In "Handbook of Philosophical Logic" M. Dunn formulated a problem of describing pretabular extensions of relevant logics (cf. M. Dunn [1984], p. 211: M. Dunn, G. Restall [2002], p. 79). The main result of this paper described in the title.

In this paper we construct an extension, ℒ, of Anderson and Belnap's relevance logic R that is classical in the sense that it contains p&p → q as a theorem, and we prove that ℒ is pretabular in the sense that while it does not have a finite characteristic matrix, every proper normal extension of it does. We end the paper by commenting on the possibility of finding other classical relevance logics that are also pretabular.

We study modal logics in the setting of varieties of modal algebras. Any variety of modal algebras generated by a finite algebra — such, a variety is called tabular — has only finitely many subvarieties, i.e. is of finite height. The converse does not hold in general. It is shown that the converse does hold in the lattice of varieties of K4-algebras. Hence the lower part of this lattice consists of tabular varieties only. We proceed to show that there is (...) a continuum of pretabular varieties of K4-algebras — those are the non-tabular varieties all of whose proper subvarieties are tabular — in contrast with Maksimova's result that there are only five pretabular varieties of S4-algebras. (shrink)