A Conventional Expansion of First-Order Belnap-Dunn Logic

This paper concerns an expansion of first-order Belnap-Dunn logic named $\mathrm{BD}^{\supset,\mathsf{F}}$. Its connectives and quantifiers are all familiar from classical logic and its logical consequence relation is closely connected to the one of classical logic. Results that convey this close connection are established. Classical laws of logical equivalence are used to distinguish the four-valued logic $\mathrm{BD}^{\supset,\mathsf{F}}$ from all other four-valued logics with the same connectives and quantifiers whose logical consequence relation is as closely connected to the logical consequence relation of classical logic. It is shown that several interesting non-classical connectives added to Belnap-Dunn logic in its studied expansions are definable in $\mathrm{BD}^{\supset,\mathsf{F}}$. It is also established that $\mathrm{BD}^{\supset,\mathsf{F}}$ is both paraconsistent and paracomplete. A sequent calculus proof system that is sound and complete with respect to the logical consequence relation of $\mathrm{BD}^{\supset,\mathsf{F}}$ is presented.