Handbook of the History of Logic: - Fordham University Faculty

Peter Abelard and His Contemporaries 135
suggests a way of understanding them which poses an impediment to grasping the
truth of the matter — which is, of course, that implication and disjunction are
correlative with conjunction, as opposed to being kinds of conjunction. Abelard’s
work on hypotheticals manifests much nascent insight into propositional logic, but
also some inherited confusion. He is able to make some useful observations about
the conditional and disjunctive connectives, and how they relate to the negative
connective. But for this to be a discussion of propositional logic in the fullest sense
there needs to be some notion of a connective for conjunction as well. What there
is instead is a way of speaking which can only impede the development of such a
notion.
There is to be found here, nonetheless, a rich discussion of the interrelationship
of conditionals, disjunctions and negations, in which Abelard’s sensitivity to the
difference between wide-scope and narrow-scope negation plays an important role.
Just as a categorical proposition can be negated as a single unit by application
of wide-scope negation, so with a hypothetical, which counts as affirmative even
if one or other of its clauses happens to be negated: “Whatever hypotheticals
are asserted — whether out of two affirmatives, such as this: ‘If it is a man it
is an animal’; or out of two negatives, such as this: ‘If it is not an animal, it is
not a man’; or out of an affirmative and a negative or vice versa, such as this:
‘If it is night it is not day’ — they all seem to assert an affirmative meaning of
entailment” [Abelard, 1970, p. 476 (8–15)]. The narrow-scope negation affecting
only the consequent in “If it is a man it is not an animal” does not negate the
hypothetical as a whole, which still affirmatively expresses a relation of following
between antecedent and consequent. If the narrow-scope negation did do this then
one would not be able to say that a relation of following is being expressed here.
Besides achieving insight into how a conditional is negated, Abelard also discusses
at length how it is related to disjunction. He receives from the Boethian
tradition the foundational insight that (∼ p∨q) is equipollent with (p ⊃ q): “If the
conjunctive is composed of two affirmatives, the disjunctive one
will consist of a negative and an affirmative” [Abelard, 1970, p. 488 (29–31)]. 102
This means that what he says about the disjunctive relation must be carefully
aligned with what he has already said about the conditional relation. This is a
good result to arrive at, of course, but it is complicated by his relevance approach
to understanding implication.
He argues against defining a disjunction truth-functionally, either inclusively as
meaning that at least one of the disjuncts is true, or exclusively as meaning that
only one of them is true. 103 The problem with these definitions is brought out in
the example, “Either I am an animal or I am an ass” [Abelard, 1970, p. 490 (24)].
Note that this proposition is true on both definitions; one of the disjuncts is true
(“I am an animal”), and in fact only one of them is true (since “I am an ass” is
false). So taken either inclusively or exclusively, the disjunction is true. Now by
the above equipollence the disjunction can be re-written as this conditional: “If I
102 Reading si for is in line 29.
103 On this point see the brief commentary in [Kneale and Kneale, 1962, p. 222].