On a Four-Valued Logic of Formal Inconsistency and Formal Undeterminedness

Belnap–Dunn’s relevance logic, \(\textsf{BD}\), was designed seeking a suitable logical device for dealing with multiple information sources which sometimes may provide inconsistent and/or incomplete pieces of information. \(\textsf{BD}\) 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 \(\textsf{BD2}\) of the four valued Belnap–Dunn logic by a classical negation. In this paper, we introduce a four-valued expansion of BD called \({\textsf{BD}^\copyright }\), obtained by adding an unary connective \({\copyright }\,\ \)which is a consistency operator (in the sense of the Logics of Formal Inconsistency, LFIs). In addition, this operator is the unique one with the following features: it extends to \(\textsf{BD}\) the consistency operator of LFI1, a well-known three-valued LFI, still satisfying axiom ciw (which states that any sentence is either consistent or contradictory), and allowing to define an undeterminedness operator (in the sense of Logic of Formal Undeterminedness, LFUs). Moreover, \({\textsf{BD}^\copyright }\) is maximal w.r.t. LFI1, and it is proved to be equivalent to BD2, up to signature. After presenting a natural Hilbert-style characterization of \({\textsf{BD}^\copyright }\) obtained by means of twist-structures semantics, we propose a first-order version of \({\textsf{BD}^\copyright }\) called \({\textsf{QBD}^\copyright }\), with semantics based on an appropriate notion of four-valued Tarskian-like structures called \(\textbf{4}\)-structures. We show that in \({\textsf{QBD}^\copyright }\), the existential and universal quantifiers are interdefinable in terms of the paracomplete and paraconsistent negation, and not by means of the classical negation. Finally, a Hilbert-style calculus for \({\textsf{QBD}^\copyright }\) is presented, proving the corresponding soundness and completeness theorems.