Intersection types for lambda-terms and combinators and their logics

It is well known that the simple types of closed lambda terms or combinators can be interpreted as the theorems of intuitionistic implicational logic (H→). Venneri, using an equivalence between the intersection type system for lambda calculus, without the universal type ω, TA∧λ, and a similar system for combinators, TA∧, shows that the types of TA∧λ are the theorems of a Hilbert-style sublogic of the → ∧ fragment of H→.; In this paper we fill a gap in the equivalence proof and introduce a new system of intersection types for lambda calculus that is weaker than TA∧λ, but has the same set of types. The new system has the advantage that the logic of types can be obtained directly from the rules - just as H→ can be obtained from simple type theory.