Handbook of the History of Logic: - Fordham University Faculty

Peter Abelard and His Contemporaries 139
conditional contained in the antecedent. 106
Abelard’s remedy lies distinguishing a “temporal” meaning of the conditional
connective from the “natural” one already discussed [Abelard, 1970, p. 472 (16–
38)]. The temporal interpretation reads “If it rains, it thunders” as “When it
rains, it thunders.” On this reading, the truth of antecedent and consequent
are not linked conditionally or causally, but only temporally, in that the two are
simply being said to be true at the same time [Abelard, 1970, p. 472 (26–29)].
They only have societas comitationis, an association of accompaniment [Abelard,
1970, p. 481 (22)], and are rendered in Latin by cum (“when”) as opposed to si
(“if”). This kind of relation is, unlike the one involved in relevance implication,
completely symmetrical, in that if the antecedent holds concurrently with the
consequent then the consequent will hold concurrently with the antecedent. It is
this temporal relation that Abelard appeals to in qualifying the above conditional
“If it is not well then it is sick.” He reads it as undergoing temporal qualification;
the extra “if” clause is read as determining the time in which the main conditional
is true, so “if it is an animal” is read as “when it is an animal.” If the temporal
qualification is applied to the whole conditional as in (i) above, or applied only to
the consequent as in (ii) above the result will, he argues, still be false [Abelard,
1970, p. 403 (19-24)]. But if it is applied only to the antecedent we get this: “If
when it is an animal it is not well then it is sick,” that is, “If it is an animal and
not well at the same time then it is sick” [Abelard, 1970, p. 403 (26–28)]. Abelard
is willing to admit that the original conditional under this qualification “perhaps
is true” [Abelard, 1970, p. 403 (29)].
This sort of qualification is referred to as constantia. Its effect is to make true
what would otherwise be a false conditional by limiting what it claims. While
the example proposition may be successfully dealt with thus, the result is, in a
broader sense, problematic. It reveals the same underlying difficulty for Abelard’s
discussion of conditionals as arises for his discussion of disjunctions. Just as his
conception of disjunction ultimately compels him to regard the two disjunctions
in ((p ∨ q) ∨ r) as operating in a different way, so his conception of the conditional
compels him to regard the two conditionals in ((p ⊃ q) ⊃ r) as operating in a
different way. This imposes a definite constraint on what Abelard can possibly accomplish
in the area of propositional logic. The fragment of this that he develops,
which treats various relationships between the conditional, disjunctive and negative
connectives, can treat these relationships only as they arise in propositional
forms of lesser complexity. Even though Abelard inherits from Boethius an ability
to identify key connectives, the account he provides of them will not support their
successive combination into the full range — or even a representative small range
— of compound propositional forms.
The fact that he sees the viable number of these propositional forms as relatively
small shapes the way he studies the deductive reasoning patterns to which they give
rise. A limited number of propositional forms will naturally give rise to a limited
number of deductive patterns, and so Abelard runs through a fairly conventional
106 This material is discussed in [Martin, 2004a, p. 178].