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Swap structures semantics for Ivlev-like modal logics

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Abstract

In 1988, J. Ivlev proposed some (non-normal) modal systems which are semantically characterized by four-valued non-deterministic matrices in the sense of A. Avron and I. Lev. Swap structures are multialgebras (a.k.a. hyperalgebras) of a special kind, which were introduced in 2016 by W. Carnielli and M. Coniglio in order to give a non-deterministic semantical account for several paraconsistent logics known as logics of formal inconsistency, which are not algebraizable by means of the standard techniques. Each swap structure induces naturally a non-deterministic matrix. The aim of this paper is to obtain a swap structures semantics for some Ivlev-like modal systems proposed in 2015 by M. Coniglio, L. Fariñas del Cerro and N. Peron. Completeness results will be stated by means of the notion of Lindenbaum–Tarski swap structures, which constitute a natural generalization to multialgebras of the concept of Lindenbaum–Tarski algebras.

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Notes

  1. This notation makes sense, provided that the set \(\varXi \) of generators will keep fixed along this paper.

  2. Avron and his collaborators use in Avron and Lev (2001), Avron (2005) and Avron and Zamansky (2011) the term legal valuation to refer to valuations over an Nmatrix.

  3. As observed in Sect. 1, Kearns restricts the valuations over the Nmatrices in order to deal with the necessitation rule.

  4. This terminology is inspired by its use in computer science to refer to states.

  5. Here \(\lnot \) represents the paraconsistent negation of a given LFI, while \({\circ }\) represents a consistency operator w.r.t. \(\lnot \). See, for instance, Carnielli and Coniglio 2016 for more information about LFIs.

  6. Recall that, in any Boolean algebra, \(a \supset b = 1\) iff \(a \le b\).

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Acknowledgements

The authors would like to thank to the anonymous referees for their insightful comments and suggestions, which helped us to improve the overall quality of the paper. Marcelo E. Coniglio was financially supported by an individual research Grant from CNPq, Brazil (308524/2014-4). Ana Claudia Golzio was financially supported by scholarship Grant from FAPESP, Brazil (2013/04568-1), and by a post-doctoral Grant from CNPq, Brazil (150064/2018-7).

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Authors and Affiliations

  1. Department of Philosophy - IFCH and Centre for Logic, Epistemology and the History of Science, University of Campinas (UNICAMP), Campinas, Brazil

    Marcelo E. Coniglio

  2. Centre for Logic, Epistemology and the History of Science, University of Campinas (UNICAMP), Campinas, Brazil

    Ana Claudia Golzio

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  1. Marcelo E. Coniglio
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  2. Ana Claudia Golzio
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Correspondence to Marcelo E. Coniglio.

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Marcelo E. Coniglio declares that he has no conflict of interest. Ana Claudia Golzio declares that she has no conflict of interest.

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Coniglio, M.E., Golzio, A.C. Swap structures semantics for Ivlev-like modal logics. Soft Comput 23, 2243–2254 (2019). https://doi.org/10.1007/s00500-018-03707-4

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