Formulae-as-types for an involutive negation

Abstract

Negation is not involutive in the λC calculus because it does not distinguish captured stacks from continuations. We show that there is a formulae-as-types correspondence between the involutive negation in proof theory, and a notion of high-level access to the stacks studied by Felleisen and Clements. We introduce polarised, untyped, calculi compatible with extensionality, for both of classical sequent calculus and classical natural deduction, with connectives for an involutive negation. The involution is due to the ℓ delimited control operator that we introduce, which allows us to implement the idea that captured stacks, unlike continuations, can be inspected. Delimiting control also gives a constructive interpretation to falsity. We describe the isomorphism there is between A and ¬¬A, and thus between ¬∀ and ∃¬.