Handbook of the History of Logic: - Fordham University Faculty

Peter Abelard and His Contemporaries 137
in play by the original three-part disjunction, not being hot does not imply being
cold. And since the conditional is false, it is also false to say, “This disjunction is
true: ‘either it is hot or it is cold.”’ Now go back to the original rendering of the
whole disjunction: “Either this disjunction is true: ‘either it is hot or it is cold’; or
it is tepid” (where the last “it” still refers to the thing whose temperature is under
question, not to the disjunction 105 ). Shorten it to: “Either this disjunction is true
or it is tepid.” Express this as a conditional: “If this disjunction is not true then
it is tepid” [Abelard, 1970, p. 493 (3)]. What we have been told so far is that the
antecedent of the conditional is true (because the disjunction expressed in it is in
fact not true). So the conditional as a whole is false when said in circumstances
where the consequent is false — in other words, when said of anything which is in
fact not tepid. What this all amounts to is that the original disjunction (“Either
it is hot or it is cold or it is tepid”) is false when said of something not tepid.
This is contrary to our intuitions, which tell us that the disjunction may well be
true under those circumstances. So the analysis which produces this result must
be incorrect.
What, then, is the correct analysis? It is one, Abelard believes, which recognizes
different kinds of disjunction as present in this case. He proposes to interpret
((p∨q)∨r) as involving one kind of disjunctive relation between p and q, and then
another kind between (p∨q) andr. For the nested disjunction Abelard reverts to a
conception of disjunction which diverges from the above conditional-based account.
It is, in fact, a truth-functional account. The nested disjunction is understood to
mean that one of p, q is true; it suffices simply that one of the disjuncts be true,
with no additional requirement that the negation of one relevantly imply the other.
The main disjunction, on the other hand, continues to be understood in accordance
with the above conditional-based account. The whole disunctive proposition then
reads “Either it is one of these or it is that”; expressed as a conditional it reads, “If
it is neither of these then it is that” [Abelard, 1970, p. 493 (14, 16)]. Abelard marks
this weakened form of disjunction by writing it as a relation between predicates
[Abelard, 1970, p. 493 (12)], saying “Either this is hot or cold,” as opposed
to “Either this is hot or this is cold.” The nested disjunction, then, is written
as predicative and interpreted as a truth functional. The main disjunction is
written as sentential and interpreted as above. The initial problem was caused
precisely because there was an implicative link at work within the nested disjuncts.
Now that the link is gone, the problem appears to be solved. “Either this is hot
or cold, or it is tepid” says that either the weaker, truth functional disjunction
holds between hot and cold, or the stronger disjunction holds between that weaker
disjunction and tepid.
Note that in (p ∨ q ∨ r) either p and q can be grouped into a nested disjunction,
or q and r. Abelard is committed by his account to accepting either pattern of
association as an equally correct reading of the original statement. He indeed
accepts this result, taking not only “Either this is hot or cold, or it is tepid”
105This disambiguation is effortless in the Latin, where disiuncta has a feminine inflection,
tepidum aneuterone.