Two pretabular linear extensions of relevance logic R

Abstract

Pretabularity is the attribute of logics that are not characterised by finite matrices, but all of whose proper extensions are. Two of the first-known pretabular logics were Dummett’s famous super-intuitionistic logic LC and the well-known semi-relevance logic RM (= R-Mingle). In this paper, we investigate Anderson and Belnap’s relevance logic R with the extra axiom: (p → q) ∨ (q → p), which we name LR, and which is (much) weaker than RM, and so is not pretabular. This means that over LR, there may be some pretabular extensions other than RM, two of which this paper presents and with which it provides axiomatizations, characteristic algebras, and Routley-Meyer semantics.