Sunday, December 2, 2007


One objection Williamson considers to his view is that not all counterpossibles are vacuously true. If this is right, then the principle NECESSITY is false:
(NECESSITY) [](A->B)->(A[]->B)
NECESSITY states that if it's necessary that A implies B, then if I were the case then B would be the case. One could hold that the principle fails when A is an impossibility. If A is an impossibility, then trivially in every possible world at which A holds, B also holds (there are no such worlds). However, even so it might still be intelligable (and correct) to say that were A to hold, B would not hold.
Williamson makes a case against this when considering a mathematician doing a reductio argument. His main strategy is to trade off truth for assertibility in such cases. Suppose you're trying to prove fermat's last theorum by using an impossible solution (S) to it to generate absurdity. You can use counter-factual reasoning to assert S []-> A, but can't assert S []->~S for the purposes of your proposed proof. He says this does not tell against the truth of S[]->~S. After all, a contradiction could be used to prove anything eventually, even if it will take more steps.
My concern is that strategies like this won't work for some logics. Suppose para-consistent logics are necessarily not correct. It still seems right to say that if they were correct then certain things would hold, and if they were correct certain other things wouldn't. After all, they're specifically designed to be able to handle such cases.

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