A FORTIORI LOGIC
CHAPTER 32 – A fortiori in various lexicons
In the present chapter, we shall investigate the treatment (or non-treatment) of a fortiori argument in general – and of the more specifically Jewish concepts of qal vachomer and dayo – in various dictionaries and encyclopedias. This is not intended to be an exhaustive survey, but should give us a good idea of how this topic is regarded and how far it is understood by the academics, writers and editors, who produce the lexicons concerned.
The Jewish Encyclopedia (henceforth, JE) is an important English-language resource for information on Judaism and Jews, which was published in 1901-6 in New York. It can nowadays be consulted online. The article in it which interests us here is called “Talmud Hermeneutics” (in vol. 12, pp. 30-33); it was apparently written by Jacob Zallel Lauterbach (1873-1942), with a bibliography by Wilhelm Bacher. This article is wide-ranging, dealing with all sorts of interpretative techniques, as well as the more logical ones enshrined in the seven rules of Hillel and the later thirteen rules of R. Ishmael.
We shall not here, needless to say, review the whole article but only focus on what it says regarding a fortiori argument, i.e. which it refers to as Ḳal wa-ḥomer, the first rule in the lists of Hillel and R. Ishmael. JE tells us that “The completed argument is illustrated in ten examples given in Gen. R. xcii.” – however, this statement is not entirely accurate, because this set of ten examples is claimed in Genesis Rabbah 92:7 to be exhaustive; JE should have pointed out that there are a lot more than ten examples of such argument in the Tanakh (at least 46, according to more recent research).
JE identifies Ḳal wa-ḥomer as “the argument ‘a minori ad majus’ or ‘a majori ad minus’;” and states that “The full name of this rule should be ‘ḳal wa-ḥomer, ḥomer we-ḳal’ (simple and complex, complex and simple), since by it deductions are made from the simple to the complex or vice versa, according to the nature of the conclusion required.” This is of course an important observation regarding the two directions of a fortiori thought; but JE does not clarify when it first appears in Jewish texts. Note also its translation/interpretation of the terms ḳal and ḥomer as respectively “simple” and “complex;” this is not the most accurate rendering of their meanings. For a start, we are not told what is qualified as “simple” or “complex,” let alone what these expressions mean. This is not terminology used by the rabbis.
Note that JE gives no unified definition for a fortiori argument, i.e. it does not attempt to uncover what the two above forms of it have in common. Nor indeed what in fact distinguishes them (namely, that the first is positive whereas the second is negative). Moreover, JE makes no attempt to clarify how and why the a fortiori argument works, i.e. to formally describe and validate it. All it tells us regarding its structure is: “The major premise on which the argument is based is called ‘nadon’, or, at a later period, ‘melammed’ (that which teaches); the conclusion resulting from the argument is termed ‘ba min hadin’, or, later, ‘lamed’ (that which learns).” It is inaccurate to call the nadon the major premise; it is more accurate to refer to it as the minor premise.
Evidently, JE is not aware of the premise that compares the major and minor terms in relation to the middle term, which deserves the name of major premise. As we saw, it does not mention the major and minor terms as such (it mentions ḳal & ḥomer and simple & complex – but it does not say that these items are terms). Also, it does not even hint at the middle term, nor realize that it has to be present (setting a threshold) in the minor premise for the conclusion to be possible. The predication in common to the minor premise and conclusion is not at all mentioned either. Furthermore, JE shows no awareness of the differences between subjectal and predicatal forms of a fortiori argument, let alone of the differences between copulative and implicational forms.
All in all, then, this article provides little information on the nature of a fortiori reasoning. It does mention the distinction, found as rules 5 and 6 in the list of Eliezer ben Jose ha-Gelili, “between a course of reasoning carried to its logical conclusion in the Holy Scriptures themselves (‘ḳal wa-ḥomer meforash’) and one merely suggested there (‘ḳal wa-ḥomer satum’).” But this is a relatively unimportant insight, as it refers to the degree to which the Biblical text is explicit or implicit rather than to the form of a fortiori argument as such.
As regards the dayo principle, JE describes it as follows:
“The process of deduction in the ḳal wa-ḥomer is limited by the rule that the conclusion may contain nothing more than is found in the premise. This is the so-called ‘dayyo’ law, which many teachers, however, ignored. It is formulated thus: dayo lavo min hadin lihiot kanidon (‘The conclusion of an argument is satisfied when it is like the major premise’).”
This description, too, is inaccurate, in that it confuses the dayo principle with what I have called the principle of deduction, i.e. the logical rule applicable to all deductive argument that “the conclusion may contain nothing more than is found in the premise.” As I have shown, the dayo principle is something else entirely: it is an ethical limitation on the inference of greater penalties for greater crimes, or relatively stringent laws, from information given in the Torah regarding lesser penalties for lesser crimes, or relatively lenient laws. It constitutes self-restraint on the part of rabbis, so as to avoid the risk of excessive punishment, or severity of duties, through erroneous human interpretation of Divine law.
JE should have realized that the reason why “the ‘dayyo’ law” could be “ignored” by “many teachers” is precisely that it is not a principle of deduction, but more like one of induction. JE also shows unawareness of the difference between inductive and deductive inference when it informs us that: “The discovery of a fallacy in the process of deduction is called ‘teshubah’ (objection), or, in the terminology of the Amoraim, ‘pirka’. The possibility of such an objection is never wholly excluded, hence the deduction of the ḳal wa-ḥomer has no absolute certainty.”
In truth, most such objections in practice refer to the content, rather than to the form, of the a fortiori argument. The deduction involved in the argument, if properly formulated, is quite certain – that is what the term “deduction” (when applicable) means. The questions raised in objections are occasionally whether the argument has indeed been correctly formulated, but more commonly whether the information or perspective it relied on is indeed reliable. Thus, we see, here again, that the author of the JE article was not too clear about many issues concerning general logic, as well as some issues relating to Talmud.
Nevertheless, the article as a whole remains very interesting historically and doctrinally. It certainly, albeit brief, contains a lot of valuable information on the topics it treats.
The Encyclopaedia Judaica (henceforth, EJ) is a more recent English-language resource for information on Judaism and Jews, which was published in 1971-2 in Jerusalem and New York; over time, this was supplemented with several yearbooks and decennial volumes; a second edition, including many major revisions and updates, was published in 2006-7. The article we shall take a look at here is called “Hermeneutics,” and is found in the first edition (pp. 367-372). It is signed by Louis Jacobs (Britain, 1920-2006). This article was retained in the second edition, perhaps with some minor modifications (since it is there cosigned by David Derovan).
Although the EJ article expounds the thirteen rules of R. Ishmael and other interpretative techniques in some detail, we shall only here be concerned with its exposition of Kal va-ḥomer, i.e. of the first of the thirteen rules. Since we have already, earlier in the present volume, devoted a whole chapter (16) to the views of Louis Jacobs regarding a fortiori argument, we need not go into great detail in the present context. It will suffice for us to briefly comment on a number of points.
To begin with, EJ defines “Kal va-ḥomer” as “an argument from the minor premise (kal) to the major (ḥomer).” This is, of course, wrong – kal and ḥomer refer not to premises but to terms within the premises and conclusion! It is true that in the a minori ad majus form of the argument the kal (minor) term is in the minor premise; but the ḥomer (major) term here intended is, not in the major premise as EJ implies, but in the conclusion. It is also to be found in the major premise, which compares the two terms, but this is not the proposition that EJ is referring to here. So this is an error, if only of inattention.
Furthermore, EJ does not mention the major premise, nor therefore the presence in it of a middle term that performs the comparison that determines which term is the major and which is the minor. Nor does EJ here mention the presence of the middle term in the minor premise and conclusion, and its crucial role in them in establishing the quantitative threshold as of which predication is possible. The subsidiary term (the predication) is likewise not highlighted. Note also that, whereas JE explicitly acknowledges both the a minori ad majus or a majori ad minus moods of a fortiori argument, EJ only considers the former (implicitly, without naming it) and ignores the latter.
So, what we find in the EJ article on the whole is a very limited understanding of the nature and scope of a fortiori reasoning. All it gives us is a very rough sketch of the paradigmatic form of such argument. Not surprisingly, it fails to distinguish between positive and negative arguments, between subjectal and predicatal ones, and between copulative and implicational ones.
However, further on, when EJ rightly strongly rejects the identification by Schwarz between a fortiori argument and Aristotelian syllogism, it offers a somewhat better definition inspired by Kunst: “in the kal va-ḥomer it is not suggested that the ‘major’ belongs in the class of the ‘minor’ but that what is true of the ‘minor’ must be true of the ‘major’”. In this statement, the “major” and “minor” more clearly refer to subjects (or even antecedents), since the phrase “what is true of” them obviously refers to a predicate (or even a consequent). Thus, EJ here may be said to point to three of the four items of a fortiori argument. But the middle term (or eventually, thesis) is still missing, and this is of course a serious lacuna.
As regards use of a fortiori argument in Judaism, EJ lists the ten examples of it found in the Bible according to the Midrash (Gen. R. 92:7), and two more examples drawn from the Mishna (Sanh. 6:5) and Gemara (Ḥul. 24a). The article then proposes Jacobs’ own theoretical distinction between “simple” and “complex” qal vachomer (which the preceding two Talmudic examples are taken to illustrate, respectively). In the “simple” type, “the ‘major’ and ‘minor’ are readily apparent,” whereas in the “complex” type, “an extraneous element… has to be adduced to indicate which is the ‘minor’ and which is the ‘major’.”
The two types are symbolically represented as follows: “Simple: If A has x, then B certainly has x;” and “Complex: If A, which lacks y, has x, then B, which has y, certainly has x.” As I have argued at length in the chapter devoted to Louis Jacobs (16), the symbolic formulae used to define the proposed distinction are very superficial, since they fail to clarify why or how the consequents should logically follow from the antecedents. In truth, a major premise is required in either case, whether it is readily apparent or takes an effort of reflection to formulate. This means that the simple type is tacitly complex, since its terms A and B are known to differ in some respect (say, in degrees of y). Thus, the distinction is formally inadequate.
Nevertheless, the “complex” form proposed by EJ does have considerable value, in that it does effectively allude to the middle term and the major premise of a fortiori argument. Here, “y” plays this mediating role, although this is just a special case of a more general form. A more general statement would have been: ‘If A, which has less y, has x, then B, which has more y, certainly has x’. In that expanded formula, the major premise that ‘B has more y than A does’ is clearly implied. The case referred to by EJ is the special case where ‘less y’ is specifically ‘no y’, and ‘more y’ is specifically ‘some y’. We do often in practice come across a fortiori statements both of the general kind (with ‘less y’ and ‘more y’) and of the special kind (with ‘zero y’ and ‘more than zero y’). Thus, EJ cannot be said to have totally ignored the middle term and the major premise – it just did not clearly acknowledge them.
As regards “the principle of dayyo (‘it is sufficient’),” EJ presents it as “a qualification of the kal va-ḥomer,” according to which “the conclusion [can] advance only as far as the premise and not beyond it.” In symbolic terms, this means: “It must not be argued that if A has x, then B has x + y. The kal va-ḥomer suffices only to prove that B has x, and it is to go beyond the evidence to conclude that it also has y.” This definition, though nice and clear, is partly inaccurate. It is a good statement of the principle of deduction for purely a fortiori argument – but that it not what the dayo principle is really about.
EJ rightly refers the principle to the Mishna Baba Qama 2:5, where it is first formulated, but it wrongly interprets its discussion in the corresponding Gemara Baba Qama 25a. The latter does not exactly or merely teach that “R. Tarfon rejects the dayyo principle in certain instances.” Rather the Gemara, relying on a baraita (known as The Baraita of R. Ishmael), advocates precisely the view that qal vachomer yields a ‘proportional’ conclusion. This is evident from the Biblical example regarded as the prototype of qal vachomer, viz. Numbers 12:14-15.
In this example, according to the Gemara, the correct inference by qal vachomer would be a punishment of fourteen days incarceration (for Miriam, for speaking out of turn concerning her brother Moses), even though only seven days incarceration is mentioned in the Torah. The dayo principle then serves to reduce the concluding penalty from fourteen days to back to seven. Thus, the Gemara’s view is that qal vachomer is essentially a crescendo argument (rather than purely a fortiori argument as EJ implies it), and that the dayo principle diminishes its conclusion ex post facto (rather than denying that the ‘proportional’ conclusion can at all be drawn as EJ implies it). Thus, EJ does not represent the dayo principle as it is actually understood in Talmudic literature.
Evidently, EJ looks upon the dayo principle as a principle of formal logic, whereas it is more precisely put an ethical principle adopted by the rabbis to ensure their deliberations do not end up with excessively severe legal rulings. Moreover, the scope of the dayo principle is clearly not as wide as EJ has it, if we refer to its genesis in the said Mishna. It is not a general principle regarding any sort of conclusion (i.e. any A, B, x and y) – but more specifically a warning against inferring a harsher penalty for a greater crime from a Biblical text that imposes a certain (gentler) penalty for a specific (lesser) crime. Even though this warning might be extended somewhat to duties in general (instead of being applied only to penalties for crimes), it is by no means as general as EJ implies.
In conclusion, while the EJ article on hermeneutics is admittedly full of valuable information, it is seen on closer scrutiny to be open to considerable criticism. As the saying goes, “the devil is in the details.” Qal vachomer and the dayo principle are certainly more intricate topics than EJ makes them out to be. And I suggest, based on my past study of the hermeneutic principles in my Judaic Logic, that EJ treatment of the other hermeneutic rules can similarly be criticized as oversimplified and not entirely accurate. Of course, one should not expect too much from a brief article in an encyclopedia.
Still, to my mind, there was a failure of adequate research by editors of this important encyclopedia. It is shocking that in the first edition (1971-72) there is no mention of the Ramchal’s 1741 contribution to understanding of qal vachomer (he managed to describe the four main moods of the argument) and that in the second edition (2006-7) there is no mention of Avi Sion’s 1995 clarification of qal vachomer (which included full formalization and validation of such reasoning).
There is in EJ, 2nd ed. (vol. 9) an article on “Interpretation” in which, under the heading of “Analogical Interpretation,” can be found a brief explanation of qal vachomer (p. 819). This article was authored by Menachem Elon. The term analogical interpretation (midrash ha-mekish) refers to “the subject matter of the first three of the 13 middot enumerated by R. Ishmael.” The first of these is “kal va-ḥomer,” which refers to “a fortiori inference, a minori ad majus or a majori ad minus.”
“The basis of this middah,” we are told, “is found in Scripture itself (Gen. 44:8; Deut. 31:27) and the scholars enumerated ten pentateuchal kallin va-ḥomarim (Gen. R. 92:7).” This information is not quite correct. Though examples of qal vachomer occur in Scripture, it does not follow that they are its “basis;” a fortiori argument is rationally evident, and does not require revelation. Moreover, the ten examples given by the cited Midrash are not all “pentateuchal,” i.e. in the Five Books of Moses (Torah), but range across the whole Jewish Bible (Tanakh); and besides, there are in fact many more instances of the argument in the latter than the ten mentioned in the Midrash.
This article defines “the rule of kal va-ḥomer” as “a process of reasoning by analogy whereby an inference is drawn in both directions from one matter to another, when the two have a common premise – i.e., it can be drawn either from the minor to the major in order to apply the stringent aspect of the minor premise also (BM 95a), or from the major to the minor in order to apply the lighter aspect of the major premise to the minor premise (Beẓah 20b).” The two examples here mentioned are not quoted in the article, but I have looked them up (in the Soncino English ed.) so as to examine them.
The example in Baba Metzia 95a reads as follows: “You can reason a minori: if a paid bailee, who is not responsible for injury and death, is nevertheless liable for theft and loss; then a borrower, who is liable for the former, is surely liable for the latter too!” This argument is indeed from minor to major, being positive subjectal in form. Its major, minor, middle and subsidiary terms are, respectively, “a borrower” (P), “a paid bailee” (Q), “responsible for injury and death” (R, ranging from zero upwards), and “liable for theft and loss” (S). Note that what EJ refers to, here, as “the stringent aspect of the minor premise” is the subsidiary term, S; more precisely, this is a stringency applicable to (i.e. predicated of) the minor term, Q (which in this case happens to be located in the minor premise) which is passed on to the major term, P (located in the conclusion).
There are in Beẓah 20b two examples of kal va-ḥomer (they are there discussed and contested, but this need not concern us here). These are indeed arguments from major to minor, being negative subjectal in form. The first example reads: “If, when it is forbidden [to slaughter to provide food] for a layman, it is permitted [to slaughter] for the Most High, then where it is permitted on behalf of a layman, it is surely logical that it is permitted for the Most High.” Its major premise is that “slaughter for a layman (P) is more restricted (R) than slaughter for the Most High (Q),” since the former is forbidden when the latter is permitted (this involves a generalization, note, from certain cases to all cases). The minor premise and conclusion predicate that P and Q, respectively, are “restricted (R) not enough to be forbidden (S).” The second example reads: “If when thy hearth is closed, the hearth of the Master is open, how much the more must the hearth of thy Master be open when thy hearth is open.” It is very similar (and indeed is presented as “the same in another form”), except that here, P is “thy hearth,” Q is “the hearth of the Master,” R is “restricted,” and S is “closed.”
With regard to these arguments, EJ remarks that they go “from the major to the minor in order to apply the lighter aspect of the major premise to the minor premise.” This tells us that the author of this article interprets the expression ‘from major to minor’ as referring not to terms but to premises; i.e. in his perspective, the “major premise” is the proposition containing the major term and the “minor premise” is the one containing the minor term. But obviously this perspective is incorrect – for the proposition containing the minor term here is not a premise but the conclusion! So this is a misinterpretation of the said expression. This reveals that the author is not very well versed in logic.
As regards the dayo principle, the article states: “Material to this rule is the principle dayo la-ba min ha-din lihyot ka-niddon (Sifra, loc. cit.; BK 25a, etc.), i.e., it suffices when the inference drawn from the argument (ha-ba min ha-din) is equal in stringency to the premise from which it is derived (the niddon), but not more so, not even when it might be argued that logically the inference should be even more stringent than the premise from which it is derived.” This formulation is potentially interesting, in that the dayo restriction is placed in opposition to possible “logical inference.” Unfortunately, the author does not discuss the implications of such opposition. If it disagrees with logical inference, then dayo is not a logical principle; in which case, we have to suppose that it is based on other considerations, perhaps moral ones. Moreover, we must ponder: is the logical inference necessarily, or only contingently, “more stringent”? If the logical inference is necessarily a crescendo, how can it be occasionally ignored by a dayo principle? And if the logical inference is only occasionally a crescendo, under what conditions does this occur, precisely? The author does not ask these questions.
Despite of such mistakes and deficiencies, the EJ article on Interpretation is on the whole, of course, very informative.
The Encyclopedia Talmudit (henceforth, ET) is a Hebrew language encyclopedia whose purpose is to “summarize all the Talmudic halakhic issues and concepts, and all the opinions of halakhic scholars, from the completion of the Talmud to modern times, on every aspect of Jewish law”. This ambitious task began over 60 years ago, and is still far from finished today (though 29 volumes have been published so far, 50 more are on the way). An English translation, called the Encyclopedia Talmudica, is also being published over time.
I looked for but nowhere found the Hebrew ET volume containing an article on qal vachomer. I did however find the volume with an article on the dayo principle (vol. 7, 1990). It is this article that I will here comment on briefly. As would be expected from an encyclopedia devoted to halakhic thought in Judaism, it simply presents the main lines of the traditional view of the dayo principle relative to qal vachomer argument. There is no novel theoretical research in it, or criticism of existing doctrines and methods.
The dayo principle is stated by the Sages in the Mishna as: dayo lavo min hadin lihiot kanidon, which can be translated as: “it is quite sufficient that the law in respect of the thing inferred should be equivalent to that from which it is derived.” ET explains this rule as teaching, with regard to inferences made by means of a fortiori, that “one cannot lay down (lehatil) more in the conclusion (halamad) than there is in the premises (hamelamed).” This is a reasonable explanation, except that it seems here intended too generally; that is to say, it does not specify what is being “laid down,” whereas (in my view) it should say that this rule specifically concerns the inference of penalties or restrictive laws (as is true of the Mishnaic example where the principle is first formulated).
ET goes on to say that “this principle of dayo is from the Torah;” and it presents its source, just as the Gemara (Baba Qama 25a) does, as the episode of Miriam’s punishment (Numbers 12:14-15). According to the Gemara, even though the Torah only mentions seven days incarceration and makes no mention of fourteen days, the penalty of seven days for offending God was not directly inferred (by purely a fortiori argument) from the seven days for offending one’s father, but was inferred indirectly (by a crescendo argument and then by application of the dayo principle) from fourteen days for offending God. After which, ET merely discusses the basis of the “fourteen days” interpolation, rather than any other (e.g. infinitely larger) quantity.
But the truth is that this narrative is a fanciful retroactive projection by the Gemara (based on a baraita). The dayo principle historically first appears in the Mishna (Baba Qama 2:5) which is being commented on by the Gemara. Here, R. Tarfon tries to prove in two different ways that the penalty for damage by an ox on private property is full payment, and his colleagues the Sages reject his two proofs, saying both times: “dayo—it is enough.” The Gemara’s explanation of this dispute is quite contrived and far from credible, as I have shown earlier in the present volume. Moreover, the Gemara only takes into consideration the first argument of R. Tarfon in its narrative. ET does not show any awareness of the logical issues involved, but accepts the traditional treatment quite unthinkingly.
ET does thereafter mention the two arguments of R. Tarfon, but only in order to illustrate the two types of dayo application. This is of course an important distinction. The first type, called “at the beginning of the law” (al techelet hadin), is illustrated by the Sages’ dayo to R. Tarfon’s second argument; while the second type, called “at the end of the law” (al sof hadin), is illustrated by the Sages’ dayo to R. Tarfon’s first argument.
Both of his arguments attempt to prove that damage by horn on private property entails full compensation; and in both instances, the Sages reject his conclusion and limit the penalty to half. R. Tarfon first argues from the facts that damage by tooth & foot in the public domain entails zero compensation, while on private property it entails full compensation, and that damage by horn in the public domain entails half compensation. Then, seeing his argument rejected, he argues from the facts that damage in the public domain by tooth & foot entails zero compensation, while by horn it entails half compensation, and that on private property damage by tooth & foot entails full compensation. Both attempts, though significantly different, are blocked by the Sages using the exact same words: dayo lavo min hadin lihiot kanidon.
ET’s explanation of the two terms seems to be as follows. The dayo objection to the second argument is characterized as “at the beginning of the law,” because it is applied to the major premise (which comes first), while the dayo objection to the first argument is characterized as “at the end of the law,” because it is applied to the minor premise (which comes last). The “beginning” type of dayo could not be applied to the first argument, because there the major premise does not mention horn damage; the “end” type of dayo could not be applied to the second argument, because there the minor premise (concerning tooth & foot) and the conclusion (concerning horn) have the same law (viz. full payment). This explanation is essentially correct in my view, though I would say more precisely that as regards the major premise, it is the generalization that precedes its formation which the dayo blocks; and as regards the minor premise, it is rather the formation of the additional premise of ‘proportionality’ that the dayo blocks.
After this, ET develops in some detail the notion, found in the same Gemara, of “nullification” of an a fortiori argument. It presents various views regarding when such nullification is possible or even necessary, and how this affects application of the dayo principle. ET also details various limitations imposed on the dayo principle in later rabbinic discourse. All this is presented with apposite examples. But the logical issues underlying such manipulations of human discourse are never raised. Thus, ET treatment of the subject-matter tends to remain on a rather superficial level, a mere presentation of traditional doctrines without any attempt to question them and dig deeper.
This, as already said, was to be expected from the sort of publication that ET is intended to be. Even so, the article can be faulted for lacking chronological information, i.e. for not tracing the historical development of ideas relating to the qal vachomer argument and the dayo principle. Moreover, it fails to present certain ideas as clearly as it might and should have, preferring to faithfully reproduce the peculiar ways of expression found in rabbinic discourse.
It is not easy to briefly define a fortiori argument, in view of its complexity and variety. Ideally, one might put it as follows (I add in the usual symbols P, Q, R, S, to facilitate the reader’s comprehension, but they would of course not be included in a lexicon): argument from the sufficiency of something lesser (Q) in some respect (R) for some attribute or consequence (S), to the sufficiency of another thing (P) greater in the same respect (R) for the same attribute or consequence (S); or from the insufficiency of something (P) greater in some respect (R) for some attribute or consequence (S), to the insufficiency of another thing (Q) lesser in the same respect (R) for the same attribute or consequence (S); or from the sufficiency of something (S) in some respect (R) for some greater attribute or consequence (P), to the sufficiency of the same thing (S) in the same respect (R) for a lesser attribute or consequence (Q); or from the insufficiency of something (S) in some respect (R) for some lesser attribute or consequence (Q), to the insufficiency of the same thing (S) in the same respect (R) for some greater attribute or consequence (P).
This definition would thus include all four (or eight) primary moods of a fortiori argument, namely the positive and negative subjectal (or antecedental) and the positive and negative predicatal (or consequental). Note that the first and fourth clauses concern argument ‘from minor (Q) to major (P)’, while the second and third clauses concern argument ‘from major (P) to minor (Q)’. What is mentioned in each case is the minor premise and conclusion; there is no need to mention the major premise (‘P is/requires more R than Q is/does’) because it is implicit in the quantification of two items as ‘greater’ and ‘lesser’. This definition applies to purely a fortiori argument; for a crescendo argument, one would have to mention the underlying proportionality (the variation of S with R in subjectal/antecedental moods, or R with S in predicatal/consequental moods). In any case, in my view it is very important to mention the middle item (R) and the fact that it is sufficiency or insufficiency of it that makes the inference possible; to fail to mention these details would miss the essence of the matter.
I am not a lexicographer, but it is obvious enough to me that the above ‘ideal’ definition would not be appropriate for a popular dictionary. It is much too technical and long-winded. For such a publication, the purpose of the ‘definition’ is to give readers a ball-park idea of the intent of the phrase. It would therefore suffice to mention the positive subjectal mood, together with a simple example. It is the paradigmatic form, and probably the most used in practice, even if (as demonstrated empirically in the present work) the other moods are also frequently used. For more academic purposes, such as a legal or philosophical dictionary, or a full-scale encyclopedia, the above full definition would seem to me worth stating, possibly together with examples. Thus, when judging the definitions proposed by various lexicons, which is what we shall attempt to do in the present chapter, we have to be understanding and flexible, and to consider the purpose and readership of the document concerned. Similar considerations apply, of course, to related topics that may be treated in certain lexicons.
In the present section we shall focus on the description and (where applicable) explanation of a fortiori argument given in certain dictionaries and encyclopedias. This is not, by far, a study of all lexicons available in libraries (which work ought to be done eventually), but refers to some of the main ones, mostly among those that are present on the Internet.
Many dictionaries and encyclopedias do not even mention a fortiori argument; some display use of it, or mention it in passing within articles on other topics, but do not have a separate entry for it. Some lexicons, of course, do have an entry for ‘a fortiori’; among these, some just give synonyms for the term, while others attempt a more descriptive, and even sometimes explicative, approach. Many give examples of the argument – some of which are correct, but some of which are not really a fortiori in form.
We shall critically examine the matter in three stages, dealing first with ordinary dictionaries and encyclopedias, intended for the public at large, then respectively with dictionaries and encyclopedias designated as legal or philosophical, which are intended more specifically for college students, academics and other professionals in those fields. Note that there are many dictionaries that I have looked into, but found nothing of interest in – these I do not mention at all. I do however mention some encyclopedias that have no entry on the subject concerning us, since that is inexcusable.
Ordinary dictionaries and encyclopedias
The Collins Dictionary defines a fortiori as meaning “for similar but more convincing reasons.” This statement mentions “more,” the indicator of quantitative comparison; but it does not tell us what makes some “similar” reasons “more convincing”! The example given, “If Britain cannot afford a space programme, then, a fortiori, neither can India,” suggests that it is because India is even poorer than Britain that it likewise cannot afford a space programme. Notice that the middle term here (poor) is not epistemic but ontical, whereas the proposed definition rather refers to an epistemic condition (conviction). Still the example is valid, its form being positive subjectal (from minor to major).
The Oxford Dictionary states that a fortiori is “used to express a conclusion for which there is stronger evidence than for a previously accepted one.” This definition differs from the preceding in that it is more empirical or objective, referring to “stronger evidence” instead of to “more conviction;” moreover, it speaks of “conclusion,” suggesting a logical process. However, it is still essentially epistemic rather than ontical. Moreover, it suggests that a fortiori argument is inductive, rather than deductive. Indeed, it confuses a fortiori argument with induction, since it only refers to comparisons between conclusions, and does not refer to the less evident “conclusion” as the premise of the more evident one. Furthermore, the example given by this dictionary is: “They reject all absolute ideas of justice, and a fortiori the natural-law position.” It sounds more high-brow, but it lacks a middle term (which would explain why the natural-law position is ‘a fortiori’ compared to the absolute ideas of justice). One might even wonder if this example is not in fact syllogistic, rather than genuinely a fortiori! Assuming the argument is a fortiori, it would be predicatal in form, since the subject is the same in premise and conclusion; whether it would be positive or negative depends on whether the author conceives rejection of all absolute ideas of justice as having more or less of the unstated middle term than rejection of the natural-law position; so the example is far from clear, anyway.
The Merriam-Webster’s Collegiate Dictionary entry for a fortiori is: “with greater reason or more convincing force – used in drawing a conclusion that is inferred to be even more certain than another;” and it proposes as example: “the man of prejudice is, a fortiori, a man of limited mental vision.” Here again, it is not said why the inferred proposition is “even more certain” or “more convincing” than the other; this makes the comparison seem like a subjective decision. Moreover, the proposed example might in fact be syllogistic, instead of a truly fortiori; for the intended argument seems to be the apodosis: ‘if a man has prejudice then, since prejudice is a sign of limited mental vision, he must have limited vision’. A more truly a fortiori interpretation of the argument would be, say: ‘since prejudice is a more advanced ailment than limited mental vision, if a man is sick enough to be prejudiced then he must be sick enough to have limited mental vision’; this would be positive predicatal (from major to minor) a fortiori argument. Clearly, the proposed example is ambiguous.
Garner’s Modern American Usage proposes a terse definition of the phrase ‘a fortiori’: “by even greater force of logic; so much the more,” without clarifying what would in this context constitute “even greater force of logic.” However, it adds to that the following two interesting remarks: “the phrase is sometimes effective, but only if the intended readers are sure to get it;” and “the phrase is used illogically when the proposition following a fortiori is no stronger than the one preceding it.” Each remark is accompanied by two illustrations drawn from actual legal discourse, which are further explained. The author of this has obviously understood a fortiori argument more than many, even if he has not attempted a proper definition.
The Macmillan British Dictionary tells us that a fortiori is “used for saying that something that is true for one case is even more true in another case.” The New World Dictionary tells us that it is “said of a conclusion that follows with even greater logical necessity than another already accepted in the argument.” In both these cases, no further explanation or example is given. We are not told why the result is “even more true,” or why it has “even greater logical necessity.” Some dictionaries are even more laconic: the Encarta Dictionary has the pro-forma “for an even stronger reason;” the American Heritage Dictionary has “for a still stronger reason; all the more;” the Random House Dictionary: has the same, plus “even more certain.” As for the Encyclopedia Britannica, it only briefly mentions (without giving an example) a fortiori argument (in contrast to a pari and a contrario arguments) in an article on rhetoric, defining it as “arguing from an accepted conclusion to an even more evident one.” Such ‘definitions’ are obviously inadequate – they do not sufficiently circumscribe the term concerned for someone who does not know anything about it.
Note finally, in passing, the definition and illustration proposed by the Christian Apologetics & Research Ministry (CARM): “argumentum a fortiori is an argument based on stronger reason. It is when an argument is proposed where the more probable of possibilities are affirmed and a conclusion is arrived at. For example, “My wife likes diamonds so she will like any diamond I give her.” This definition is ambiguously worded: it is not clear whether it intends that the “conclusion arrived at” is, or is derived from, the “affirmed more probable possibilities.” In any case, whether the latter refers to a premise or to the conclusion, we are not told why some possibilities are more probable and worthy of being affirmed. Moreover, the example given is clearly not a fortiori argument, but mere syllogism.
Legal dictionaries and encyclopedias
Two online French-language websites have interesting definitions of a fortiori argument in legal contexts. The Faculty of Law of Université Paris Descartes has: “a fortiori interpretation consists in extending a rule to a hypothesis not foreseen by it;” for example: “if a regulation forbids the presence of dogs in accommodations, it will be admitted that the presence of bears is equally proscribed.” The legal dictionary of the Montreal law firm of Lecours-Hébert has: “Latin expression: with stronger reason… aims at cases for which the field of application of a legal norm concerns one or more situations or conventions that were not originally aimed at.” Of course, these definitions do not tell us just how the application of a rule may be enlarged, e.g. the example given would have been more enlightening if it was explained that bears cause more damage than dogs. Comparatively, the definitions given in English-language websites concerned with law are far less informative.
The dictionary of the Encyclopedia of Law Project states that a fortiori is “applied to the argument that, because of the concession or establishment of a given proposition, another included in it is by the greater reason true.” This definition is not very clear, and no example is proposed which might clarify it. It says that because some proposition is given, having already been conceded or established, another proposition “included in” it is “by greater reason” true. But this does not tell us what is meant here by these two phrases; a fortiori is not like syllogism about inclusion, nor even (despite its etymology) about comparison of reasons. The Law.com dictionary  states that the expression a fortiori “applies to a situation in which if one thing is true then it can be inferred that a second thing is even more certainly true,” and gives as example: “if Abel is too young to serve as administrator, then his younger brother Cain certainly is too young.” The example is clearly a fortiori argument (negative subjectal in form, from major to minor), and should have inspired a more precise definition than the vague one proposed.
Webster’s New World Law Dictionary proposes: “To draw an inference that when one proposition is true, then a second proposition must also be true, especially if the second is included in the first; and gives as example: “if a 19 year old is legally an adult, then a 20 year old is, too.” Note that this example is similar to the preceding (though positive subjectal in form, from minor to major); and indeed the definition is similar, except for the clause: “especially if the second is included in the first.” The latter clause is inappropriate here, since it suggests syllogistic rather than a fortiori reasoning. Cornell University Law School Legal Information Institute puts it more succinctly as: “If a particular fact is true, then one can infer that a second fact is also true.” Here, there is happily no added clause about inclusion; but there is no explanation or justification of the claim that “one can infer” – not even an example. How, then, is one expected to distinguish this form of reasoning from any other? A species definition, Aristotle taught, must contain both a genus and a differentia; all lexicographers should know that!
West’s Encyclopedia of American Law states, concerning a fortiori: “this phrase is used in logic to denote an argument to the effect that because one ascertained fact exists, therefore another which is included in it or analogous to it and is less improbable, unusual, or surprising must also exist;” no illustration is proposed. This definition of a fortiori argument looks a bit more sought out than usual, but upon further scrutiny it is obvious that its author did not grasp the reasoning involved. Although the clause “[is] analogous to it” could pass for a fortiori reasoning (since quantitative comparison is analogy of sorts), the clause “which is included in it” would only be appropriate for syllogistic reasoning. Also, in any case, the inference from something “ascertained” to another “less improbable” is not the essence of a fortiori argument.
Worth mentioning here is the definition of a fortiori given in a newspaper article on a particular law case: “In case of absence of a text on the case at hand, the search for the presumed intent of the legislator is required by deduction, including reasoning by analogy, that is to give the case at hand the same treatment given to a past case when they are similar and united in the same causation; or the reasoning a fortiori, that is to apply to the case terms considered in another case to have a stronger causality than the case at hand, in other words when the conditions for the case at hand are more suitable for applying the law than those stipulated by the legislator, which is a logical device that emphasizes causation of stronger cases for application to weaker ones.” The latter article is noteworthy for its attempt to differentiate a fortiori argument from mere analogical (or a pari) argument, with reference to “causation.” This clearly refers to “the conditions for… applying the law,” which may be more “suitable” in some cases than in others. This is, of course, one possible major premise for a fortiori argument. It does not constitute a general definition of such argument, but can be used as a general model for inference from a legal text.
Philosophical dictionaries and encyclopedias
I have already in the preceding chapter (31.4) analyzed the interesting definition of a fortiori argument proposed by Pierre André Lalande, perhaps in collaboration with other authors, in his 1926 opus, Vocabulaire technique et critique de la philosophie. So I will just repeat his definition in the present context, with a very brief comment. It went (my translation): “Inference from one quantity to another quantity of similar nature, larger or smaller, and such that the first cannot be reached or passed without the second being [reached or passed] also.” What needs to be noted here is that: this definition refers explicitly to quantitative comparison, and suggests both inference from minor to major and that from major to minor; also, it hints at the middle term, by specifying that the quantities are “of similar nature;” and it alludes to a threshold which needs to be “reached or passed” for the inference to occur. However, though very good, this definition has imperfections, notably the non-mention of the subsidiary term (i.e. that which is being inferred); and the failure to distinguish subjectal and predicatal arguments and draw attention to negative moods.
Another interesting definition from a French source is that of the Encyclopédie Philosophique Universelle (my translation): “A fortiori argument rests on the following schema: x is y, whereas relatively to the issue at hand z is more than x, therefore a fortiori z is y.” This definition is also very good, in that it comprises the major premise (z is, relatively to the issue at hand, more than x), the minor premise (x is y) and the conclusion (z is y) of a fortiori argument. However, it is deficient in not having a symbol (say, w) for the middle term, being content to call it “the issue at hand;” and for failing to mention this term in the minor premise and conclusion, as the threshold condition underlying the inference. That is to say, it should have had: z is more w than x is, and x is w enough to be y; therefore, z is w enough to be y. This is, of course, positive subjectal argument (from minor to major), i.e. the simplest form of a fortiori. Missing here are negative subjectal as well as positive and negative predicatal forms of it; also, implicational forms.
It is interesting to compare these last two definitions. The latter has some improvement over the former, in that it includes a subsidiary term (y) and clarifies things by introducing symbols. But the latter fails to mention important elements of the former, namely the possibility of argument from major to minor and also the fact that there is a threshold condition for the predications in minor premise and conclusion. Nevertheless, taken together, these French-language definitions are clearly valuable contributions to the field, and far superior to those given in English-language publications that I have looked into.
The online edition of the Oxford Dictionary of Philosophy simply defines a fortiori as a “phrase used for ‘all the more’ or ‘even more so’;” and it gives as example: “if all donkeys bray, then a fortiori all young donkeys bray.” This is definition merely by synonym; and the example offered is not correct, being syllogism rather than a fortiori argument. It is shocking that such a prestigious publication should display such poverty of thought. A Dictionary of Philosophy, which I have a copy of, does no better. It similarly defines a fortiori as “a phrase used to signify ‘all the more’ or ‘even more certain’;” and it gives as example another mere syllogism: “If all men are mortal, then a fortiori all Englishmen – who constitute a small class of all men – must also be mortal.” Another Dictionary of Philosophy, found online, just says, without giving an example: “A fortiori: a phrase signifying all the more; applied to something which must be admitted for a still stronger reason.”
Some Internet sites do somewhat better. The philosophical dictionary provided by Philosophy Pages explains the argument by saying: “we are bound to accept an a fortiori claim because of our prior acceptance of a weaker application of the same reasoning or truth;” and it gives as illustration: “Frank can’t run to the store in less than five minutes, and the restaurant is several blocks further away than the store. Thus, a fortiori, Frank can’t run to the restaurant in less than five minutes.” The Fallacy Files Glossary explains a fortiori as a “phrase is used when arguing that what is true of a given case because it possesses a certain attribute will certainly be true of another case which has more of the relevant attribute;” and proposes as an example: “Suppose that Tommy is Timmy’s older brother. We can argue that if Tommy is too young to see a certain movie, then a fortiori Timmy is too young, as well, since he is younger than Tommy.”
Both of the preceding examples are valid ‘from minor to major’ a fortiori arguments. But note that while the latter is positive subjectal, the former is negative predicatal. I doubt that the author of the Philosophy Pages entry realized that he was using a relatively unusual form of the argument; in any case, his explanation is not very accurate, being rather vague and epistemic. The Fallacy Files explanation is a lot better, because it is ontical and refers to a comparison between two cases, one having “a certain attribute” and the other “more of the relevant attribute.” The two “cases” are the minor and major terms; the “attribute” they have in common to different degrees is the middle term, and the phrases “what is true” and “will certainly be true” refer to the subsidiary term – so all four required terms are present in it. However, the idea of a threshold value of the middle term is not sufficiently highlighted, although suggested somewhat by the conjunction “because.” Also, of course, this definition refers to just one mood of a fortiori argument, and fails to mention the other three (or seven); moreover, it only refers to purely a fortiori argument, and fails to mention a crescendo argument.
It is sad to see that some reputed major encyclopedias of philosophy with an Internet presence do not have an entry for a fortiori argument, or do not at least mention it as a form of reasoning in articles on related subjects, even if the phrase is used in some articles; I here refer to: the Stanford Encyclopedia of Philosophy (SEP), the Internet Encyclopedia of Philosophy (IEP) and the Concise Routledge Encyclopedia of Philosophy. The Oxford Companion to Philosophy is likewise silent regarding a fortiori argument.
Wikipedia, the online “free encyclopedia that anyone can edit,” is very often a useful source of information, which I for one happily look into when I need a quick answer to a question. However, as can be expected from the fact that it is open to all volunteers irrespective of their real knowledge of what they are talking about, it is often enough inaccurate, and therefore should always be viewed with caution (double checking information in other venues).
A fortiori argument. A case in point is the Wikipedia article on a fortiori argument. Although this article tries to be more informative than those on the same topic in many dictionaries and encyclopedias, it is still inadequate. The article first defines the term as argument “from a stronger reason,” which merely literally translates its Latin name: a fortiori ratione. An example is then given: “if it has been established that a person is deceased (the stronger reason), then one can with equal or greater certainty argue that the person is not breathing. This example is valid a fortiori argument (positive predicatal in form, going from major to minor). There follows three sections, devoted to usage, meaning and prevailing circumstances of use.
As regards “usage,” we are told that “in the natural sciences and in social and other human sciences where statistics plays a large role, the phrase is used to mean ‘even more likely’ or ‘with even more certainty’.” This definition is incorrect: a fortiori argument is not inference of a more likely conclusion from a less likely premise; it is impossible through any sort of deduction, let alone of induction, to infer more certainty from less certainty. Moreover, this definition is inconsistent with the suggestion in the previous paragraph that the premise is to be called “the stronger reason.”
Furthermore, the example given here, viz. that if two or more phenomena are observed to be conjoined for a certain amount of time, they will be “a fortiori” be present as much or more of the time, since they might occur separately as well as in conjunction (my paraphrase), is not really an a fortiori argument but rather a syllogistic one, being about the inclusion of a smaller set in a possible larger one. Moreover, this example, being ‘from minor to major’ (i.e. from a smaller portion of time to a possibly larger one), differs from the previous example, and the author of the article fails to take stock of the difference out loud and modify his initial definition accordingly.
Further on, another definition is proposed: “In classical logic, ‘a fortiori’ is a signal indicating an attempt to justify an inferential step by claiming that the point being proven follows ‘from a[n even] stronger [claim]’ or has been stated ‘by means of [an even] stronger [assertion]’. That is, the phrase indicates that a) a proposition previously given or proven in the argument contains and implies a variety of ‘weaker’ or less contentful propositions and b) the proposition being proven is only one of the propositions contained and implied.” This definition is also wrong. It is more akin to the first than to the second, in that the argument is thought here to go from a ‘stronger’ claim to a ‘weaker’ one; but it is also in part comparable to the second definition, in that the argument is here thought to be about inclusion (note the idea that the premise ‘contains’ the conclusion).
In the section on “meaning,” the argument is presented as: “a proof in which one demonstrates a claim by invoking as proof an already proven, stronger claim.” And the example offered here is: “if it is forbidden to ride a bicycle with an extra passenger, it is also forbidden to ride a bike with fourteen extra passengers.” This example is a valid a fortiori argument, but not for the reason given. It is valid because riding with fourteen passengers is more dangerous than riding with only one passenger, and not because the prohibition of riding with one passenger is a “stronger claim.” The reasoning here is from minor to major, and not from major to minor. The article says a few more things under the said three headings, but I do not repeat them here or comment on them as they are not directly relevant to our subject.
The article also informs us that “there are two types of the a fortiori argument: a maiore ad minus: ‘from greater to smaller’ [and] a minore ad maius: ‘from smaller to greater’.” If we follow the hyperlinks on these expressions, we find their definitions to be as follows. The former “describes a simple and obvious inference from a claim about a stronger entity, greater quantity, or general class to one about a weaker entity, smaller quantity, or specific member of that class;” and the latter “denotes an inference from smaller to bigger.” The author believes that the former argument is “more universally known” and “also usually has a broader usage” and “is incomparably more general.” Moreover, whereas he offers examples of the former, he apparently cannot think of one for the latter.
Here again, there is some confusion with syllogistic reasoning, in the application of the term ‘from major to minor’ to arguments “from general to particular” (e.g. “what holds for all X also holds for one particular X”) or to those “from the whole to the part” (e.g. “if the law permits a testator to revoke the entirety of a bequest… then the law also permits a testator to revoke the portion of a bequest”). Traditionally, the term is only used for a fortiori argument, i.e. in relation to arguments “from stronger to weaker” (e.g. “if one may safely use a rope to tow a truck… one may also use it to tow a car”).
Thus, to conclude, the author (or is it authors?) of the Wikipedia article on a fortiori argument has some idea of a fortiori argument, but not a very clear idea. He does not realize that the argument has four main moods: the positive and negative subjectal and the positive and negative predicatal, and that both the first and fourth of these go ‘from minor to major’ and both the second and third of them go ‘from major to minor’. He does not mention the negative moods at all. He also does not mention the corresponding four implicational (antecedental or consequental) moods. He also does not raise the issue of proportionality (a crescendo argument).
Moreover, contrary to what the author imagines, the positive predicatal mood (which goes from major to minor) is not the most typical or the most used. If we look at actual usage statistics, we see that the positive subjectal mood (which goes from minor to major) is the most typical and the most used. In Plato’s works, there are apparently none of the former and at least 9 of the latter. In Aristotle’s work, there are 5 of the former and 50 of the latter. In the Tanakh (Jewish Bible), there are 15 of the former and 14 of the latter. In the Mishna, there is only one of the former and 32 of the latter. In the Christian Bible, there are 3 of the former and 28 of the latter. In the Koran, there is only one of the former and none of the latter. Need one say more? The author obviously did not research the matter.
It is interesting to note that in another page of Wikipedia, listing Latin phrases, a fortiori is described as: “often used to lead from a less certain proposition to a more evident corollary.” This of course refers to ‘from minor to major’ argument, demonstrating that another author has another opinion than the one above. Of course, this definition is also inadequate since it does not refer to argument ‘from major to minor’, and moreover since it wrongly considers the premise as “less certain” and the conclusion as “more evident.”
It should also be noted that the main Wikipedia article on a fortiori argument seems to be based on very little research. It only mentions two obscure references, namely: “Grabenhorst, Thomas K.: Das argumentum a fortiori (Verlag Peter Lang, 1990)” and “Schneider: Logik für Juristen, S. 158ff.” Note also that both these books are in German. I have not found further information on the second reference, but regarding the first, it is described as “a pilot study using the practice of decision justification.” If the author of the Wikipedia article truly based his understanding of a fortiori argument on these two books, then they cannot have been too informative on the subject!
Note additionally: I came across a book in Amazon.com, entitled A Fortiori Argument: Rhetoric, Truth-Value, Argument (paperback, 96 pages), edited by Lambert M. Timpledon, Miriam T. Marseken, Susan F. Surhone, and published in late 2010 by Betascript. This is billed as “high quality content in Wikipedia articles,” and upon further research I found that Betascript was a label created in 2010 that “specialize[s] in publishing and selling Wikipedia articles in printed form via print on demand.” I do not know how they managed to make a 96p book out of the brief article we have above reviewed; but in any case, based on this review, it is very doubtful that it has “high quality content”!
Kal va-chomer. Wikipedia has a brief definition of kal va-chomer, i.e. of a fortiori argument as used in Jewish literature, in its article on Halakha, as follows: “We find a similar stringency in a more lenient case; how more so should that stringency apply to our stricter case!” This is a correct rendition of rabbinic formulations of the principle; but missing here is the other half of such formulations, viz. if a leniency is applicable to a relatively stringent case, then it should also apply to a more lenient case.
However, the main discussion of kal va-chomer is to be found in the article on Talmudic Hermeneutics. This article is a copy-paste of the article on the same subject in the Jewish Encyclopedia (JE), which we have already reviewed in the first section of the present chapter – to which some examples and explanations have been added. We need only here highlight the JE statement: “the process of deduction in the kal wa-chomer is limited by the rule that the conclusion may contain nothing more than is found in the premise. This is the so-called ‘dayyo’ law, which many teachers, however, ignored.” “It is formulated thus… ‘The conclusion of an argument is satisfied when it is like the major premise’.” It should here be stressed that the author of the statement imposes his own interpretation, confusing the dayo (sufficiency) principle with the principle of deduction.
The Wikipedia article offers the following illustration for this principle: “If a parent will punish his or her child with a minor punishment should the latter return home with scuffed shoes, surely the parent will punish his or her child with a major punishment should the latter return home with scuffed shoes, ripped pants and a torn shirt.” And it comments: “This is an illogical deduction; although it might be a fair speculation, it cannot be proven with logic. All that can be proven is at least the result of the lesser offense.” All this is not quite correct. Although purely a fortiori argument logically demands an identical conclusion, a crescendo argument (which is a fortiori argument with an additional premise about proportionality) logically allows for a ‘proportional’ conclusion.
Moreover, a crescendo argument is frequent in the Tanakh (6 cases out of 46), in the Mishna (some 10 cases of 46) and in the Gemara (not yet counted, but note for a start the discussion in Baba Qama 25a-b). The point is that the dayo principle is not a logical rule, as both JE and Wikipedia wrongly assume it to be, but an ethical one, addressed to rabbis (legislators and judges) attempting to infer greater penalties for greater crimes, from lesser penalties for lesser crimes specified in the written Torah.
 See at: www.jewishencyclopedia.com/articles/14215-talmud-hermeneutics. The JE articles on Talmud www.jewishencyclopedia.com/articles/10739-methodology and on Bible Exegesis: www.jewishencyclopedia.com/articles/3263-bible-exegesis are also worth reading in this context.
 Two remarks in this article are worth noting in passing. First, that “neither Hillel, Ishmael, nor Eliezer ben Jose ha-Gelili sought to give a complete enumeration of the rules of interpretation current in his day.” Second, that “The antiquity of the rules can be determined only by the dates of the authorities who quote them; in general, they can not safely be declared older than the tanna to whom they are first ascribed. It is certain, however, that the seven middot of Hillel and the thirteen of Ishmael are earlier than the time of Hillel himself, who was the first to transmit them. At all events, he did not invent them, but merely collected them as current in his day, though he possibly amplified them. The Talmud itself gives no information concerning the origin of the middot, although the Geonim regarded them as Sinaitic…. This can be correct only if the expression means nothing more than ‘very old’, as is the case in many Talmudic passages. It is decidedly erroneous, however, to take this expression literally and to consider the middot as traditional from the time of Moses on Sinai.”
 Online at: www.jewishvirtuallibrary.org/jsource/judaica/ejud_0002_0009_0_08805.html. The only difference I have spotted in the section about qal vachomer in the 2nd ed. is the added statement: “Not all of the thirteen principles are based on logic as is the kal va-ḥomer. Some are purely literary tools, while the gezerah shavah is only valid if received through the transmission of a rabbinic tradition.” This statement is interesting in that it explicitly characterizes qal vachomer as “based on logic.”
 In a 1972 paper, “The Qal Va-Ḥomer Argument in the Old Testament” (Bulletin of the School of Oriental and African Studies, 35:221-227. Cambridge University Press), Jacobs writes more accurately: “The argument runs: if A is so then B must surely be so; if the ‘minor’ has this or that property then the ‘major’ must undoubtedly have it.”
 A. Schwarz in Hermeneutischer Syllogismus in der talmudischen Litteratur (1901); A. Kunst in Bulletin of the School of Oriental African Studies, 10 (1942), 976-91.
 Maybe the libraries I looked in did not buy that volume; or maybe it has not been published yet; or maybe it has not been written yet – I do not know.
 I do not know who its author(s) is (or are). I hired someone to translate it for me. Actually, he only translated the main text, and ignored the footnotes. He was a Hebrew speaker, but (to put it mildly) not very good at English.
 To my knowledge ET does not state precisely who first made the distinction; this information is historically important and must be sought. I suspect offhand the discovery was made by some Tosafist, though I have not found out just who and in what commentary precisely.
 Perhaps some of that may appear in the ET article on qal vachomer, if there already is or ever is one. I would be curious to see, in particular, if ET there mentions the important contribution of the Ramchal to the categorization and thence understanding of qal vachomer argument. See my essay on this topic in an earlier chapter of the present book (9.10).
 As of late May 2013.
 On CD (version 2.5), 2000.
 By Bryan A. Garner (New York: Oxford UP, 2009). See full entry at: books.google.ch/books?id=FwmQpyibKkAC&pg=PA28&dq=fortiori&hl=en&sa=X&ei=h1KjUabRIMmGhQfQnYGoAg&redir_esc=y#v=onepage&q=fortiori&f=false.
 At: www.macmillandictionary.com/dictionary/british/a-fortiori. The same definition is found at: www.merriam-webster.com/dictionary/a+fortiori.
 Webster’s New World College Dictionary (Cleveland, Ohio: Wiley, 2010).
 See: www.memidex.com/fortiori; www.yourdictionary.com/a-fortiori; dictionary.infoplease.com/a-fortiori. Note that mimidex.com gives as example: “if you are wrong then, a fortiori, so am I.” This example is truly no example; it gives no indication of the reasoning involved. Apparently, these people imagine that the mere use of the words ‘a fortiori’ is evidence that a fortiori argument is involved – this is far from true.
 At: www.britannica.com/EBchecked/topic/64/a-fortiori. Similarly in a CD edition (2004) that I have. The 1911 edition of this encyclopedia describes a fortiori argument as: “if A is proved less than B, and is known to be greater than C, it follows a fortiori that C is less than B without further proof.” See full statement at en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/A_Fortiori. This is of course not a fortiori argument, but simply the mathematical argument that if B > A and A > C, then B > C.
 At: www.droit.univ-paris5.fr/HTMLpages/recherche/griad/regle.html. My translation. The example given is valid positive subjectal (from minor to major) a fortiori argument. Note also the definitions given here for a pari argument (extending the rule to a hypothesis not foreseen by it but judged to be analogous) and for a contrario argument (limiting the rule to the foreseen hypothesis so as to infer the contrary solution in all other hypothesis). “Hypothesis” here, of course, refers to a situation or type of case.
 At: www.dailystar.com.lb/Law/Aug/13/Children-of-foreign-father-declared-Lebanese.ashx. Reference is there made to “Raymond Farhat, Introduction à l’étude du droit, Beryte ed., 62.” I detect in this article’s terms of reference an influence of Islamic hermeneutics, which is not surprising since it comes from Lebanon, even if not all Lebanese are Moslem.
 Paris: PUF, 1990. I did not look for the name of the author of this definition; but it is clear that he deserves to be acknowledged by name.
 London: Pan, 1975.
 At: www.fallacyfiles.org/glossary.html. See also the discussion of “if X is good then more X is better,” which is posted at: www.fallacyfiles.org/archive032009.html#03212009. This is really not an issue of a fortiori argument, but concerns pro rata argument. Given the major premise that ‘good (say, for Y)’ varies proportionally with ‘X’, then indeed it would follow that if for some value X1 of ‘X’ the corresponding value of ‘good for Y’ is Y1, then for a greater (or lesser) value X2 of X the corresponding value of ‘good for Y’ would be Y2 (the value of which to be calculated using the exact ratio between the terms). This is generally true; but a problem arises when the major premise is only true within a limited range of values. Then, of course, the reasoning becomes fallacious. Note also that if we did try to interpret “if X is good (or bad) then more X is better (or worse)” as an a fortiori argument, it would be positive predicatal in form, so that it would be fallacious to reason with it from minor to major. So the flaw inherent in “if X is good then more X is better” is formally explicable.
 Oxford: Oxford UP, 1995.
 At: en.wikipedia.org/wiki/Argumentum_a_fortiori. In my experience, when you try to correct errors in a Wikipedia, you may find your efforts erased; so, I never waste my time trying to teach anything in that forum. Note that the article here examined is that posted end May 2013; someone may of course make changes to it in the future. Hopefully, some changes will be made to it by someone, based on the comments in the present work.
 The statistics are developed in greater detail elsewhere in the present work. Note that I here lump positive antecedental cases together with positive subjectal cases. There are no positive consequental cases for the texts mentioned, to my knowledge so far.