On the expressive power of certain expansions of Belnap's four-valued logic

Abstract

The paper is devoted to closed classes of functions of four-valued logic. We present the following results:

(1) The basic operations of the logic obtained by expanding the four-valued De Morgan algebra by the conflation operator generate a closed class of all functions that simultaneously preserve classical truth values and are self-dual with respect to conflation. This class is precomplete in the class of all functions that preserve classical truth values.

(2) There are exactly two closed classes between the closed class generated by the basic operations of von Wright's truth logic and the class of all functions that preserve classical truth values. Each of them is a class of all functions that simultaneously preserve classical truth values and one of the three-element supersets of the set of classical truth values.

(3) The basic operations of tetravalent modal logic obtained by expanding the
four-valued De Morgan algebra by the necessity operator generate a closed class
of all functions that simultaneously preserve classical truth values, are self-dual with respect to conflation, and also preserve both three-element supersets of the set of classical truth values. We show that this class is precomplete in the class of all functions that simultaneously preserve classical truth values and are self-dual with respect to conflation. In addition, we demonstrate that between this class and the closed class generated by the operations of von Wright's truth logic, there is exactly one closed class.

Thus, we obtain a seven-element lattice consisting of all possible four-valued extensions of tetravalent modal logic that preserve classical truth values.