Here we give completeness proofs for many quantified modal logics, using a variant of the method of maximally consistent sets. Although the previous chapter already established completeness for many quantified modal logics using the tree method, there are good reasons for covering the method of maximally consistent sets as well. First, this is the standard approach to obtaining completeness results, so most students of modal logic will want some understanding of the method. Second, the tree method applied only to those systems for which it was shown how to convert a tree into a proof. The method of maximally consistent sets applies to more systems, though it has limitations described below in Section 17.2.

How Quantifiers Complicate Completeness Proofs

One might expect that proving completeness of quantified modal logic could be accomplished by simply “pasting together” standard results for quantifiers with those for propositional modal logic. Unfortunately, it is not so easy. In order to appreciate the problems that arise, and how they may be overcome, let us first review the strategies used to show completeness for propositional modal logic with maximally consistent sets. Then it will be possible to outline the difficulties that arise when quantifiers are added.

The basic idea behind completeness proofs that use maximally consistent sets is to show that any argument H / C that is not provable has a counterexample.