Ternary relations and relevant semantics

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 i LT 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&...

• Type: Journal
• Year: 2004
• Where: APAL
• Authors: Robert K. Meyer
• Keywords: APAL 2004, Central Ternary Relation, Logic, Substructural Logics