Metamathematics of Modal Logic

<p dir="ltr">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. </p><p dir="ltr">The two kinds of model are intimately related. Frames may be constructed from algebras through various lattice representation theorems. Algebraic models may be obtained as subset lattices of frames. Furthermore, the syntactical frame constructions in the Henkin style that are now widely employed in set-theoretic semantics may be mirrored on the algebraic level to produce representations of lattices.</p>