Many different systems of quantified modal logic have been presented in this book, each one based on the minimal system fK. In the next few chapters, we will show the adequacy of many of these logics by showing both their soundness and completeness. When S is one of the quantified modal logics discussed, and the corresponding notion of an S-model has been defined, soundness and completeness together amount to the claim that provability-in-S and S-validity match.

1. (Soundness) If H ⊢S C then H ⊧S C.

2. (Completeness) If H ⊧S C then H ⊢S C.

This chapter will be devoted to soundness and to some theorems that will be useful for the completeness proofs to come. Some of these results are interesting in their own right since they show how the various treatments of the quantifier are interrelated. Sections 15.4–15.8 will explain how notions of validity for the substitution, intensional, and objectual interpretations are shown equivalent to corresponding brands of validity on truth value models – the simplest kind of models. This will mean that the relatively easy completeness results for truth value models can be quickly transferred to substitution, intensional, and objectual forms of validity. Readers who wish to study only truth value models may omit those sections.

Two different strategies will be presented to demonstrate completeness for truth value models. Chapter 16 covers completeness using a variation on the tree method found in Chapter 8. The modifications needed to extend the completeness result to systems with quantifiers are fairly easy to supply.