Truth values algebras and proof normalization

Abstract

We extend the notion of Heyting algebra to a notion of truth values algebra and prove that a theory is consistent if and only if it has a B-valued model for some non trivial truth values algebra B. A theory that has a B-valued model for all truth values algebras B is said to be super-consistent. We prove that super-consistency is a model-theoretic sufficient condition for strong normalization.