On the existence of finite models and decision procedures for propositional calculi

Abstract

1. Introduction. In this paper we consider certain general properties of propositional calculi. Two forms of the definition of a finite model of such a calculus are discussed, these forms differing in the manner prescribed for the satisfaction of the rules of the calculus. The methods of definition are shown to be equivalent for the application of the finite model method for the demonstration of the unprovability of formulae in the calculus. It is further proved that, although the decidability of a calculus follows from the existence of a finite model counter-example for each unprovable formula of the calculus, the converse result is not true.