Mixed Yablo Paradoxes

The collection of infinite Yabloesque sequences that contain both infinitely many Y-all sentence and infinitely many Y-exists sentences, however, is a much larger collection. It is what is called
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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,. . .

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News source: OUPblog » Philosophy

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