The Review of Symbolic Logic

Research Article

THE MODAL LOGIC OF STONE SPACES: DIAMOND AS DERIVATIVE

GURAM BEZHANISHVILIa1 c1, LEO ESAKIAa2 c2 and DAVID GABELAIAa2 c3

a1 Department of Mathematical Sciences, New Mexico State University

a2 Department of Mathematical Logic, A. Razmadze Mathematical Institute

Abstract

We show that if we interpret modal diamond as the derived set operator of a topological space, then the modal logic of Stone spaces is K4 and the modal logic of weakly scattered Stone spaces is K4G. As a corollary, we obtain that K4 is also the modal logic of compact Hausdorff spaces and K4G is the modal logic of weakly scattered compact Hausdorff spaces.

(Received January 28 2009)

Correspondence:

c1 DEPARTMENT OF MATHEMATICAL SCIENCES, NEW MEXICO STATE UNIVERSITY, LAS CRUCES, NM 88003. E-mail: gbezhani@nmsu.edu

c2 DEPARTMENT OF MATHEMATICAL LOGIC, A. RAZMADZE MATHEMATICAL INSTITUTE, M. ALEKSIDZE STR. 1, TBILISI 0193, GEORGIA. E-mail: esakia@hotmail.com

c3 DEPARTMENT OF MATHEMATICAL LOGIC, A. RAZMADZE MATHEMATICAL INSTITUTE, M. ALEKSIDZE STR. 1, TBILISI 0193, GEORGIA. E-mail: gabelaia@gmail.com