Monday, June 11, 2007

Curry's Paradox

......Let A = ‘C, if A,’ where C is a contradiction. E.g. “If this statement is true then 0 = 1.” If indicative conditionals were material (if, from a falsity, anything followed) we would be able to get C even from a false A and then, since we’d also get C from a true A (as that is just what A says), we would have C, rather paradoxically; but are they? If they are suppositional (if ‘C, if A’ asserts nothing if A is not true) then we could only deduce (correctly) that since if A were true then C would be true, hence A is not true. Still, the subjunctive version remains puzzling, as follows.
......Let S = ‘If S were true, something that is not true would be true.’ Much as before, S is not true because if it were true then (its antecedent being true) it would be true that something that is not true is true (which is a contradiction). But S therefore seems to be saying that if something (i.e. S) that is not true were true, then something that is not true would be true, which appears to be correct. So although S is certainly not true, it therefore appears to be true. Still, since that appearance must be deceptive, we might resolve this paradox along the lines of the Liar paradoxes, as follows.
......Let L = ‘L is not true.’ By the definition of ‘not,’ L is either true or not true, and if L is true then L is (as it says) not true, whence L is not true. But since we know that L is not true, L therefore appears (paradoxically) to be saying something true. The best explanation for that seems (in my opinion) to be that, whilst another expression (i.e. not L) could be sensibly used to assert that L is not true (as that did), L cannot itself say that because it is nonsense. So perhaps the best explanation for why S does not appear to be nonsense is that it resembles (very closely) a true expression?

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