Handbook of the History of Logic: - Fordham University Faculty

Peter Abelard and His Contemporaries 141
compound conditional forms just considered, Abelard lists the mediate conditional
forms [Abelard, 1970, pp. 516 (16)–530 (26)]. These involve conjunctions of
conditionals: “If it is A then it is B; ifitisBthen it is C.” The inference pattern
arising from these is an amalgam of what now goes by the name of “hypothetical
syllogism” and modus ponens: “If it is A then it is B; ifitisBthen it is C. It
is A. Therefore it is C.” The work done by (what is now called) hypothetical
syllogism is rather understated in this arrangement, which seems to present the
whole inference as just another version of modus ponens — modus ponens applied
to a conjunction of conditionals, instead of a single one. In that case the mediate
conditional inferences are brought into neat alignment with the others, as involving
their own specialized versions of modus ponens (and, in the other half of the cases,
modus tollens). But this is at the expense of not fully articulating the distinct
presence of (what is now called) hypothetical syllogism. (ii) The list of hypothetical
syllogisms is rounded out by considering what now goes by the name of “disjunctive
syllogism.” The corresponding inference patterns are systematically generated, as
above, by listing a full range of substitutional forms. Start with “Either it is an
A or it is a B,” negate either or both disjuncts to produce an additional three
forms of the disjunction, and thus generate four disjunctive forms. The denial of
either the first or the second disjuncts in each of those four disjunctions creates a
distinct inference pattern for asserting the truth of the remaining disjunct. The
product of the analysis is of course eight such patterns [Abelard, 1970, pp. 531
(33)–532 (21)]. No compounded disjunctive inference patterns are considered.
Summing the variant moods arising from conditionals and disjunctions, we get
a substantially sized list, to be sure. But of course very little ground is actually
covered in this exercise, which, because of the proliferation of the forms considered,
falsely conveys the impression that much ground has been covered. The inference
patterns are generated by what are often negligible variations in the ways of applying
negation. And they cannot include larger, more intricately compounded
structures because Abelard’s relevance account of the connectives simply raises
too many difficulties for this. Even one level of nesting between hypotheticals
requires invoking a secondary, temporal conditional; and even one level between
disjunctions requires invoking a secondary, predicate disjunction. The result is a
taxinomically-oriented presentation whose taxonomy leaves something to be desired.
If nothing else, the above account of Abelard’s work on entailment gives the lie
to this pre-scholarly story about medieval logic: that it accomplishes very little
in the area of propositional logic because of a lack of interest in the area, a lack
of interest due to the period’s preoccupation with Aristotle, who had no interest
in propositional logic at all. Abelard has a great deal of interest in the area
and accomplishes much. If we take him at his word and view the discussion of
topical theory as a preparation for the discussion of hypothetical syllogisms, then
we can fairly say that about half of the Dialectica is given over to discussion of
propositional logic. The limitation in Abelard’s accomplishment here is neither one
of interest nor insight. It lies in a basic heuristic problem for propositional logic