www.TheLogician.net © Avi Sion - all rights reserved
© Avi Sion
All rights reserved
© Avi Sion, 1995. All rights reserved.
4. QAL VACHOMER.
In the previous chapter, we considered the formal, deductive aspects of
a-fortiori argument. In the present chapter, we shall relate our findings to
past Jewish studies in this field, and also consider certain more inductive and
Jewish logic has long used and explicitly recognized a
form of argument called qal
vachomer (meaning, lenient and stringent). According to Genesis
Rabbah (92:7, Parashat Miqets), an authoritative Midrashic work, there are ten samples of
such of argument in the Tanakh: of which four occur in the Torah (which dates
from the 13th century BCE, remember, according to Jewish tradition), and another
six in the Nakh (which spreads over the next eight or so centuries). Countless
more exercises of qal vachomer
reasoning appear in the Talmud, usually signaled by use of the expression kol
sheken. Hillel and Rabbi Ishmael ben Elisha include this heading in their
respective lists of hermeneutic principles, and much has been written about it
English discourse, as we saw in the previous chapter, such arguments are called a-fortiori
(ratione, Latin; meaning,
with stronger reason) and are usually signaled by use of the expression all
the more. The existence of a Latin, and then English, terminology suggests
that Christian scholars, too, eventually found such argument worthy of study
(influenced no doubt by the Rabbinical precedent).
But what is rather interesting, is that modern secular treatises on formal logic
all but completely ignore it - which suggests that no decisive progress was ever
achieved in analyzing its precise morphology. Their understanding of a-fortiori
argument is still today very sketchy; they are far from the formal clarity of
for instance the example given in an otherwise quite decent Dictionary
of Philosophy: "If all men are mortal, then a
fortiori all Englishmen - who constitute a small class of all men - must
also be mortal". This is in fact not an example of a-fortiori argument, but
merely of syllogism,
showing that there is a misapprehension still today. Or again, consider the
following brief entry in the Encyclopedie
"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. It is not a logically valid argument, since it
depends not on the form but on the content (Ed.)". The skeptical evaluation made in this case is clearly
only due to their inability to apprehend the exact formalities; yet the key is
not far, concealed in the clause "relatively to the issue at hand".
Many dictionaries and encyclopaedias do not even mention a-fortiori.
Qal vachomer logic was
admittedly a hard nut to crack; it took me two or three weeks to break the code.
The way I did it, was to painstakingly analyze a dozen concrete Biblical and
Talmudic examples, trying out a great many symbolic representations, until I
discerned all the factors involved in them. It was not clear, at first, whether
all the arguments are structurally identical, or whether there are different
varieties. When a few of the forms became transparent, the rest followed by the
demands of symmetry. Validation procedures, formal limitations and derivative
arguments could then be analyzed with relatively little difficulty. Although
this work was largely independent and original, I am bound to recognize that it
was preceded by considerable contributions by past Jewish logicians, and in
celebration of this fact, illustrations given here will mainly be drawn from
formalities of a-fortiori logic are important, not only to people interested in
Talmudic logic, but to logicians in general; for the function of the discipline
of logic is to identify, study, and validate, all forms of human thought. And it
should be evident with little reflection that we commonly use reasoning of this
kind in our thinking and conversation; and indeed its essential message is well
known and very important to modern science.
seems obvious at the outset, is that a-fortiori logic is in some way concerned
with the quantitative and not merely
the qualitative description of phenomena. Aristotelean syllogism deals with
attributes of various kinds, without effective reference to their measures
or degrees; it serves to classify attributes in a hierarchy of species and
genera, but it does not place these attributes in any intrinsically numerical
relationships. The only "quantity" which concerns it, is the extrinsic
count of the instances to which a given relationship applies (which makes a
proposition general, singular or particular).
is very interesting, because - as is well known to students of the history of
science - modern science arose precisely through the growing awareness of
quantitative issues. Before the Renaissance, measurement played a relatively
minimal role in the physical sciences; things were observed (if at all) mainly
with regard to their qualitative similarities and differences. Things were, say,
classed as hot or cold, light or heavy, without much further precision. Modern
science introduced physical instruments and mathematical tools, which enabled a
more fine-tuned pursuit of knowledge in the physical realm.
argument may well constitute the formal bridge between these two methodological
approaches. Its existence in antiquity, certainly in Biblical and Talmudic
times, shows that quantitative analysis was not entirely absent from the thought
processes of the precursors of modern science. They may have been relatively
inaccurate in their measurements, their linguistic and logical equipment may
have been inferior to that provided by mathematical equations, but they surely
had some knowledge of quantitative issues.
In the way of a side note, I would like to here make some comments about
the history of logic. Historians of logic must in general distinguish between
several aspects of the issue.
(a) The art or practise of logic:
as an act of the human mind, an insight into the relations between things or
ideas, logic is part of the natural heritage of all human beings; it would be
impossible for us to perform most of our daily tasks or to make decisions
without some exercise of this conceptual power. I tend to believe that all forms
of reasoning are natural; but it is not inconceivable that anthropologists
demonstrate that such and such a form was more commonly practised in one culture
than any other,
or first appeared in a certain time and place, or was totally absent in a
(b) The theoretical awareness and teaching
of logic: at what point in history did human beings become self-conscious in
their use of reasoning, and began to at least orally pass on their thoughts on
the subject, is a moot question. Logic can be grasped and discussed in many
ways; and not only by the formal-symbolic method, and not only in writing. Also,
the question can be posed not only generally, but with regard to specific forms
of argument. The question is by definition hard for historians to answer, to the
extent that they can only rely on documentary evidence in forming judgements.
But orally transmitted traditions or ancient legends may provide acceptable
(c) The written science of Logic,
as we know it: the documentary evidence (his written works, which are still
almost totally extant) points to Aristotle (4th century BCE) as the first man
who thought to use symbols in place of terms, for the purpose of analyzing
various eductive and syllogistic arguments, involving the main forms of
categorical proposition. Since then, the scope of formal logic has of course
greatly broadened, thanks in large measure to Aristotle's admirable example, and
findings have been systematized in manifold ways.
Some historians of logic seem to equate the subject exclusively with its
third, most formal and literary, aspect (see, for instance, Windelband, or the Encyclopaedia
Britannica article on the subject). But, even with reference only to Greek
logic, this is a very limiting approach. Much use and discussion of logic
preceded the Aristotelean breakthrough, according to the reports of later
writers (including Aristotle). Thus, the Zeno paradoxes were a clear-minded use
of Paradoxical logic (though not a theory concerning it). Or again, Socrates'
discussions (reported by his student Plato) about the process of Definition may
be classed as logic theorizing, though not of a formal kind.
Note that granting a-fortiori argument to be a natural movement of
thought for human beings, and not a peculiarly Jewish phenomenon, it would not
surprise me if documentary evidence of its use were found in Greek literature
(which dates from the 5th century BCE) or its reported oral antecedents (since
the 8th century); but, so far as I know, Greek logicians - including Aristotle -
never developed a formal and systematic study of it.
dogma of the Jewish faith that the hermeneutic principles were part of the oral
traditions handed down to Moses at Sinai, together with the written Torah - is,
in this perspective, quite conceivable. We must keep in mind, first, that the
Torah is a complex document which could never be understood without the mental
exercise of some logical intuitions. Second, a people who over a thousand years
before the Greeks had a written language, could well also have conceived or been
given a set of logical guidelines, such as the hermeneutic principles. These
were not, admittedly, logic theories as formal as Aristotle's; but they were
still effective. They do not, it is true, appear to have been put in writing
until Talmudic times; but that does not definitely prove that they were not in
use and orally discussed long before.
regard to the suggestion by some historians that the Rabbinic interest in logic
was a result of a Greek cultural influence - one could equally argue the
reverse, that the Greeks were awakened to the issues of logic by the Jews. The
interactions of people always involve some give and take of information and
methods; the question is only who gave what to whom and who got what from whom.
The mere existence of a contact does not in itself answer that specific
question; it can only be answered with reference to a wider context.
case in point, which serves to illustrate and prove our contention of the
independence of Judaic logic, is precisely the qal
vachomer argument. The Torah provides documentary evidence that this form of
argument was at least used at the time
it was written, indeed two centuries earlier (when the story of Joseph and his
brothers, which it reports, took place). If we rely only on documentary
evidence, the written report in Talmudic literature, the conscious and explicit discussion of such form of argument must be dated to at least the
time of Hillel, and be regarded as a ground-breaking discovery. To my knowledge,
the present study is the first ever thorough analysis of qal
vachomer argument, using the Aristotelean method of symbolization of terms
(or theses). The identification of the varieties of the argument, and of the
significant differences between subjectal (or antecedental) and predicatal (or
consequental) forms of it, seems also to be novel.
Our first job was to formalize a-fortiori arguments, to try and express them in symbolic
terms, so as to abstract from their specific contents what it is that makes them
seem "logical" to us. We needed to show that there are legitimate
forms of such argument, which are not mere flourishes of rhetoric designed to
cunningly mislead, but whose function is to guide the person(s) they are
addressed to through genuinely inferential thought processes. This we have done
in the previous chapter.
With regard to Hebrew terminology. The major, minor and middle terms are
called: chomer (stringent), qal (lenient),
and, supposedly, emtsa'i
(intermediate). The general word for premise is nadon (that which legalizes; or melamed,
that which teaches), and the word for conclusion is din (the legalized; or lamed,
the taught). I do not know what the accepted differentiating names of the major
and minor premises are in this language; I would suggest the major premise be
called nadon gadol (great), and the minor premise nadon katan (small). Note
also the expressions michomer leqal (from
major to minor) and miqal lechomer
(from minor to major).
I have noticed that the expression "qal
vachomer" is sometimes used in a sense equivalent to "kol sheken" (all the more), and intended to refer to the minor
premise and conclusion, respectively, whatever the value of the terms that these
propositions involve (i.e. even if the former concerns the major term, and the
latter concerns the minor term), because the conclusion always appears more
'forceful' than the minor premise. This usage could be misleading, and is best
us now, with reference to cogent examples, check and see how widely applicable
our theory of the qal vachomer
argument is thus far, or whether perhaps there are new lessons to be learnt. I
will try and make the reasoning involved as transparent as possible, step by
step. The reader will see here the beauty and utility of the symbolic method
inaugurated by Aristotle.
a-fortiori arguments generally seem to consist of a minor premise and
conclusion; they are presented without a major premise. They are worded in
typically Jewish fashion, as a question: "this and that, how much more so
and so?" The question mark (which is of course absent in written Biblical
Hebrew, though presumably expressed in the tone of speech) here serves to signal
that no other conclusion than the one suggested could be drawn; the rhetorical
question is really "do you think that another conclusion could be drawn?
the absence of a major premise, it is well known and accepted in logic
theorizing that arguments are in practise not always fully explicit (meforash,
in Hebrew); either one of the premises and/or the conclusion may be left tacit (satum,
in Hebrew). This was known to Aristotle, and did not prevent him from developing
his theory of the syllogism. We naturally tend to suppress parts of our
discourse to avoid stating "the obvious" or making tiresome
repetitions; we consider that the context makes clear what we intend. Such
incomplete arguments, by the way, are known as enthymemes
(the word is of Greek origin).
missing major premise is, in effect, latent
in the given minor premise and conclusion; for, granting that they are intended
in the way of an argument, rather than merely a statement of fact combined with
an independent question, it is easy for any reasonably intelligent person to construct the missing major premise, if only subconsciously. If the
middle term is already explicit in the original text, this process is relatively
simple. In some cases, however, no middle term is immediately apparent, and we
must provide one (however intangible) which verifies the argument.
such case, we examine the given major and minor terms, and abstract from them a
concept, which seems to be their common factor. To constitute an appropriate
middle term, this underlying concept must be such that it provides a
quantitative continuum along which the major and minor terms may be placed.
Effectively, we syllogistically substitute two degrees of the postulated middle
term, for the received extreme terms. Note that a similar operation is sometimes
required, to standardize a subsidiary term which is somewhat disparate in the
original minor premise and conclusion.
are logically free to volunteer any credible middle term; in practise, we often
do not even bother to explicitly do so, but just take for granted that one
exists. Of course, this does not mean that the matter is entirely arbitrary. In
some cases, there may in fact be no appropriate middle term; in which case, the
argument is simply fallacious (since it lacks a major premise). But normally, no
valid middle term is explicitly provided, on the understanding that one is easy
to find - there may indeed be many obvious alternatives to choose from (and this
is what gives the selection process a certain liberty).
Let us begin our analysis with a Biblical sample of the simplest form of qal
vachomer, subjectal in structure and of positive polarity. It is the third
occurrence of the argument in the Chumash,
or Pentateuch (Numbers, 12:14). Gd
has just struck Miriam with a sort of leprosy for speaking against her brother,
Moses; the latter beseeches Gd to heal her; and Gd answers:
If her father had but spit in her face, should she not hide in shame
seven days? let her be shut up without the camp seven days, and after that she
shall be brought in again.
we reword the argument in standard form, and make explicit what seems to be
tacit, we obtain the following.
disapproval (here expressed by the punishment of leprosy)" (=P) is more
"serious disapproval" (=R) than "paternal disapproval (signified
by a spit in the face)" (=Q);
paternal disapproval (Q) is serious (R) enough to "cause one to be in
isolation (hide) in shame for seven days" (=S),
Divine disapproval (P) is serious (R) enough to "cause one to be in
isolation (be shut up) in shame for seven days" (=S).
that the middle term (seriousness of disapproval) was not explicit, but was
conceived as the common feature of the given minor term (father's spitting in
the face) and major term (Gd afflicting with leprosy). Concerning the subsidiary
term these propositions have in common, note that it is not exactly identical in
the two original sentences; we made it uniform by replacing the differentia
(hiding and being shut up) with their commonalty (being in isolation). More will
be said about the specification "for seven days" in the subsidiary
term (S), later.
A good Biblical sample of negative subjectal qal
vachomer is that in Exodus, 6:12
(it is the second in the Pentateuch). Gd tells Moses to go back to Pharaoh, and
demand the release of the children of Israel; Moses replies:
Behold, the children of Israel have not hearkened unto me; how then shall
Pharaoh hear me, who am of uncircumcised lips?
argument may be may be construed to have run as follows:
children of Israel (=P) "fear Gd" (=R) more than Pharaoh (=Q) does;
they (P) did not fear Gd (R) enough to hearken unto Moses (=S);
the more, Pharaoh (Q) will not fear Gd (R) enough to hear Moses (S).
again, we were only originally provided with a minor premise and conclusion; but
their structural significance (two subjects, a common predicate) and polarity
were immediately clear. The major premise, however, had to be constructed; we
used a middle term which seemed appropriate - "fear of Gd".
our choice of middle term. The interjection by Moses, "I am of
uncircumcised lips", which refers to his speech problem (he stuttered),
does not seem to be the intermediary we needed, for the simple reason that this
quality does not differ in degree in the two cases at hand (unless we consider
that Moses expected to stutter more with Pharaoh than he did with the children
of Israel). Moses' reference to a speech problem seems to be incidental - a
rather lame excuse, motivated by his characteristic humility - since we know
that his brother Aaron acted as his mouthpiece in such encounters.
any case, note in passing that the implicit intent of Moses' argument was to
dissuade Gd from sending him on a mission. Thus, an additional argument is
involved here, namely: "since Pharaoh will not hear me, there is no utility
in my going to him" - but this is not a qal
The first occurrence of qal vachomer in the Torah - and perhaps historically, in any extant
written document - is to be found in Genesis,
44:8 (it thus dates from the Patriarchal period, note). It is a positive
predicatal a-fortiori. Joseph's brothers are accused by his steward of stealing
a silver goblet, and they retort:
Behold, the money, which we found
in our sacks' mouths, we brought back unto thee out of the land of Canaan; how
then should we steal out of thy lord's house silver or gold?
to our theory, the argument ran as follows:
will agree to the general principle that more "honesty" (=R) is
required to return found money (=P) than to refrain from stealing a silver
yet, we (=S) were honest (R) enough to return found money (P);
you can be sure that we (S) were honest (R) enough to not-steal the silver
again, the middle term (honesty) was only implicit in the original text. The
major premise may be true because the amount of money involved was greater than
the value of the silver goblet, or because the money was found (and might
therefore be kept on the principle of "finders keepers") whereas the
goblet was stolen; or because the positive act of returning something is
superior to a mere restraint from stealing something.
There is no example of negative predicatal a-fortiori in the Torah; but I will
recast the argument in Deuteronomy, 31:27, so as to illustrate this form. The original
argument is in fact positive predicatal in form, and it is the fourth and last
example of qal vachomer in the
For I know thy rebellion, and thy stiff neck; behold, while I am yet
alive with you this day, ye have been rebellious against the Lrd; and how much
more after my death?
may reword it as follows, for our purpose:
"self-discipline" (=R) is required to obey Gd in the absence of His
emissary, Moses (=P), than in his presence (=Q);
children of Israel (=S) were not sufficiently self-disciplined (R) to obey Gd
during Moses' life (Q);
they (S) would surely lack the necessary self-discipline (R) after his death
this case, note, the middle term was effectively given in the text;
"self-discipline" is merely the contrary of disobedience, which is
implied by "stiff neck and rebelliousness". The constructed major
premise is common sense.
have thus illustrated all four moods of copulative qal
vachomer argument, with the four cases found in the Torah. For the record, I
will now briefly classify the six cases which according to the Midrash occur in
the other books of the Bible. The reader should look these up, and try and
construct a detailed version of each argument, in the way of an exercise. In
every case, the major premise is tacit, and must be made up.
I, 23:3. This is a positive antecedental.
12:5. This is a positive antecedental (in fact, there are two arguments with the
same thrust, here).
15:5. This is a negative subjectal.
11:31. This is a positive subjectal.
9:12. This is a positive antecedental (if at all an a-fortiori, see discussion
in a later chapter).
following is a quick and easy way to classify any Biblical example of qal vachomer:
What is the polarity of the given sentences? If they are positive, the
argument is a modus ponens; if
negative, the argument is a modus tollens.
Which of the sentences contains the major term, and which the minor term?
If the minor premise has the greater extreme and the conclusion has the lesser
extreme, the argument is a majori ad minus;
in the reverse case, it is a minori ad
Now, combine the answers to the two previous questions: if the argument
is positive and minor to major, or negative and major to minor, it is subjectal
or antecedental; if the argument is positive and major to minor, or negative and
minor to major, it is predicatal or consequental.
Lastly, decide by closer scrutiny, or trial and error, whether the
argument is specifically copulative or implicational. At this stage, one is
already constructing a major premise.
will here only give one example of the more complex, implicational form of qal
vachomer. It is described in the Encyclopaedia Judaica (8:367), as follows:
"It is stated in Deuteronomy 21:23 that the corpse of a criminal executed
by the court must not be left on the gibbet overnight, which R. Meir takes to
mean that Gd is distressed by the criminal's death. Hence, R. Meir argues:...
If Gd is troubled at the shedding of the blood of the ungodly, how much
more at the blood of the righteous!
is evidently a positive antecedental argument; verbalized more fully, it would
be stated as follows:
shedding of the blood of the righteous" (=P) is generally more
"troubling" (=R) than "the shedding of the blood of the
"the blood of the ungodly is shed" (Q), then Gd is to some extent
"troubled" (R), specifically to the extent of enacting the law in
Deuteronomy 21:23 (=S);
if "the blood of the righteous is shed" (P), then Gd is to some extent
"troubled" (R), to an extent not here specified but at least similar
to the previous (S).
middle term (trouble) is in this case given in the original text. It is not
expressed identically in the major premise (troubling) and the other
propositions (troubled), but this is a turn of language which is easily
remedied. The major premise could have been expressed more elliptically as
"P implies, for any subject, that he will be troubled, more than Q
implies". Note the absence of an explicit consequent (subsidiary thesis) in
the conclusion, and our use of the clause "to some extent"; more will
we said about this later.
will have occasion to discuss other examples of implicational qal
vachomer, drawn from the Gemara, in a later section.
Rabbinical logicians raised an important question in
relation to certain qal vachomer
arguments. For instance, in the argument about Miriam (which we analyzed in the
previous section), the minor premise posits a punishment of seven days for a
relatively lesser crime, and the conclusion likewise posits a punishment of
seven days for a relatively greater crime. Why only seven days? they wondered;
should not the punishment be more, proportionately
to the severity of the crime? A reasonable question.
the sample argument is of Divine origin, some Rabbis postulated that it suggests
a universal logical rule, namely that the conclusion of a qal
vachomer can never go further than the minor premise, in the specification
of the measure or degree of the terms involved.
They called this, the dayo (sufficiency)
principle (see Baba Qama, 2:5). Other
Rabbis, like R. Tarphon (in Baba Qama,
25a), did not concur, but regarded a proportionate inference as permissible, at
least in some cases. For my part, I would like to say the following.
the argument concerning Miriam, it can easily be countered that Gd sentenced her
in the conclusion to only seven days incarceration out of sheer mercy, though
she might have been strictly-speaking subject to infinitely more; and that in
any case, the seven days mentioned in the minor premise are not known through
natural human insight, but equally through Divine fiat. Thus, this example does
not by itself resolve the issue incontrovertibly.
we compare, for instance, the argument made by R. Meir (also previously
mentioned), we see that going beyond the given quantity is intuitively quite
reasonable. Here, the minor premise is that Gd is (to some unspecified degree)
troubled by the blood of a criminal, but the intended conclusion is that He is
troubled even more (to a greater, though also unspecified degree) by the blood
of an innocent. It has to be so, because the concrete expression of the distress
of Gd, in the first case, is that the court must remove the criminal's corpse
before nightfall; the implied obligation, in the second case, cannot be the
same, since the court would not execute an innocent - it is rather a general
prescription that good people be treated still better than bad people.
however, that in both our examples, the quantitative factor at issue may be made
to stand somewhat outside the regular terms of the a-fortiori argument as such.
In both cases, 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.
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.
we may acknowledge the dayo principle as correct, provided it is understood as being a
minimal position. It does not insist on the quantitative equality of the
subsidiary or middle term (as the case may be) in the conclusion and minor
premise, nor does it interdict an inequality; it merely leaves the matter open
for further research. A-fortiori argument per
se does not answer the question; it is from a formal point of view as
compatible with equality as with inequality. To answer the question, additional
information and other arguments must be sought. This is a reasonable solution.
speaking, what is needed ideally is some mathematical formula which captures the
concomitant variation between a term
external to the a-fortiori argument as such (e.g. amount of punishment), and a
term of variable value implicit in the a-fortiori (e.g. severity of the sin).
This formula then stands as the major premise in a distinct argument, whose
minor premise and conclusion contain the indefinite term at issue in the
a-fortiori argument (the middle or subsidiary term, as the case may be, to
repeat) as their common subject, and the said external term's values as their
is no guarantee, note well, that the variation in the major premise will be an
arithmetical proportionality; it could just as well be an inverse
proportionality or a much more complex mathematical relationship, even one
involving other variables. This is why the a-fortiori argument as such cannot
predict the result; its premises lack the information required for a more
refined conclusion. In some cases, the concomitance is simple and well known,
and for this reason seems to be an integral part of the a-fortiori; but this is
an illusion, the proof being that it does not always work, and in more complex
cases a separate judgement must be made.
us now analyze the issue underlying the dayo
principle in more formal terms. Consider a positive subjectal a-fortiori, whose
subsidiary term (S) is a conjunction of two factors, a constant (say, K) and a
variable (say, V); and suppose V is a function (f)
of the middle term (R), i.e. that V = f(R)
in mathematical language. On a superficial level, the argument is simply as
is more R than Q,
Q is R enough to be S;
P is R enough to be S.
"R enough" is a threshold, it is not a fixed quantity. In the case of
the minor premise, involving Q, the value of R is Rq, say; whereas, in the case
of the conclusion, involving P, the value of R is Rp, say; and we know from the
major premise that Rp is greater than Rq. Looking now at S, it is evident that
if it consists only of a constant (K), it will be identical in the minor premise
and the conclusion. But, if S involves a variable V, where V is a function of R,
then S is not necessarily exactly the same in both propositions. If V = f(R)
represents a straightforward linear relationship, then Vp = f(Rp)
will predictably be proportionately greater than Vq = f(Rq);
but if V = f(R) represents a more
complicated relationship, then Vp = f(Rp)
may be more or less than Vq = f(Rq),
or equal to it, depending on the specifics of the formula.
comments can be made with regard to the other valid moods of qal
vachomer. Note in any case that all this is well and good in principle; but in practise, we may not be able to provide an appropriate and
accurate mathematical equation. Some phenomena are difficult and even impossible
to measure; we may know that they somehow vary, but we may have no instruments
with which to determine the variations, precisely or at all.
The physical sciences acquired enormous prestige, because they
concentrated their efforts on accessible phenomena (at least until the advent of
sub-atomic physics, where according to the Heisenberg Principle precise
determinations become in principle impossible, in view of the influence of
available experimental means on the matter observed). Measurement is currently
more difficult in biological or psychological contexts. In the still more
abstract realm of ethical and legal discussions, not to mention purely spiritual
issues, objective means are well-nigh non-existent, and we have to refer to
Biblical hints or intuitive conventions to establish scales.
Nevertheless, if that is a consolation, what is of interest to us here is
the essential similarity in principle
- with regard to the formal logic involved - between all human endeavors in the
pursuit of knowledge. The proverbial superiority of modern physical sciences, in
view of their powers of measurement, is relative and incidental. Their
epistemological tools are no different than those of any other discipline. Other
disciplines may be equally "scientific", in the root sense of the
word, which refers to knowledge acquired through the strictest methodology; they
are not totally incapacitated by the strictures which their peculiar
subject-matter imposes on them.
The subject-matter of physics is relatively easy of access, so that it
can measure more and achieve greater precision; other domains are progressively
more difficult to deal with, and so the (in the widest sense) scientific
endeavors which concern them are bound to be accordingly limited. But the
requirements of objectivity of attitude, open-mindedness to new data,
carefulness in reasoning, and honesty, are the same throughout; and this is what
counts in evaluating any body of knowledge.
The formalization of a-fortiori argument has been found
difficult by past logicians for various reasons. (a) The complexity and variety
of the propositional forms involved. (b) There are many varieties of the
argument. (c) Known samples are usually incompletely formulated. (d) Known
samples often intertwine a mixture of purely a-fortiori and other forms of
deductive inference. (e) The deductive and inductive issues were not adequately
separated. We will clarify these matters in the present section.
far, our goal has been to discover the essential form(s) of a-fortiori argument.
We found the various kinds of premises and conclusion which ideally constitute
such movements of thought. As in all formal logic, the conclusion follows
from the premises; if the premises are true, then the conclusion is true.
The presentation of a form of argument as valid does not in itself guarantee the
truth of the premises. If any or all of the premises are not true, then the
conclusion does not follow; the conclusion may happen to be false too, or it may
be true for other reasons, but it is in any case a non sequitur.
understanding of the relationship of premises and conclusion is not a special
dispensation granted to our theory of a-fortiori, but applies equally well to
all inference, be it eductive, syllogistic or otherwise deductive, or even
inductive. In all cases, the question arises: how
are the premises themselves known? And the answer is always: by any of
the means legitimatized by the science of logic. A premise may be derived from
experience by inductive arguments of various kinds, or be a logical axiom in the
sense that their contradictories are self-denying, or even be Divinely revealed;
or it may be deductively inferred in one way or another from such relatively
primary propositions (whether they are a posteriori or a priori,
to use the language of philosophers).
issue has been acknowledged in the literature on Talmudic logic, through the
doctrine of objection
(in Hebrew, teshuvah;
in Aramaic, pirka). A given a-fortiori argument, indeed any argument, may be
criticized on formal grounds, if it is shown not to constitute a valid mood of
reasoning. But it may also be objected to on material grounds, by demonstrating
one or both of its premises is/are wholly or partly false, or at least open to
serious doubt. The deduction as such may be valid, but its inductive backing (in
the widest sense) may be open to doubt.
for examples the Biblical samples of qal
vachomer we have used as our illustrations.
the argument concerning Miriam, we were given two sentences, neither of which is
in itself obvious. Assuming that the Biblical verse as a whole is indeed
intended as an argument, and not as two unrelated assertions, we may regard the
first as a Divinely guaranteed truth and use it as our minor premise, but the
second must somehow emerge as a conclusion. However, the major premise, which we
ourselves construct to complete the argument, is in principle not indubitable.
The one we postulated happens to seem reasonable (i.e. appears to be consistent
with the rest of our knowledge); but it is conceivable that some objection could
eventually be raised concerning it (say, that Gd attaches more importance to
sins against parents than to sins against Himself).
the next argument, by Moses, the major and minor premises are both known by
empirical means. The former is a generalization, based on the past behavior
patterns of the children of Israel and Pharaoh; and the latter is a statement
concerning more recent events. These propositions happen to be true, so that the
conclusion is justified, but they might conceivably have been factually
inaccurate, in which case an objection could have been raised.
argument made by Joseph's brothers is much more open to debate. The steward
might have argued that they returned the money they found out of some motive(s)
other than the sheer compulsion of their honest natures: (a) to liberate their
brother Simeon, which had been kept hostage (see Genesis, 42:24 and 43:23); or
(b) because the famine in Canaan forced them to come back to Egypt (see 43:1);
or even (c) because they feared eventual pursuit and retaliation; or simply (d)
because the silver cup, being a tool for divining purposes, had more value than
the sacks full of money, and thus tempted them to take more risks.
accept the brothers' argument, because we believe that their honesty proceeded
from their exceptional fear of Gd (irrespective of any more down to earth
concerns), but it is not unassailable. Clearly, the empirical foundations of the
major premise are rather complex, and an additional complication is the rather
abstract psycho-ethical concept (namely, honesty) it involves. With regard to
the minor premise, about the restitution of money - that was a straightforward
observation of a singular physical event. In any case, this example well
illustrates the inductive issues which may underlie an a-fortiori argument.
the case of the argument by Moses concerning the stiff-neck and rebellion of the
children of Israel, the major premise might be construed as a generalization
from common experience. We know that children are less well behaved in the
presence of their parents or school-teachers than in their absence, and
similarly that people follow their leaders more strictly when their leaders'
backs are not turned - and on this basis, the postulated major premise seems
reasonable. But it might well be argued that though this is more often than not
true, it is not always true (the children of Israel are indeed requested by
Moses to make it untrue!) - and thus put the whole argument in doubt, or at
least make it probable rather than necessary. As for the minor premise, it could
be viewed as an overly severe evaluation of the behavior of the children of
Israel - there is a subjective aspect to it.
comments can be made with respect to Rabbi Meir's argument, demonstrating its
possible weaknesses. We need not belabor the matter further. All this goes to
prove, not as some logicians have
claimed that a-fortiori argument is in principle without formal validity, but
that it is often difficult to find solid material grounds for its effective
exercise. It is thus understandable why Rabbinical legislators have usually
regarded qal vachomer arguments as
insufficient in themselves to justify a law,
unless supported by the authority of tradition.
would like now, in the way of a final illustration and test of our theory, to
analyze an a-fortiori argument given in the Encyclopaedia Judaica.
It is drawn from the Talmud (Chulin,
24a), which bases the argument on certain passages of the Torah (Leviticus,
21:16-21; and Numbers, 8:24-25). The argument seems complicated, but it is
simply, as we shall see, positive antecedental in form; I quote:
If priests who are not disqualified for service in the Temple by age, are
disqualified by bodily blemishes; then Levites, who are disqualified by age,
should certainly be disqualified by bodily blemishes.
clue to a solution is in the verb involved; we notice that the central issue under discussion
is the threshold of disqualification from Temple service. Our middle term (R),
then, must be a concept with many different degrees (say,
"unfitness"), such that there is a cut-off point along it, which
signifies the occurrence of disqualification; this is effectively the subsidiary
term (S), which will be the consequent of our minor premise and conclusion.
bodily blemishes" (=P) implies more "unfitness for Temple
service" (=R) than "being past a certain age" (=Q);
a Levite reaches that age (Q), he is sufficiently unfit (R) that "he is
all the more, if a Levite has bodily blemishes (P), he is sufficiently unfit (R)
to be disqualified (S).
that the antecedents of the minor premise and conclusion, respectively, contain
the minor and major terms, which cause the requisite degree of unfitness for
disqualification. We see that this argument is identical in form to that of R.
Meir, which we previously analyzed. What distinguishes it, however, is the way
we construct the major premise. In the R. Meir argument, no explicit source is
given for the major premise; but in the present example, we do have some
additional data with which to justify our major premise.
a-fortiori argument as such makes no mention of the priests; it only concerns the
Levites. The logical utility of the statements in the original text about
priests, is to serve as a springboard from which we can leap to the needed major
premise. The two propositions "priests are not disqualified by old
age" and "priests are disqualified by bodily blemishes", provide
us with the Scriptural grounds for an inductive generalization to the
proposition "Bodily blemishes more easily disqualify than old age",
which in turn becomes our major premise.
argument by analogy is involved, when we move from the case of priests, to all
cases (all Temple servers), including eventually the case of Levites. This
argument is not formally unassailable; the Torah might well have made a fine
distinction, and allowed Levites with bodily blemishes to serve in the Temple
(in view of their distinctive functions there). Two subjects can always have
opposite predicates, without doing violence to logic. However, since
the Torah does not in fact make such a distinction, we may reasonably
generalize as the Rabbis did.
to summarize, not all of the Talmudic passage under discussion constituted an
a-fortiori argument. The first section, concerning priests, was
not an inherent part of the qal vachomer inference per se, but served as the
premise of a preliminary inductive argument (namely, a generalization) which
established the major premise of the qal
vachomer as such. Only the second section, about Levites, belongs within the
qal vachomer process proper.
In this context, I would like to criticize and reject the theory of qal
vachomer arguments proposed by the author, L. Jacobs (presumably), of the
aforementioned Encyclopaedia Judaica article. He rightly (together with Kunst)
dismisses the claim by some researchers (notably, A. Schwartz), that they may be
identified with syllogistic reasoning; for the latter serve only the eventual
purposes of subsumption of individuals in classes, or classes in
classes-of-classes. However, Jacobs' own analysis of the topic is also faulty.
Jacobs' effort at formalization is not only an inadequate
oversimplification, but also contrary to reason. He claims that the (above
mentioned) argument of R. Meir (which he labels "simple") can be
formalized as "if A has x, then certainly B has x"; but this explains
nothing, it does not tell us why the inference is at all possible, because it is
too vague. Similarly, he formalizes the argument about the priests and Levites
(which he contrasts as "complex") as follows: "if A, which lacks
y, has x, then B, which has y, certainly has x"; but this is absurd! The
arrow is pointing in opposite directions in the antecedents (in one case against
y, in the other case towards y), and then it flips over and points in the same
direction in the consequents (toward x)!
Clearly, more precise formal tools, more careful logic and more
perspicacious linguistic analyses, were needed to solve the mystery of qal
vachomer. I believe that the theory I have proposed offers a definitive
An important test of our general forms of qal
vachomer, is their applicability to the
formulation of a-fortiori argument traditionally made in the Rabbinic literature.
Some logicians, like R. Luzatto (also known as the Ramchal), have a pretty large
concept of qal vachomer, which
includes any kind of scale of comparison as the effective middle term.
However, most authors seem to limit their concept to one specific kind of middle
term, namely the concept of 'legal restriction'. Thus, for instance, R. Chavel
describes the argument as follows:
A form of reasoning by which a certain stricture applying to a minor
matter is established as applying all the more to a major matter. Conversely, if
a certain leniency applies to a major matter, it must apply all the more to the
is even clearer, as the following quotation shows. (Note that we are effectively
dealing with a scale of modality, and with nesting of modalities within
Any stringent ruling with regard to the lenient issue
must be true of the stringent issue as well;
any lenient ruling regarding the stringent issue must
be true with regard to the lenient matter as well.
special formulations are easily assimilated by our general theory of qal
vachomer argument, as follows:
P generally implies more 'stringency for the practitioner' (=R) than Q
nonetheless, Q is stringent (R) enough to imply 'the practitioner subject
to a certain restriction (or not-subject to a certain liberty)' (=S),
all the more, P is stringent (R) enough for this same ruling to apply
P generally implies more 'stringency for the practitioner' (=R) than Q
nonetheless, P is not stringent
(R) enough to imply 'the practitioner subject to a certain restriction (or
not-subject to a certain liberty)' (=S),
all the more, Q is not
stringent (R) enough for this same ruling to apply (S).
that both arguments are antecedental in form, and one is expressed positively
and the other negatively. The extreme theses (P, Q) are legal rulings; their
middle thesis (R) is the magnitude of burden, let us say,
they impose on a practitioner, and their subsidiary thesis (S) is a third legal
clause, itself evaluated as burdensome to a certain degree. If the smaller
burden (Rq) includes the subsidiary (Rs), then so does the larger (Rp); and by
contraposition, if the greater burden excludes the subsidiary, then so does the
lesser. Note, for the sake of symmetry, that we could conceive of similar
formulas in which the middle thesis (R) is 'leniency for the practitioner',
provided the subsidiary thesis (S) likewise changes in polarity, becoming 'the
practitioner is subject to a certain liberty (or not-subject to a certain
formulas may be objected to, firstly, on the ground of their limited concept:
they are conceived specifically in relation to the severity or laxity of ethical
propositions (legal rulings, in Rabbinical terminology),
whereas a-fortiori is a much wider process, applicable to non-ethical
propositions. Secondly, and more radically, these formulas involve a middle
thesis ('burdensomeness', say) too vague
and diffuse to enable a sure conclusion: the major premise must be general,
and such generality can only be known by generalization or enumeration. If by
generalization, the conclusion is at best probable; if by enumeration, we are
begging the question (i.e. we had to know the desired conclusion beforehand).
a law P may be burdensome in many respects and another law Q may be burdensome
in many respects, and P may well be burdensome in numerically more respects than
Q is burdensome; even so, the burdens of P may or may not include all the burdens of
Q, and indeed the burdens of P and Q may not overlap at all! In other words, in
principle (i.e. formally), the inference is not necessary without further
specifications which somehow guarantee that the burdens of P include all those
of Q. That is, the laws under discussion here, P and Q, have certain implicit
material relations which must be brought out into the open.
the above mentioned Rabbinical formulations of a-fortiori argument, are not only
limited in scope (to ethical theses), but they cannot be considered as having
formal validity (i.e. invariably guarantee inference). They are at best broad
guidelines, which may occasionally be found inapplicable. Indeed, the Rabbis
were aware of this problem, and did occasionally object to attempted such
inferences by one of their colleagues, and claim that a stringency of Q did not
necessarily apply to P or a leniency of P did not necessarily apply to Q.
Effectively, they invalidated the major premise, denying it to be general and
making it at best probable, by apposition of an acknowledged exception; and by
this means, they inhibited application of qal
vachomer reasoning to S, the new case under consideration.
There are already, in the Christian Bible, examples of a-fortiori,
some of which are analyzed by H. Maccoby in The
Mythmaker: Paul and the Invention of Christianity. The author mentions
Paul's fondness for the argument, but shows him to have lacked knowledge of
the 'dayo principle' (see further on), concluding that his use of the
form was more akin to the rhetoric of Hellenistic Stoic preachers (pp.
It could be said that there is an a-fortiori movement of thought
inherent in syllogism, inasmuch as we pass from a larger quantity (all) to a
lesser (some). But in syllogism, the transition is made possible by means of
the relatively incidental extension of the middle term, whereas, as we have
seen, in a-fortiori proper, it is the range of values inherent to the middle
term which make it possible.
Vol. 1, p. 51, my translation.
I must report that near the end of writing this book, I uncovered a
much better definition of a-fortiori argument by Lalande, in the Vocabulaire
technique et critique de la philosophie. He writes (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]
also." Note, however, that this definition fails to specify that the
positive movement from large to small is predicatal, while that from small
to large is subjectal; and it ignores negative moods altogether, as well as
differences between copulative and implicational forms. Lalande adds that
the argument is of legal origin, quoting the Latin rule "Non debet, cui plus licet, quod minus est non licere" (p. 32).
I have an impression, for instance, that modern French discourse
involves more use of a-fortiori than modern English discourse. To what
extent that is true, and why it should be so, I cannot venture to say.
It is interesting to note in any case, that Josephus Flavius claims
that a disciple of Aristotle, called Clearchus, wrote a book, which is no
longer extant, in which he reports a meeting between Aristotle and a Jew,
during which presumably ideas were exchanged. What ideas were exchanged, and
whether this story is fact or legend, I do not know (see Bentwich).
I wish to make an acknowledgement at this stage. My special interest
in a-fortiori argument was aroused back in 1990 by a Vancouver, B.C.,
lawyer, Mr. Daniel Goldsmith. I had written an article on "Jewish
Logic" which was gradually published in a local Jewish paper called
"World of Chabad". One
reader, Mr. Goldsmith, wrote to me suggesting that I pay special attention
to a-fortiori argument, as a form of reasoning which was particularly Jewish
and which had not so far received much formal treatment. I resolved at the
time to follow this suggestion, and the present essays on the subject are
The principle is stated as din
leba min hadin lihiot benadon. Note that J.E.
translates this as "the conclusion of an argument is satisfied when
it is like the major premise"; but what they mean by 'major premise' is
what we here, more precisely, name 'minor premise'.
We can now mention the Rabbinical theory that the dayo
principle has subdivisions, and may be applied either 'to the lenient
case' or 'to the strict case'. This idea was based on a limited
comprehension of a-fortiori argument, due precisely to the Rabbis' failure
to distinguish between the deductive process itself and its inductive
precedents. An example provided by Bergman (p. 128) is the following
"If shen and regel,
which cause no liability when committed on public property, nevertheless
give rise to full liability if committed on the injured party's property,
then keren, for which there is [half]
liability if it occurred on public property, certainly should cause full
liability if committed on the property of the injured party" (B.Q.,
25a): on the basis of the dayo
principle, one cannot so argue, he says, but must conclude on half liability, for keren
on private property, i.e. no more than keren
invokes on public property. I agree with his conclusion, but for other
reasons: we have 3 givens: (1)
relatively unintentional damages by animals (shen
and regel) in the public domain
imply owner's liability to pay none of the damages; whereas, (2) the same on
private property imply his liability to pay all of the damages; and (3)
relatively intentional damages by animals (keren)
in the public domain imply liability to pay half of the damages. The first
two givens serve to induce the major premise of our actual
a-fortiori: make the comparison 'for unintentional acts, acts committed on
private property imply more liability than those committed in the public
domain', then generalize to 'for all acts, the same', then educe the new
particular 'for intentional acts, the same'. This result, combined with the
third given, which serves as minor premise, form the a-fortiori argument
proper, whose conclusion is 'intentional acts on private property imply
liability to pay half the damages'. The application of the dayo
principle involved in this last stage (subsidiary term: 'having to pay half
damages') is perfectly regular, and requires no special new division, note
We might also mention a description proposed by Maccoby, "if
something is known about one thing which has a certain quality in relatively
'light' form, then it must be true 'all the more so' of some other thing
that has the same quality in a relatively 'heavy' form". This
description is incomplete in various ways, but at least does not limit
itself to legal issues.
P. 27, n. 106.
The indefinition and apparent subjectivity of the concepts of
'lenient' and 'stringent' (or synonyms to the same effect) is important to
note. They seem to refer to subjective/emotional reactions to laws; i.e.
whether a law is felt by people as a further hardship or as a release from
duty. If we suppose more formal definitions (see ch. 13), and regard every
law - positive or negative, i.e. an imperative or a prohibition - as
"stringent", and every absence of law - i.e. ethical contingency,
permission and exemption - as a "leniency", then we must be very
careful in this context, as modal logic is involved, which has special
syllogistic behaviour-patterns (notably, one cannot draw a conclusion from a
first-figure major premise which is not positively or negatively necessary).
This matter requires further study, in relation to Rabbinical formulations
of a-fortiori argument concerning "leniency".
Perhaps a word should be said about this difference in terminology.
In philosophy, a distinction is made between ethical and legal propositions
as follows. Ethics is the broader concept, which includes law. Law refers
specifically to ethical propositions enforced by society (or the ruling
segment thereof); some ethical propositions are not considered so
enforceable (though supposedly inferred from nature). The Rabbis are clearly
aware of this functional distinction, but tend to regard all ethical
propositions as "laws", because they view them as ultimately
enforced by Gd if not by society.