Handbook of the History of Logic: - Fordham University Faculty

Peter Abelard and His Contemporaries 133
concludes that the topic from opposites should be rejected.
Why does he regard that logical form as self-contradicting? [(p&q) ⊃∼(p&q)]
is a substitution instance of the simpler formula (p ⊃∼p). Abelard takes it as selfevident
that conditionals of this form are false, because, he believes, no proposition
can imply its own negation. Of course it makes sense that he should think this,
given his understanding of what counts as a true conditional. If the meaning of
the consequent must be contained in the meaning of the antecedent in order for
the conditional to be true, then any case where the consequent simply denies the
antecedent will represent the most obvious case of non-containment, and so the
most obvious case of a false conditional. The form (p ⊃∼ p) represents a direct
challenge to Abelard’s way of thinking about the conditional, and he naturally
regards any move in argument which gives rise to a conditional of that form as
fallacious. Rejecting this form (a pre-requisite for giving it the diagnostic role that
Abelard does) is in fact typical of the approach to relevance logic now known as
“connexivism”; a system of connexivist logic is characterized by its not accepting
this formula as true. 99 Needless to say, any such system represents a substantial
departure from familiar systems of two-valued propositional logic; these interpret
a case where a proposition implies its own negation as simply deriving from the
fact that it is false. This connexivist principle is derived from Boethius, who
himself extracts it from Aristotle’s Prior Analytics: “It is impossible that the
same thing should be necessitated by the being and by the not-being of the same
thing” [Aristotle, 1984a, p. 91 (57b3-4)] — in other words, no proposition can be
implied both by another and by the negation of that other. 100 The corollary of
this, of course, is that no proposition can be implied by its own negation. It is clear
from contemporary testimony that the acceptance of the connexivist principle was
regarded as an important issue [Martin, 1987a, pp. 378–379], and indeed, as we
shall see in the next section, it was to become a major point of contention between
Abelard and others.
The upshot of the discussion of the topics from opposites and immediates is
that they yield no maximal propositions and serve to confirm no entailments. The
kind of entailments they would confirm if this result were otherwise would be ones
where one of the clauses of the conditional posited its predicate and the other
clause removed its predicate. In other words, they would confirm conditionals of
mixed quality, with an affirmative antecedent and negative consequent, or vice
versa. The implication of Abelard’s result is that no such conditionals will be
confirmed by topical theory, and that the sorts of conditional entailments that
will be confirmed will be jointly composed out of affirmative clauses or jointly
composed out of negative ones. 101 Presumably these limitations establish the
99For the notion of connexivism see [McCall, 1966]; this source is cited in [Martin, 1987a, p.
381, note 10].
100The origin of this principle is discussed in [Martin, 1987a, p. 378–379]. See also [McCall,
1966, pp. 415–416].
101Martin has recently suggested a constraint on Abelard’s rejection of mixed quality conditionals
as successful entailments: that this rejection applies only in cases where the negated clause
is subject to extinctive negation, not separative [Martin, 2004a, p. 196, note 48].