Abstract
This paper concerns an expansion of first-order Belnap-Dunn logic
named BD$^{\supset,F}$.
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 BD$^{\supset,F}$ 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 BD$^{\supset,F}$.
It is also established that BD$^{\supset,F}$ is both paraconsistent and
paracomplete.
A sequent calculus proof system that is sound and complete with respect
to the logical consequence relation of BD$^{\supset,F}$ is presented.