The proof theory and semantics of intuitionistic modal logic

TLDR: This thesis investigates the intuitionistic modal logics that arise when the semantic definitions in the ordinary meta-theory of informal classical mathematics are interpreted in an intuitionistic meta- theory that no longer satisfy certain intuitionistically invalid principles.

Abstract: Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary meta-theory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic meta-theory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are self-justifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic first-order logic. It is also established that, in many cases, the natural deduction systems induce well-known intuitionistic modal logics, previously given by Hilbert-style axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their susceptibility to proof-theoretic techniques. Strong normalization (and confluence) results are proved for all of the systems. Normalization is then used to establish the completeness of cut-free sequent calculi for all of the systems, and decidability for some of the systems. Lastly, techniques developed throughout the thesis are used to establish that those intuitionistic modal logics proved decidable also satisfy the finite model property. For the logics considered, decidability and the finite model property presented open problems.