Handbook of the History of Logic: - Fordham University Faculty

Peter Abelard and His Contemporaries 143
via responses to objections against it, or via objections to competing theorems,
the presupposition being that its content and purpose are already clear. The extreme
case of this is the widespread use of counter-exemplification as a means of
assessing inferential forms. An example of a form is given, and then juxtaposed
with a series of proposed counter-examples (instantiae), the entire commentary
on the form being essentially this list consisting of example and counter-examples
[Iwakuma, 1987, p. 438], with little or no supporting commentary. The late twelfth
century logicians drew on basically the same logical corpus as Abelard, with the
exception that in mid-century Aristotle’s Sophistical Refutations was brought into
currency. The instantiae technique was doubtless encouraged by resulting interest
in the theory behind fallacy identification; but relentless use of the technique
served to make the logical literature of this period all the more inaccessible to the
uninitiated.
That said, the richness of the resulting debate is undeniable. Its genesis often lies
in the reaction of Abelard’s rivals to controversial elements in his own philosophical
views, the ensuing discussion sharpened by the involvement of the corresponding
schools. An outstanding example is the debate arising from Abelard’s handling
of the topics “from opposites” and “from immediates.” Abelard’s argument for
rejecting these as legitimate topics was adapted by Alberic to pose a problem for
Abelard’s whole approach to the conditional.
Alberic’s argument starts from any obvious and self-evident conditional, such
as “If Socrates is a man, Socrates is an animal.” Here are the steps, as suggested
in a text originating from Alberic’s school [de Rijk, 1967b, pp. 65 (35)–66 (4)],
and reconstructed thus:
(i) If Socrates is a man and not an animal, Socrates is not an animal
(ii) If Socrates is not an animal, Socrates is not a man
(iii) If Socrates is not a man, it is not the case that (Socrates is a man and not
an animal)
(iv) Therefore, if Socrates is a man and not an animal, it is not the case that
(Socrates is a man and not an animal). 108
(iv) follows from the other three propositions by two applications of hypothetical
syllogism. So the logic of this argument is unimpeachable. If the argument is to be
shown unsound, one of its three premises must be shown to be false. But (i) is a
tautology, self-evidently true by simplification. (ii) follows from the original, selfevident
conditional by contraposition. And (iii) just contraposes “If Socrates is a
man and an animal, Socrates is a man,” itself a tautology, self-evidently true by
simplification. So (iv) evidently follows. One can start from any true conditional
and derive from it, through parallel steps, a proposition having the same logical
form as (iv).
108 See [Martin, 1986, pp. 570–571], [Martin, 1987a, pp. 394–395], [Martin, 1987b, p. 433],
[Martin, 1992, p. 122] and [Martin, 2004a, pp. 191–192].