There exist exactly two maximal strictly relevant extensions of the relevant logic R

Abstract

In [60] N. Belnap presented an 8-element matrix for the relevant logic R with the following property: if in an implication A → B the formulas A and B do not have a common variable then there exists a valuation v such that v (A → B) does not belong to the set of designated elements of this matrix. A 6-element matrix of this kind can be found in: R. Routley, R.K. Meyer, V. Plumwood and R.T. Brady [82], Below we prove that the logics generated by these two matrices are the only maximal extensions of the relevant logic R which have the relevance property: if A → B is provable in such a logic then A and B have a common propositional variable.

Varieties of De Morgan monoids: Minimality and irreducible algebras

“…The first assertion of the next theorem has unpublished antecedents in the work of relevance logicians. A corresponding result for 'relevant algebras' is reported in [66,Prop. 5], but the claim and proof below are simpler.…”

“…The main motivation for RA is that it algebraizes the logic R. The algebraization process for R t and DMM carries over verbatim to R and RA, provided we use (12) as a formal device for eliminating all mention of e. Further work on relevant algebras can be found in [18,21,37,38,52,54,65,66].…”

“…(For one generator, the indicated bounds are built up using all six of the inequivalent implicational one-variable formulas of R, determined in [42].) The result is attributed in [66] to Meyer and to Dziobiak (independently).…”

There exists an uncountable set of pretabular extensions of the relevant logic R and each logic of this set is generated by a variety of finite height

“…Unfortunately, the pretabular logics described in this paper do not solve all questions concerning pretabularity of /?-logics: in particular the logics described above are not extensions of any of the two maximal strictly relevant extensions of the logic R (cf. K. Swirydowicz [1999]).…”

Structural Completeness in Relevance Logics

“…4.11]. A finitely generated relevant algebra A has a least and a greatest element, which form a Boolean subalgebra [26,Prop. 5], and A itself is a reduct-not merely a subreduct-of a De Morgan monoid.…”