Yablo's paradox

Abstract

Stephen Yablo has given an ingenious liar-style paradox that, he claims, avoids self-reference, even of an indirect kind, one that is, in fact, 'not in any way circular' (Yablo 1993, his italics). He infers that such circularity is not necessary for this kind of paradox. Some others have agreed.1 The point of this note is to demonstrate that self-referential circularity is involved in Yablo's paradox. I shall also show that Yablo's paradox has exactly the same structure as all the familiar paradoxes of set theory and semantics. To put the discussion into context, think, first, of the standard Liar paradox, 'This sentence is not true'. Writing T as the truth predicate, then the Liar sentence is one, t, such that t = '-iTt'. The fact that 't' occurs on both sides of the equation makes it a fixed point of a certain kind, and, in this context, codes the self-reference. In this particular case, the existence of the fixed point is obvious, due to the use of the demonstrative 'This sentence', but in general one often has to work quite hard to show the existence of fixed points. For example, if we take naming to be implemented by g6delization, we have to prove the existence of a certain number. Specifically, we show, by a diagonalization argument, that if a(x) is any formula of one free variable, x, there is a number, n, such that n is the code of the formula a(n) where n is the numeral of n or, at least, of one logically equivalent to it. n is the fixed point.2