From Belnap-Dunn four-valued logic to six-valued logics of evidence and truth. (English) Zbl 07871658

Coniglio, Marcelo E.; Rodrigues, Abilio

Stud. Log. 112, No. 3, 561-606 (2024).

Summary: The main aim of this paper is to introduce the logics of evidence and truth \(LET_K^+\) and \(LET_F^+\) together with sound, complete, and decidable six-valued deterministic semantics for them. These logics extend the logics \(LET_K\) and \(LET_F^-\) with rules of propagation of classicality, which are inferences that express how the classicality operator \({\circ }\) is transmitted from less complex to more complex sentences, and vice-versa. The six-valued semantics here proposed extends the 4 values of Belnap-Dunn logic with 2 more values that intend to represent (positive and negative) reliable information. A six-valued non-deterministic semantics for \(LET_K\) is obtained by means of Nmatrices based on swap structures, and the six-valued semantics for \(LET_K^+\) is then obtained by imposing restrictions on the semantics of \(LET_K\). These restrictions correspond exactly to the rules of propagation of classicality that extend \(LET_K\). The logic \(LET_F^+\) is obtained as the implication-free fragment of \(LET_K^+\). We also show that the 6 values of \(LET_K^+\) and \(LET_F^+\) define a lattice structure that extends the lattice L4 defined by the Belnap-Dunn four-valued logic with the 2 additional values mentioned above, intuitively interpreted as positive and negative reliable information. Finally, we also show that \(LET_K^+\) is Blok-Pigozzi algebraizable and that its implication-free fragment \(LET_F^+\) coincides with the degree-preserving logic of the involutive Stone algebras.

© Springer Nature B.V. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

MSC:

03-XX

Mathematical logic and foundations

Keywords:

paraconsistency; paracompleteness; logics of evidence and truth; logics of formal inconsistency; swap structures; twist structures; Nmatrices; involutive Stone algebras; crystal lattice

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