......But, ordinary modal logic operates in a space of descriptions that are either true or else false. So perhaps we should not have supposed, to begin with, that C is true, because C is true only insofar as it is not true (because insofar as it is true we get a contradiction) and so it is as true as not. That is a consistent possibility because it is as true as not that a contradiction follows from a statement that is as true as not (since it would follow from a statement that was false). And it resolves the paradox because ordinary logic breaks down with propositions that are only as true as not (e.g. see the Sorites paradox, higher-order vagueness, and the revenge problem for this resolution of the Liar paradox).

......Curry's paradox shows that we should not even

*suppose*the truth of some conditionals that are as true as not. The grammatical structure of C is therefore less logical than that of, say, A, the claim that A is not true, even though you could paraphrase A as the claim that if A is true then pigs fly. We suppose, for the sake of argument, the truth of claims like A when we reason about the Liar paradox, but we should not do that with the paraphrase. Cf. the peculiarity of the Liar, that A claims that A is true, since it claims that it is not true that A is not true.

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