negative (major to minor):
Mood (1): P is a more serious (R) matter than Q;
and Q is serious (R) enough to be subject to stringency S;
therefore, P is serious (R) enough to be subject to stringency S.
Mood (4): P is a more serious (R) matter than Q;
and P is serious (R) not enough to be subject to stringency S;
therefore, Q is serious (R) not enough to be subject to stringency S.
Note that the negative predicate “is not subject to stringency” is subsequently read positively as “is subject to leniency.” Also note that the subsidiary term S, whatever its level of severity, is identical in minor premise and conclusion, which suggests that no proportionality or dayo principle is invoked by the rabbis in these two cases.
Now, given the validity of mood (1), it follows that mood (8) is invalid, since it advocates the exact opposite conclusion. Likewise, given the validity of mood (4), it follows that mood (7) is invalid. The remaining moods cannot be similarly validated or invalidated; so their putative conclusions are all equally undetermined. This means that we may without contradiction come across the positive conclusion, or alternatively the negative one, in conjunction with the same premises. This is how I would present the eight moods; and it is clear from this presentation why the rabbis generally list only two of them – because they are the only two valid moods in this set of eight.
Wiseman, on the other hand, classifies the moods numbered (1) to (5) as “normally acceptable,” and moods (6) to (8) as “unacceptable.” He does this merely intuitively, without any formal process of justification – which is why he makes some errors, as we shall now show. Moods (7) and (8) are, I agree, “unacceptable,” because they are formally invalid, since their conclusions contradict those of moods (4) and (1), respectively.
But what of mood (6) – what is wrong with that? Nothing at all, since it is quite conceivable and often happens that a stringency applies to all cases of a kind, the more serious and the less serious, indiscriminately. Indeed, mood (1) attests that stringency may be general. Moreover, if Wiseman considers mood (6) as “unacceptable,” why not consider mood (2) as equally so? Symmetry would demand that both get the same treatment. There is, of course, also nothing wrong with mood (2), since it is quite conceivable and often happens that a leniency applies to all cases of a kind, the less serious and the more serious, indiscriminately. Indeed, mood (4) attests that leniency may be general.
The reason why Wiseman, “on grounds of fairness and justice,” refuses mood (6) and accepts mood (2), is that he views, as it were, the former as passing stringency downhill and the latter as passing leniency uphill, and he believes, on the basis of an alleged general rabbinic injunction to “Be lenient in judgment,” that stringency should not be propagated, whereas leniency may be propagated. He argues that “punishment should fit the crime, rather than undervalue severer cases with minor penalties or commit injustice when minor misdemeanors receive overly severe judgements. To treat serious crimes lightly or light crimes too harshly undermines the very notion of fairness and justice (unless deeper, ethically right and justifiable reasons override).”
But this interprets these two arguments as generating respectively increased stringency or leniency, whereas all they are doing is discovering respectively a stringency or leniency that is already there. It is of course very often true that different degrees of a crime deserve proportional punishments, measure for measure; but it is not always true – sometimes, the punishment is the same for all cases. For that reason, logic cannot impose proportionality indiscriminately, but must allow for equality. In any case, since in fact the given premises do not formally determine the said conclusions, there is no passing on of stringency or leniency in these two moods.
The next question is: why does Wiseman consider moods (3) and (5) as “acceptable”? The reason he does so is that he wants to legitimatize ‘proportional’ a fortiori argument, contra Maccoby who would “disqualify” such reasoning. And indeed, proportionality is not formally excluded. But this does not mean that a fortiori argument per se ever implies it; it just means it is compatible with it. So Wiseman’s “acceptable” should not here be taken to mean valid. Nevertheless, Maccoby was wrong to regard these two arguments as formally invalid.
Notice that moods (6) and (5) have the same premises, yet the first yields an ‘equal’ conclusion (same stringency for the less serious matter) while the other yields a ‘proportional’ one (less stringency for the less serious matter). Similarly, moods (2) and (3) have the same premises, yet the first yields an ‘equal’ conclusion (same leniency for the more serious matter) while the other yields a ‘proportional’ one (less leniency for the more serious matter). This is why both moods in each pair must be classed as undetermined, because either outcome is logically possible.
From this analysis, we see the informality of Wiseman’s approach to the moods of a fortiori argument and how such informality can lead to errors. As for Wiseman’s concerns in this context, regarding proportionality and the dayo principle, more will be said further on.
One of Wiseman’s main theses is that a fortiori argument ranges in reliability from the inductive to the deductive. As he puts it in his abstract: “While the argument aspires to be true and it can be deductively valid in those cases where heritable properties recur, it is more likely to be inductively probable.” He expends quite some effort arguing in favor of induction; and he is right to do so, because many people still today do not realize the important role it plays in our lives. Induction produces various degrees of certainty, from low to high; and deduction is its summit:
“Although fallible, induction is a broad, sufficiently reasonable form of argument. Indeed, because induction can cover deduction when 100% true or false, it is more inclusive in scope. On the other hand, one can view deduction as an ideal pattern that induction tries to approach”. “Up to 50% of actual occurrence, we lack sufficient assurance… At 50%, we are undecided. For basic confidence, we need more than a 50% probability. Above 50%, the argument is more likely true than not” .
Moreover, the inductive probability of any proposition may change over time: “Although we start with some sense or estimate (of it being less likely, equally, or more likely than not), compared experience or compiled statistics can revise our earlier, partial intuitions or partial data accordingly”. So well and good; all this is well known.
However, when we consider what Wiseman more specifically means by deductive and inductive a fortiori argument, we must express strong disappointment. It is evident from his definitions of a fortiori argument that his meaning is mainly the following:
Wiseman’s deductive a fortiori argument:
Wiseman’s inductive a fortiori argument:
A is more (of something) than B;
A is more (of something) than B;
and B has f;
and B probably has f;
therefore, A surely has f (or more).
therefore, A should have f (or more).
As we have already pointed out, Wiseman’s deductive form of a fortiori argument is invalid, since it lacks the crucial clause in the minor premise and conclusion about the sufficiency of the middle term (i.e. that B has enough of it, and therefore A has enough of it too). In other words, as it stands, its conclusion does not necessarily follow from its premises. As I have explained in an earlier chapter (3.3), an inductive argument must be based on a valid deductive argument to be declared valid from the point of view of formal logic. Since the proposed deductive form is invalid, the proposed inductive form must also be declared invalid. Strictly-speaking, then, both arguments are formally non sequiturs, and cannot be called deductive or inductive inference. We might call both arguments ‘inductive’ in a very loose sense of the term, but this is more misleading than useful.
In the said earlier chapter (3.3), based on my earlier researches, mainly in Future Logic, I list four types of formal induction: (a) generalization and particularization, (b) adduction, (c) induction based on deduction, and (d) induction not based on deduction. Wiseman’s proposed “inductive a fortiori argument,” and indeed his proposed “deductive a fortiori argument,” clearly both belong under the last category (d): they are “induction” in a very weak and unreliable sense of the term. He gives no evidence of awareness of what truly constitutes deduction and induction, and how exactly they are related to and distinguished from each other. That is to say, he does not show understanding of the formal conditions of validity for alleged deductive and inductive arguments, since his deductive form and consequently his inductive form cannot be formally validated.
He is evidently unaware of formal induction, since he shows no awareness of the formalities of the first three types of induction in my list. When he refers to inductive argument, he seems to intend argument which, though structurally sound, has uncertain premises so that the conclusion is likewise uncertain (we may refer to this as doubt concerning the content of the argument). But he does not realize that the deductive argument he has in mind as underpinning such induction is in fact structurally unsound (so that there is in fact doubt in the form of the argument). So, his inductive a fortiori argument cannot be said to be justified by his deductive a fortiori argument, because the latter is itself deficient and so unable to justify anything!
Wiseman often speaks of a fortiori argument as either deductive or inductive, as if these were two distinct forms of the argument. He repeatedly speaks of an “inductive form” of a fortiori argument. But nowhere does he actually explicitly define such a form, distinguishable from the deductive form. What he actually proposes, as we have seen, is a single, vague form of argument, which may in some unspecified circumstances turn out to be deductive, and in some unspecified circumstances turn out to be inductive. This argument of his somewhat resembles strict a fortiori argument to some extent, but is manifestly not an accurate statement of it. Thus, his theory is in fact that one and the same form of a fortiori argument ranges from inductive to deductive applications. This is not formal logic, since there is no formal differentia by means of which we can tell a valid from an invalid argument, and a deductive from an inductive argument.
For him, “a fortiori” argument is typically this: given that A > B, then, if B has f, A should have f. The range of a fortiori argument is, then, the range of meaning of the qualifier “should.” This may or may not indicate the inner certainty that its outward use suggests; i.e. sometimes it is used in a strong sense, and sometimes in a rather skeptical sense. It may be literally indicative of “a virtual 100% likelihood,” in which case “should” becomes “surely;” or it may merely express our current expectation, even though we know that the probability is less than maximal, signifying merely inductive confidence.
As an example of inductive a fortiori argument he gives us the following: “If this little apple is sweet, would a bigger one of the same family also be sweet?” The argument he is rhetorically considering is: If this little apple is sweet, then probably a bigger one of the same family would also be sweet. Now, the reason why such an argument would strike us as unconvincing is because the given information is far from sufficient to construct a valid a fortiori argument. Let us consider this example more precisely, by making explicit the aimed at standard format a fortiori argument (although, of course, Wiseman does not do that):
The second apple (P) is bigger (R) than the first apple (Q);
so, if the first apple (Q) is big (R) enough to be sweet (S),
then second apple (P) is big (R) enough to be sweet (S).
Now, what information would we need to justify this argument? The major premise can be readily granted, assuming that we are able to measure the two apples compared (and so relate them as major and minor terms, P and Q). What is open to debate is the minor premise. We cannot construct such a proposition (which is of ‘suffective’ form) merely from the known facts that the apple has some size and tastes sweet: we must additionally show that sweetness (the subsidiary term, S) is causatively related to size (the middle term, R); or more precisely, that below a certain minimum size (say, Rq), sweetness is imperceptible. Only given such empirically determinable information (as to how much R is enough) could we then dare to infer the conclusion. Thus, the putative conclusion is not inductive because of some flaw in the process, but because of some deficiency in the minor premise. Moreover, from such purely a fortiori argument we could only deduce that the bigger apple is sweet; to claim it to be sweeter (than the smaller one), we would need additional empirical information to the effect that sweetness increases with size.
Thus, the reason why the proposed conclusion seems doubtful is that we are not given any empirical evidence that justifies the required minor premise. Wiseman says, “We cannot deduce it; but inductively, we can assume or test it;” and he refers to our varying “confidence levels.” But he does not tell us what tests we should carry out. Anyway, the truth is, as regards the said minor premise, we would not normally assume it, being given merely that the apple has some size and tastes sweet; to do so would be very presumptuous. To assume the minor premise, we would need to have made some observations or even deliberate experiments, and found that apples smaller than a certain size are not sweet; in that case, our confidence in it would be proportional to how many apples were tested (since the initial conjunction of size with sweetness or sourness might be coincidental). At that stage, we could use the minor premise to deduce the conclusion, and our confidence in the latter would reflect our confidence in the former.
Note this well: we cannot generalize from “the first apple has some size and tastes sweet” to “the first apple has enough size to taste sweet,” because the former proposition is entirely positive, whereas the latter proposition has an implicit negative component, viz. “if an apple does not have a certain minimal size, it will not taste sweet”. Generalization can only proceed along the same polarity – it cannot produce information concerning the opposite polarity. Many people, still today, do not realize this important inductive principle. If the opposite polarity is assumed without specific justification, that is mere speculation – the very lowest type of induction. It is for this reason precisely that the above example seems particularly unconvincing – its minor premise is merely speculative rather than based on generalization. But all this has nothing to do with the form of the argument sought for, which is clearly a fortiori.
From this example we see that what Wiseman means by “inductive a fortiori argument” is not something formally distinct from “deductive a fortiori argument,” but merely an incompletely formed a fortiori argument, a merely ‘potential’ a fortiori argument. That is, some elements are known which could eventually crystalize as an actual a fortiori argument, but they do not in the present context of knowledge suffice to do so. We can perhaps foresee the valid, deductive a fortiori argument that we wish to formulate on the horizon – but we do not have all the information needed to formulate it – so, instead, we only claim an “inductive a fortiori argument,” even if in fact there is no genuine a fortiori argument at all, yet. Thus, Wiseman applies the label “a fortiori” to some arguments without formal justification, on the presumption that they might eventually turn out to be a fortiori arguments.
As already suggested, the reason why Wiseman imagines that such inchoate argument may be characterized as “inductive a fortiori” is that he is wrongly defining a fortiori argument, omitting to mention the crucial factor of sufficiency (or insufficiency) of the middle term in the minor premise and conclusion, and thus making the concept more inclusive than it really is. This is evident in the definitions of a fortiori argument that we listed at the beginning. These definitions are far too vague for any formal development and validation.
A fortiori argument is essentially deductive; if it is not deductive, then it is not really a fortiori argument, or at least not yet so. To be characterized as a fortiori, an argument must be well-formed in accord with the standards of a fortiori logic. ‘Well-formed’ means that all the relevant elements are clearly laid out, and the process is fully understood and validated. In practice, it is true, very often what we have before us is an argument of unclear content (with some elements left tacit), and therefore of uncertain form, so that the putative conclusion must be regarded as at best tentative and in need of review. Although strictly-speaking this does not qualify as valid a fortiori argument, or even as at all a fortiori argument, it is considered as ‘in the same ball park’, and conventionally labeled as ‘a fortiori’ so as to put a handle on it for discussion purposes.
In this sense, we might like Wiseman speak of an inductive a fortiori argument, when what we have before us is something that more or less looks like an a fortiori argument, but which may in fact not be one at all, or at least not be a well-formed one. Such argument being structurally deficient, it would be more accurate to speak in such cases of an attempt at a fortiori argument. The speaker aims for that ideal form, but does not yet fully succeed to attain it for lack of some needed information. Nevertheless, ultimately, only a fully explicit and obviously valid a fortiori argument, which can confidently be called ‘deductive’, can rightly be called a fortiori.
Needless to say, we can legitimately speak of an inductive a fortiori argument, when the attempted thought-process is clearly a fortiori deduction, but the premises have some material uncertainty. Since the premises, though correctly structured, have content that is less than 100% sure, the conclusion is accordingly unsure. There is no formal difference between such argument and deductive argument, note well – the difference is only in the factual reliability of the premises. In this sense, since human knowledge is rarely absolutely certain, we might say that almost all a fortiori arguments formulated in practice are effectively inductive, even if the relation between premises and conclusion is formally quite deductive.
Moreover, note well, the underlying principle, viz. that the probability of the conclusion reflects that of the premises, is not specific to a fortiori argument, but concerns all forms of argument, including categorical syllogism. A categorical syllogism yields a deductive conclusion if both its premises are 100% sure, whereas it yields an inductive conclusion if one or both of its premises are less than 100% sure. The same is true of apodosis, dilemma or what have you.
If, for instance, as is usually the case, the premises (of whatever deductive argument) are based on generalization from experience, then the conclusion, even though it follows necessarily from the premises, must be regarded as also inherently empirical in status. We simply must not confuse the deductive character of the process with the usually inductive status of its conclusion. Thus, there is no need to justify such reasoning in the context of a fortiori argument specifically; it is a general problem of logic with a general solution.
But Wiseman’s inductive a fortiori argument is not like that, since the deductive a fortiori argument it is based on is itself invalid. The problem with his inductive argument is not that it has material uncertainty; it is rather that it has formal uncertainty. Wiseman at one point seems to sense that the overly skeletal way he has defined a fortiori argument makes the relation between premises and conclusion very tenuous. In an attempt to forestall incorrect inductive inferences, he suggests that “we can strengthen the relevancy relation of the premises to allow only those that suit a sound, deductive a fortiori.” He explains this concept in a footnote as follows:
“The relevancy demand makes the argument a restricted deduction. Here, the premises (P’s) are to be relevantly true and the conclusion (C) true for validity; if not, the claim is unacceptable. To be relevantly valid as an argument, it is not good enough to merely avoid the case of true P’s and a false C, we want to disallow cases of false P’s, and allow only true (P’s) with true C. The relevancy condition, rather than general acceptability of the argument form, requires that the premises be inseparably related to the conclusion for the C to be valid.”
I am not sure just what he means, here. Is it possible that he does not realize that the validity or invalidity of an argument has nothing to do with the truth or falsehood of its premises? All we can say is that if the argument form is valid, then true premises yield a true conclusion; and if the premises are true and the conclusion is false, then the argument form must be invalid. An argument with false premises may still be valid; an argument with true premises may still be invalid. We cannot judge the argument form on the basis of the truth or falsehood of its conclusion, either. The argument form may be valid, yet have a false conclusion in a given case because of a false premise or two; the argument form may be invalid, yet have a true conclusion in a given case, whether its premises are true or false. We can say that of all argument forms, not just a fortiori.
I suspect that what Wiseman has in mind when he worries about “relevancy” is the practice that I have identified in the opening chapter (1.1) as the fallacy of two middle terms. The example I give there is, effectively: ‘Since humans are more powerful than horses, it follows that if horses are powerful enough to run at a certain speed, humans must be powerful enough to do so too’. What is wrong with this argument is that the superiority intended in the major premise relates to mental capacities, whereas that intended in the minor premise and conclusion relates to physical capacities: so the argument involves two distinct middle terms. Here, obviously, the major premise is not relevant to the rest of the argument. But Wiseman’s approach is here again too approximate to pinpoint the problem.
Whatever the case, Wiseman’s above remark betrays his lack of clarity as to what exactly constitutes an “a fortiori argument.” He does not see the exclusive connection between its premises and conclusion, and so is at a loss as to how to establish it. He apparently thinks it is all a matter of tinkering. This is also evident in his next words:
“With the a fortiori, mere superior over inferior relation of general commonality is fine as far as it goes, but it is insufficient to grant certainty for the conclusion. The less has to imply the more and vice-versa. This would narrow down the a fortiori to just those sorts of inclusions that work. By requiring a relevant relation and a true conclusion, we assure the argument‘s validity and increase its potential soundness… However, if the conclusion is well known, there is little point in making the argument. As such, relevancy may be too strong a requirement.”
All this is very surprising, coming from someone acquainted with the treatment of a fortiori argument in my Judaic Logic. It is of course true that the major premise alone cannot guarantee the conclusion; we also need the minor premise for that. And the required minor premise is not simply about possession of some feature (the subsidiary term, S), but about satisfying a certain condition (a threshold magnitude of the middle term, R). In that case, the conclusion will be very specific and relevant, and follow necessarily; and seeing this is the point of the whole exercise.
Wiseman considers that a fortiori induction has two “sub-forms,” namely “abduction” and “conduction.” These, together with analogy, are instruments of “practical,” as against scientific, induction:
“Induction expands the scope of reasonable answers to cover less universal and more particular issues. So too, when the greater rigour and assurance of scientific induction are unattainable, one can settle on largely successful, practical inductions or its sub-forms. These include the abductive (best choice), conductive (likelihood), and analogical (possibilities), sometimes combined.”
All three of these processes concern the “transfer” of information from one context to another, on the basis of indices insufficient for deduction.
The highest form of such induction, in Wiseman’s view, is “abduction,” because in its case the available alternatives are known and their relative ranks have been determined, so we choose “the overall best alternative.” Thus, if some item B has some desirable feature and another item A has more of that feature, we would choose A as “that much better than” B. The reasoning he describes is, I agree (if we insert appropriate corrections into his formula), related to a fortiori argument in that we think: if the minor term B has enough of the decisive feature to be desirable, then the major term A has enough of the decisive feature to be desirable. This is a positive subjectal argument, with the “decisive feature” as the middle term (R) and “desirability” as the subsidiary term (S) (although Wiseman does not explicitly distinguish the latter two terms).
However, I do not see that we can conclude from the given premises, as Wiseman seems to, that A is proportionately more desirable than B, and thus automatically choose A over B. There may well be a ceiling for the decisive feature above which it is excessive and not so desirable. As common wisdom has it, more is not always better; i.e. there is sometimes too much of a good thing. To draw a valid proportional conclusion, we would need an additional premise about proportionality. So, we could say that what makes the above “abduction” an inductive argument is only the speculative assumption of proportionality; for otherwise, the argument (if duly corrected as above suggested) would be deductive.
“Conduction,” next in line in Wiseman’s list, is comparatively a more epistemic form of reasoning, since it suggests the “more likely” choice for some purpose. Here, we are not sure whether A and B have the desirable feature, or to what extent they do; but we do know that A is more likely to have it than B, so we choose A over B. This thought process also, I agree (provided it is more correctly formulated), involves a fortiori thinking, except that the middle term (R) is now “likely to have the decisive feature,” and the subsidiary term (S) is now only “likely desirability” (i.e. we somewhat arbitrarily pass on the likelihood of the middle term to the subsidiary term, for obviously something is actually desirable only on condition that it actually has the decisive feature). Obviously, with such unsure middle and subsidiary terms we cannot expect a very sure conclusion; but it is arguably better than no conclusion at all if we really must make a choice.
In any case, the conclusion here again can only be non-proportional, since Wiseman does not provide us with an additional premise about proportionality. So “conduction” is inductive argument in three ways. First, in view of the probabilistic character of its middle term; second, in view of the rather arbitrary passing over of the probability of the middle term to the subsidiary term in the minor premise (which makes the whole argument highly speculative); and thirdly, because of the assumption that the predicate in the conclusion will surpass the predicate in the minor premise in a proportionate manner, making a definite choice possible (which is also speculation, since not based on additional information).
The third and least reliable form of induction, in Wiseman’s account, is mere “analogy,” which “proposes some like feature from a given case that might apply to the new.” More specifically, this means: “If A is more or less like B,” and B has some feature, then A should have that feature. This is not, of course, a fortiori argument – and Wiseman does not claim it to be. Wiseman does, moreover, point out that we should not “just assume that superficial likeness to the given case will lead to the feature‘s reoccurrence” in the similar case, and he enjoins us to “try to defeat the argument.” A claim is “much more likely to be true,” if it has “weathered genuine attacks” than if it remains “untested.” Well and good.
However, I do not see how this method (analogy) fits in with the preceding two: they are aids in decision-making, but in what way does analogy facilitate choice? Maybe Wiseman has in mind a sort of a pari argument: just as we chose B because of the decisive feature, then we should choose A because of it. But what of cases when we have to choose between A and B? Presumably, we would prefer B, which is a given, to A which is merely inferred by analogy (this is the thinking behind the rabbinical dayo principle, incidentally). But in that case, why bother making an analogy at all?
Anyhow, what should especially be noticed in this context is that Wiseman seems to opt for proportional conclusions in the two processes involving a fortiori argument that he describes. His “abduction” involves a proportional a fortiori argument whose additional premise concerning proportionality is not given, but merely speculatively assumed. His “conduction” involves a proportional a fortiori argument whose middle and subsidiary terms are both unsure, and which also involves an unsure proportionality. He thus “infers” choices to be made a bit hastily, without underscoring the additional assumptions they involve.
It is interesting that Wiseman conceives of induction as either abduction or conduction or analogy. This was inevitable, considering that Wiseman’s vague definition of a fortiori argument, which made impossible the formal validation of his deductive form and thence all the more of his inductive form. Since his deductive form was itself too vague to be validated, his inductive form could not be claimed to be ‘induction based on deduction’, but only at best characterized as ‘induction not based on deduction’. Indeed, even his so-called deductive form can only be so characterized. Thus, it is no accident that Wiseman’s conception of induction refers to only the very weakest forms of induction – forms which are more guesswork that serious inference.
The strongest form of induction, according to Wiseman, is “abduction.” This term is explained as follows in (for instance) a Wikipedia entry:
“Abduction is a form of logical inference that goes from data description of something to a hypothesis that accounts for the data. The term was first introduced by the American philosopher Charles Sanders Peirce (1839–1914) as ‘guessing’. Peirce said that to abduce a hypothetical explanation a from an observed surprising circumstance b is to surmise that a may be true because then b would be a matter of course. Thus, to abduce a from b involves determining that a is sufficient (or nearly sufficient), but not necessary, for b.”
Judging by the definition here proposed, Pierce simply meant guessing or surmising what the explanation for something might be. This is technically mere speculation, of course. Effectively, the first possible explanation of a phenomenon that comes to mind is cheerfully relied on; at least, until some other explanation pops in one’s head. However, judging by the way people commonly use the term “abduction” nowadays, including Wiseman, it is taken to refer to the best guess in a given context of information – i.e. to the most credible hypothesis given an amount of information too small to make a definite prediction.
And how is the truly best guess to be determined? Ideally, by ‘the scientific method’, of course – i.e. the formulation, testing, confirmation, comparison and elimination of competing hypotheses. There is no other reliable way. But Pierce only mentioned the positive aspect of this process, and missed out on (or at any rate did not sufficiently emphasize) the crucial negative aspect, namely that the hypothesis might upon further testing be eliminated either because some contrary fact is discovered or because another hypothesis is found more appropriate in the current context of knowledge. What he apparently had in mind, then, was rather ‘intuitive’ guesswork, without checks and balances, with no trial and therefore no error (i.e. no awareness of error) and therefore no correction of error.
Thus, the correct usage for the term abduction is with reference to what we normally refer to as ‘making an educated guess’. It is inaccurate to use the word abduction to refer to the intricacies of the scientific method. For the latter, and we should use the older appellation: adduction. In plain English, we adduce, i.e. we add, we put forward for consideration, evidence or arguments in support of or in opposition to some thesis or counter-thesis. Such thinking is not mere guesswork, but thoroughly pondered judgment. We rigorously pursue the objectively most fitting hypothesis in our current context of knowledge, and indefatigably look out for improvements.
In any case, Wiseman does correctly use the term abduction in the present context, since as we have seen his alleged inductive (and indeed deductive) a fortiori argument provides a very weak connection, if any at all, between the premises and conclusion. Therefore, he has to resort to guesswork to produce the semblance of a connection. The best method available for this purpose is abduction, because this bases a choice on some factual givens. In the absence of the latter, i.e. when the uncertainties involved are even greater, conduction is used instead. Both these methods inevitably involve guesswork, since they lack a firm formal foundation.
Even so, as already shown, Wiseman does not clearly discern the logical objects he refers to by means of the words abduction and conduction. His understanding of the thought processes he so labels is imprecise, due to insufficient analysis of what is logically going on in in them. Moreover, his claim that induction is either abduction, conduction or analogy is inaccurate. There are many more, and much better, means of induction than those he here lists. And those processes are more complex than mere conjectures, and most importantly they can be formally validated.
The second of Wiseman’s main theses is that a fortiori argument may have proportional as well as non-proportional conclusions. Although I read a bit of Wiseman’s work as soon as I got hold of it, enough to see where he was heading, I quickly decided to stop reading it, so as not to be overly influenced by it, until I had almost finished writing my own work. What I did note in my first cursory reading was that Wiseman is throughout very concerned with the issue of ‘proportional’ a fortiori argument.
My explanation for this common practice in my earlier work, Judaic Logic, was that, although a fortiori argument as such can only yield a conclusion with a subsidiary term equal in quantity to that in the minor premise, there might well in some cases be additional information that would allow, whether deductively or inductively, proportional inference; however, such inference would occur after the a fortiori argument, rather than as part of it. I there put it like this:
“It is not the quantitative difference between the major and minor terms which is at issue; that is already given (or taken for granted) in the major premise. What is at issue is a quantitative evaluation of the remaining terms, the middle term and the subsidiary term, as they appear in the minor premise and conclusion. According to our theory, the outward uniformity of these terms in those propositions is a formal feature of a-fortiori argument. But this feature does not in itself exclude variety at a deeper level. Such specific differences are side-issues which the a-fortiori argument itself cannot prejudge. It takes supplementary propositions, in a separate argument, which is not a-fortiori but purely mathematical in form, to make inferences about the precise quantitative ramifications of the a-fortiori conclusion.”
Wiseman, on the other hand, considers that a proportional conclusion is, in some cases at least, as intrinsic to the a fortiori process as an equal conclusion. As he wrote back to me, in response to my congratulations for his work: “I believe there is one major disagreement, although that is at the core of the thesis – that proportionality must be accounted for, not just equality to the given.” Although I thought my statement on the matter in Judaic Logic sufficiently clear and accurate, I took Wiseman’s criticism to heart, and resolved to respond more fully to his concern eventually.
I very early had the idea of coining the expression a crescendo argument to describe ‘proportional’ a fortiori argument. However, it is only when I was several months into the writing of the present book that it occurred to me to try and give a precise form to such argument. It was then that I had the good idea to explicitly formulate the ‘additional premise’ about proportionality, with which a fortiori argument would formally become a crescendo argument. It should be noted that, contrary to what I had assumed in my earlier work, this additional premise is not just an add-on at the end of the ‘proportional’ a fortiori argument, but is part and parcel of it, since the pro rata argument involving the said additional premise uses an implication of the minor premise and produces a quantitative change in the purely a fortiori conclusion. That is, a crescendo argument intertwines purely a fortiori argument and pro rata argument; it is not a mere succession of two such arguments.
This finding may now look simple, but it took me a while to realize it and properly develop it, and it had an enormous effect on my subsequent perceptions. I was forced because of it to spend several weeks laboriously rewriting much of what I had written until that point. For a start, my assessment of Maccoby’s views had to be radically reviewed. Thus, although my notion of a crescendo argument as against purely a fortiori argument was developed independently, I can say with gratitude that Wiseman’s loudly expressed concern had a direct role in the findings by stimulating this research.
Wiseman is of course right to refuse to brush off a crescendo argument. This is evident in a Judaic context: of the 46 a fortiori arguments found in the Tanakh (Jewish Bible), 6 are proportional (see Appendix 1); likewise, of the 46 a fortiori arguments found in the Mishna (the earlier compendium of Jewish law), 10 are proportional (see Appendix 2). It is also evident in a larger context: for instances, of the 15 a fortiori arguments found in Plato’s works, 4 are proportional, and of the 80 a fortiori arguments found in Aristotle’s works, 8 are proportional (see Appendix 4). Thus, ‘proportional’ a fortiori argument is quite present in human discourse. Moreover, the ratio being about the same in either context (respectively 17% and 13%), we can say that Judaism is not special in respect of use of a crescendo argument, compared say to Greek philosophy.
Let us now look more closely at what Wiseman says in theoretical terms about proportionality. His thesis seems to be that a fortiori argument is essentially ‘proportional’, although it gives an ‘equal’ conclusion in specific cases where the ratio happens to be 1:1, or as a minimum in undocumented cases. He says or implies this several times. For instance, in his abstract: “Logically, the a fortiori’s conclusion can be either limited to the same feature given in one of its premises or else proportioned to it in a way that suits both premises. Mathematically, the same outcome is just one possible ratio.” Again, later: “Overall, proportions [are] more natural, while fixing the conclusion to the level of the given is an exception in everyday situations”; or again, further on:
“Concerning the amount of the conclusion, logic on its own is neutral; it is satisfied with equality (as an interim minimum) and proportionality (as something more, which may or may not be exactly decidable). We have to decide which choice may be better in the actual context or accept both as reasonable, alternate solutions. Yet equality is just a special state of a more general proportionality.”
But I have in the present study definitively shown on formal grounds that the reverse is true; that is: purely a fortiori argument is the norm, whereas a crescendo argument occurs only in specific cases where an additional premise (whether explicitly put forward or tacitly intended) informs us that there is proportionality, so that a complementary pro rata argument can be constructed. Although (as already mentioned) I agree with Wiseman that a crescendo argument is often used in practice and may not be written off as illogical, the theoretical emphasis must in my view still be put on purely a fortiori argument.
The fact is, as we shall now show, Wiseman nowhere succeeds in formalizing, let alone formally demonstrating, his idea that, as regards a fortiori argument, ‘proportionality’ is the rule and ‘equality’ is the exception.
To formalize or buttress his thesis, Wiseman proposes “a treatment of a fortiori argument that utilizes quantificational predicate logic,” abbreviated to “QPR”. Using symbolic logic he tries to demonstrate that “in those cases that have the feature or property in question continuously, in that it is hereditary or ancestral or fixed… a feature of a category has the property throughout and so is necessarily true, universally or for a clearly defined range in which it holds.” But, looking at his formulae, all I see is syllogistic reasoning, i.e. ‘given a general proposition, it applies to all particular cases’. Although he puts this in the redundant form of ‘given a general proposition, if it applies to this particular case, then it applies to that particular case’, this is still not distinctively a fortiori reasoning.
As an example, he takes “circles.” First he tries to “prove” that if a little circle has a circumference, then a bigger circle has one too. Well, this is obvious by syllogistic reasoning: given that all circles have a circumference, then all circles do have a circumference, whether they are bigger or smaller! The mention of relative sizes does not turn a syllogism into an a fortiori argument, and is quite redundant. One could equally well argue that if a bigger circle has a circumference, then a smaller circle has one too; or even that if a blue circle has a circumference, then a red circle has one too. Moreover, the conclusion here that both bigger and smaller circles have a circumference does not imply that these circumferences are at least equal (if not proportional) – whereas in truly a fortiori argument such minimal equality is indeed implied.
Next, he tries to “prove” that, given one circle is bigger than the other, the circumference of the bigger is proportionately greater than the circumference of the smaller. Here, we might ask him how he defines a “circle” – for surely this all-inclusive concept is not something separate from that of circumference. I would have thought that the problem posed is one to be solved by mathematics, not pure logic. Geometry refers to the radius (r), and teaches us that the circumference (c) of a circle is equal to 2πr. Therefore, yes: a bigger circle (radius) has a proportionately greater circumference than a smaller one. Also the converse is true; i.e. a smaller circle has a proportionately lesser circumference than a bigger one. What we have here is mere pro rata argument. We are not arguing in a fortiori style from minor to major, or even from major to minor, but just applying given ratios.
Wiseman does not apparently regard his proof of proportionality as determining, since he sums up somewhat lamely with: “both the basic fact of a circumference… and the greater amount… are equally valid conclusions.” How so? If these are deductions, as they seem intended to be, how can their different conclusions be reconciled? Obviously, their premises must and do differ. In any case, it is evident that his attempted logical demonstrations have foregone conclusions – that is, they teach us nothing new but merely apply (in the tortuous fashion dear to modern logicians, who are all bureaucrats at heart) a principle already declared at the outset. This is, if not question begging, a very simple sort of syllogistic thinking which does not significantly clarify a fortiori argument.
I would say that Wiseman felt obliged to try out a “QPR” treatment – even though just a few pages before he has shown his acquaintance with my Judaic Logic exposé of a fortiori logic – simply because there is a belief out there, in today’s academia, that putting things in symbolic terms somehow, magically, gives them greater credence. But as I keep repeating throughout this volume, the opposite is true. Symbols are bound to largely confuse those who resort to them, and prevent them from seeing reality clearly. They may be used to present results ex post facto in an abbreviated manner, but they should never be used before a thorough and convincing conceptual analysis has already obtained these very results. They are mere means, not ends in themselves.
In any case, Wiseman’s formula for a fortiori argument predictably cannot be proved by “QPR,” or any other symbolic hanky-panky, for the simple reason that the formula is wrong – i.e. its premises logically do not imply its conclusions. Wiseman’s “QPR” treatment shows yet again that he does not realize that the concept of sufficiency (or insufficiency) is one of the defining features of a fortiori argument. Without the specification that the subject needs to “be (or not-be) R enough to” have the predicate, there is in fact no valid a fortiori argument. It is crucial. Even if Wiseman claims to “have relied upon [Sion] extensively throughout”, he has not fully internalized this sine qua non; and this causes him many problems.
For example, he refers to the “transitivity argument: A > B, B > C, so A > C” as “a special a fortiori”. But this argument is not at all a fortiori! I could construct one from it for him, as follows: A is greater than B; so, if B is big enough to be ‘greater than C’, then A is big enough to be ‘greater than C’. The middle term here is the unspecified respect in which A is greater than B; and the subsidiary term is, note well, not the third subject ‘C’ but the predicate ‘greater than C’. But the two arguments are formally very different logical entities, because the “transitivity argument” lacks the said crucial feature.
If we look at Wiseman’s definition, on the same page, of “the more typical a fortiori argument,” we see the same deficiency. Such argument, he tells us, “has a new feature that it claims to transfer to the other case. This more frequent, a fortiori argument form says that if the lesser has a regularly associated factor, the greater should surely or probably have it too, minimally (A > B; B has f; A should have f).” Note well the failure to mention sufficiency of a middle term (or for that matter, the middle term).
This omission is the main cause of his difficulties with regard to ‘proportional’ a fortiori argument. Nowhere in his treatment, note well, does Wiseman manage to give a credible formal expression to his (understandable) belief that a fortiori argument is essentially ‘proportional’, even if it is often more specifically ‘equal’ either because the applicable proportion happens to be even or because information is lacking to assume more than this even proportionality.
He does not manage it, because his working definition of a fortiori argument (i.e. the formula “A > B; B has f; A should have f”) is too vague. In his mind, such vagueness is broadness; it is what makes possible greater variety of conclusion, ranging from the inductive to the specifically deductive, and from the proportional to the specifically equal. But in fact, such vagueness is what prevents him from identifying and developing the precise conditions of proportionality. He is thus condemned to deal with the matter by general appeals to reasonableness, concluding “We have to decide which choice may be better in the actual context or accept both as reasonable, alternate solutions”.
We have already examined Wiseman’s theory of the moods of a fortiori argument. There is admittedly some formalism in it, since it refers to the terms and their relations in abstract ways (i.e. using words like “the lesser,” “the greater,” etc.). But remarkably, nowhere in it does he show under what precise circumstances the conclusion from a given premise might change. For instance, the two moods: “If the lesser has the feature, surely (or very likely) the greater has the same, equal feature,” and “If the lesser has the feature, surely (or very likely) the greater has more of the feature,” are placed side by side, with no formal explanation as to why the same premise has one conclusion one time and another conclusion another time, let alone any validation of these conclusions.
It seems that Wiseman thinks that the difference cannot be expressed in formal ways, but only through logical insight on a case by case basis. As he puts it, “Such inductive, a fortiori arguments involve empirical matters thought or known to be reliable.” But that is not a teaching of formal logic – it is merely material logic. And the truth is, we refer to material (or informal) logic when we are unable to clearly discern the formal logic involved.
Let us now look and see at Wiseman’s treatment of the dayo principle, which I think is the central concern of his whole treatise.
Maccoby’s dayo. Wiseman expends much effort contesting Hyam Maccoby, because the latter is on record as an uncompromising opponent of ‘proportionality’. Wiseman is of course right, and Maccoby wrong; but the question we must ask is whether the reasons Wiseman gives are sufficient to defeat Maccoby’s viewpoint. Wiseman correctly reproves him, by pointing out that the latter nowhere proves his position, i.e. nowhere formally invalidates ‘proportional’ a fortiori argument. But the problem is that Wiseman’s own approach is not sufficiently formal to definitely validate ‘proportionality’, even if only under certain specified conditions. For this reason, he has a hard time overcoming Maccoby’s harder stance once and for all. He is forced to resort to reasonable examples, and even (to some extent) to appeal to authorities.
Maccoby’s showcase for a fortiori argument was: given that “a moderately good child deserves one sweet,” it is “correct” to infer that “a very good child deserves one sweet,” and “incorrect” to infer that “a very good child deserves two sweets.” As I have shown in the chapter devoted to Maccoby (22.1-2) this example is a mixture of true and false ideas. For a start, it lacks the crucial clause about sufficiency of the middle term, without which no formal validation of the argument is possible. Construed as a purely a fortiori argument, the example should have been formulated as follows:
A very good child (P) is more deserving (R) than a moderately good child (Q),
and, a moderately good child (Q) is deserving (R) enough to get one sweet (S);
therefore, a very good child (P) is deserving (R) enough to get one sweet (S).
Secondly, while it is true that, given only these two premises, only the “one sweet” conclusion is logically permissible – it is also true that given an additional premise about proportionality, the “two sweets” conclusion becomes logically permissible. The argument would then have the following form:
A very good child (P) is more deserving (R) than a moderately good child (Q),
and, a moderately good child (Q) is deserving (R) enough to get one sweet (S);
and, reward in sweets to children (S) are to be proportioned to their deserts (R);
therefore, a very good child (P) is deserving (R) enough to get two sweets (S).
Actually, unless we can provide a precise mathematical formula for the concomitant variation between S and R, the conclusion should rather read, more indefinitely, “more than one sweet;” or we should admit that the exact quantity is somewhat subjectively assessed. In any case, the two above arguments, the purely a fortiori and the a crescendo, can be shown valid (as done in part I of the present volume). Therefore, as I said, Maccoby had some formal justification for his advocacy of non-proportional argument; but he was also formally unjustified in his categorical denial of proportional argument.
Now, let us compare Wiseman’s stance. First, like Maccoby, he does not include the crucial clause about sufficiency of the middle term – a grave deficiency. Second, like Maccoby, he does not realize that the difference between proportional and non-proportional a fortiori argument lies in whether or not we have additional information regarding proportionality, i.e. in the above example the third premise: “reward in sweets to children (S) are to be proportioned to their deserts (R).” Thus, Wiseman is as unable to prove his position as Maccoby is unable to prove his. They are both too informal in their approach to be able to settle the matter. Third, both Maccoby and Wiseman stray by focusing on “rewards” for good deeds, whereas the Mishnaic dayo principle is more specifically concerned with penalties for offenses.
Wiseman’s position could be considered closer to the truth, since he admits of both non-proportional and proportional a fortiori arguments, even if he does so on the basis of vague definition. On the other hand, Maccoby is more accurate, in that he claims his non-proportional argument as a deduction – whereas Wiseman views both types of a fortiori argument as essentially inductive (even if he admits that under some unspecified conditions they may become deductive). Maccoby fails to perceive under what conditions non-proportional deduction can become proportional deduction. Wiseman fails to perceive the formal conditions for both non-proportional and proportional deduction, so that his reasoning is always relative to material examples. This is confirmed by his concluding remark that “philosophically, Maccoby‘s denial of degrees was untrue; even if his argument was formally valid, it was still factually unsound”.
He proposes a “counterexample” to Maccoby’s good children example. He argues that “Although the dayo recognizes the value of persons as equal, still, it is often right to reward a better performance,” as in the case of two students who get “differing grades or rewards for varied performances.” While he does not deny that Maccoby’s egalitarianism might be appropriate in some cases, e.g. to encourage behavior “pure and free of any ulterior motive” or to discourage “mere posturing or outward show for the sake of reward” or to avoid “quarrels,” he insists that “by the principle of fairness or justice… much better work surely deserves more” reward. Sometimes, when the focus is on an “essential quality” held in common by the subjects, we opt for a single lowest common denominator as predicate; other times, when the focus is on the “relative performance” of the two subjects, we opt for quantitatively different predicates.
But though this counterexample may seem reasonable enough as argument in favor of the ‘measure for measure’ principle, it is not really relevant to the issue of proportionality in a fortiori argument. The issue is not whether proportionality exists in nature and morality, which no one denies, but whether it can be inferred by means of a fortiori argument, and if so under what precise formal conditions. Even though Wiseman asks the crucial question “how could one credibly scale exact rewards to ill-definable levels of goodness?” – he does not in fact ever take it to heart and adequately answer it. Yet this is the crux of the matter, and ultimately the raison d’être of the rabbinical dayo principle. Wiseman does not realize that this principle was not formulated in ignorance of the measure for measure principle, but precisely in order to impede its overenthusiastic use.
The Mishnaic source. Moreover, Wiseman has much in common with Maccoby with respect to the interpretation of Mishna (Baba Qama 2:5), from which the rabbinic dayo principle apparently sprung. In this documentary source, the majority of rabbis (the Sages) seem to reprove a member of their assembly (R. Tarfon) for engaging in ‘proportional’ a fortiori argument, saying ‘dayo’, i.e. it is enough to conclude with the given quantity, it is not legitimate to infer a proportionate quantity. This is interpreted by both Maccoby and Wiseman as a blanket interdiction by the Sages to engage in proportional a fortiori argument. Although Maccoby looks upon the Sages’ dayo principle as supportive of his view of it as logical, whereas Wiseman considers it as unreasonable and unfair – they basically agree that this interpretation of the Sages’ intention is correct. However, it is clear from this interpretation that neither Maccoby nor Wiseman has sufficiently studied the relevant sources.
Maccoby sustained the apparent position of the Sages, by declaring the non-proportional conclusion to be the only strictly logical conclusion. However, Maccoby based this judgment on a partial reading of the relevant Mishna, namely on only the first argument of R. Tarfon and the Sages’ first dayo objection. He did not notice the second argument of R. Tarfon and the Sages’ second dayo objection, which are significantly different and cannot be read the same way. Furthermore, he summarily dismissed the Gemara’s treatment of the matter as the work of an ignorant Amora, having failed to notice that the Gemara attributes its view of a fortiori argument to a baraita, i.e. a Tannaic source, one as old as the Mishna.
Wiseman has likewise not grasped the significance of the second argument of R. Tarfon and the Sages’ renewed objection to it. He somewhat agrees with Maccoby that, following the Mishnaic Sages, Judaic logic is principally, if not exclusively, in favor of non-proportional argument. However, he gives more credence to the deviations from that norm in the Tanakh, the Mishna, the Gemara and other rabbinic sources, since of course this confirms his own theory. However, it is evident that Wiseman has not actually studied the relevant Gemara (Baba Qama 25a). He too makes no mention of the baraita there presented, “How does the rule of qal vachomer work?” – which seems to claim the natural conclusion of all qal vachomer to be proportional, a position very close to Wiseman’s.
The truth is, if we look dispassionately at the data, both ‘proportional’ and ‘non-proportional’ a fortiori arguments are used in Judaism. I have above provided some statistics relating to the Tanakh and the Mishna; I expect comparable figures obtain in the Gemara. The two arguments of R. Tarfon in this context are frankly ‘proportional’, although the second can also be read as ‘non-proportional’ (whereas the first cannot). The Gemara’s position (at least ad loc) is consciously ‘proportional’, and its author goes to great lengths to demonstrate that the Sages have the same belief. However, the Sages’ position is objectively subject to various interpretations. The most likely interpretation in my view is that their dayo objections were not intended to reprove R. Tarfon for any logical error, but merely meant to institute an ethical principle.
Now, Wiseman ultimately upholds at the same idea, that the dayo principle is more ethically than logically motivated. Nevertheless, not having thoroughly studied the Mishna and Gemara, not to mention later commentaries on them and other relevant sources, he is unable to fully substantiate his position. His position is right, but not solidly founded. He adheres to this idea simply because it makes sense to him, as a way to prevent the overly extreme judgment of Maccoby, who contrary to evident common practice rejects all ‘proportional’ a fortiori argument offhand as inherently illogical.
The rabbis’ motive. Contrary to what Wiseman imagines, the dayo principle is not on the rabbis’ part a dogmatic rejection of pro rata reasoning as such, but merely a humble admission that they dare not use such uncertain reasoning when human destinies are at stake. It is not an act of injustice on their part, but of justice. They do not deny that proportionality exists; they merely deny that they have the cognitive means to identify it correctly in the course of Torah interpretation, and so prefer to take no chances to make a wrong estimate. If they had the means to predict exactly what the right proportional penalty should be for each fault, they might well edict such penalties; but there is no exact science in these matters, so they prefer to rest content with the minimum penalty specifically mentioned in the Torah.
Thanks to the restraint enjoined by the dayo principle, the rabbis are sure not to offend God by handing down an unjust sentence. They prefer to sin on the side of caution and mercy, than to sin through excess of harshness and vengefulness. In both cases there is injustice, but more so in the latter than the former. True, some guilty persons may thus get away with less punishment than they deserve; but that is not as bad as causing even one innocent person to receive more punishment than he deserves. God can, if He later chooses to do so, more easily redress the wrong of insufficient punishment than that of excessive punishment. Earthly rabbinical justice is, in any case, not the end of the matter; God may also have His say, in this world or the next.
This is what the dayo principle is all about. It does not limit the extension of a law to analogous cases, but prohibits a fortiori inference based on ‘measure for measure’ or on generalization. Inference is permitted, but boldness is not.
Wiseman admits that: “in many situations it is safer to go with what you know than a more doubtful and perhaps, partly arbitrary, proportional conclusion, especially as relative differences between things are often too vague, hard to assess, or irrelevant to satisfy the need for a definite result”. Moreover, he is well aware that “the Rabbis did not want to cause undue or unjust suffering”. Yet, he is so intent on justifying his ideas concerning proportionality and the dayo principle that he does not realize the finality of these insights, and their decisive role in explaining the rabbis’ perplexity and their preference in principle for dayo over proportionality.
Measure for measure. It is only recently, when looking through Wiseman’s paper in order to review it, that I found out he mentioned the principle of ‘measure for measure’ before me. I did not realize this till then. Wiseman hinted it when he wrote to me, after his book was finished: “The issue of mercy that you noted was strengthened.” But the truth is, although in my Judaic Logic, following traditional commentaries, I had pointed out that, in the example of Miriam (Num. 12:14-15), the penalty imposed on her was “out of sheer mercy” less severe than might be intuitively and logically expected, I made no reference there to the principle of ‘measure for measure’ (midah keneged midah). It is only in the course of writing the present book that I referred to this principle as having a role in understanding the dayo principle. That I came upon this insight independently of Wiseman, though he preceded me with it, is clear from the different emphasis we place on it.
The principle of measure for measure plays a much lesser role in Wiseman’s treatment of the dayo principle, than it does in my present study. He appeals to the former principle in order to combat what he perceives as excessive reliance on the latter by the rabbis – whereas, in my account, the dayo principle is intended by the rabbis as a restraint on the measure for measure principle. Note well the difference. Wiseman rightly states that measure for measure “equitably relates a consequence to an action,” implying that “identical acts have equal results, and differing acts have correspondingly scaled results.” But he does not realize that this thought is precisely the tacit premise of proportionality in halakhic a fortiori arguments that the rabbis sought to harness when they formulated the dayo principle. They did not make that decision unaware of the morality of measure for measure, but instituted the dayo principle as a further refinement in morality in view of the uncertainties inherent in inference.
In his analysis of the Miriam episode, Wiseman refers to the dayo principle to raise the question as to why she received a greater punishment than Aaron did for apparently the same crime. In this context, he is taking the principle to mean that ‘for the same fault, the same punishment is due’. This seems to classify the dayo principle as a special case of the measure for measure principle, whereas the dayo principle is clearly not an application of that principle, but on the contrary a limitation on its applications. However, I think this is just equivocation on Wiseman’s part – he tends to use the word dayo in variegated senses. He does in the main treat the dayo and measure for measure principles as antithetical.
For him, the measure for measure principle is an argument against the dayo principle, which he thinks was intended in too sweeping a manner. Whereas for me, the dayo principle is an argument against the measure for measure principle, which the Sages considered potentially too sweeping.
Gezerah shavah. Wiseman tries to argue that if a qal vachomer argument is prevented by the dayo principle from drawing a proportional conclusion, it would be functioning as no more than a gezerah shavah. The latter term refers to the second of the rabbinical hermeneutic principles (qal vachomer being the first), and translated literally means “equal judgment.” This consists in argument by analogy based on similar wording or meaning, making possible the inference of one law (not explicitly given in the Torah) from another (explicitly given in the Torah). Wiseman makes no mention of the third hermeneutic principle, binyan av, which is also an argument by analogy of sorts, passing from common causal characteristics to common effects or vice versa. He also does not mention inferences from context, based on the textual proximity of topics (heqesh, semukhim, meinyano and misofo).
Wiseman develops this argument in an effort to show Maccoby’s preference for non-proportionality to be a dead end. Briefly put, if this rule was true, “the QC would then reduce to a GS” (“QC” and “GS” are Wiseman’s abbreviations for qal vachomer and gezerah shavah respectively). Again, “the same, given dayo, makes the Mishnaic QC function like a strict, Scriptural GS. This effectively erases the separate natures of the religious QC and the Scriptural GS as individual, hermeneutic rules, as promoted by the same Mishnaic authorities.” Again, “Then the QC and GS are interchangeable when each result is the same. For all its acceptable, Jewish uses, therefore, such a religious QC reduces to a GS and does not really function as a unique rule of Biblical interpretation as so assumed. If they are distinct rules, equality is problematic.”
But this viewpoint is, of course, wrong. A fortiori argument has to do specifically with comparisons of quantity and with thresholds – considerations which are absent in gezerah shavah and other midot. A crescendo (i.e. proportional a fortiori) argument and purely (i.e. non-proportional) a fortiori argument are both quite different from any other sort of argument by analogy (whether verbal/semantic, causal/effectual, or spatial/temporal). It is not the quantification of the subsidiary term that makes a fortiori argument different from other analogical arguments, it is the whole peculiar structure of it which makes it unique. An a crescendo argument to which the dayo principle is applied does not thereby cease to be an a fortiori argument; it just becomes a purely a fortiori argument, which still retains the special structure that differentiates it from other analogies.
Certainly, quantity is not by itself indicative of a fortiori argument: one could conceivably (but I do not know if this is ever actually done) effect a gezerah shavah or other analogy with reference to a quantitative term without intending a qal vachomer. Wiseman does not see all that, because his concept of the a fortiori argument is too vague. His attempt to involve other hermeneutic principles in the discussion reflects this lack of clarity. It is just an expedient for polemical purposes; it has no formal standing.
Moreover, he is mistaken with regard to both qal vachomer, or more precisely the dayo principle relating to it, and gezerah shavah, in thinking of them as general principles of inference which could be applied within the Jewish Bible. This is evident, for instance, when he says that “Malachi 1:6 could be a QC, although on the face of it is a GS”. No! A statement in the Tanakh is neither a dayo nor a gezerah shavah. These are rabbinic hermeneutic principles through which the rabbis, who come long after the Scriptures, try to infer Jewish laws from the Torah (maybe from the Nakh too, in the limit). They are not to be confused with use of a fortiori argument, whether proportional or non-proportional, or of argument by analogy, within the narrative.
Furthermore, strictly speaking, later rabbis are not allowed to apply the hermeneutic techniques, originally intended for inferences from Scripture (i.e. the Torah, or more broadly perhaps the whole Tanakh), to earlier rabbinic tradition, i.e. to the “oral law.” I quoted Bergman (who refers to Rashi on Shabbat 132a) to this effect in my Judaic Logic: “The Oral Law cannot be interpreted with any of the thirteen hermeneutic rules.” Even if this restriction is maybe not always obeyed in practice, it is an officially accepted one (at least for serious legal matters, i.e. halakhah – for homiletic purposes, i.e. haggadah, there is much more leeway). Yet Wiseman (following Samely) assumes “the Mishnaic QC” to be “a tool to uncover existing or hidden truths of Scripture or tradition” (my emphasis).
Sometimes, Wiseman seems to consider that the Mishna Baba Qama 2:5 Sages intended the dayo principle as a general truth, applicable in the way of a rule of natural logic to all a fortiori argument whatever its content, whether secular or religious. In this, he reflects the view of his rival commentator, Maccoby, except that unlike the latter he does not agree with such a sweeping rule and seeks to combat it by means of rational reflection and examples.
Other times, Wiseman seems to consider that these Sages intended dayo principle as a general truth for Judaism specifically. In this, he perhaps reflects the view I (wrongly) took for granted in my earlier work, Judaic Logic. Here again, Wiseman responds in a combative manner, opposing the Sages on rational grounds, mainly by appeal to the ‘measure for measure’ principle, and by adducing examples from the Tanakh, the Mishna, and the Gemara, to show that even within Judaism such generality is not generally believed or adhered to. In his view, “both the same precedent and ratios” are “possible conclusions of an a fortiori,” so that “either way, enduring moral truths” can be “applied to new cases”.
Wiseman’s position, then, is that one way or the other the dayo principle is overextended, and he makes every effort to circumscribe its scope. This aim of his discourse is, to my mind, commendable. But, as should be evident, I do not agree with all the means he uses. Moreover, although I agree that the Sages’ dayo principle has been interpreted too broadly, either as a logical generality or as a Judaic one, I very much doubt that the Sages themselves ever intended either of these interpretations. I believe that their intent was from the start very specifically halakhic, to interdict the inference, by rabbinical legislators and judges, of a greater penalty for a greater offense merely on the basis of analogical reasoning (or more specifically, a fortiori argument).
Wiseman, on the other hand, regards the Mishna Sages as having themselves intended a larger scope for the dayo principle. This is evident in the way he often directs his criticism in large part towards them, rather than towards subsequent interpreters of their thought. It is also implied in his frequent use of the word “dayo” as synonymous with ‘equal’ (i.e. non-‘proportional’) a fortiori argument. This is a rather funny usage, suggesting that any ‘equal’ conclusion is indicative of dayo application, even though by his own admission ‘equal’ conclusions are sometimes justified, whether because the proportionality involved happens to be ‘equal’ or because lack of information renders impossible the conclusion of a more accurate proportion. Moreover, dayo is supposed to be a principle that obstructs an attempt at proportional conclusion, not one that preempts any such attempt; it comes after, not before, the velleity of proportional reasoning.
That some of Wiseman’s beliefs are in contradiction is evident, and I submit he does not clearly and convincingly resolve the contradictions and arrive at a consistent thesis. This may in part be due to the great quantity of material and issues that needed to be taken into consideration and dealt with. But it is also in large part due to his failure to develop the subject-matter in a sufficiently thorough and systematic manner. Though he tries to do so, he does not have all the needed methodological and logical tools in hand to get the job done. Moreover, he has not sufficiently studied the relevant texts; and of course, this affects his views. For example, because he is aware only of the first dayo objection, and not of the second, he is unable to immediately refute Maccoby’s idea that the dayo principle is a general truth, or to adequately analyze the Gemara and the Miriam story mentioned in it.
Wiseman also deserves criticism for his tendentious interpretation of many Biblical and Talmudic examples. He freely reads meanings into given texts that are simply not there, which suggests that he has not fully understood the difference between rhetoric for polemical purposes and impartial scientific discourse, or between fiction and fact. When analyzing a narrative, one should seek to determine what the characters in it are actually described as thinking, saying or doing, and not merely project what one thinks they might have or ought to have thought, said or done. Rigorous textual analysis must precede and be consciously distinguished from speculative interpolations.
Wiseman’s motive is of course primarily to show that there are many examples of proportional a fortiori argument in the Torah and the Nakh, and in the Mishna and Gemara. And indeed there are many examples – but he does wrong in trying to artificially produce many more. The reason he does that is that he desires to prove that proportional a fortiori arguments are more frequent than non-proportional ones. But this more specific thesis is not confirmed by the bare facts, so he tries to incline the data in the desired direction. Even within rabbinic hermeneutics, a distinction is made between explicit and implicit a fortiori arguments; he does not heed this warning, and indulges in freewheeling interpretation. He does admit this somewhat, e.g. when he writes: “Some of the claims around the Mishnaic era are my reading into these issues”; but he shows little restraint in practice, especially when interpreting Biblical examples. We shall now examine some examples of this excess.
As regards Biblical instances of qal vachomer, Wiseman refers to 25 “main” cases. His list is based on the one I developed in my Judaic Logic. He examines these cases one by one, with a view to “evaluate” whether its intent is: proportional (P), “dayo” (D), i.e. non-proportional, or a “tie” (T), the latter apparently referring to cases which are both or uncertain (i.e. D and/or P). Note his telling use of the word “dayo” to refer to any non-proportional a fortiori argument, rather than to a principle that turns certain proportional arguments into non-proportional ones.
The result of his detailed analysis seems to be that, of the 25 cases surveyed, 14 are P, 8 are D and 3 are T. From these figures he infers that, in the Tanakh, proportional qal vachomer arguments are more frequent than non-proportional ones. And he uses this finding to demonstrate that the dayo principle cannot be as universal, or at least as widespread in Judaic contexts, as the rabbis seem to think. And more broadly, he generalizes it to suggest that proportional a fortiori argument is more common in human discourse than non-proportional a fortiori argument.
My own assessment (in Appendix 1) is that, of 46 Biblical a fortiori arguments, only 6 are a crescendo. This means that non-proportional qal vachomer arguments are much more frequent than proportional ones; but it still supports the thesis that there are, in both the Torah and the Nakh, some samples of a crescendo argument. More specifically, of the 30 arguments considered by Wiseman, only 4 are a crescendo. My accounting thus differs from Wiseman’s in 13 cases (including the 3 “ties”). This great disparity shows that Wiseman’s interpretation are, to put it mildly, much less conservative than mine. Let us now look more closely at some of Wiseman’s readings.
As regards the “ties,” Wiseman concedes the non-proportional interpretation, but regards the proportional one as equally if not more reasonable. For instance, for Ex. 6:12, he favors the reading that Moses expects a greater rejection from Pharaoh than he got from the Israelites, though he admits the possibility of equal rejection as the “default” expectation, speculating that the Israelites’ rejection will incite greater rejection by Pharaoh. Similarly for 1 Sam. 23:3 and the three cases in Job. But Wiseman does not understand that such material speculations are beside the point: what matters is formal logic, i.e. what conclusions the given premises logically allow. That other possibilities exist no one denies – but the question to ask is what the specific data at hand implies.
In some cases, a proportional conclusion is manifestly justified by the language used in the given text. Thus, in Gen. 4:24, Lemekh explicitly argues from a premise of seven to a conclusion of seventy-seven; in 1 Sam. 14:29-30, Jonathan predicts “a much greater slaughter;” and in 2 Sam. 12:18, David’s servants argue from his “not hearkening to our voice” to his “do[ing] himself some harm.” These are three arguments clearly intended as proportional, proceeding from a lesser to a greater quantity. In Esth. 9:12, the issue is debatable, since the premise mentions 500 dead, whereas the conclusion is a mere question; but let us grant this ellipsis as signifying an expectation of many more dead (since this interpretation has become a venerable tradition). Yet, even if Wiseman rightly classifies these cases as proportional, his analysis of them is far from credible.
In other cases, Wiseman classifies arguments as definitely proportional without any basis in the text. Thus, for Deut. 31:27, whereas Moses only predicts the Israelites will continue to rebel after his death, Wiseman subjectively quantifies the prediction, claiming things are bound to go from bad to worse “because people tend to move away from God,” and he cites subsequent events as proof. Again, in 1 Kings 8:27, Solomon argues that if the heavens are not big enough to contain God, then an earthly house is not big enough to do so; this being a negative argument, there is no call for a proportional reading; yet Wiseman gets into a discussion about the relative sizes of God, the heavens and an earthly building, and concludes that the proportionality (not just the a fortiori) is “impeccable.” He treats such arguments as springboards for his own preachments or speculations, instead of focusing on what the speaker is actually saying. Even if what Wiseman says is true in the abstract, it does not follow that it is directly pertinent to the issue of whether the given argument is proportional or not.
Next, look at 2 Kings 5:13, where Naaman’s servants advise him to obey Elisha’s injunction to immerse in the Jordan river so as to cure his leprosy; the premise is that Naaman would have done whatever Elisha asks if the latter had asked for difficult things, and the conclusion is that Naaman should do whatever Elisha asks even if the latter asks for something easy. The consequence is essentially the same in both cases, viz. doing whatever Elisha asks to do. Yet Wiseman constructs another a fortiori argument, which refers not to the act of obedience, but to Naaman’s pride – and since doing the easier thing requires more humility, Wiseman “opt[s] for proportionality.”
Next, consider 2 Kings 10:4, in which the rulers of Jezreel reflect: if the two kings were not strong enough to resist Jehu, then we are not strong enough to do so. Here again, the conclusion is obviously identical to the premise: there was and will be no resistance to Jehu. But Wiseman imagines how much more death and destruction would befall the speakers if they did resist, and so concludes with proportionality. Thus, whenever he can spin some proportional change as conceivably occurring in the background, he judges the argument as proportional. He does not pay attention to the terms of the given argument, but engages in fiction writing, adding elements that have no direct relevance to the logical form of the actual text under scrutiny.
Wiseman fancifully misreads even obvious arguments. He judges Ps. 78:20 as a proportional argument, because “meat, water and manna” is better than just “water and manna;” but that is not the given argument, which is that since it is more difficult to draw water from a rock than to provide food, the fact that God did the former is proof that He can do the latter. Again, Wiseman judges Ps. 94:9-10 as proportional arguments, because the Creator has “superior ability” compared to humans; but this is not the given arguments, which are that since God is powerful enough to create sense-organs, He is powerful enough to see and hear, and since He has the majesty to chastise nations, He can well reprove individuals. Note that these arguments are positive predicatal, major to minor in form.
Wiseman also tendentiously misinterprets Prov. 11:31, which says “Behold, the just man shall be recompensed on earth: and likewise the wicked and the sinner,” as proportional – arguing that since a positive action is worth more than a negative one, it follows that if a positive action has positive consequences, a negative action must have negative consequences (at least, this is how I understand his symbolic statement “If A+, then C+; A+ > A–; so for A-, then C–”). But this is not the argument intended in the text; and anyway it is not a valid a fortiori argument, because it is positive subjectal in form and yet goes from major to minor, and because the predicate is not the same in the premise and conclusion. The argument has to be something like: If the just man is imperfect enough to be recompensed on earth, then the wicked and sinner are imperfect enough to be recompensed on earth; and this form is definitely non-proportional, since no quantity is mentioned.
Prov. 21:27, which reads “If the sacrifice of the wicked is an abomination; how much more brought with a wicked intent?” is interpreted by Wiseman as proportional, because “the bad intent of the evil perpetrator makes his offering an even more unacceptable sham, rather than just a reinforcement of the same degree of badness.” Here again, his assumption is that the degrees in the subject must be reflected in the predicates. But this is not the given argument, which is rather: If the sacrifice of the wicked brought with a ‘sincere’ intent is abominable enough to be rejected, then the sacrifice of the wicked brought with a wicked intent is abominable enough to be rejected. The sacrifice will in either case be rejected (or be “unacceptable”). There are no degrees to rejection: it either happens or does not. The predicate is the same in both premise and conclusion. We could well imagine that the rejection in one case will involve some public ceremony of humiliation, and in the other case an even more humiliating ceremony – but nothing to that effect is remotely hinted at in the text.
Wiseman misinterprets the arguments in Jer. 12:5 as proportional, just because they could conceivably be so. Even if it is very reasonable to suppose that one will suffer proportionately more in more difficult circumstances, it does not follow that one can infer the greater suffering from the lesser one, unless one accepts the premise of proportionality as a given – and in this case nothing in the text suggests it (and on empirical grounds, one may well not suffer more, so there is no reason to expect such a premise); in other words, the intent may well not be quantitative. Anyhow, God (the speaker of those lines) is not forewarning Jeremiah about suffering, but telling him in a metaphorical way that, in view of his human limitations, he could not possibly grasp the answers to his questions as to why the wicked prosper. The arguments are thus essentially negative predicatal, minor to major, in form.
Finally, consider Dan. 2:9, where Nebuchadnezzar demands: “Tell me the dream, and I shall know that you can declare its interpretation to me.” Wiseman reconstructs this to read: “Were I to tell you the dream… I could not be certain that your version was correct; so it is all the more necessary that you tell me both the dream and its meaning, so that I can trust your interpretation.” He views this as proportional with reference to the degrees of certainty involved: “If I tell, then answer A1 is uncertain, but if I do not tell, answer A2 is certain.” But once again, this is not the argument directly intended. Nebuchadnezzar is clearly arguing: if Daniel has powers sufficient to tell the dream, then he has powers sufficient to interpret it. It is the cognitive powers required of Daniel for the two feats which are compared. Note that this is a positive predicatal, major to minor argument.
We have thus looked at all of Wiseman’s proposed reinterpretations, where he opts for proportionality, and found them all unconvincing. Some are incomprehensible and some are in manifest error. Some of his arguments are not even a fortiori in form. He invents alleged a fortiori arguments out of hand, with terms not directly given or implied in the text. He ignores the forms as well as the contents in the original text – reading negative arguments as positive ones, or predicatal arguments as subjectal ones, so that almost all his readings are predictably positive subjectal. He hardly notices the difference between minor-to-major and major-to-minor argument, or realize its logical significance.
Wiseman tried to obtain a maximum number of proportional examples, in order to buttress in larger thesis that proportional a fortiori argument is the rule and non-proportional a fortiori argument is the exception. He also sought thereby to buttress his narrower thesis against the rabbinical the dayo principle. But this effort was unnecessary, because the few demonstrable examples of proportional argument that there are suffice to prove the point – viz. that there are evident departures from the dayo principle, conceived as a general injunction applicable to all Jewish matters if not universally. These departures prove, not that the Sages’ dayo principle is often ignored in practice, but that those who interpret it too broadly are wrong.
Properly understood, the dayo principle has no application in Biblical passages or in their interpretation. If (as I now suggest) the dayo principle is an ethical-legal restriction, designed to obstruct the inference of penalties for offenses that are not explicitly found in the Torah from penalties for offenses that are explicitly found in the Torah, then we have no reason to expect the dayo principle to be used in the Torah. It might conceivably be used in the Nakh (the later books of the Bible), with reference to a Torah law and penalty – say, in a story where a king or some judges are trying to derive a law from the Torah (I have not tried to find out whether such use in fact occurs). But there is no reason to expect a ubiquitous dayo principle, i.e. to regard as Wiseman tends to all non-proportional a fortiori arguments as having been subjected to the principle, and all proportional a fortiori arguments as having erroneously escaped it.
Wiseman offers two examples of Mishnaic qal vachomer. The first is not from the Mishna, but is attributed to a 2nd cent. CE Tanna (a Mishnaic Sage), R. Shimon bar Yohai, in the Talmud Yerushalmi. It reads: “Not even a bird perishes without the will of heaven, how much less a man.” I would interpret this argument as follows: A man is more precious to God than a bird; a bird has sufficient worth to be incapable of perishing without God’s decree; therefore, a man has sufficient worth to be incapable of perishing without God’s decree (positive subjectal, minor to major). There is no possibility here of proportionality: “cannot perish” means what it says and does not allow of degrees. If R. bar Yohai intended otherwise, he would have worded his minor premise more softly. Since no proportionality is attempted, the dayo principle cannot be called upon to oppose such attempt.
The second example is from the Mishna (Sanhedrin 6:5), and reads: “If God is troubled at the shedding of the blood of the ungodly, how much more at the blood of the righteous!” I interpret this argument as follows: God has for the righteous more concern than He has for the wicked; if for the wicked God has concern enough to be troubled at the shedding of their blood, then for the righteous God has concern enough to be troubled at the shedding of their blood (positive subjectal, minor to major). As it stands, the argument is purely a fortiori; but it could easily have been made a crescendo by adding the premise that the degree to which He troubled at the shedding of someone’s blood is proportional to the degree of His concern for him. But even if proportionality was intended, it is hard to see how the dayo principle might be called upon to oppose it, since the conclusion does not infer a Torah penalty but merely informs that the blood of the righteous is (or is more) troubling to God.
Wiseman understands these two arguments somewhat similarly, even if less clearly. For the first, he remarks: “As a physical maximum, death is the severe end of life. We have little reason to seek some decision procedure here, or that the dayo is offered as a universal norm (although the end is clearly the same), or that one needs to know anything other than that the prior example elicits a similar result.” And for the second, he remarks: “Despite our inability to account for the greater upset that God might feel, the most sensible view is that of an increase;” to which he adds: “why not a justly proportional conclusion in differing cases?” although he admits: “One does see a distinction between conceptual acceptance of this possibility versus practical punishment.”
He says many more things, some of which I would criticize, but I will focus on just these insights. What they show is that Wiseman is well aware that the ‘equal’ conclusion (which he misleadingly characterizes as “dayo”) is sometimes appropriate (as in the first example), while in other cases it is open to discussion (as in the second example). It is also interesting to see that he makes a distinction between “conceptual” expectation of proportionality and the “practical” difficulty of imposing punishment on that basis, though he does not pursue this thought as far as he ought to have. He sums up: “I simply propose that the dayo cannot be exclusive in every case, let alone in religious ones, especially when dealing with many practical issues that would make some outcomes unfair or even unbelievable. It is normal that a more severe crime deserves a more severe outcome, although an equal outcome might work.”
This is true enough. The problem, as we have seen, is that his approach to the a fortiori argument is too vague to be able to formally predict when an equal or proportional outcome is the more appropriate. He is only able to deal with this issue on material grounds, on grounds of apparent “reasonableness.” The other problem is his confusion of the term “dayo” with that of “equal” conclusion. Although former implies the latter, it does not follow that the latter implies the former. Moreover, while according to the measure for measure principle “a more severe crime deserves a more severe outcome,” the dayo principle is precisely aimed at interdicting that way of thinking when it is an attempt to infer a harsher penalty (albeit for a worse offense) from a lighter Torah penalty (for a lighter offense). This is what the Mishna Sages consciously intended; it was not a thoughtless dictate. Logically, the Sages’ position is unassailable; they break no law of logic in advocating their dayo principle. Yet Wiseman keeps trying to talk them out of it, as it were!
With regard to qal vachomer in the Gemara, it is unfortunate that Wiseman nowhere in his paper attempts a study of the crucial passage in Baba Qama 25a-b, which comments on the Mishna Baba Qama 2:5. He mentions this sugya of the Talmud a few times, but has obviously made no effort to actually look at it and study it. He relies entirely on hearsay concerning it, mainly Maccoby’s; and Maccoby of course is an unreliable source, since he himself did not study it attentively. As a result, Wiseman constructs a somewhat fanciful view of what the Amoraim (i.e. the rabbis of the Gemara) said about qal vachomer and the dayo principle. Though he occasionally mentions “examples” of Amoraic proportional a fortiori argument, I have not found any cited in his paper. Nevertheless, let us consider his views.
Wiseman imagines that the Amoraim advocated proportional a fortiori argument. This seems accurate, since a baraita is cited to that effect in the Gemara we just mentioned. He thinks the Amoraim did so in opposition to the Mishna Sages who instituted the dayo principle (the “majority”) and in support of their opponent, R. Tarfon (the “minority”). This is inaccurate; the Gemara actually seeks to justify the dayo principle (“Is not dayo of Biblical origin?”) and to show that R. Tarfon believes in it too (“Does R. Tarfon really ignore [it]?”). Wiseman does not, however, claim that the Amoraim totally denied the dayo principle; he sees them as rationally circumscribing it. There is truth in that; the Gemara makes it more conditional, in an effort to reconcile the opposing views in the Mishna.
Additionally, Wiseman suggests that the dayo principle was instituted and accepted in Mishnaic times because the surrounding dangers made unity, preservation and transmission imperative. On the other hand, the Amoraim, living in less disturbed times, could afford to be more open to the extension of laws through proportional a fortiori argument. But this view is, in my opinion, incorrect. The Mishnaic Sages instituted the dayo principle not out of fear of external forces, but out of fear of God – the desire to avoid punishing people more harshly than specifically permissible under Torah law. And (to my knowledge) the Amoraim never theoretically reprove them for it, nor markedly deviate from it in practice.
Furthermore, Wiseman implies that the early rabbis who instituted the dayo principle did so out of rigidity of mind, rejecting the nuances made possible through proportional a fortiori argument, whereas the later rabbis were freer and more precise thinkers. But the Tannaim were just being morally responsible, very careful not to make mistakes through fanciful extrapolations. Moreover, it cannot truly be said that the Amoraim, who considered themselves their humble disciples, were “freer” than the Tannaim in the interpretation of existing laws or formulation of new ones. Certainly Wiseman does not substantiate this claim in a systematic manner. In particular, he gives no examples of Amoraim freely using proportional a fortiori argument in halakhic contexts.
The problem here is, of course, one of perception. When Wiseman thinks of the dayo principle, he thinks of it as intended by the rabbis who instituted it very broadly, at least within the sphere of Judaism. Then, when he sees (or hears about) later rabbis apparently ignoring this broad dayo principle and indulging in proportional qal vachomer, it seems to him to constitute a reversal of previous policy. But in fact, dayo was from the start not intended as so broad a principle, but one limited to specific inferences (as earlier described). So, before claiming that later rabbis sometimes ignore dayo, it is necessary to demonstrate that they ignore this limited principle, and not a conjectured broader one. It may be that they sometimes do; but Wiseman has not empirically demonstrated it.
Wiseman expends considerable effort criticizing the Torah and subsequent commentaries for not judging Miriam and her brother Aaron with the same leniency (or severity), regarding the events narrated in Numbers 12. Miriam and Aaron are there said to have spoken against their brother Moses, apparently for having married a foreign woman, and seemingly suggesting that their own level of prophesy was equal to his. God summons them and reproves them. When He withdraws, Miriam is struck with “leprosy;” but there is no mention of Aaron being so afflicted too. Thereafter, Miriam is condemned to a seven-day imprisonment outside the camp; but again Aaron is not so punished. As Wiseman puts it:
“More severity falls on Miriam than on Aaron. If Aaron could get away with just a lesser rebuke, why did Miriam get more: a rebuke, leprosy, and seven days of separation? Not even Aaron could fathom why she got these heavier strokes. Nor did it appear right to Moses, who accepted Aaron’s plea for their sister and asked God for a lessening (or healing). In other words, both men thought that she had received considerably more for no apparent reason, perhaps reasoning that a rebuke should be enough (dayo) for her too.”
To Wiseman, who interprets the episode as “a challenge to Moses’ leadership” by both of siblings, this apparent double-standard is unreasonable and unfair. According to him, Miriam should have received the same light sentence as Aaron, by a dayo inference from the lighter sentence against him; alternatively, we might add (though Wiseman does not say it), Aaron should have received the same severe sentence as Miriam, by the same reasoning.
Wiseman acknowledges that the argument in the Torah concerning Miriam is from seven days for offense against a father to seven days for offense against God (or Moses, in his view). But he wonders why a similar inference is not made for Aaron too. He complains that past commentators seem to ignore the “disparities” between their sentences. All this makes God seem “inconsistent”. Wiseman wonders whether perhaps God exempted Aaron because of his privileged position as High Priest, objecting that his priestly duties could have been temporarily carried out by his sons.
Wiseman has evidently not properly studied this episode. As Rashi reads the story, Miriam is mentioned before Aaron to indicate a leadership role on her part in this incident. Also, they mentioned Moses’ wife to criticize him for causing her chagrin by no longer having marital relations with her, and they mentioned their own status to point out that although they had not given up on marital relations their prophetic powers were not diminished. God later replies to them that He ordered Moses to stay away from his wife, because he needed to be readily available at all times to receive His instructions – so there was no sin on his part, and they were wrong to criticize him.
Wiseman should have realized that if Aaron was punished less severely than Miriam it was not because of unjust favoritism, but simply because their sins were different. Wiseman does momentarily admit the possibility that Miriam “bore greater guilt” than Aaron, since she is “mentioned first as instigator, gossip”; but he does not linger on that thought. Reading the story, the first explanation that comes to mind is that Miriam’s sin was to talk, whereas Aaron’s sin was to listen to her talk. And indeed, this is the explanation offered by a Midrash, according to a rabbi I questioned. It is true that listening to “evil speech” (lashon hara) is considered as sinful as proffering it; nevertheless, a difference in degree may be assumed between the two acts (one being passive, the other active).
I would add that Aaron seems to have confessed and repented more readily, since it is he alone who says to Moses: “we have done foolishly… we have sinned.” Notice, too, that he altruistically speaks on her behalf, as well as his own. Moreover, he appealed to Moses for his sister’s release from the disease, whereupon Moses uttered his famous prayer, “El na refa na lah!” Miriam was perhaps too shocked and afraid to say anything for herself, let alone for Aaron. In sum, there are many indices that could explain the difference in judgment besides the difference in initial sin.
So much for the material aspects of the case. What needs clarifying now are the formal aspects. The first thing to notice is that Wiseman introduces an argument of his own, namely, “If Aaron could get away with just a lesser rebuke, why did Miriam get more: a rebuke, leprosy, and seven days of separation?” This seems reasonable to him, because he assumes their sins to have been the same. His answer seems to be that their different treatment is a Divine fiat: “In essence, Divine revelation required more [for Miriam] than Aaron’s judgement, not the same.” Although human logic expects an equal sentence for the same offense, “God’s-eye-view trumped human thinking that took Aaron’s, lesser precedent as governing.”
He then asks: “If this actual, prior given is not binding, how can the dayo arise from Miriam’s case?” By this he means: if in the above argument the dayo principle is not applied (since Miriam’s sentence differs from Aaron’s, contrary to expectation), how can the dayo principle be derived (as the Gemara teaches) from this episode? Although he admits that the Torah refers to an inference from (a theoretical) offense against a father to an offense against God, he argues that this “still does not explain the unequal sentences between the co-participants.” Why, he asks, does the same argument “not work with Aaron as well?”
Wiseman proposes an accounting of the penalties involved. Miriam is obviously more severely punished than Aaron, since he just gets a rebuke while she gets that plus leprosy and seven days quarantine. In his opinion, “Aaron‘s rebuke for disrespect, as the greater person, should have been sufficient (dayo) for Miriam, the lesser;” and he is indignant that “The greater one gets less and the lesser more!” His answer to the question of equity is that “mercy in judgement” may be involved, and possibly “major, social concerns too, such as education.” But as regards basing the dayo principle on this episode, he expresses strong doubt: “Logical necessity is not evident, however, in this dayo as a QC paradigm.”
In reply, let us first note that Wiseman’s first argument (that since Aaron’s fault was equal to Miriam’s, her punishment should by the dayo principle have been equal to his) is not in fact an application of the dayo principle. It is an egalitarian (a pari) a fortiori argument, as follows: ‘Assuming Aaron’s sin was the same in gravity as Miriam’s, it follows that if Aaron’s sin was not grave enough to merit leprosy and imprisonment, then Miriam’s sin was not grave enough for the same penalties’. No dayo application is called for, or even possible, in this reasoning. Moreover, it does not reflect the intent of the dayo principle.
Wiseman evidently wrongly imagines the dayo principle to function laterally as well as vertically. Laterally, meaning: by transferring a law from one case to another on the same plane. As against vertically, meaning: by applying a more ancient and authoritative source to a more recent quandary. In truth, as just explicated, the lateral function consists in a pari a fortiori argument; the law used is one and the same in both cases (viz. Aaron and Miriam). The vertical function, if we are to illustrate it by means of the episode in Numbers, consists in a minor to major (positive subjectal) a fortiori argument from the ‘more ancient and authoritative’ case of a penalty for offense against a father, to the ‘more recent’ quandary of the penalty for offense against God. The premise about a father is the given law (here given as human instinct, let us say) and the conclusion about God is the derived law (which human instinct could not predict).
The dayo principle as it emerges in Mishna Baba Qama 2:5 is aimed at the vertical function. Here, the penalty for some fault is clearly spelled out in the Torah, and the rabbis (as legislators and judges) need to know what penalty to apply to a comparable but not identical fault, which is not explicitly dealt with in the Torah. They argue by analogy, and are perhaps tempted to apply a proportional penalty, in accord with their intuition of ‘measure for measure’. However, the dayo principle – i.e. the precedent set in the said Mishna example – teaches them not to indulge in such pro rata reasoning, but to stick to the given penalty.
This Mishnaic dayo principle may be illustrated and derived analogically from the Miriam case given in the Torah, but only to the extent that we accept the idea floated above that the penalty for offense against a father is obvious to all, whereas the penalty for offense against God is not as obvious. Clearly, we are not in that Torah example literally arguing from Torah law to ‘rabbinically derived’ law, since both penalties are in fact Torah given in that example. But we can say, as the Gemara effectively does, that this argument from the more obvious case (offense against father) to the less obvious case (offense against God) is by analogy an illustration of and a justification for the Mishnaic dayo principle, even though the latter strictly only applies to inference from a Torah-given penalty (obvious, since explicit in the Torah) to a ‘rabbinically derived’ penalty (less obvious, since not explicit in the Torah).
In truth, as Wiseman also has noticed, the argument used as the Torah illustration and justification of the dayo principle is not entirely non-proportional. Comparing the theoretical precedent of an offending daughter punished by her father with Miriam’s punishment by God, Wiseman rightly points out that “being spit upon and getting leprosy” involve “a quantitative change of progressive increase.” Indeed, if the punishments are not exactly identical, it cannot be said that this argument has a conclusion equal to the minor premise. The punishment is the same only as regards the seven days of isolation; but as regards the being spit upon and getting leprosy, “the difference in kind and degree stand out.”
Wiseman suggests that maybe these different treatments were necessary to obtain “the same level of shame” (implying that it is easier to feel shame for offending one’s father than for offending God – a debatable assumption). But “still, a psychological effect is one thing, the actual stroke quite another.” Wiseman’s conclusion is that the attempted grounding of the dayo principle is not so sure. As he puts it, “The Rabbis claim that this passage is a QC argument, although the actual, Biblical text is not that clear.” The said objection is quite valid in my opinion. But Wiseman’s conclusion is not inevitable.
As I have pointed out in my own analysis, we can only argue that the Miriam story is a fitting a fortiori argument, i.e. a purely a fortiori argument as I take it to be, or an a crescendo argument coupled with an application of dayo as the Gemara takes it to be, if we gloss over the said difficulty, and focus only on the seven days isolation penalty (ignoring also the difference between the voluntary isolation of the daughter and the enforced isolation of Miriam).
We can justify that, I believe, by pointing out that the leprosy part of Miriam’s punishment was Divinely carried out, whereas the seven days isolation part is to be carried out by the earthly court of law. Since the dayo principle is intended for use by earthly judges, only this aspect of the Divine decree in the Miriam story needs be taken into consideration. Moreover, the leprosy part is already a done deed at the time that the a fortiori argument is pronounced; so the argument can only concern the not yet actual part, i.e. the sentence of seven days isolation of Miriam which Moses is ordered to execute.
Thus, contrary to Wiseman’s view, the Miriam episode can be credibly used to illustrate the dayo principle, and even somewhat ground it Biblically, if it is read carefully. But it must be said that we cannot correctly perceive the dayo principle’s place in the Miriam episode without referring to the Mishna formulation. So, strictly speaking, the Mishna formulation is conceptually prior to the Biblical interpretation. That is, the Miriam episode can be cited, but it is not by itself a solid foundation.
In his concluding remarks, Wiseman argues that the Miriam qal vachomer “could not qualify as a paradigm.” It “turned out to have a mixed conclusion—at once the same, proportioned, and restrained. Her case was isolated and unique.” Her judgment involved nuances that could not be adequately explained by the dayo principle. Instead, it was conceptually preferable to refer to God’s mercy which “could moderate or modify the prior norm of proportional justice when it advanced the overall good.”
Wiseman’s final conclusion that “the dayo is non-universal” remains true, even if the reasons he gives for that conclusion are not all entirely convincing and even though he does not clearly identify its intended scope. It is obviously wrong to consider the dayo principle as a general rule of reasoning, applicable universally or even only in Judaic contexts. That the principle is much more narrowly intended is clear from its initial, Mishnaic formulation. Its Biblical grounding is possible, but not simple.
On other commentators. Wiseman devotes a good deal of his paper to detailing and analyzing the views of various, mostly contemporary, logicians or commentators, on the subject of a fortiori argument, myself included. This is a very important service that he has rendered to our field of study, bringing to wider attention many contributions to it and stimulating debate. Thanks to his mentions of Samely, for example, I discovered the work of that author and was moved to comment extensively on it. Again, Wiseman’s translations of articles by Avraham and Brachfeld made possible my evaluation of these authors’ ideas.
I have followed Wiseman’s lead, and tried to say something about as many past contributors as I could, including many from past and present that he does not mention. My approach differs from his somewhat, however, in that I am much more critical than he has been. For instances, with regard to the Avraham and Brachfeld articles, Wiseman is content to report their views, whereas I am quick to point out that Avraham wrongly defines a fortiori argument and Brachfeld makes no attempt to define it.
Although Wiseman does on occasion offer value-judgments, his treatment of most commentators with the exception of Maccoby is overall not very critical. We could say that Wiseman is open-minded to almost all approaches, regarding them all as valuable contributions in various respects. But that open-mindedness is only due to the vagueness and uncertainty of his own perceptions: if he fully realized the logical conditions for a fortiori argument, he would be like me less tolerant of deviations from the norm.
I have already described and commented on Wiseman’s treatment of Schwarz in the chapter devoted to the latter (14.4). His thoughts on Ostrovsky, Daube, Guggenheimer, Cohen, Avraham, and Brachfeld are incidentally apparent in the sections I devote to these authors in a later chapter (31). His reaction to Maccoby is described in the discussion of the dayo principle in the present chapter; as there made evident, he strongly disapproves of Maccoby’s position.
As regards Alexander Samely, Wiseman can be said to approve considerably of his views. The reason for that is, I think, that Samely’s definitions of a fortiori argument are, like his own, sufficiently vague and uncertain as to allow for ‘inductive’ and ‘proportional’ a fortiori arguments. As I have shown in the chapter dedicated to Samely’s work (23 – see my full analysis there), the latter’s approach leaves much to be desired, involving as it does both important gaps and significant errors, in both form and content. Therefore, for Wiseman to lean on Samely to support certain ideas is not very secure.
Wiseman presents Samely’s qal vachomer formula (“somewhat simplified,” i.e. in if–then form) as “If norm n, belonging to category N that is lower on scale X, has predicate A, then norm m, of category M, higher on scale X, logically has predicate A too.” To which he adds the comment: “I take his term ‘logically’ to mean ‘with good reason’, rather than as a deductive truth.” The reason why Wiseman does this is that, like Samely, he is unable to explain how the putative conclusion follows from the given premise(s). This being the case, he (they) must claim this conclusion to be, albeit not deductive, at least a reasonable expectation (i.e. inductive). By referring to Samely, Wiseman hopes to buttress his own idea that a fortiori argument as he conceives it may be claimed inductive.
As regards proportionality, although Samely states the conclusion in ambiguous terms, as “therefore: To norm m feature A should apply even more,” Wiseman interprets this explicitly as possible proportionality, saying it could be A or A+ (i.e. more than A). I would say that Samely’s ambiguity is deliberate, because he does not know exactly what to logically conclude. Wiseman’s more explicit rendering is justified by the example under consideration, which does suggest an a crescendo conclusion. But Wiseman is equally unable to formally differentiate the two possible conclusions. He does try to do so, with reference to a diagram; but since this makes no mention of the crucial clause “R enough to be” it is not a credible model. My point here is that although Wiseman’s position here is more definite than Samely’s, he is still relying on Samely’s tergiversation to buttress his own allowance for proportionality.
Wiseman, mirroring Samely, then offers the following justification: “In general, Samely urges caution in pressing a formal, logical schema on the Rabbinic QC, so that the rhetorical and informal nature of this language, culturally and historically based, can express itself. By allowing language its initial, creative freedom and vagueness, one does not force language artificially into overly restricted or truncated forms at the outset. Inasmuch as logical systems eventually can and do expand the range of things they cover, this more permissive approach has clear benefits for later advances in logic.” This is of course just spinning an excuse for a cop-out. Obviously, the “later advances in logic” would allow us to judge the logical validity of historical examples – so why not effect these advances and judge?
Finally, let us look at Wiseman’s references to Avi Sion, i.e. to my own past work on a fortiori argument (in my Judaic Logic). Although he devotes a number of pages of his book to the study of my work, and he there seems to have understood it, in the rest of his work he in fact completely ignores crucial findings of mine, notably the factor of sufficiency (or insufficiency) of the middle term which makes possible the formal validation of such argument. He defines a fortiori argument without this crucial factor, and thus makes the argument seem much more vague and uncertain. He does this, as we have seen, in order to lend support to his beliefs in inductive and proportional a fortiori argument.
But I had previously clearly formalized and validated a fortiori argument – specifically with the crucial factor “R enough (or not enough)” – thus definitively showing its deductive status! How could he then claim such argument to be inherently inductive, and only in special (unspecified) cases deductive? Moreover, I had explained that a proportional conclusion was not justified from the given two premises, although it might be justified given additional information. How then could Wiseman maintain the possibility of a proportional conclusion from the two premises without specifying an additional premise?
Although Wiseman is not entirely wrong in speaking of “Sion and Abraham’s contention that the QC was often inductive and not just deductive”, my view of the inductiveness of many arguments presented as a fortiori is not like Wiseman’s, or for that matter Avraham’s. Whereas they imagine that a fortiori argument as such can be inductive as well as deductive, my own view is that although many concrete arguments intended as a fortiori by their speakers have inductive status and value, as regards a fortiori argument as such, as an abstract logical entity, it is clearly deductive. They effectively think that “inductive a fortiori argument” has a distinct form or is of uncertain form; whereas I insist that the inductiveness of some such arguments in practice is simply due to uncertainties in the content of one or more of their premises and therefore in their conclusion.
I think Wiseman presented my ideas on proportionality, as laid out in my Judaic Logic, with (on the whole) accuracy and fairness, and he was right to demand that more stress be laid on the possibility of ‘proportional’ a fortiori argument. However, as already shown, his own attempt to give formal basis to such a fortiori argument is not successful. Here again, the main reason for his failure is his appeal to a very vague definition of a fortiori argument, devoid of a middle term or at least devoid of the issue of sufficiency (or insufficiency) of the middle term. The a crescendo argument that I have developed in the present study, comprising a purely a fortiori argument combined with a pro rata argument, is the correct and complete formal solution to the problem wisely posed by Wiseman.
As regards the dayo principle, Wiseman considers that his view “largely agrees with Sion[’s].” That is somewhat true, but not entirely so. At the time I wrote Judaic Logic, I assumed the rabbinical dayo principle to have a broader intent than I do today. I think it seemed to me, as it did to Maccoby, to be essentially a logical principle. However, I differed significantly from Maccoby in admitting that given additional information a proportional conclusion might be drawn. Thus, it seems that I regarded the dayo principle as a rule intended more specifically for Judaic contexts, although I could see it is not always adhered to even there.
However, I did not (as I do today) further circumscribe the restriction, by making a distinction between punishments and rewards, or by specifying that the premise of such argument had to be a Torah given penalty, because I had not studied the relevant Mishna and Gemara texts. Wiseman too has evidently not studied the relevant texts. His treatment of the dayo principle seems partly influenced by mine, in its uncertainty as to the exact scope intended by the rabbis who instituted it. But his treatment differs from mine in that it is strongly focused on opposing and rebutting Maccoby’s rigid view. And of course, he does this in ways that are too informal to be definitely convincing.
Funnily enough, Wiseman at one point effectively accuses me of informality! Referring to my ordinary-language method of formalization and validation of a fortiori argument, Wiseman says: “Since he verbally explains the argument in an implicational form without symbols, I make a number of assumptions to symbolize them in propositional logic to show its possible deductive validity and certainty;” to which he adds in a footnote: “Sion also alludes to mathematical formalizations without showing them”. Here, Wiseman is confusing formalization and symbolization.
Personally, I see no profit in replacing “If P then Rp” (or in other words, “P implies Rp”) with “P → Rp” or with “p → r1”! As far as I can see, all the good it does is save space on paper or on computer screens. At the cerebral level, on the other hand, there is a costly slowing down of thought. Such symbolization puts the reader unnecessarily one or two steps behind, since every time he sees an arrow he has to mentally translate it into a verbal “if-then” (or “implies”) and likewise if the letters are changed, he must remember that the letters p and r1 mean the same as the previous letters P and Rp. Tell me, how does this visual and verbal reshuffle “show” the “possible deductive validity and certainty” of the given text?
Symbolic logic is a myth, which many people fall for – but I refuse to. Ordinary language is already symbolic enough; nothing much is gained and much is lost by re-symbolizing it. Such activity is at best trivial, and more likely harmful. Instead of shedding light and facilitating, it obscures and misleads. That is why I avoid excessive symbolization. If I were to further symbolize my findings, it would not add any credence to them. All I would succeed in doing is make my work seem more inscrutable, and therefore more impressive to vain people who confuse inscrutability with depth. My goal in writing logic is not to entertain people, however, but to teach them to think more accurately and more efficiently.
Concluding remarks. Wiseman’s goals in writing his dissertation were ambitious. His opening remark in his abstract is: “This study proposes to clarify the a fortiori argument’s components, structure, definitions, formulations, and logical status, as well as the specific conditions under which it is to be employed, both generally and in a Jewish context.” And further on, in his introduction, he states: “In sum, the overall purpose of the thesis is to advance the a fortiori’s place as an acceptable reasoning method;” or again, in his concluding remarks, “The purpose of this dissertation has been to determine the reasonableness of the a fortiori argument and its parallel Jewish QC.”
But, as we have seen in the present review, although he can be said to have advanced the cause of a fortiori argument, and this is a very commendable achievement, he did not demonstrate a thorough understanding of the nature of the argument, and therefore could not establish its rationality. Even though he was well aware of my formal definition and validation of a fortiori argument, he inexplicably opted for a vaguer definition that could not possibly be validated, since it made no mention of a ‘threshold’ for predication. He did so with two main motives in mind: to allow for “inductive a fortiori argument” and for “proportional a fortiori argument.” But though he tried, he did not manage to work out credible formal supports for these ideas.
Wiseman’s desire to expand and innovate in the field of a fortiori logic is praiseworthy; the problem is only that he was not equal to the task. We might here apply his own words: “Past and current good practices are preferable to poorer, new ones, especially where shortsighted tampering lacks sufficient prior knowledge or skill”.
Let me clearly state that my criticism of some of Wiseman’s views is not criticism of his general attitudes. Unlike certain other contemporary commentators we have looked at, he does not come across as posturing and pretentious, but exhibits sincere interest in getting at the truth without prejudice. Moreover, “inductive a fortiori” and “proportional a fortiori” are worthy ideas. It is only the inadequacies of his explanations of them that I have here criticized. These ideas have an important place in a fortiori logic, but not exactly the place he assigned them.
As regards the dayo principle, Wiseman was quite right in having sensed that this rule taken to extremes, as many commentators (and notably Maccoby) take it, is rationally untenable. And he was quite right in looking for ways and means to diminish its apparent scope. His arguments in this respect, that proportional a fortiori argument is often theoretically reasonable and often used in practice, are largely correct; the problem is only in accurately determining just how far to push back the dayo restriction. If Wiseman had studied the pertinent texts (Mishna, Gemara, etc.) more closely, he might well have attained a more precise (narrower) statement of the principle.
Not having done so, he misconstrues the intent of the Mishna rabbis who formulated the principle. He thinks their purpose was “to limit reason to prescribed bounds” and “curtail the range of possible opinions;” to “cajole, coerce, and control” out of “religious and political paternalism” and in defense of “orthodox Judaism”. All this is groundless interpolation on his part. The Mishna rabbis stuck to the revealed quantity (i.e. the penalty given in the Torah for a lesser offence) to avoid the risk of excessive extrapolation. It was not arrogance of power, but kindness and humble admission of their human limits that motivated them.
But anyway, Wiseman’s projections of dubious motives onto the rabbis are not of great moment. What matters is the bottom line, viz. precisely delimiting the legitimate application of the dayo principle. He does succeed in delimiting it somewhat, at least by declaring it to be a Judaic (even specifically Mishnaic) principle rather than (as Maccoby presents it) a logical one. But he does not attain the needed pinpoint precision, i.e. the reference specifically to penalties given in the Torah.
 Waterloo, Ont.: University of Waterloo, 2010. The whole book (249p. A4) is posted in the website of the University of Waterloo, in Ontario, Canada, at: uwspace.uwaterloo.ca/bitstream/10012/5038/1/A Contemporary Examination of the A Fortiori Argument Involving Jewish Traditions.pdf.
 Of course, Schwarz had long before devoted a whole book to qal vachomer argument (and another to gezerah shavah, for that matter), but I was not aware of his work till I came across Wiseman’s.
 Notably those of Lenartowicz and Koszteyn (2002), of Abraham, Gabbay and Schild (2009), of Goltzberg (2010), and of Schumann (2010).
 See pp. 2, 47, 52, 66, 69, 140.
 On p. 44 and p. 213, respectively. I marvel at the reference to a “heritable” property. Wiseman appears, on p. 92-3, to explain the concept with reference to mathematical recursion theory, naming Frege and Russell; but in the present context this is just a smokescreen. Do properties come with a label saying “I am heritable”?! How is this feature to be determined, exactly? He does not say. I view the use of this adjective as an attempt to give the impression that his definition formally accounts for deductive cases. It is a hidden circular argument, a sleight of hand. Surely, it is precisely the function of a fortiori argument to determine under what logical conditions properties are “heritable.”
 See pp. 37-40. Also, p.68, p. 130.
 On p. 64.
 On p. 68. On p. 40, he states that “the stronger can have or do what the weaker cannot or does not, because the stronger passes the weaker’s (current) maximum value or threshold.” But he does not take this momentary insight any further. He does not tell us precisely what the “maximum value or threshold” refers to. Instead, he offers as vague explanation an illustration: “human babies can have baby teeth, but they lack adult teeth.” He apparently means that babies have lesser means of survival than adults – until they get strong teeth, at least. But he does not clarify or develop the idea of a threshold.
 As Diagram 1.1, in chapter 1, in the section on Validation.
 On p. 83.
 This is putting the argument in positive subjectal form. In positive predicatal form, it would read: ‘If I am strong enough to lift a kilo with one finger, then I am strong enough to do so (or more) with my entire hand.”
 That is, only given that ‘Jill is (strong enough to be) beautiful’ could we infer from the major premise that ‘Jack is (strong enough to be) beautiful’.
 Pp. 81-7.
 P. 82, text and footnote.
 P. 64.
 Pp. 36-40. Most of the moods I mention here merge together two of his moods, for brevity’s sake.
 This doctrine is on pp. 113-4.
 Understanding the Talmud, p. 88.
 P. 108. He refers to “Pirkei Avot (Sayings of the Fathers),” but does not cite chapter and verse. I have looked for but not found this precise sentence. Perhaps he had in mind 1:6, where Joshua ben Perahyah says “Judge all men charitably” (dan et kol haadam lekhaf zekhut). If so, the meaning is not the same; this is obviously intended as a general ethical recommendation, and not as an injunction intended specifically for rabbinical legislators and judges. Wiseman again mentions “be lenient” on pp. 168, with some other citations.
 P. 47.
 P. 49.
 Footnote on p. 49.
 I mention all that, for my part, in my first book, Future Logic, in chapters 21, 46, 64, and elsewhere.
 In truth, he does once mention the induction by elimination of alternatives, on p. 150: “When two possible answers exist for an argument, in order to conclude a deduction validly and soundly, one must eliminate the other alternate. Similarly, for an induction, one must greatly diminish the alternate’s likelihood. If neither is accomplished, the disjunct remains.” He is saying this to criticize Maccoby for only considering the possibility of a non-proportional conclusion from a fortiori argument, without having first convincingly demonstrated the impossibility of a proportional conclusion. However, Wiseman does not make use of this principle anywhere else, and in particular not in his own theory of induction.
 See pp. 9, 15, 39, 50, 71, 146, 213, 221. Sometimes he has it in the plural, as “inductive forms” – in such cases he has in mind the “sub-forms” of inductive a fortiori argument (which I discuss further down).
 Indeed, he writes, on p. 146: “Inductive forms of the QC as analogical approximations are reasonable and, although tentative, can cover areas of nature, experience, and thought beyond the current reach of strict, formal logic” (my italics) – meaning that though he talks of “inductive forms” he does not see “strict, formal logic” as currently able to handle them.
 For the “a fortiori” formula, see p. 69. For the grades of sureness, see p. 59. Needless to say, I agree that we often say that we are sure of something when we well know that we are in fact not so sure; what I am contesting is that such subjective notions can be used as logical differentiae.
 In all fairness, I had not realized this clearly when I wrote my Judaic Logic; it is only in the course of the present, new study that I explicitly specified the negative aspect of suffective propositions.
 A similar example is found on p. 58: “Mountain A is greater than mountain B; B has trees; so most likely, A has trees.” Here, no reason is apparent why one should expect the size of a mountain to have anything to do with its tree cover. There is indeed some likelihood that any given mountain will have trees, since some mountains do; but there is no certainty it will, since some mountains don’t. The degree of likelihood depends on the relative proportion of those two possibilities, worldwide or in the geographical region under consideration. But anyway, it is not through an a fortiori argument that this conclusion is arrived at. It does not logically proceed from the given premises. Without the “enough” factor, no claim can be made to a fortiori argument.
 And of course it is this fact that makes possible the rabbinical dayo principle – which is an interdiction to generalize particular givens in certain contexts.
 P. 59.
 Elsewhere (in a footnote on p. 66) Wiseman states: “Some Rabbis attacked the premises’ truths, to reject the conclusion, which is an incorrect formal disproof procedure.” But to my mind, it is quite okay to attack the premises of an argument to at least put in doubt its conclusion. The conclusion is not thereby proved false (whether the argument is valid or invalid), unless the negation of the premises formally implies (as may happen) the negation of the conclusion. But the conclusion is indeed put in doubt, assuming (as is often the case) that those premises were its only known source of credibility at that stage (i.e. that there is no alternative route for proving it).
 It is amusing to note that in the very act of ignoring the suffective form of the minor premise and conclusion, here, Wiseman is expressing himself by means of a (negative) suffective, viz. “it is insufficient to grant certainty for the conclusion.”
 P. 71. These processes are detailed in pp. 51-2. Note that I often resort to paraphrase so as to keep the exposition short, because Wiseman often expresses his thoughts through examples.
 Wiseman elsewhere (pp. 57-8) shows he is well aware that there may be limits, going so far as to propose a “fallacy of going beyond a limit.” But here he forgets it, apparently; or maybe he considers that such forgetfulness is part of the “abductive” movement of thought.
 This would explain Pierce’s takeover of a word which originally (1666) meant “unlawful carrying away of a woman for marriage or intercourse” (Merriam-Webster’s Collegiate Dictionary); note the qualification “unlawful.” That is, maybe he conceived of abduction as kidnapping of information in support of some arbitrary theory. (I am of course engaging in “abduction” as I emit this guess!)
 According to Merriam-Webster’s Collegiate Dictionary, “to adduce” means “to offer as example, reason or proof in discussion or analysis;” and adduction refers to “the act or action of adducing.” These words are dated as being from the 14th or 15th century. My above definition differs in adding in the negative aspects, which are essential to the scientific approach. I suspect the reason some people nowadays prefer the newer word “abduction” is that it is more abstruse and therefore more impressive to pseudo-intellectuals.
 That Wiseman thinks that the terms abduction and adduction mean the same is suggested on p. 134, where he writes: “…like Sion’s majority vote or abductive decision (adductive as he calls it).” On the other hand, in a footnote on p. 150 he writes: “To cite the thing to which an argument or set of facts points is to adduce it. The process is an adduction. To take the stronger argument or set of facts is an abduction. Both are inductive in nature.” Neither of these views is correct.
 As it turned out, there was very little danger of that; but before having read Wiseman’s whole book I could not know it.
 Chapter 4.3.
 Even though a fortiori argument in general is, I would wager, more frequent (per block of discourse) in Judaism than elsewhere.
 P. 63, footnote. To which he adds, “although in some Jewish contexts (the Mishnah’s), equality is the norm.”
 P. 93.
 See pp. 91-3. Wiseman additionally treats a fortiori argument with “two particular premises” in an appendix (E.4) on pp. 245-6. What he has in mind is presumably quantification of a fortiori argument, judging by his comments on p. 21, where he compares it to syllogism. He there seems to think that only a fully universal a fortiori argument would be valid. I deal with this topic fully in an earlier chapter of the present work (3.2).
 Note moreover that although, given “All X are Y,” it does formally follow that “if this X (e.g. a smaller instance of X) is Y, then that X (e.g. a larger instance of X) is Y,” this merely means that if the universal “All X are Y” is true, then the particular “this X is Y” is true and the particular “that X is Y” cannot be false. The relation of implication from “this particular X is Y” to “that particular X is Y” must not be construed as independent of the universal given “All X are Y;” strictly speaking, that implication is no more than a conjunction. The illusion of independence makes the proposed inference essentially fallacious.
 As Wiseman earlier (on p. 67) explicitly admits, saying: “The relative sizes are superfluous facts once we know that all examples have the relevant characteristic (feature or judgement) anyway.” Wiseman, by the way, there wrongly interprets the inference from “All apples are fruit” to “little apples are fruit” or “big apples are fruit” as one from a universal (A) to a particular (I) proposition (the latter being “formed directly” from the former); in truth, the inference involved is first figure AAA syllogism: ‘Since all apples are fruit, and little (or big) apples are apples, then little (or big) apples are fruit’.
 As Wiseman later suggests in his earlier quoted summary, where he refers to an “interim minimum.”
 Wiseman does mention the radius earlier on (p. 67), but not here. And more important, he does not mention the c=2πr formula, i.e. that the proportionality in this case is established through geometrical theorems and not through his proposed a fortiori argument.
 This is also evident from his diagrams on p. 64, and his statements there that “we could reverse course” and “the situation can be reversed.” Applying ratios signifies pro rata argument.
 Even if he claims to “generalize” each conclusion from an arbitrary case to all cases, this is just conventional language and technique – he is really thereby intending deduction not induction. That is, his result is meant to be not less than 100% sure, on abstract grounds.
 P. 65.
 P. 69.
 See also Wiseman’s attempt to differentiate a fortiori argument from ordinary analogy, on p. 63.
 On the next page, Wiseman formulates the following example: “If A is a bigger cheat than B, and we hold B guilty, surely we also hold A guilty to some degree.” Here, the middle term (“cheat”) is indeed specified in the major premise, but it is still lacking in the minor premise and conclusion (which should specify that if B is enough of a cheat to be held guilty, then so is A).
 Note that since, as we have seen, Wiseman distinctively believes a fortiori inference can in principle proceed from major to minor as well as from minor to major, the ‘equal’ conclusion has to be for him not a minimum assumption but rather a balanced one (i.e. the middle ground between greater and lesser).
 P. 93. He proposes, on p. 65, a triad of “principles” that “operate in an a fortiori conclusion,” namely “the Precedent [Principle] (represented by the dayo), the Practical Revision [Principle] (when an adjustment is better), and Proportionality [Principle] (to scale the variation appropriately).” But these notions are too loosely defined to serve as effective guides, whether logical or heuristic, through which one might judge which way to opt in practice; they could only at best serve to classify one’s judgments ex post facto.
 In The Philosophy of the Talmud (London: Routledge Curzon, 2002.)
 P. 211.
 See pp. 106-7, 137-140.
 P. 108.
 P. 112.
 See pp. 111-3.
 See for instance p. 208, where he argues in favor of “a circumspect dayo principle applicable in the light of a normal, proportional justice, guided by a ‘Measure-for-Measure’ principle.”
 He does mentions “context” as one of the bases of gezerah shavah; but that is an error.
 Mainly in pp. 148,153-7.
 He quotes various other commentators who have regarded a fortiori argument as an argument by analogy of sorts: Maccoby, Samely, and Moshe Weiss (whose paper “The Gezera Shava and the Qal-VaChomer in the Explicit Discussions of Bet Shammai and Bet Hillel” I have unfortunately not been able to find). He could have mentioned me; I too see it as analogical. But the issue is not one of authority – it is a formal one.
 See footnote to p. 127. Note that the argument here is neither an a fortiori nor an argument by analogy. It is: Fathers are generally honored and I am their Father, therefore I would expect them to honor Me (syllogism) – yet they do not (contrary to rational expectation).
 P. 127. He is perhaps also influenced by Daube in this matter; see footnote to p. 133. See also p. 218, where Wiseman describes the dayo principle as “pegging the conclusion to the given, Rabbinic tradition.”
 P. 213. Notice his reference to “enduring moral truths” generally, and not specifically to Torah penalties for offenses.
 Certainly, they deserve criticism for being insufficiently clear about their intentions. Or at least, the compiler(s) of the Mishna left unsaid many things that it would have been wise to say out loud. It is very doubtful that the Sages were actually as laconic as they are made to appear in the written record.
 In the Mishna concerned, the Sages formulate their dayo objections in reaction to R. Tarfon’s attempts at proportional reasoning.
 He pays no heed to the interpretations proposed in chapter 6 of my Judaic Logic.
 P. 11.
 See pp. 173-194. Wiseman additionally mentions Esth. 7:4 and Isa. 49:15 as “implied QC”, and as an “abductive comparison” Gen. 29:19 – but, frankly, I see no intended a fortiori argument in these passages. He also suggest that Gen. 3:1-5, in view of its use of the keywords af-ki/lo, may be intended as qal vachomer. But I see no such velleity here either: the serpent’s argument seems to be (in paraphrase): since God did not say ‘you shall not eat of any tree of the garden’, then you may eat of this tree; to which Eve rightly replies ‘He said we may eat of all trees except this one’; thus, the serpent attempted a fallacious inference from ‘not all are forbidden’ to ‘this one is not forbidden’.
 See e.g. the first table in chapter 5. Wiseman acknowledges the source of his list in a footnote, saying “much of this is Sion’s.” My list actually has 31 cases; but he counts some sets of cases as single cases (namely, the 2 cases in 1 Kings 8:27 and 2 Chronicles 6:18, the 3 cases in Job, the 3 cases in Psalms 94, and the 2 cases in Jeremiah); also, he leaves out one case (namely, 1 Samuel 21:6) without any explanation – presumably just an oversight due to inattention, because he does not explain the lapse; and he adds on the case in Esther 9:12, which I had there intentionally left out as doubtful, even though it is traditionally counted as a fortiori.
 These figures are my own accounting, based on the contents of the cells in his table (“Diagram 9”). The totals given by Wiseman in the same table are 12-15 P, 7-10 D and 3 T, due to some cases being counted more than once. Moreover, in a comment below the table, he has the score as 12 P and 8 D. He gives no explanation for the inconsistencies.
 Additionally, Wiseman draws a distinction between arguments with “natural” content, and those which are “revelatory” in content. Results: of the 25 arguments, 13 (8 P, 3 D, and 2 T) are natural; and 12 (6 P, 5 D, and 1 T) are revelatory. This goes to show that the distribution (of P, D and T cases) is about the same in both areas.
 Namely, Genesis 4:24, 1 Samuel 14:29-30, 2 Samuel 12:18, and Esther 9:12.
 His reading of Gen. 4:24 is particularly confused.
 Of his 25 cases, 14 are in fact (by my readings) major-to-minor (6 P, 5 D, 3 T). More precisely, 10 are positive subjectal, 5 are negative subjectal, 9 are positive predicatal, and 1 is negative predicatal. Wiseman does not notice these formal differences, or reflect on their implications.
 I must confess that, in my earlier work Judaic Logic, I wrongly perceived the dayo restriction as intended as general to Judaism, although I considered such general restriction as unjustified and not always obeyed. For these reasons, in my analysis of Biblical a fortiori argument there, I remarked in the three cases I identified as proportional a fortiori argument: “Dayo ignored.” Wiseman may have been influenced by these errors of mine, although he took them farther by reading proportionality into many more arguments.
 Pp. 114-7. Wiseman also mentions (on pp. 119-125): the qal vachomer in Mishna Yadayim 4:7 (which I deal with in Appendix 2), as an example of how such arguments are “neutralized;” and the one in the name of Hillel in the Gemara Pessachim 66a (which I quote in the section on Gary Porton, further on), which is a baraita (i.e. claimed as Tannaic, but not given in a Mishna), as an example of how a gezerah shavah “comes to the aid of” a qal vachomer. And maybe others still. I will not however, for a lack of space, analyze and evaluate his every statement on this topic.
 Zevachim 9:1, according to Wiseman (I have not checked).
 On pp. 159, 170, 219.
 Which he variously calls: “ratios,” pp. 109, 150, 171; “proportional QC’s,” pp. 111, 210; “explicitly non-dayo QC’s,” p. 149; “scaled QC’s” p. 154; “degrees,” pp. 157, 168; “proportion,” p. 161; “QC‘s with degrees,” p. 170; “scaled examples,” p. 219.
 The argument is “reassessed” by them (p. 99); “the minority Mishnaic view (and the greater Amoraic latitude) differs from the majority one” (p. 99); the Amoraim “strengthened” the minority view (pp. 111, 149); “The Amoraic addition of proportional examples may well express their disagreement with the arbitrary dayo fiat” (p. 159); “the Amoraim sought a solution that brought harmony back” (p. 160); “One Rabbinic view can correct another” (p. 161).
 “Like the later Amoraim, we should opt for a better, QC solution that allows for any rational result: whether a degree, sameness, or a compromise” (p. 156); “All told, the Amoraic position is preferable, for a simple weakening of the strong dayo to a principle alongside proportion removes its arbitrary character” (pp. 160-1); “Because the Amoraim concluded some QC‘s as ratios, they understood that the dayo was not a Divine truth about all QC’s” (p. 161); they “saw the Mishnah’s dayo claim as applicable under specific conditions only, while their own view worked under others” (p. 170).
 P. 170. “Minority compliance may have been for reasons other than the dayo’s partial sensibleness: they submitted because tradition was important, majority decisions were preferable, unity in difficult times was critical, their keen protests were recognized, and they feared excommunication.” “What served the purposes of the Tannaim—preservation and transmission—was not the same as those of the Amoraim—classification and extension. So one expects that the freer Amoraim could systematize and generalize more than their predecessors, who needed to remember, record, and reiterate the tradition in times of persecution.”
 “Without stating why they [the Amoraim] shifted from the Tannaic dayo norm, they simply added scaled examples, to readjust the imbalance so that it correlated with the Tanach’s earlier range of proper QC’s” (p. 219).
 P. 196.
 In a footnote, he speculates that Miriam and Aaron had “personal and perhaps political motives.” There was “a tinge of jealousy” in their criticism of their “younger brother,” whom they had respectively rescued and helped in the past. They may have “wanted a more egalitarian, less hierarchal structure,” which could be “construed as an attack upon God‘s theocratic rule through Moses, desiring a wider form of participation, although not necessarily a call for democracy.” He is here confusing this story with that of Korach.
 P. 213.
 It is interesting to note in passing that this tradition, that Moses abstained from sexual relations with his wife to be ready for prophesy at all times, was apparently known to Philo of Alexandria, i.e. in the period of the Mishna. This is according to E. Starobinski-Safran, who cites “Mos. II, 68-69” (in a fn. on p. 169). I do not know whether earlier evidence of this tradition has been found; but it seems doubtful Philo was its author, as the rabbis do not ever, to my knowledge, refer to him for information or ideas.
 “And,” Wiseman adds, “even bully of her weak-willed brother Aaron.” In fact, though Miriam is indeed mentioned first, and as a gossip (it is she who found out about Moses’ abstinence from marital relations), she is nowhere implied to have bullied Aaron: the latter accusation is a bit of drama added on by Wiseman!
 I did not ask him for exact references, because this explanation seemed to me the likely one even before I asked. The writer of the Midrash obviously had the same thought, long before.
 Some rabbis (e.g. R. Shlomo Ganzfried in his Kitzur Shulchan Aroukh) say that listening to lashon hara is worse than speaking it; but in the present context we may take the more lenient view as applicable. In practice, it is not always possible to avoid hearing lashon hara, and one cannot always readily reprove the speaker.
 He says “their father,” but this is not in the text; and he says “Moses,” but this is not the usual reading.
 I was told that, according some Midrash, Aaron was indeed momentarily afflicted with “leprosy,” while Miriam’s “leprosy” disappeared soon after, as soon as Moses prayed for her. So the difference in their punishments was mainly in the seven days of imprisonment, which Aaron was spared but Miriam had to endure.
 Significantly, as already pointed out, Wiseman does not realize that he could equally well have argued the other way: ‘if Miriam’s sin was grave enough to merit leprosy and imprisonment, then Aaron’s sin was grave enough for the same penalties.’ This means that the major premise, which claims the sins of both to be equal in gravity, does not by itself tell us whether to argue from Aaron to Miriam or vice versa.
 P. 212.
 Wiseman on several occasions refers to Chaim Hirschensohn, author of Beirurei HaMidot (Jerusalem: Haivri Press, 1929). But his comments (at least those Wiseman mentions) seem to all relate to the conditions and limits traditionally set on a fortiori reasoning, so there is not much for me to say about him.
 Additional comments on Maccoby can be found in the section on Daube in a later chapter (31.5).
 Analogy 4.2 as Samely calls it (in his appendix, pp. 413-4). Wiseman, p. 129.
 Actually, Wiseman allows for three possible conclusions: “A (the same), A+ (more), or A- (less),” in accord with his very ‘open’ approach. I do not mention his third, out of acute embarrassment at the very thought! In any case, Samely surely does not intend this additional alternative.
 See my Appendix 2, Mishna Makkot 3:15 (b).
 Pp. 81-7.
 P. 12. Similarly, on p. 35, “as Sion observes, the Rabbis approach was more inductive than deductive.”
 Obviously, I do not mean that all moods are valid. I refer here to the valid moods of it, like the minor-to-major positive subjectal, and do not deny there are invalid moods, like the major-to-minor positive subjectal.
 Wiseman does make one error in his presentation, on p.86. He misquotes me as saying “Though P is R enough to be Q,” whereas I say: “Though P is more R than Q.” This is obviously just an error of momentary inattention. I immediately spotted it, but others may not – so Wiseman should fix it. All the more so, since he tries to construct an argument of his own on the basis of this wrong reading!
 Except where, on p. 84, Wiseman wrote that my method “does not fully capture the significance of the two related items…; instead it focuses on the mutually same feature.” I understand that this was a criticism designed to make room for a proportional conclusion. But in fact, the room was already there; as he well knew, since he refers to my openness to ‘proportionality’ on several occasions. For instance, on p. 94, “Sion allows either a proportion or the same amount as possible, although his method tends to favour the latter as truer.” My conclusion was formally correct for purely a fortiori argument, and did not exclude the possibility of a crescendo argument.
 P. 17.
 P. 215.
 Pp. 216-8.
 Fanciful too, in my opinion, is Wiseman’s assumption that the Amoraim were basically at odds with the Tannaim with respect to the dayo principle. E.g. on p. 137, where he says: “a trend is evident through time, with an increasing contrast between the Mishnah’s limit on reason and the Gemara’s greater latitude.”