On a four-valued logic of formal inconsistency and formal underterminedness

TL;DR

This paper introduces a four-valued logic BD2 that incorporates a primitive weak consistency operator, enabling it to handle both inconsistency and underdetermined information, with formal semantics and a first-order extension.

Contribution

It redefines BD2 using a primitive weak consistency operator, establishing it as both a logic of formal inconsistency and underdeterminedness, with semantics and a first-order version QBD2.

Findings

Semantic characterization via twist-structures

Interdefinability of quantifiers in QBD2

Soundness and completeness of the calculus

Abstract

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.