Handbook of the History of Logic: - Fordham University Faculty

Peter Abelard and His Contemporaries 125
He supplies explicit definitions for both. The topical differentia is defined as
“that thing (res) in whose relation to another thing the strength of the entailment
resides” [Abelard, 1970, p. 263 (7–8)]. Since he speaks of a thing, not a word, we
must understand the differentia in the above example as man, not “man” — or,
more precisely, as man taken as a species, not “man” taken as a species. Abelard
has presumably resolved to his satisfaction the metaphysical problems lurking in
this use of general terms, so these need not detain us. Considering man’s relation to
animal prompts our taking man as a species, at which point the differentia is clear
to view, and the maximal proposition on the way. It all starts with the relation
of man to animal, and so it is in this relation that “the strength of entailment
resides.”
The account of the maximal proposition is more involved. Abelard defines it
as “that proposition which, containing the sense of many entailments, reveals a
common method of proof which the differentiae of those entailments possess —
owing to their having the force of the same relation” [Abelard, 1970, p. 263 (12–
14)]. 79 This is a densely packed formulation. The key feature of the above maximal
proposition (“Whatever the species is predicated of the genus is predicated too”)
is its standing as a general expression of a whole series of entailments: “If it is a
man it is an animal,” “If it is a pearl it is a stone,” “If it is a rose it is a flower,”
“If it is red it is a colour,” and so on. These entailments have different terms
(man and animal, pearl and stone, and so on), but are alike in that the terms are
all related as species and genus. We can see each entailment as arising through
the substitution of similarly related terms into the maximal proposition, which
in this manner can be seen as “containing” the sense of each. This is why the
maximal proposition plays a role in confirming each through a “common method
of proof.” This, as we have seen, starts by construing terms under the guise of the
appropriate relation: man taken as species, pearl taken as species, rose taken as
species, red taken as species, and so on. So construed they become the differentiae,
each of which gives rise to a method of proving the conditional from which they
arise — and it is the same method in each case. The maximal proposition reveals
what this shared and common method is. 80
This is the sense in which maximal propositions play for conditionals the role
that syllogistic rules play for syllogisms: the role of reliably identifying which forms
mark genuine entailments. The rule for Barbara (“If anything 〈A〉 is predicated
of something 〈B〉 universally, and if something else 〈C〉 is subject to that subject
〈B〉 universally, that subject 〈C〉 is also subject to the predicate 〈A〉 in the same
way, that is universally” [Abelard, 1970, p. 237 (7–9)]) confirms as genuine entailments
all of those syllogisms which arise from uniform substitution of terms in
its open places (marked by “A”, “B”, and “C”.) The rule “Whatever the species
is predicated of the genus is predicated too” works in much the same way to con-
79 On this passage see [Martin, 2004, pp. 174–175].
80 Note that this account of the essentials of Abelard’s topical theory is based on texts from the
Dialectica. Certain aspects of the account change in his Super Topica glossae [Abelard, 1969b].
For a summary of changes see [Green-Pedersen, 1977, pp. 135–137].