Handbook of the History of Logic: - Fordham University Faculty

138 Ian Wilks
[Abelard, 1970, p. 493 (12)] but also “Either this is hot, or it is cold or tepid”
[Abelard, 1970, p. 493 (19)] as allowable readings of the original. Of course on his
account these cannot be viewed as having exactly the same meaning, because in
the former the merely truth functional relation exists between the first and second
disjuncts (hot and cold), while in the latter it exists between the second and third
(cold and tepid). However subtle, this must be taken as making a difference in the
meaning of the whole.
How we group the disjuncts in a proposition of the form (p ∨ q ∨ r) is intuitively
thought not to make a difference to the meaning of the whole compound disjunction,
but Abelard’s analysis suggests otherwise. So this is itself a counter-intuitive
result. His initial account of stronger, non-truth-functional disjunction makes it
difficult to incorporate that connective into nested contexts. As we shall now see,
he encounters a similarly counter-intuitive result with the conditional, where again
he must have recourse to a secondary and weaker version of the connective in order
to deal with nested propositional structures.
Consider the conditional “If it is not well then it is sick.” This proposition is
false, given that anything, including rocks, can fail to be well; but only living things
(in Abelard’s view) can be sick. So failing to be well does not imply sickness. But
failing to be well among animals does seem to imply sickness, so it seems plausible
that if we write the condition of being an animal into the conditional in question
then the result will be true. It is therefore tempting to apply a qualifying “if”
clause: “If it is an animal, it follows that if it is not well then it is sick.” Abelard
argues, however, that the result of such qualification, whether it takes this form
or others, will not be true, given the requirements for implication. He considers
three cases, none of which he finds satisfactory. (i) “If it is an animal, it follows
that if is not well then it is sick.” This attempt attaches the qualification to the
whole conditional, and makes that conditional the consequent in the larger one
that results from the attachment. But we know that the conditional standing
as the consequent taken by itself is false (“If it is not well then it is sick”). So
if the antecedent is true then we have a true antecedent and a false consequent
— rendering false the compound conditional taken as a whole when said of an
animal [Abelard, 1970, p. 402 (14–15)]. (ii) “If it is not well it follows that
if it is an animal it is sick.” Here the qualifying “if” clause is attached to the
consequent of the original. But “If it is an animal, it is sick” cannot be true
since there are many animals that are not sick; animality and sickness are not in
the right relation to make that conditional true. So again here the consequent
is false, and so if the antecedent is true then the compound conditional as a
whole is false [Abelard, 1970, p. 402 (21–23)]. (iii) “(If it is an animal then
it is not well) that it is sick.” This attempt attaches the qualifying
“if” clause to the antececent of the original conditional. But what we have here
is the claim that a categorical proposition follows from a conditional one. This
claim cannot be true, [Abelard, 1970, p. 402 (26–27)], presumably because of
the existential presupposition expressed in the categorical, which is absent from