Improved Decision Procedures for Pure Relevant Logic

Abstract

We study Church’s “weak implicational calculus” of 1951, which is the pure implicational part R→ of the relevant logic R. We investigate and develop, again following Church, the effect of adding propositional quantifiers to R→; this allows also the specification of a minimal, a De Morgan and finally a Boolean negation. The bulk of the paper deals with finite model properties for various fragments of R, including R→. A modification of an argument due to Saul Kripke is the chief tool in this project, yielding an Infinite Division Prinicple (IDP) for “Church monoids” that facilitates our finitization. Thus systems intermediate between R→ and LR (non-distributive R) have the finite model property and are hence model-theoretically decidable. A concluding section shows how to define various logical particles using Church’s propositional quantifiers, which turns what are conservative extension results at the quantifier-free level into axiomatic extensions when quantifiers are present.