A Second Pretabular Classical Relevance Logic

Abstract

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 (Swirydowicz in J Symb Log 73(4):1249–1270, 2008), for the classical relevance logic $\hbox {KR} = \hbox {R} + \{(A\,\, \& \sim A)\rightarrow B\}$ there has been known so far a pretabular extension: ${\mathcal L}$ (Galminas and Mersch in Stud Log 100:1211–1221, 2012). 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 ${\mathcal M}$, and which is a neighbor of ${\mathcal L}$, in that it is an extension of KR. Also in this section, an algebraic semantics, ‘${\mathcal M}$-algebras’, will be introduced and the characterization of ${\mathcal M}$ to the set of finite ${\mathcal M}$-algebras will be shown. In Section 3, the pretabularity of ${\mathcal M}$ will be proved.