Sunday, December 21, 2008

The impossibility of a naturalistic semantics

The stuff below may be old hat. But it's fun.

A naturalistic semantics would give an account of the truth of a naturalistically acceptable sentence (i.e., sequence of symbols) in scientific terms (I am not asking for the naturalistic semantics to give an account of the truth of non-naturalistic sentences, though that might be letting the naturalist off too easy). It would, thus, give a naturalistically acceptable predicate (perhaps a very logically complex one) T such that a naturalistically acceptable sentence s satisfies T if and only if s expresses a truth. Thus, for instance, T will be such that "A dog is running" satisfies T if and only if a dog is running.

A complete naturalistic semantics is impossible, and its impossibility can be shown in a way parallel to the proof of Goedel's first incompleteness theorem. (I am now thinking of ways of generalizing the incompleteness theorem to something very, very general. This is just one application.) Any syntactically permissible combination of naturalistically acceptable terms, logical constants, and quantification over naturalistically acceptable entities (and that should include sentences, since we can model these mathematically as sequences of symbols) should be a naturalistically acceptable sentence. Let P be any naturalistically acceptable predicate such that the sentence

  1. Every sentence s satisfying P fails to satisfy T
is in fact the one and only sentence that satisfies P. (For instance, P might simply specify the time and place at which (1) is written.) Then (1) is a naturalistically acceptable sentence, and so (1) is true iff (1) satisfies T. It follows from the fact that (1) is the one and only sentence that satisfies P that (1) is true iff (1) is false, which is absurd.

I think a case can be made from this that there is no naturalistically acceptable property equivalent to truth. This is a good argument against naturalism.

(A challenge is to show that this does not lead to a paradox for the non-naturalist. I think there is a principled way in which one can count as nonsense sentences that directly or indirectly talk of their own truth. But (1) doesn't do that—it talks of the sentence s's satisfying T, where T is some natural predicate, not truth.)

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