Adding Negation to Lambda Mu

TL;DR

The paper addresses integrating negation into a classical logic computational framework by extending the $λμ$-calculus to form ${\cal L}$ with explicit negation as a type constructor. It introduces a reduction system augmented with two new rules, proves subject reduction, and establishes confluence via Aczel's generalisation of parallel reduction. Using Girard's reducibility candidates, it shows strong normalization for typeable terms and, leveraging principal typing, demonstrates robust type assignment within ${\cal L}$. The work provides a principled, confluent, and strongly normalizing foundation for representing classical proofs as typed terms with explicit negation, with potential implications for proof assistants and formalizations of classical reasoning in computation.

Abstract

We present $\cal L$, an extension of Parigot's $λμ$-calculus by adding negation as a type constructor, together with syntactic constructs that represent negation introduction and elimination. We will define a notion of reduction that extends $λμ$'s reduction system with two new reduction rules, and show that the system satisfies subject reduction. Using Aczel's generalisation of Tait and Martin-Löf's notion of parallel reduction, we show that this extended reduction is confluent. Although the notion of type assignment has its limitations with respect to representation of proofs in natural deduction with implication and negation, we will show that all propositions that can be shown in there have a witness in $\cal L$. Using Girard's approach of reducibility candidates, we show that all typeable terms are strongly normalisable, and conclude the paper by showing that type assignment for $\cal L$ enjoys the principal typing property.