Page 23 - A Fortiori Logic: Innovations, History and Assessments

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We could in fact say that all a fortiori arguments are tacitly implicational. The thin line between
copulative and implicational argument becomes evident when we reword a typical copulative
argument in implicational form, as follows:

P (= something being p) implies more R (= r in it) than Q (= something being q) does,
and
Q (= something being q) implies enough R (= r in it) to imply S (= it to be s);
therefore, P (= something being p) implies enough R (= r in it) to imply S (= it to be s).

This argument is obviously a special case of the preceding one. Here, instead of four subjects (A, B, C,
D), we only have two (or even just one). They are unspecific (i.e. not labeled A and B, as earlier
done), in the sense that they each refer to ‘something’ (i.e. anything – the intent is general, not
particular) that is solely defined by the predicate initially attached to it (viz. p, q, respectively). The
‘something’ that is intended in P and the ‘something (else)’ intended in Q are here distinct objects,
note (although, as we have already seen, they could well in some cases be one and the same subject).
Each of them is subject to a different measure or degree of the middle predicate ‘r’ (whence r is ‘in
it’). And each of them is or turns out to be subject to the subsidiary predicate ‘s’. The case shown (here
again) is the positive antecedental mood; the same can obviously be done with the positive predicatal
mood, and with the negative forms of both of these.
Looking back at the way I came upon these various argument forms when I wrote Judaic Logic, I
remember first discovering the copulative forms and later, finding them insufficient to account for all
examples of a fortiori argument I came across, I developed the implicational forms. In a sense, they
were conceived as generalizations of the corresponding copulative forms. Indeed, I overgeneralized a
bit, because I did not realize at the time that the notion that a thesis may “imply more” of another
thesis is logically untenable. Much later, I started wondering whether ‘hybrid’ arguments signified
additional types, besides the copulative and implicational. It is only recently that I better understood
the relationships between the various forms of argument as above described. So the present account
amends past errors and uncertainties.
I should also here mention the following special case, where the major premise “P implies more A to
be B than Q does” means “P implies that a number x of A are B, and Q implies a that number y of A
are B, and x > y.” The change in magnitude involved in this case is not in the subject A or the
predicate B inherent in the middle thesis, but in the quantifiers of A. So the middle thesis is not, as
might be thought, about “how much A is B,” or even “how much B A is,” but about the frequency of
occurrence of ‘A being B’. In such case, the proposition could be stated less ambiguously as “P
implies more instances of A to be B than Q does.” The frequency involved may be extensional, as
here; or it could have to do with another mode of modality, i.e. more often in time or place, or in more
circumstances or contexts.
Moreover, though I have here presented the middle thesis R as a single categorical proposition, it
should be kept in mind that R could contain a compound thesis, i.e. it could involve a complex set of
variable factors.
In conclusion, when in formulating implicational a fortiori argument we refer to the middle thesis ‘R’,
the intention is more precisely ‘something in R’, meaning ‘some term(s) in thesis R’ or even ‘some
modal qualifier in thesis R’. That is, when we say: ‘implies more R’ or ‘more R is required to imply’
or ‘implies enough R’ – we must be understood to mean: ‘implies more of something in R’ or ‘more of
something in R is required to imply’ or ‘implies enough of something in R’, respectively. Though I
will continue to use the abridged formulae, these more elaborate formulae will be tacitly intended.
More will, of course, be said about implicational a fortiori argument as we proceed.

3. Validations

Validation of an argument means to demonstrate its validity. An argument is ‘valid’ if, given its
premises, its conclusion logically follows. Otherwise, if the putative conclusion does not follow from
the given premises or if its denial follows from them, the argument is ‘invalid’. If the putative
conclusion is merely not implied by the given premises, it is called a non sequitur (Latin for ‘it does
not follow’); in such case, the contradictory of the putative conclusion is logically as compatible with