Handbook of the History of Logic: - Fordham University Faculty

132 Ian Wilks
as appropriate for topical use. Rejection, as noted, can happen because the principle,
while true, does not play the particular justifying role expected of a given
topic. Or it can happen because the principle gives rise to false conditionals (such
as “If Socrates is a rational, mortal animal then he is a man”). Or it can happen
not by direct reasoning in this way but by indirect reasoning, where Abelard shows
that justificatory use of a proposed maximal proposition leads to contradiction.
This is the form the argument takes when he rejects all the proposed maximal
propositions for the topics “from opposites” and “from immediates.” And it is
here that we find what has so far emerged as perhaps the most interesting of all
the sections in Abelard’s discussion of topical theory. 97 Here, in order to derive
the perceived contradiction, he articulates insights that stand at the very heart of
his relevance logic.
Opposites (white, black) are to contraries as immediates (white, non-white) are
to contradictories. If one opposite is present then the other must be absent, and
likewise with immediates. But the absence of one opposite does not imply the
presence of the other since they may both be absent. By contrast, the absence of
one immediate does imply the presence of the other. So a conditional licensed by
the topic “from opposites” is “If it is a man it is not horse” [Abelard, 1970, p. 393
(19)], since nothing can be both at the same time, but something can be neither. 98
A conditional licensed by the topic “from immediates” is “If it is not healthy it is
sick,” since, presumably, anything which can meaningfully and truthfully be said
not to be the one must be said to be the other. Abelard’s rejection of both topics
proceeds similarly, so I will focus on the one “from opposites.”
The principle common to the maximal propositions falling under this topic is
that “when the one is posited, the other is removed” [Abelard, 1970, p. 393 (6)].
Consider the conditional, “If Socrates is a man, he is not a stone” [Abelard, 1970,
p. 395 (10)]. Being a man and being a stone are in opposition to each other since
nothing can be both, while many things are neither. So the topic “from opposites”
tells us that the conditional is true; positing man entails removing stone. But note
that the same topic also tells us that this conditional is true: “If Socrates is a stone
he is not a man” [Abelard, 1970, p. 395 (15)], for the same reason as before (i.e.,
nothing can be both a stone and a man). So we have what are presumed to be
two true conditionals: “If Socrates is a man he is not a stone” and “If Socrates is
a stone he is not a man.” Abelard now makes one conditional out of the two by
conjoining their antecedents into one antecedent, and conjoining their consequents
into one consequent: “If Socrates is a man and a stone, Socrates is not a man
and a stone” [Abelard, 1970, p. 395 (17)]. This move is intuitive and logically
unobjectionable. Its product may be formalized thus: (p&q) ⊃∼ (p&q). But
this represents a logical form which Abelard regards as self-contradicting, so he
97 The significance of Abelard’s treatment of these topics was first brought to light by Christopher
Martin in [Martin, 1986, pp. 569–570], [Martin, 1987a, pp. 386–392], and [Martin, 1987b,
pp. 432–433]. Further treatments are found in [Martin, 1992, pp. 117–122] and [Martin, 2004a,
pp. 189–191]. My own account here is based on these sources.
98 The corresponding maximal proposition is: “If one opposite is predicated of something, then
the other opposite is removed from that same thing” [Abelard, 1970, p. 393 (24–25)].