Handbook of the History of Logic: - Fordham University Faculty

Peter Abelard and His Contemporaries 131
wall” fails the containment condition for entailment, even though it passes the
necessity test [Abelard, 1970, p. 346 (18–28)]. Abelard also argues that attempts
to re-formulate the maximal proposition for this topic in such a way as to make
it acceptable fail; they fail because they yield variants of the proposition which
now fall under a different topic. One such variant is “Whatever wholly agrees with
the whole, that is, according to its individual parts, agrees with each one of those
parts”; but this variant licenses universal-particular inferences, not whole-part
ones, and hence does not qualify as a topic “from an integral whole.” Abelard
draws the same conclusion of “If anything is predicated of the whole it is also
predicated to all the parts of it taken together” [Abelard, 1970, p. 344 (6–24)]. 93
(iii) Under the topic “from an equal” Abelard accepts four maximal propositions
for what he calls “equality of predication” (that is, co-extensionality): “Whatever
one equal is predicated of, the other is too”; “From whatever one equal is
removed, the other is too”; “Whatever is predicated of one equal is predicated of
the other”; and “Whatever is removed from one equal is removed from the other”
[Abelard, 1970, p. 349 (28–32)]. He also accepts two maximal propositions for
what he calls “equality of inference,” where in this case what are called “equals”
are a given conditional and its contrapositive: “When one equal is posited, the
other is too”; “When one equal is denied the other is too” [Abelard, 1970, p. 351
(31-32)]. These latter two maximal propositions simply affirm the equivalence of
contrapositives. 94 (iv) Under the topic “from antecedent or consequent” Abelard
considers the status of various basic rules of propositional logic, such as “If the
antecedent is posited then the consequent is posited” [Abelard, 1970, p. 365 (18)].
While not denying the truth of these, he argues that they are not genuine maximal
propositions, and so should be re-expressed in terms of other propositions which
are. For example, the rule just cited has no explanatory power, he argues, because
it does not actually prove the truth of conditional; to apply the rule at all in a
given case one already needs to accept one proposition as antecedent and the other
as consequent — in which case one has already in effect accepted the truth of the
conditional the rule is supposed to be justifying [Abelard, 1970, p. 365 (24–26)].
What actually demonstrates the relation between antecedent and consequent, he
says, is consideration of how species relates to genus; he therefore refers the reader
to the topic “from species” [Abelard, p. 365 (27–28)]. By this sort of reasoning
all of the principles associated with the topic “from antecedent or consequent” are
rejected as maximal propositions. 95
The above gives the barest hint of the intricacy of Abelard’s argumentation in
this section. 96 He takes up a long series of what we would call logical and semantic
principles and accepts or rejects these, sometimes after painstaking re-formulation,
93The topic “from the integral whole” is discussed in [Martin, 2004a, pp. 188–189] and [Stump,
1989, pp. 100–101].
94The topic “from an equal” is discussed in [Stump, 1989, pp. 103-106].
95The topic “from antecedent or consequent” is discussed in [Stump, 1989, pp. 106–107].
96A summary of maximal principles is provided by de Rijk in [Abelard, 1970, pp. lxxxiii–
lxxxix]. For a comment on the general organization of topical theory into three divisions (intrinsic,
extrinsic and mediate) see [Martin, 2004a, p. 198, note 65].