A Minimal Propositional Type Theory

Abstract

Propositional type theory, first studied by Henkin, is the restriction of simple type theory to a single base type that is interpreted as the set of the two truth values. We show that two constants (falsity and implication) suffice for denotational and deductive completeness. Denotational completeness means that every value of the full set-theoretic type hierarchy can be described by a closed term. Deductive completeness is shown for a sequent-based proof system that extends a propositional natural deduction system with lambda conversion and Boolean replacement.