Page 24 - A Fortiori Logic: Innovations, History and Assessments

18 A FORTIORI LOGIC

the given premises as the putative conclusion is. If a contrary or the contradictory of the putative
conclusion is positively implied by the given premises, the putative conclusion is called an absurdity
(lit. ‘unsound’) or more precisely an antinomy (lit. ‘against the laws’ of thought).
The validity of an argument does not guarantee that its conclusion is true, note well. An argument may
be valid even if its premises and conclusion are in fact false. Likewise, the invalidity of an argument
does not guarantee that its conclusion is false. An argument may be invalid even if its premises and
conclusion are in fact true. The validity (or invalidity) of an argument refers to the logical process, i.e.
to the claim that a set of premises of this kind formally implies (or does not imply) a conclusion of that
kind.
A material a fortiori argument may be validated simply by showing that it can be credibly cast into any
one of the valid moods listed above. If it cannot be fitted into one of these forms, it is invalid – or at
least, it is not an a fortiori argument. The validations of the forms of a fortiori argument may be carried
out as we will now expound. Invalid forms are forms that cannot be similarly validated. Obviously,
material arguments can also be so validated; but the quick way is as just stated to credibly cast them
into one of the valid forms. Once the forms are validated by logical science, the material cases that fit
into them are universally and forever thereafter also validated.
One way to prove the validity of a new form of deduction is through the intermediary of another,
better known, form of deduction. Such derivation is called ‘reduction’. ‘Direct’ reduction is achieved
by means of conversions or similar immediate inferences. If the premises of the tested argument imply
those of an argument already accepted as valid, and the conclusion of the latter implies that of the
former, then the tested argument is shown to be equally valid. ‘Indirect’ reduction, also known as
reduction ad absurdum, on the other hand, proceeds by demonstrating that denial of the tested
conclusion is inconsistent with some already validated process of reasoning.
It works like this: Suppose A and B are the two (or more) premises of a proposed argument, and C is
its putative conclusion. If the C conclusion is correct, this would mean that (A + B) implies C; which
means that the conjunction (A + B + not-C) is logically impossible. Let us now hypothetically suppose
that C is not a necessary implication in the context of A + B; i.e. that not-C is not impossible in it. In
that case, we could combine not-C with one of the premises A or B, without denying the other. But we
already know from previous research that, say, (A + not-C) implies not-B; which means that the
conjunction (A + not-C + B) is logically impossible. Therefore, we must admit the validity of the
newly proposed argument. Note that the two stated conjunctions of three items are identical except for
the relative positions (which are logically irrelevant) of the items conjoined.
Analysis of constituents
8
The validation procedures are accordingly uniform for copulative and implicational a fortiori
arguments. They are based on analysis of the meanings of the propositions involved in such argument,
i.e. on reduction of these more complex forms to simpler forms more studied and better understood by
logicians.
The following are the two main reductions needed for validation of the earlier listed copulative
arguments. The major premises (characterized as “commensurative” because they compare measures
or degrees) of subjectal and predicatal arguments are always positive and have the following
components:
The subjectal major premise, “P is more R than (or as much R as) Q is,” means:
P is R, i.e. P is to a certain measure or degree R (say, Rp);
Q is R, i.e. Q is to a certain measure or degree R (say, Rq);
and Rp is greater than (or equal to) Rq (whence: Rp implies Rq).
The predicatal major premise, “More (or as much) R is required to be P than to be Q,” means:
Only what is at least to a certain measure or degree R (say, Rp) is P;
only what is at least to a certain measure or degree R (say, Rq) is Q;
and Rp is greater than (or equal to) Rq (whence: Rp implies Rq).
We could more briefly write the first two components of the predicatal major premise as exclusive
implications: ‘If and only if something is Rp, then it is P’ and ‘If and only if something is Rq, then it is

8 See my Judaic Logic chapter 3, section 2 – ‘Validation Procedures’ – for additional details on this topic.
However, note well, there are significant changes in the present treatment.