9 First-order modal logic

Abstract

First-order modal logics are modal logics in which the underlying propositional logic is replaced by a first-order predicate logic. They pose some of the most difficult mathematical challenges. This chapter surveys basic first-order modal logics and examines recent attempts to find a general mathematical setting in which to analyze them. A number of logics that make use of constant domain, increasing domain, and varying domain semantics is discussed, and a first-order intensional logic and a first-order version of hybrid logic is presented. One criterion for selecting these logics is the availability of sound and complete proof procedures for them, typically axiom systems and/or tableau systems. The first-order modal logics are compared to fragments of sorted first-order logic through appropriate versions of the standard translation. Both positive and negative results concerning fragment decidability, Kripke completeness, and axiomatizability are reviewed. Modal hyperdoctrines are introduced as a unifying tool for analyzing the alternative semantics. These alternative semantics range from specific semantics for non-classical logics, to interpretations in well-established mathematical framework. The relationship between topological semantics and D. Lewis's counterpart semantics is investigated and an axiomatization is presented.