A Systematic Presentation of Quantified Modal Logics

this paper is an attempt at providing a systematic presentation of Quantified Modal Logics (with constant domains and rigid designators). We present a set of modular, uniform, normalizing, sound and complete labelled sequent calculi for all QMLs whose frame properties can be expressed as a finite set of first-order sentences with equality. We first present CQK, a calculus for the logic QK, and then we extend it to any such logic QL. Each calculus, called CQL, is modular (obtained by adding rules to CQK), uniform (each added rule is clearly related to a property of the frame), normalizing (frame reasoning only happens at the top of the proof tree) and Kripke-sound and complete for QL. We improve on the existing literature on the subject (mainly, [21]) by extending the class of logics for which such a presentation is given, and by giving a new proof of soundness and completeness.