Handbook of the History of Logic: - Fordham University Faculty

592 Mikko Yrjönsuuri
else as well. The first part of this proof is relatively understandable, and we can
still use ‘a’ as a name for the sentence at issue (although it must now be taken as
a name for an arbitrary sentence satisfying the criterion of self-reference, or any
sentence for which the first premise (1) is true). 20
Proof.
(1) ‘a’ signifies only that ‘a’ is not true Assumed
(2) ‘a’ is not true Assumed
(3) It is not wholly as ‘a’ signifies (2), (Def. of truth)
(4) It is not the case that ‘a’ is not true (1), (3), (T1.2)
(5) ‘a’ is true (4)
(6) ‘a’ signifies itself to be true (1), (2)–(5), (P2)
Most steps of this argument seem clear enough. In essence, (3) expresses what
it is for ‘a’ not to be true. 21 Then at step (4), ‘the case that ‘a’ is not true’ is
substituted for ‘wholly as ‘a’ signifies’ since that is all that ‘a’ signifies according to
(1). Then at step (5) the double negation is eliminated. At step (6), Bradwardine
applies his principle (P2), which is doing the crucial work.
A reader may wonder about the line (2) and the application of the principle
(P2). Is line (2) assumed for the sake of conditional proof? The lines (2)–(5)
would then infer “a’ is true’ from “a’ is not true’. The consequence would then
be used at (6). This interpretation seems a good suggestion, especially in view of
applying the principle (P2) on (1) to achieve (6). However, again it seems that it
is (P2)* that is needed, since line (2) appears to be what ‘a’ signifies and not the
examined sentence ‘a’ itself. 22 The reasoning seems to run from ‘a’ :¬T ‘a’ (cf.
(1)), proving (on steps (2)–(5)) the consequence ¬T ‘a’ → T ‘a’ and using it with
the principle (P2)* to achieve ‘a’ :T ‘a’.
The second part of the proof is more difficult. What if the examined sentence
means something else as well? Bradwardine proceeds as follows. Let that other
signification be ‘B is C’, and run the reasoning similarly. Now it is as follows: 23
20 In Stephen Read’s translation, the text runs as follows: “Let A be a proposition signifying
itself not to be true, whether signifying anything else or not. If not, then it follows from the
assumption that A is not true, by definition (D1), that it is not wholly as is signified by A, and
by A is signified only that A is not true (we supposed), so by thesis (T1.2) it follows that it is
not the case that A is not true, that is, that A is true, whence by postulate (P2) A signifies itself
to be true.” [Bradwardine, internet, 45]. The proof has been analyzed in [Spade, 1981, 120–125;
Read, 2002, 211–212], but my analysis differs slightly from both. See also [Read, forthcoming,
8–10].
21 Bradwardine refers to his definition D1 (see footnote 19).
22 There seems to be no sufficient reason for assimilating ‘a’ with“a’ is false’, since Bradwardine
quite explicitly presents the reasoning as something that applies to any sentence ‘a’ signifying
its own falsity, and “a’ is false’ is certainly only one such.
23 The text continues from footnote 19: “If, on the other hand, A signifies more than that it