First-order modal logic
Bertrand Russell, Herbrand's Theorem, and the Assignment Statement
AISC '98 Proceedings of the International Conference on Artificial Intelligence and Symbolic Computation
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First-order modal logic, in the usual formulations, is not sufficiently expressive, and as a consequence problems like Frege's morning star/evening star puzzle arise. The introduction of predicate abstraction machinery provides a natural extension in which such difficulties can be addressed. But this machinery can also be thought of as part of a move to a full higher-order modal logic. In this paper we present a sketch of just such a higher-order modal logic: its formal semantics, and a proof procedure using tableaus. Naturally the tableau rules are not complete, but they are with respect to a Henkinization of the "true" semantics. We demonstrate the use of the tableau rules by proving one of the theorems involved in Gödel's ontological argument, one of the rare instances in the literature where higher-order modal constructs have appeared. A fuller treatment of the material presented here is in preparation