On Natural Deduction for Herbrand Constructive Logics I: Curry-Howard Correspondence for Dummett's Logic LC

TL;DR

This paper develops a natural deduction and Curry-Howard correspondence for Dummett's logic LC, introducing a lambda calculus with parallel computation and communication, and proves normalization and Herbrand disjunction extraction.

Contribution

It introduces a typed lambda calculus with an operator for Dummett's axiom, establishing normalization and Herbrand disjunction extraction for Dummett's logic LC.

Findings

Typed calculus is normalizing.

Proof terms for existential formulas reduce to Herbrand disjunctions.

Curry-Howard correspondence established for Dummett's logic LC.

Abstract

Dummett's logic LC is intuitionistic logic extended with Dummett's axiom: for every two statements the first implies the second or the second implies the first. We present a natural deduction and a Curry-Howard correspondence for first-order and second-order Dummett's logic. We add to the lambda calculus an operator which represents, from the viewpoint of programming, a mechanism for representing parallel computations and communication between them, and from the viewpoint of logic, Dummett's axiom. We prove that our typed calculus is normalizing and show that proof terms for existentially quantified formulas reduce to a list of individual terms forming an Herbrand disjunction.