Handbook of the History of Logic: - Fordham University Faculty

Peter Abelard and His Contemporaries 109
contradictoriness. Contradictories are supposed to divide completely the conceptual
space that they mutually imply. They can neither be true at the same time
nor false at the same time, so that the truth of one properly implies the falsity of
the other. This relation is formalized in Aristotle’s famous square of opposition,
where the diagonally opposed propositions are the ones that are contradictorily
opposed:
A: All S is P E: No S is P (= Not some S is P )
I: Some S is P O: Not all S is P .
The E- and O-propositions here are rendered extinctively, so as to contradict,
respectively, the I- and A-propositions. Say the O-proposition were rendered separatively:
“Some S is not P .” 50 This version, Abelard argues, does not contradict
the corresponding A-form, because both could be false together. The reason lies in
the further issue of whether the subject term can be assumed to name something
that really exists — that is, whether it carries existential presupposition. If the
negation comes after the subject term then the proposition as a whole presupposes
that there is something corresponding to the subject term. But if the negation
comes before the subject term then the presupposition disappears. Here is the key
claim:
But it is one matter to disjoin things from each other, another to show
that they do not agree with each other. For one who treats of things
as being disjoined from one another takes them as if they exist; but
one who treats of things as not agreeing with one another does not
suggest their existence any more than their non-existence . . . Hence
the preposed negation has the one import and the interposed negation
the other [Abelard, 1970, p. 177 (12–15, 16–17)].
For P to be disjoined from S suggests removal of P from S, and therefore presents
S as some existing thing from which the removal is made. Existential presupposition
is attached to the subject term (and to the predicate term too, since “Some S
is not P ” is equivalent to “Some P is not S”). But when the negation is put at the
front of the proposition, what is suggested is only the disagreement of subject and
predicate terms, not removal of one from the other. Mere disagreement does not
carry with it the suggestion of an existing thing from which removal is made. 51
It is this existential presupposition in an O-proposition with separative negation
that makes it fail to contradict its corresponding A-proposition. Say no humans
exist. Then the A-proposition “All humans are mortal” is false, because it presupposes
that humans do exist. But “Some humans are not mortal” is false too, since
it presupposes that humans do exist. Two propositions capable of being false together
are not contradictories. So the separative version of the O-proposition fails
50 Note that separatively negating the A-form to yield “All S is not P ” would not express what
is needed, and would, in fact, be a clumsy way of expressing the E-form
51 Note that the existential presupposition issue is distinct from the issue of having a subject
term with personal signification. “The chimaera is a mythical beast” is a personal proposition,
even though its subject term fails to name an actual thing.