A Term Assignment for Dual Intuitionistic Logic

We study the proof-theory of co-Heyting algebras and present a calculus of continuations typed in the disjunctive–subtractive fragment of dual intuitionistic logic. We give a single-assumption multiple-conclusions Natural Deduction system NJ for this logic: unlike the best-known treatments of multiple-conclusion systems (e.g., Parigot’s λ−μ calculus, or Urban and Bierman’s term-calculus) here the term-assignment is distributed to all conclusions, and exhibits several features of calculi for concurrency, such as remote capture of variable and remote substitution. The present construction can be regarded as the construction of a free co-Cartesian closed category, dual to the familiar construction of a free Cartesian-closed category from the syntax of positive intuitionistic logic. Here duality is extended from formulas to proofs and it is shown that every computation in our calculus of continuations is isomorphic to a computation in the simply typed λ-calculus. An informal interpretation of this system in the framework of the logic for pragmatics is suggested as a calculus of refutations in the logic of conjectures. §1. Preface. This paper is a contribution to the proof-theory of co-Heyting algebras. A co-Heyting algebra is a (distributive) lattice C such that its opposite C is a Heyting algebra. In C the operation of Heyting implication B → A is defined by the familiar adjunction, thus in the co-Heyting algebra C subtraction (i.e., co-implication) A r B is defined dually C ∧B ≤ A C ≤ B → A A ≤ B ∨ C A r B ≤ C Thus we have an intuitionistic propositional formal system, where formulas are built from atoms and a constant for falsity using disjunction and subtraction, which is the dual of a fragment of minimal (positive intuitionistic) logic. Proofs can be represented in a sequent calculus LJrg, where sequents are restricted to a single formula in the antecedent; here the rules for subtraction are precisely dual to the rules for implication Thanks to Stefano Berardi, Corrado Biasi, Tristan Crolard, Arnaud Fleury, Nicola Gambino, Maria Emilia Maietti, Kurt Ranalter, Edmund Robinson and Graham White for their help and cooperation at various stages of the project.