Page 18 - A Fortiori Logic: Innovations, History and Assessments

12 A FORTIORI LOGIC

We can summarize all information about predicatal argument as follows: “Given that more or as much
R is required to be P than to be Q, it follows that: if S is R enough to be P, then S is R enough to be Q;
and if S is R not enough to be Q, then S is R not enough to be P; on the other hand, if S is R not
enough to be P, it does not follow that S is R not enough to be Q; and if S is R enough to be Q, it does
not follow that S is R enough to be P.” In this summary format, we resort to nesting: the major premise
serves as primary antecedent, and the valid minor premises and conclusions appear as consequent
conditions and outcomes, while the invalid moods are expressed as non-sequiturs.
For example: granted that it takes more strength (R) to lift 50 kilos (P) than 30 (Q): if someone (S) can
lift 50 kilos, then surely he can lift 30; and if he can’t lift 30, then he can’t lift 50. Needless to say, the
conditions are presumed identical in both cases; we are talking of the same handle, on the same day,
and so on. If different conditions are intended, the argument may not function correctly. The a fortiori
argument is stated categorically only if there are no underlying conditions. Obviously, if there are
conditions they ought to be specified, or at least we must ensure they are the same throughout the
argument.

Thus, to summarize, there are four valid moods of copulative a fortiori argument: two subjectal
moods, in which the major and minor terms (P and Q) are the logical subjects of the three propositions
concerned; and two predicatal moods, in which the major and minor terms (P and Q) are the logical
predicates of the three propositions concerned. The major premise is always positive, though it differs
in form in subjectal and predicatal arguments. In each of these types, there are two variants: in one, the
minor premise and conclusion are positive; and in the other, they are negative. The positive and
negative versions in each case are obviously closely related – the minor premise of the one is the
negation of the conclusion of the other, and vice versa; that is, each can be used as a reductio ad
absurdum for the other.
Note well the order in which the major and minor terms (P and Q) appear in the four moods: in the
subjectal moods they are subjects; and in the predicatal ones they are predicates. It follows that in the
two subjectal moods, the subsidiary term (S) is a predicate; and in the two predicatal moods, it (S) is a
subject. The middle term (R), however, is a predicate in both premises and the conclusion of all the
moods, note well. In subjectal moods it is a predicate of the major and minor terms (P and Q); in the
predicatal moods it is a predicate of unspecified subjects in the major premise and a predicate of the
subsidiary term (S) in the minor premise and conclusion, the subsidiary term being one instance of the
unspecified subject-matter of the major premise.
The difference between subjectal and predicatal moods is called a difference of structure. The
difference between positive and negative moods is called a difference of polarity. The difference
between moods that go “from minor to major” and those that go “from major to minor” is called a
difference of orientation. Sometimes this difference of direction is stated in Latin, as “a minori ad
3
majus” and “a majori ad minus” . Note that the “from” term may be the minor or major and occurs in
the minor premise; and the “to” term is accordingly the major or minor, respectively, and occurs in the
conclusion. Notice the variations in orientation in accord with the structure and polarity involved.
In sum, these four valid moods are effectively four distinct figures (and not merely moods) of a fortiori
argument, since the placement of their terms differs significantly in each case. This is clearly seen in
the following table:

Figure/mood +s –s +p –p
major premise PQR PQR RPQ RPQ
minor premise QRS PRS SRP SRQ
conclusion PRS QRS SRQ SRP
Table 1.1

3
I notice that that the Soncino Talmud does not apparently use the term a fortiori as a general term, but
distinguishes between a minori and a fortiori (instead of a majori). Maybe this was an error. In any case, in my
opinion, such usage should be avoided as it would leave us with no general term. The term a fortiori is needed as
a common label for all forms of the argument. Whereas a majori means from the major (term to the minor term),
a fortiori means with stronger (reason); so these expressions are not equivalent.