Handbook of the History of Logic: - Fordham University Faculty

140 Ian Wilks
catalogue of such patterns, presumed to be comprehensive. The treatment of this
material is — like the treatment of argument forms in modal and tense logic —
taxonomic rather than axiomatic. The intent is a list, not a reduction to principles.
The discussion of deductive patterns arising from the conditional is, naturally,
focused on reasoning “through the positing of the antecedent” and “through the
destruction of the consequent”: modus ponens and modus tollens respectively.
These rules end up being variously specified, according to the character of the
conditional under question.
Abelard starts with the four simple conditional forms: “If it is A it is B”; “If it
is A it is not B”; “If it is not A it is B” and “If it is not A it is not B” [Abelard,
1970, p. 498 (27–30)]. 107 From these are generated eight moods of hypothetical
syllogism, four arising from modus ponens, four from modus tollens [Abelard,
1970, pp. 501 (30)–505 (7)]. The first mood applies “It is A” to“If it is A it is B.”
The second applies “It is A” to“IfitisA it is not B.” The third applies “It is
not A” to “If it is not A it is B.” The fourth applies “It is not A” to “If it is not
A it is not B.” The fifth to eighth proceed in parallel fashion to apply the denial
of the consequent to each of the four forms. One notices the inefficiency of this
presentation, since there are, after all, really only two rules at work. The moods
strike one as overly specified.
But some need to make modus ponens and modus tollens case-specific emerges
when Abelard considers the compound conditional forms [Abelard, 1970, p. 505
(9)–516 (14)]. Compound conditionals involve nesting a conditional, either in
the consequent, or in the antecedent, or both. As we have seen, for Abelard
nested conditionals themselves involve a second kind of conditionality — that of
the temporal conditional. The rules of modus ponens and modus tollens are not
quite enough to capture the truth conditions of these compounds, and need to be
adapted to accommodate the presence of two kinds of conditional relation. So a
compound like “If it is A, then when it is B it is C” falls under this version of
modus ponens: “When the antecedent is posited, its consequences will be at the
same time” [Abelard, 1970, p. 506 (4–5)]. A compound like “If, when it is A, it
is B, then it is C” falls under a different version: “When the antecedent exists
with whatever is posited, any consequent of it is posited” [Abelard, 1970, p. 509
(8–9)]. Compound conditionals with temporal qualification in both antecedent
and consequent will need another version again.
Abelard accordingly tailors modus ponens and modus tollens to handle different
kinds of content. This is an impediment to recognizing the merits of a very
different approach: keep the rules the same for all cases and develop a more sophisticated
account of propositional substitution. For all his innovations at a more
foundational level of propositional logic, Abelard is evidently not ready to take this
step.
The final tally of moods for the hypothetical syllogism gives us one hundred and
twenty-eight. These include two further kinds of case. (i) Besides the simple and
107 The variables “A” and“B” are present in Abelard, and are derived from Boethius. (Upper
case forms are used here, not the lower case forms in the text, for typographical clarity.)