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From Belnap-Dunn Four-Valued Logic to Six-Valued Logics of Evidence and Truth

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Abstract

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.

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Acknowledgements

A preprint of this article can be found in [24]. We want to thank the anonymous referees by their comments and suggestions about our manuscript. These comments and corrections helped us to improve the overall quality of the manuscript. The authors acknowledge support from the National Council for Scientific and Technological Development (CNPq, Brazil), research grants 306530/2019-8, 310037/2021-2, and 408040/2021-1. The second author also acknowledges support from Minas Gerais State Agency for Research and Development (FAPEMIG, Brazil), research grant APQ-02093-21.

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  1. Institute of Philosophy and the Humanities (IFCH), University of Campinas (UNICAMP), Campinas, SP, Brazil

    Marcelo E. Coniglio

  2. Department of Philosophy, Federal University of Minas Gerais (UFMG), Belo Horizonte, MG, Brazil

    Abilio Rodrigues

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Presented by Walter Carnielli; Received September 26, 2022.

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Coniglio, M.E., Rodrigues, A. From Belnap-Dunn Four-Valued Logic to Six-Valued Logics of Evidence and Truth. Stud Logica (2023). https://doi.org/10.1007/s11225-023-10062-5

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