The Semantics of Entailment Omega

This paper discusses the relation between the minimal positive relevant logic B+ and intersection and union type theories. There is a marvellous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the tweaking B ∧T of B+, which saves implication → and conjunction ∧ but drops disjunction ∨. The filter models of the λ-calculus (and its intimate partner Combinatory Logic CL) of the first author and her co-authors then become theory models of these calculi. (The logician’s Theory is the algebraist’s Filter.) The coincidence extends to a dual interpretation of key particles – the subtype≤ translates to provable →, type intersection ∩ to conjunction ∧, function space → to implication and whole domainω to the (trivially added but trivial) truth T. This satisfying ointment contains a fly. For it is right, proper and to be expected that type union ∪ should correspond to the logical disjunction ∨ of B+. But the simulation of functional application by a fusion (or modus ponens product) operation ◦ on theories leaves the key Bubbling lemma of work on ITD unprovable for the ∨-prime theories now appropriate for the modelling. The focus of the present paper lies in an appeal to Harrop theories which are (a) prime and (b) closed under fusion. A version of the Bubbling lemma is then proved for Harrop theories, which accordingly furnish a model of λ and CL.