Algebraic Semantics

Abstract

In the preceding chapter, we introduced and investigated at some length several classes of algebraic structures, claiming that there exists a correspondence between the diagram of logics in Table 2.2 and the diagram of algebras in Table 5.1. Our present task will be to show that our claim was sound. In fact, we shall prove completeness theorems for most of the Hilbert-style calculi of Chapter 2 using the algebraic structures of Chapter 5. Subsequently, we shall see that — at least in some cases — such classes of structures are even too large for our purposes: due to the representation results of Chapter 5, in fact, the theorems of the logics at issue coincide with the formulae which are valid in a smaller (and usually much easier to tinker with) class of structures. In a few lucky cases, it will be sufficient to consider a single manageable structure, just as it happens for classical logic (even though this structure may not be just as simple and wieldy). Finally, we shall quickly browse through some applications of algebraic semantics to the solution of purely syntactical problems concerning our substructural calculi.