ABSTRACT

We can generalize this to produce longer indirectly self-referential loops:

These will have to be alternately true and false: TFTF . . . or FTFT . . . Since S1 has to be true if Sn is true, and false if Sn is false, we get a paradox in the cases where n is even, since in those cases S1 and Sn will have different truth values. (If n is odd, the list is paradoxical in the way the truth-teller is, since there is no way of choosing between the two alternative assignments. But set those cases aside here.)

Stephen Yablo’s paradox involves an infinite sequence of sentences:

Yablo claims that, unlike other versions of the liar, this paradox does not involve self-reference, since each sentence is about those following it and no sentence is about itself. But each sentence seems to be implicitly self-referential, since ‘all of the following sentences’ has in each case to be understood as ‘all the sentences following this one’. (Yablo actually refers to the following sentences by ‘for all k > n’, where n is the current subscript to Y, but selfreference is still arguably implicit there.)

But, whether self-referential or not, we clearly have a paradox here. See the entry on The Liar for possible resolutions: on the third approach canvassed there, none of the sentences in the infinite sequence can be allowed to express a statement.