A Class of Implicative Expansions of Belnap-Dunn Logic in which Boolean Negation is Definable

Belnap and Dunn's well-known 4-valued logic FDE is an interesting and useful non-classical logic. FDE is defined by using conjunction, disjunction and negation as the sole propositional connectives. Then the question of expanding FDE with an implication connective is of course of great interest. In this sense, some implicative expansions of FDE have been proposed in the literature, among which Brady's logic BN4 seems to be the preferred option of relevant logicians. The aim of this paper is to define a class of implicative expansions of FDE in whose elements Boolean negation is definable, whence strong logics such as the paraconsistent and paracomplete logic PL4 and BN4 itself are definable, in addition to classical propositional logic.