Ternary relations and relevant semantics

Abstract

Modus ponens provides the central theme. There are laws, of the form A→ C. A logic (or other theory) L collects such laws. Any datum A (or theory T incorporating such data) provides input to the laws of L. The central ternary relation R relates theories L, T and U, where U consists of all of the outputs C got by applying modus ponens to major premises from L and minor premises from T. Underlying this relation is a modus ponens product (or fusion) operation ∘ on theories (or other collections of formulas) L and T, whence RLTU iff L∘ T⊆ U. These ideas have been expressed, especially with Routley, as (Kripke style) worlds semantics for relevant and other substructural logics. Worlds are best demythologized as theories, subject to truth-functional and other constraints. The chief constraint is that theories are taken as closed under logical entailment, which clearly begs the question if we are using the semantics to determine which theory L is Logic itself. Instead we draw the modal logicians’ conclusion—there are many substructural logics, each with its appropriate ternary relational postulates. Each logic L gives rise to a Calculus of L-theories, on which particular candidate logical axioms have the combinatorial properties expected from the well-known Curry–Howard isomorphism (with improvements by Dezani and her fellow intersection type theorists.). We apply their bubbling lemma, updating the Fools Model of Combinatory Logic at the pure → level for the system B ∧ T . We make fusion ∘ an explicit connective, proving a combinator correspondence theorem. Having taken relevant → as a left residual for ∘, we explore its right residual mate → r. Finally we concentrate on and prove a finite model property for the classical minimal relevant logic CB, a conservative extension of the minimal positive relevant logic B + .