The lattice of all 4-valued implicative expansions of Belnap–Dunn logic containing Routley and Meyer’s basic logic Bd

Abstract The well-known logic first degree entailment logic (FDE), introduced by Belnap and Dunn, is defined with $\wedge $, $\vee $ and $\sim $ as the sole primitive connectives. The aim of this paper is to establish the lattice formed by the class of all 4-valued C-extending implicative expansions of FDE verifying the axioms and rules of Routley and Meyer’s basic logic B and its useful disjunctive extension B$^{\textrm {d}}$. It is to be noted that Boolean negation (so, classical propositional logic) is definable in the strongest element in the said class.