**Amir Hossein Sharafi, Asadollah Fallahi**

Iranian Institute of Philosophy

**Abstract**. One way of investigation on properties of a class of logics, such as tabularity and pretabularity, is to describe the lattice of those logics. This investigation can be done in algebraic or relational aspects. In algebraic viewpoint, we study the properties of the lattice of corresponding varieties. This helps us to invoke general results of the algebraic theory of lattices of varieties. Tabular varieties (corresponding to tabular logics) are generated by a finite algebra and pretabular ones are those which are not tabular but their pure subvarieties are tabular. Our goal is to specify the number of pretabular subvarieties of the variety corresponding to the classical relevance logic (KR). Then we would have found the number of pretabular extensions of classical relevance logic. We also want to know the number of subvarieties of these pretabular varieties as their height. For instance, Maksimova could prove that the pretabular extensions of intuitionistic logic Int and modal logic S4 are finite (three and five, respectively) [6, 7]. But in [2] Blok showed that the number of pretabular extensions of K4 is uncountable. Although all of these logics have uncountable height [3, 2], Swirydowicz deduced that relevance logic R has uncountable pretabular extensions by finite heights [8]. Furthermore, two pretabular extensions of KR have been derived, one by Galminas and Mersch in [5] and another by Fallahi in [4]. Finding a way to estimate the number of pretabular extensions of classical relevance logic (KR) and their heights has leaded us to make a KR-frame inspiring from the recession frame introduced in [1], and the relational frame used in [8]. This KR-frame gives us the idea that there are uncountable pretabular extensions for KR with finite height.

**References.**

[1] Blok. W. J. The Lattice of Modal Logics: An Algebraic Investigation, the journal of smbolic logic, 45, 221-236, 1980.

[2] Blok. W. J. Pretabular Varieties of Modal Algebras, Studia Logica, 39: 101-124, 1980.

[3] A. Day, Varieties of Heyting algebras, unpublished manuscript.

[4] Fallahi, A., A Second Pretabular Classical Relevance Logic, Studia Logica, 106, pp. 191-214, 2018.

[5] Galminas, L., and J. G. Mersch, A Pretabular Classical Relevance Logic, Recent Developments related to Residuated Lattices and Substructural Logics, Studia Logica 100: 1211–1221, 2012.

[6] Maksimova, L. L., ‘Pretabular extensions of Lewis’s logic S4’, Algebra i Logika 14(1): 28–55, 1975.

[7] Maksimova, L. L., ‘Pretabular superintuitionistic logics’, Algebra i Logika 11(5): 558–570, 1972.

[8] Swirydowicz, K., 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, The Journal of Symbolic Logic 73 (4): 1249-1270, 2008.