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

Studia Logica:1-42 (forthcoming)

Abstract

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 $${\copyright }\,\$$ which is a consistency operator (in the sense of the Logics of Formal Inconsistency, _LFI_s). In addition, this operator is the unique one with the following features: it extends to $$\textsf{BD}$$ the consistency operator of LFI1, a well-known three-valued _LFI_, still satisfying axiom _ciw_ (which states that any sentence is either consistent or contradictory), and allowing to define an undeterminedness operator (in the sense of Logic of Formal Undeterminedness, _LFU_s). Moreover, $${\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.

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2024-05-04

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