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Monday, May 12, 2014

A Fuzzy Argument for Crisp Negation? (Maybe.)

Here's an argument that might be taken to show that 'not,' in English, does not express a fuzzy negation function; otherwise we have a paradox. (I'll assume throughout that talk of "degrees of truth" still makes sense for other, especially formal, languages or metalanguages.) Consider sentence S1:

S1: This sentence is half true.

S1 is true, i.e. has the value 1, if and only if it is half true, i.e. has the value 1/2. Grant for the sake of argument that we'll set aside "fuzzy dialetheism", that a statement could have multiple degrees of truth. It follows that S1 is neither true nor half true. The only value that it seems to make sense to assign to it is 0.

Now consider S2:

S2: This sentence is not half true.

S2 would seem to be true, i.e. have the value 1, if and only if it is not half true, i.e. iff it does not have the value 1/2. (If S2 did have the value 1/2, it would seem to be false, and hence also have the value 0; and we have ruled out it's having two different truth degrees.) Since it can't both have the value 1 and some value other than 1, it would seem that the only value that it makes sense to assign to it is 1.

But here's the catch: In standard fuzzy logic, p and ~p are equivalent when they both take the value 1/2. So if I were to say of some statement p that it is half true, that would, assuming that 'not' is fuzzy, be in some sense equivalent to saying that it is also half false. But S2 (so the argument goes) is equivalent to the negation of S1. It does make sense then that they have opposite truth values, given the definition of negation in fuzzy logic; yet given what I've said about the equivalence of half-truth and half-falsity their content would appear to be the same, and thus that they should accordingly have the same truth value! Strange; nay, even paradoxical, no?

Well, honestly, I'm not sure. What do you think?

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