Two Approaches to Substructural Modal Logic

Abstract

Two semantic approaches to substructural modal logics may be distinguished. Firstly, one takes Kripke frames hS, Ri, where R is a binary relation on the non-empty set S, and extends the frame to a model by adding an evaluation function from atomic formulas to a residuated lattice. More generally, one may use any lattice of truth values. The standard approach here is to define logics semantically by fixing one specific lattice of truth values.