Yablo's paradox is a logical paradox published by Stephen Yablo in 1993 that is similar to the liar paradox. Unlike the liar paradox, which uses a single sentence, this paradox uses an infinite sequence of statements, each of which refers 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, although no statement directly refers to itself.
Paradox and analysis
Yablo's paradox arises from considering the following infinite set of sentences:
- (S1): for all k > 1, Sk is false
- (S2): for all k > 2, Sk is false
- (S3): for all k > 3, Sk is false
The paradox can be analyzed as follows. First, suppose that some statement Si is true. Then it follows from the statement of Si that every statement later in the sequence is false, and in particular that Si+1 is false. Hence, since Si+1 is false, there is some j>i+1 such that Sj is true. But, because j is also greater than i, this means that Si must have been false. This is a contradiction, so the original assumption that Si is true must be wrong. Thus Si must be false for every i. But this means, in particular, that Si is false for every i>1, and thus S1 is true. This is paradoxical, because the analysis has already shown that S1 cannot be true.
The analysis shows that there is no consistent way to assign truth values to the statements in the paradox. Moreover, none of the sentences refers to itself, but only to the subsequent sentences; this leads Yablo to claim that his paradox does not rely on self-reference. However, this claim is disputed.