Metamathematics of Modal Logic

Abstract

"The formal study of symbols of systems either in their relation to one another (syntax) or in their relation to assigned meanings (semantics) is called metamathematics ..." (R. Feys and F.B. Fitch, Dictionary of Symbols of Mathematical Logic, North Holland 1969.)

The techniques employed in the semantic analysis of non-classical propositional languages fall roughly into two kinds. The first of these, the algebraic method, uses lattices with operators to interpret languages. Each formula induces a polynomial function on the appropriate algebras, with propositional variables ranging over elements of the lattice, and the logical connectives corresponding to its algebraic operators. The other approach is sometimes called model-theoretic, but is probably better described simply as set-theoretic semantics. Here the models, or frames, consist of sets carrying structural features other than finitary operations, such as neighbourhood systems and finitary relations. In this context formulae are interpreted as subsets of the model, in a manner constrained by its particular structure.