Saturday, November 15, 2008

Curry's Paradox

A sentence is true insofar as it describes reality, ordinarily (and adequately enough here), so consider the following sentence:
(C)......If this sentence is true then so is the sentence (S).
Suppose that (C) is true. We would have, not only that (C) was true but also—from (C)’s definition—that from (C) being true we would have the truth of (S), so we would have, consequently, that (S) is true. That is, if (C) is, as we supposed, true then (S) is true. But that means—from (C)’s definition—that (C) is true.
......It seems that, logically, (C) is true, which is a paradox because (C) cannot be true, of course, because (S) was arbitrary, and some sentences are certainly false. If there was (as there seems) nothing wrong with our logical steps, then there was something wrong with our original sentence. And note that (C) says (of itself) that contradictions—e.g. (S) = “2 + 2 = 5”—follow from its truth, so essentially it says (of itself) that it is not true. That is, (C) is a Liar-style sentence, and the traditional resolution of such sentences is that they are senseless, a bit like such nonsense as “I met a man who wasn’t there” (can be).
......The propositional content of a Liar sentence such as “This sentence is not true,” which I shall call ‘(L),’ is essentially the same as that of the clearly empty “Disobey this command!” Such an utterance from one’s superior would naturally prompt the thought: What command? And note that just as (L) can seem true—since it seems to say that (L) is not true, and (L) is senseless so it is not—paradoxically because it is not, so similarly (C) can seem true: If (C) is false then it is false that from a falsehood—that (C) is true—we might deduce a contradiction—via all sentences being true—whereas such a deduction is plausibly alright.

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