The "Relevance" of Intersection and Union Types

Notre Dame Journal of Formal Logic 38 (2):246-269 (1997)
Abstract
The aim of this paper is to investigate a Curry-Howard interpretation of the intersection and union type inference system for Combinatory Logic. Types are interpreted as formulas of a Hilbert-style logic L, which turns out to be an extension of the intuitionistic logic with respect to provable disjunctive formulas (because of new equivalence relations on formulas), while the implicational-conjunctive fragment of L is still a fragment of intuitionistic logic. Moreover, typable terms are translated in a typed version, so that --typed combinatory logic terms are proved to completely codify the associated logical proofs
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