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An Algebraic View of Super-Belnap Logics

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Abstract

The Belnap–Dunn logic (also known as First Degree Entailment, or FDE) is a well-known and well-studied four-valued logic, but until recently little has been known about its extensions, i.e. stronger logics in the same language, called super-Belnap logics here. We give an overview of several results on these logics which have been proved in recent works by Přenosil and Rivieccio. We present Hilbert-style axiomatizations, describe reduced matrix models, and give a description of the lattice of super-Belnap logics and its connections with graph theory. We adopt the point of view of Abstract Algebraic Logic, exploring applications of the general theory of algebraization of logics to the super-Belnap family. In this respect we establish a number of new results, including a description of the algebraic counterparts, Leibniz filters, and strong versions of super-Belnap logics, as well as the classification of these logics within the Leibniz and Frege hierarchies.

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Acknowledgements

Adam Přenosil gratefully acknowledges the support of the Grant P202/12/G061 of the Czech Science Foundation. The research of Umberto Rivieccio has been supported by EU FP7 Marie Curie PIRSES-GA-2012-318986 project and by CNPq process 482809/2013-2 of the Brazilian National Council for Scientific and Technological Development.

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Authors and Affiliations

  1. Departament de Filosofia, Universitat de Barcelona, Montalegre 7, 08001, Barcelona, Spain

    Hugo Albuquerque

  2. Institute of Computer Science, Czech Academy of Sciences, Pod Vodárenskou věží 271/2, Prague, Czechia

    Adam Přenosil

  3. Department of Informatics and Applied Mathematics, Federal University of Rio Grande do Norte, Campus Universitário Lagoa Nova, Natal, RN, Brazil

    Umberto Rivieccio

Authors
  1. Hugo Albuquerque
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  2. Adam Přenosil
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  3. Umberto Rivieccio
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Correspondence to Adam Přenosil.

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Albuquerque, H., Přenosil, A. & Rivieccio, U. An Algebraic View of Super-Belnap Logics. Stud Logica 105, 1051–1086 (2017). https://doi.org/10.1007/s11225-017-9739-7

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