Swap structures semantics for Ivlev-like modal logics

  • Marcelo E. ConiglioEmail author
  • Ana Claudia Golzio


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.


Swap structures Modal logics Non-deterministic matrices Lindenbaum–Tarski algebra 



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).

Compliance with ethical standards

Conflict of interest

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

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Philosophy - IFCH and Centre for Logic, Epistemology and the History of ScienceUniversity of Campinas (UNICAMP)CampinasBrazil
  2. 2.Centre for Logic, Epistemology and the History of ScienceUniversity of Campinas (UNICAMP)CampinasBrazil

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