Completeness and Soundness Results for 𝒳 with Intersection and Union Types

Abstract

With the eye on defining a type-based semantics, this paper defines intersection and union type assignment for the sequent calculus X, a substitution-free language that enjoys the Curry-Howard correspondence with respect to the implicative fragment of Gentzen's sequent calculus for classical logic. We investigate the minimal requirements for such a system to be complete i.e. closed under redex expansion, and show that the non-logical nature of both intersection and union types disturbs the soundness i.e. closed uder reduction properties. This implies that this notion of intersection-union type assignment needs to be restricted to satisfy soundness as well, making it unsuitable to define a semantics. We will look at two confluent notions of reduction, called Call-by-Name and Call-by-Value, and prove soundness results for those.