Natural deduction systems for some modifications of Kleene's and Dunn — Belnap's logics

Kleene introduced the notions of regular logical connective and regular logic as well as considered three-valued examples of such logics. Finn and Komendantskaya studied functional properties of regular three-valued logics. Base ourselves upon their results, we present four-valued analogues of Kleene's three-valued logics. The first four-valued generalization of Kleene's three-valued logics (more exactly, of strong Kleene's logic) is Dunn–Belnap's logic. Two different orders (truth and information ones) can be defined on the set of truth values of this logic (we follow Belnap's semantics). Using them, one can define two sets of logical connectives. Only one of them (which is based on truth order) is presented in Dunn–Belnap's logic itself. Fitting considers two sets at the same time. We study logic (we call it Belnap–Fitting's logic) which have the connectives based on information order. Using these connectives (more exactly, we substitute them into Finn and Komendantskaya's equations instead of the connectives of strong Kleene's logic), we obtain a new class of four-valued logics which are analogues of regular three-valued ones. All the elements of this class are formalized via natural deduction systems.