7 Modal decision problems

Abstract

Publisher Summary This chapter analyzes the development of modal logic within the research framework formulated for a historical analysis of mathematical modal logic. Modal logic is decidable when interpreted on the class of all models. By changing the interpreting class of models, the logical validities can be changed, and decidability is typically lost when the structural conditions come too close to “danger zones” such as tiling patterns, arithmetic, or other structures allowing for Turing machine computation. The chapter examines how the properties as decidability, the finite model property, and finite axiomatizability are distributed across the lattice of normal modal logics. The emphasis is on providing general results and drawing attention to important open questions. Thomason's explication of the semantical part of the research programme is discussed. The appropriate refinement of the syntactical part of the research programme for normal modal logics and solutions to it given by Chagrov and Thomason are described. The logics axiomatized by formulas satisfying certain syntactical constraints, in particular, Sahlqvist formulas, uniform formulas, and modal reduction principles are considered. The Big Research Programme for the class of all tense logics of linear flows of time is also described.