## Mixed Yablo Paradoxes

The Yablo Paradox (Yablo, Stephen 1993) is an infinite sequence of sentences of the form:
S1: For all m > 1, Sm is false.
S2: For all m > 2, Sm is false.
S3: For all m > 3, Sm is false.
: : : :
Sn: For all m > n, Sm is false.
: : : :
Loosely put, each sentence in the Yablo sequence ‘says’ that all of the sentences ‘below’ it on the list are false. The Yablo sequence is paradoxical – there is no coherent assignment of truth and falsity to the sentences in the list – as is shown by the following informal argument:
Proof of paradoxicality: Assume that Sk true, for arbitrary k. Then, for every m > k, Sm is false. It follows that Sk+1 is false. In addition, it follows that, for every m > k + 1, Sm is false. But given what Sk+1 says (that every sentence ‘below’ it is false), it follows that Sk+1 is true. Contradiction. Thus, Sk cannot be true, and must be false. Since k was arbitrary,. . .

News source: OUPblog » Philosophy