Handbook of the History of Logic: - Fordham University Faculty

144 Ian Wilks
The problem with this result of course is that, as we have seen, (iv) is on
Abelard’s account obviously false. (iv) is of the form [(p&q) ⊃∼ (p&q)]; this is
a substitution instance of the simpler formula (p ⊃∼ p). According to Abelard’s
theory of conditional entailment, no proposition can entail its own negation (since
no proposition can contain its own negation). So it is important for Abelard that
(iv) be false. Yet it has just been proven true. We have no clear record of his
response to this problem, but the approaches of the above-named rivals are known
in outline.
Alberic considers the content of the premises (i)–(iii) not adequate to motivate
use of hypothetical syllogism. What motivates this argument form, in Alberic’s
view, is not the fact that antecedents and consequents are related in a certain
way (i.e., related so that the consequent of one conditional is the antecedent of
another), but the fact that predicates are related in a certain way. Remember that
the conditionals studied by these logicians typically have a shared subject term
between antecedent and consequent (“If Socrates is a man then Socrates is mortal”);
in the above argument, “Socrates” is in all the antecedents and consequents
as subject, and so the inferences made are interpreted by Alberic as depending on
relationships between the predicates. In his view, hypothetical syllogism demands
a sequence of predicates such that the first’s being predicated of the subject is
the cause of the second’s being predicated of the subject (as Socrates’ being a
man is the cause of his being mortal), and such that the second’s being predicated
is likewise the cause of the third’s. In Alberic’s view this relationship does not
exist here. The required causal relation between predicates does not hold in the
sequence “being a man and not being an animal,” “not being an animal” and “not
being a man.” In particular, the conjunction of predicates in “being a man and
not being an animal” is self-contradicting, and cannot be represented as a cause
of anything. So the linkage does not exist which would allow the hypothetical syllogism
to work. Alberic’s conclusion, then, is that the argument is invalid because
the conditions for legitimate use of hypothetical syllogism do not exist [Anonymous,
1967, pp. 64 (31)–65 (3, 10-12); Martin, 1987a, p. 396; Iwakuma, 2004, pp.
325–326].
From Gilbert of Poitiers and his school, and Robert of Melun and his, we get an
approach which simply questions the truth of conditionals like (i), not the validity
of the argument as a whole. This is an important distinction, because what Alberic’s
approach essentially does is cease to view the above argument as an exercise
in propositional logic at all, construing it instead as an exercise in term logic. By
contrast, the approach of these other two schools accepts hypothetical syllogism
as an inferential relation between propositions, not terms. The problem raised by
the Porretani for conditionals like (i) echos Alberic’s concern for causal relevance,
but this time the relevance problem is expressed in terms of the relation of antecedent
to consequent. Their key principle is that the content of the antecedent
must have relevance to the content of the consequent, in that the truth of the
predication in the former can be seen as causing the truth of the predication in
the latter. “If Socrates is a man and an ass, then Socrates is a man,” is seen as