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).
......So, maybe Curry's paradox is showing us that we should not even suppose the truth of some conditionals that are as true as not. Of course, C does just that, with its "if C". But that just means that its grammatical structure might be less logical than that of, say, A, the claim that A is not true (which basically says that if A is true then pigs fly). That might explain how all the steps of our fallacious argument could have been logical, while the Liar paradox needs us to presuppose bivalency.

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