Quantified Modal Logics: One Approach to Rule (Almost) them All!

We present a general approach to quantified modal logics that can simulate most other approaches. The language is based on operators indexed by terms which allow to express de re modalities and to control the interaction of modalities with the first-order machinery and with non-rigid designators. The semantics is based on a primitive counterpart relation holding between n-tuples of objects inhabiting possible worlds. This allows an object to be represented by one, many, or no object in an accessible world. Moreover by taking as primitive a relation between n-tuples we avoid some shortcoming of standard individual counterparts. Finally, we use cut-free labelled sequent calculi to give a proof-theoretic characterisation of the quantified extensions of each first-order definable propositional modal logic. In this way we show how to complete many axiomatically incomplete quantified modal logics.