Semantical Approach to Cut Elimination and Subformula Property in Modal Logic

Abstract

This is a short survey of semantical study of cut elimination and subformula property in modal logics. Cut elimination is a basic proof-theoretic notion in sequent systems, and subformula property is the most important consequence of cut elimination. A special feature of our presentation is its unified semantical approach to them based on Kripke models. Along the same lines as Takano’s works on subformula property, these properties, together with finite model property, will be discussed as modifications of standard construction of canonical Kripke models. These semantical approaches will be compared with algebraic approaches in modal logics, which often take the forms of various kinds of embedding theorems. In the last part of the paper, an attempt is made to clarify connections between semantical approach to cut elimination and algebraic one.