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

Marcelo E. Coniglio; G. T. Gomez-Pereira; Martín Figallo

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arXiv · 2212.01677

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

Marcelo E. Coniglio·G. T. Gomez-Pereira·Martín Figallo

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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.

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BibTeX RIS

Marcelo E. Coniglio, G. T. Gomez-Pereira, Martín Figallo. 2022-12-03. On a four-valued logic of formal inconsistency and formal underterminedness. https://arxiv.org/abs/2212.01677

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