Page 22 - A Fortiori Logic: Innovations, History and Assessments

16 A FORTIORI LOGIC

(or is not) a consequent of P and Q; while in consequental argument, R is (or is not) a consequent of S.
Do not be misled by the fact that R is placed to the left of P and Q in the major premise of
consequental a fortiori arguments. The thesis R does not there play the role of antecedent of P and Q
(i.e. it does not imply them). The theses P, Q and R are there all consequents of some unstated
antecedents; and thesis S is a specified instance of such unstated antecedent (in the positive case) or
not so (in the negative case).
Variation of the middle thesis. Concerning the middle thesis R, the sense in which it is quantitatively
variable (i.e. that more or less of it can be implied) needs to be clarified. A proposition as such does
not have degrees; so it would be incorrect to imagine that the proposition R as a whole has degrees. A
thesis (e.g. Rp) is not a quantity, and so cannot be “greater” than another thesis (e.g. Rq). Therefore,
when in the major premises of implicational a fortiori argument we say that “more of thesis R” is
implied or required, we must refer to a variation in the predicate and/or in the subject within thesis R.
This insight can be better understood if we formulate an implicational a fortiori argument in such a
way that the categorical propositions inherent in it are made explicit. This can be done with
antecedental and consequental arguments of whatever polarity. Consider for instance the following
case, which is doubtless the most frequent:

P (= A is p) implies more R (= C is r) than Q (= B is q) does, and
Q (= B is q) implies enough R (= C is r) to imply S (= D is s).
So, P (= A is p) implies enough R (= C is r) to imply S (= D is s).

Here, I have shown each of the four categorical propositions as involving four different subjects (A, B,
C, D) with four different predicates (p, q, r, s). The middle thesis R is here taken to mean that ‘C is r’.
The variation of R may in this light be understood in various ways. In the most frequent case, the
subject C is constant and it is the predicate r within R that is variable, C being r p in thesis Rp and C
being r q in thesis Rq (r p > r q). Comparatively rarely, the predicate r is constant and it is the subject C
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within R that is variable, Cp being r in thesis Rp and Cq being r in thesis Rq (Cp > Cq) . In more
complex cases, both the subject C and the predicate r might conceivably vary, Cp being r p in thesis Rp
and Cq being r q in thesis Rq. The important point is that the resultant R theses can reasonably be said
to satisfy the condition that Rp > Rq.
As regards language, the major and minor theses might in practice be stated in gerundive form, as ‘A
being p’ and ‘B being q’, while the subsidiary term might more naturally be stated in the infinitive
form, as ‘D to be S’. For the middle thesis, we might say ‘more r in C’ to signify that it is the predicate
that varies, or ‘more C to be r’ to signify that it is the subject that varies. Quite often in practice,
people do not state the whole middle thesis, but only the most relevant term in it – i.e. the variable
predicate (usually) or subject (rarely). Thus, instead of saying in the major premise “implies more R,”
they might say “implies more r” or “implies more C”; and likewise, instead of saying in the minor
premise and conclusion “implies R enough,” they might say “implies r enough” or “implies C
enough.”
Strictly speaking, of course, this is inaccurate, because a lone term cannot be implied (or imply). The
logical relation of implication concerns whole theses, never mere terms. But since this confusion
occurs in everyday discourse, it is well to be aware of it and to take it into consideration. Thus, when
in practice we encounter an a fortiori argument with whole theses as major and minor items, and a lone
term as middle item, we should not think that this exemplifies a ‘hybrid’ type of argument which is
partly copulative and partly implicational. Formally, such a construct is still implicational argument,
except that the middle thesis is not entirely spoken out loud; i.e. either its subject or its predicate is left
tacit. In the same way, the subsidiary thesis is sometimes incompletely stated. To validate such partly
formulated arguments, we of course need to specify the intended unspoken term(s).

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That we have to acknowledge the possibility that the subject varies in magnitude will be evident further
on, when we consider predicatal a crescendo (i.e. proportional a fortiori) argument. There it is manifest that this
is logically possible and occurs in practice. As regards the assumption that Rp (Cp is r) > Rq (Cq is r) is implied
when Cp > Cq (rather than when Cp < Cq) – this seems reasonable to me at this time, though some uncertainty
persists.