Denotational Semantics of the Simplified Lambda-Mu Calculus and a New Deduction System of Classical Type Theory

TL;DR

This paper develops a denotational semantics for a simplified lambda-mu calculus, revealing an infinitely layered Boolean domain structure, and introduces a new sequent calculus for classical type theory that aligns with this semantic framework.

Contribution

It provides the first denotational semantics for a simplified lambda-mu calculus with an infinitely layered Boolean domain and proposes a new sequent calculus reflecting this semantic structure.

Findings

Domains contain infinitely many ranks with Boolean structures

Absurdity type is identified as the truth value type

New sequent calculus (CTS) aligns with the semantic model

Abstract

Classical (or Boolean) type theory is the type theory that allows the type inference $\sigma \to \bot) \to \bot => \sigma$ (the type counterpart of double-negation elimination), where $\sigma$ is any type and $\bot$ is absurdity type. This paper first presents a denotational semantics for a simplified version of Parigot's lambda-mu calculus, a premier example of classical type theory. In this semantics the domain of each type is divided into infinitely many ranks and contains not only the usual members of the type at rank 0 but also their negative, conjunctive, and disjunctive shadows in the higher ranks, which form an infinitely nested Boolean structure. Absurdity type $\bot$ is identified as the type of truth values. The paper then presents a new deduction system of classical type theory, a sequent calculus called the classical type system (CTS), which involves the standard logical operators such as negation, conjunction, and disjunction and thus reflects the discussed semantic structure in a more straightforward fashion.