Lattices of Four-valued Modal Logics

Abstract

In this paper four lattices of four-valued modal logics are considered. Different algebraic structures are the basis for construction, then these structures are gradually extended by the endomorphisms and constant functions. In the first case, the lattice of extensions of Boolean algebra $B^2$ is formed, then the lattice of extensions of De Morgan algebra $DM4$ is constructed. In both cases the different modal logics appear, the properties of which are described and compared. The lattice with tetravalent modal logic $TML$ is considered individually. Finally, the first two lattices are joined and the class of basic four-valued modal logics is singled out. This class consists of$_$-modal system of _ukasiewicz, logic $V2$ of Sobochin__ski and von Wright’s truth-logic $Tr$$^{\prime\prime}$. Special attention should be paid to the logic $Tr$, which is functionally equivalent to the logic $V2$ and occupies a central place in the final lattice. This logic is the only of the above-considered logics which has the Craig’s interpolation property, and besides is an excellent candidate for the role of propositional logic of truth. In conclusion its axiomatization is presented.