The Semantics of Entailment Omega

Abstract

This paper discusses the relation between the minimal positive relevant logic B$_{+}$ and intersection and union type theories. There is a marvelous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the tweaking B$\wedge$T of B$_{+}$, which saves implication $\to$ and conjunction $\wedge$ but drops disjunction $\vee$. The filter models of the $\lambda$-calculus (and its intimate partner Combinatory Logic CL) of the first author and her coauthors 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 $\leq$ translates to provable $\to$, type intersection $\cap$ to conjunction $\wedge$, function space $\to$ to implication, and whole domain $\omega$ 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 $\cup$ should correspond to the logical disjunction $\vee$ of B$_{+}$. But the simulation of functional application by a fusion (or modus ponens product) operation $\circ$ on theories leaves the key Bubbling lemma of work on ITD unprovable for the $\vee$-prime theories now appropriate for the modeling. 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 $\lambda$ and CL.