Saturday, October 27, 2012

Curry's paradox

Consider C, which says that if C then B, where B says that black is white. Suppose that C is true. Then it is true that C implies B. So if C is true, then B is true. But therefore C is true. So B is true. That in a nutshell is Curry's paradox. Every step in that argument that black is white appears to be a logical step.
......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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