Handbook of the History of Logic: - Fordham University Faculty

Treatments of the Paradoxes of Self-reference 591
(2) ‘a’ signifies that what is uttered by Socrates is false P2
(3) ‘a’ signifies that ‘a’ is false T1.2
(4) ‘a’ signifies itself to be false P2
(5) ‘a’ signifies itself to be true T2
(6) ‘a’ is false T2
The edge of the argument really comes from the fact that if a similar reasoning is
applied to ‘b’, or the proposition put forward to you in the disputation, it will not
go through. The beginning of the reasoning works, of course, in the same way:
(1)* ‘b’ signifies that Socrates utters a falsehood
(2)* ‘b’ signifies that what is uttered by Socrates false P2
(3)* ‘b’ signifies that ‘a’ is false T1.2
But after this line it is clear that the introduction of the pronoun ‘itself’ would be
fallacious. For ‘a’ isnot‘b’, it is a different token of the same type. And because
there is no self-reference, Bradwardine’s second thesis (T2) cannot be used. The
true claim that Socrates utters a falsehood gets separated from the false one.
Bradwardine’s solution of the paradox amounts thus to exact logical way of
spelling out the fact that whereas ‘a’ signifies something about itself, there is no
such self-reference for ‘b’. Thus, the meanings of ‘a’ and ‘b’ differ, and they differ
so that ‘b’ means almost everything ‘a’ does, except that ‘a’ additionally means
that ‘a’ is true. To be sure, ‘b’ does not mean that ‘a’ istrue,anditdoesnot
mean that ‘b’ is true.
Even if this is the core of the solution, working it out in an acceptable way was
not a simple project. When we look at Bradwardine’s successors, we find that in
general they did not buy Bradwardine’s (T2). There were exceptions, though. At
least Albert of Saxony and to some extent John Buridan, as we will see, accepted
(T2) in virtue of accepting the stronger claim that all sentences signify their own
truth, but this seems not to have been Bradwardine’s intention. He appears to
say that it is only the paradoxical sentences that signify their own truth. 19
Bradwardine’s proof of the crucial thesis (T2) is difficult to spell out, but it is
really the most important part of the solution, since without proof the thesis just
is not acceptable. With the proof, Bradwardine can rely only on (P2), which is
much easier to accept as it stands. To prove (T2), he first proves that any sentence
which signifies that it itself is not true also signifies that it itself is true. Then he
extends the result to sentences that signify their own falsehood. As the last part
he proves the final bit that because these sentences signify their own truth, they
are false. The crucial part of the logic is to found in the first part, which works
with the assumption that the sentence either does or does not mean something
19 The agreements and differences between Bradwardine, Buridan and Albert of Saxony have
been spelled out in detail by Stephen Read in [Read, 2002].