The completeness of quantified modal logics can be shown with the tree method by modifying the strategy used in propositional modal logic. Section 8.4 explains how to use trees to demonstrate the completeness of propositional modal logics S that result from adding one or more of the following axioms to K: (D), (M), (4), (B), (5), (CD). In this chapter, the tree method will be extended to quantified modal logics based on the same propositional modal logics. The reader may want to review Sections 8.3 and 8.4 now, since details there will be central to this discussion. The fundamental idea is to show that every S-valid argument is provable in S in two stages. Assuming that H / C is S-valid, use the Tree Model Theorem (of Section 8.3) to prove that the S-tree for H / C closes. Then use the method for converting closed S-trees into proofs to construct a proof in S of H / C from the closed S-tree. This will show that any S-valid argument has a proof in S, which is, of course, what the completeness of S amounts to.

The Quantified Tree Model Theorem

In order to demonstrate completeness for quantified modal logics, a quantified version of the Tree Model Theorem will be developed here. This will also be useful in showing the correctness of trees for the quantified systems.