Cut-free Gentzen calculus for multimodal [formula omitted

Abstract

This paper extends previous work on the modal logic CK as a reference system, both proof-theoretically and model-theoretically, for a correspondence theory of constructive modal logics. First, the fundamental nature of CK is discussed and compared with the intuitionistic modal logic IK which is traditionally taken to be the base line. Then, it is shown, that CK admits of a cut-free Gentzen sequent calculus G - CK which has (i) a local interpretation in constructive Kripke models and (ii) does not require explicit world labels. Finally, the paper demonstrates how non-classical modal logics such as IK , C S 4 , CL , or Masiniʼs deontic system of 2-sequents arise as theories of CK , presented both as special rules and as frame classes.