  Belnap-Dunn's relevance logic, BD, was designed seeking a suitable logical
device for dealing with multiple information sources which sometimes may
provide inconsistent and/or incomplete pieces of information. BD is a
four-valued logic which is both paraconsistent and paracomplete. On the other
hand, De and Omori while investigating what classical negation amounts to in a
paracomplete and paraconsistent four-valued setting, proposed the expansion BD2
of the four valued Belnap-Dunn logic by a classical negation.
  In this paper, we reintroduce the logic BD2 by means of a primitive weak
consistency operator $\copyright$. This approach allows us to state in a direct
way that this is not only a Logic of Formal Inconsistency (LFI) but also a
Logic of Formal Underterminedness (LFU). After presenting a natural
Hilbert-style characterization of BD2 obtained by means of twist-structures
semantics, we propose a first-order version of BD2 called QBD2, with semantics
based on an appropriate notion of partial structures. We show that in QBD2,
$\exists$ and $\forall$ are interdefinable in terms of the paracomplete and
paraconsistent negation, and not by means of the the classical negation.
Finally, a Hilbert-style calculus for QBD2 is presented, proving the
corresponding and soundness and completeness theorems.
