Labelled Modal Logics: Quantifiers

In previous work we gave an approach, based on labelled natural deduction, for formalizing proof systems for a large class of propo-sitional modal logics that includes K, D, T, B, S4, S4:2, KD45, and S5. Here we extend this approach to quantiied modal logics, providing for-malizations for logics with varying, increasing, decreasing, or constant domains. The result is modular with respect to both properties of the accessibility relation in the Kripke frame and the way domains of individuals change between worlds. Our approach has a modular metatheory too; soundness, completeness and normalization are proved uniformly for every logic in our class. Finally, our work leads to a simple implementation of a modal logic theorem prover in a standard logical framework.