Pretabular logics are those that lack finite characteristic matrices, although all of their normal proper extensions do have some finite characteristic matrix. Although for Anderson and Belnap’s relevance logic R, there exists an uncountable set of pretabular extensions :1249–1270, 2008), for the classical relevance logic \\rightarrow B\}\) there has been known so far a pretabular extension: \. In Section 1 of this paper, we introduce some history of pretabularity and some relevance logics and their algebras. In Section 2, we introduce (...) a new pretabular logic, which we shall name \, and which is a neighbor of \, in that it is an extension of KR. Also in this section, an algebraic semantics, ‘\-algebras’, will be introduced and the characterization of \ to the set of finite \-algebras will be shown. In Section 3, the pretabularity of \ will be proved. (shrink)

We exhibit infinitely many semisimple varieties of semilinear De Morgan monoids (and likewise relevant algebras) that are not tabular, but which have only tabular proper subvarieties. Thus, the extension of relevance logic by the axiom $$(p\rightarrow q)\vee (q\rightarrow p)$$ ( p → q ) ∨ ( q → p ) has infinitely many pretabular axiomatic extensions, regardless of the presence or absence of Ackermann constants.

KR is Anderson and Belnap’s relevance logic R with the addition of the axiom of EFQ: \ \rightarrow q\). Since KR is relevantistic as to implication but classical as to negation, it has been dubbed, among many others, a ‘classical relevance logic.’ For KR, there have been known so far just two pretabular normal extensions. For these pretabular logics, no simple axiomatizations have yet been presented. In this paper, we offer some and show that they do the job. We also (...) introduce some Routley–Meyer semantic conditions for these axioms. All these may facilitate realizing our desire to discover other classical pretabular extensions over KR, if such extensions exist. (shrink)