8 Modal consequence relations

Abstract

Publisher Summary This chapter describes the basic concepts from universal algebra and basic logical notions such as consequence relations, rules, the deduction theorem, and interpolation. It also focuses at modal consequence relations and the structure of the lattice they form and the notion of a splitting. The notion of consequence is a fundamental logical concept and in the setting of modal logic, it can be defined in a number of different ways. The consequence relations from an algebraic perspective; the global consequence relations, and the connection between semisimple varieties of modal algebras and weak transitivity are outlined in the chapter. The chapter the reviews results establishing that the lattices of polymodal and polyadic logics can be naturally embedded into the lattice of monomodal logics preserving and reflecting a good deal of properties. An algebraic characterization of interpolation and ways of establishing interpolation for logics, Beth-definability, and fixed point theorems are also reviewed. The chapter then outlines some most important ideas, such as local versus global consequence, reducing multimodal consequence to monomodal consequence, interpolation theorems, and the admissibility of rules.