Yablo's paradox

Yablo's paradox is a logical paradox published by Stephen Yablo in 1993. It is similar to the liar paradox. Unlike the liar paradox, which uses a single sentence, this paradox uses an infinite sequence of statements, each referring to the truth values of the later statements in the sequence. Analysis of the statements shows there is no consistent way to assign truth values to all the statements. Since each sentence does not refer to itself, Yablo claims his paradox is "not in any way circular." However, Graham Priest disputes this.

Consider the following infinite set of sentences:

Assume there is an n such that S_n is true. Then S_{n + 1} is not true, so there is some k > n + 1 such that S_k is true. But S_k is not true because S_n is true and k > n. Assuming S_n is true implies a contradiction, S_k is true and not true, so our assumption is absurd. We must conclude for each i, the sentence S_i is not true. But if S_i is not true, it is also true, so we have the paradox that each sentence in Yablo's list is true and not true.

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