${\rm R}$-mingle and beneath. Extensions of the Routley-Meyer semantics for ${\rm R}$.
Jon Michael Dunn · Notre Dame Journal of Formal Logic · 1979
This note presupposes the notation, terminology, and results of [7].There Routley and Meyer (in their section called "R-mingle and beyond.Extensions of the semantics'') give a semantical postulate p7 0 x < a vy < a.This postulate has certain advantages over p7.x First it is more natural in that the characteristic axiom scheme for RM, A -* (A -* A), is negation-free, whereas the specific mission of the *-operation in the Routley-Meyer semantics is to provide for the treatment of negation. 2The second advantage is that Sem(l) generalizes in certain natural ways, as shall be shown, so as to provide semantical characterizations of certain natural subsystems of R.2 Semantics for RM Recall that upon defining A o B = ~(A -• ~B), we get a "consistency" connective that "imports" and "exports" (cf.[7]).This allows us to take the characteristic RM axiom in the form Syn(l) A o A^ A.Soundness Theorem If \^A, then A is valid in all r.m.s.satisfying Sem(l) {for short, all "rm.m.s.").Proof.In view of Theorem 2 in [7], it suffices to verify that Syn(l) is valid *Thanks are due for partial support to NSF grant GS-33708, and also to R. K. Meyer for many helpful conversations providing background.