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Default Modal Systems as Algebraic Updates

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Dynamic Logic. New Trends and Applications (DaLi 2020)

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Abstract

Default Logic refers to a family of formalisms designed to carry out non-monotonic reasoning over a monotonic logic (in general, Classical First-Order or Propositional Logic). Traditionally, default logics have been defined and dealt with via syntactic consequence relations. Here, we introduce a family of default logics defined over modal logics. First, we present these default logics syntactically. Then, we elaborate on an algebraic counterpart. We do the latter by extending the notion of a modal algebra to acommodate for the main elements of default logics: defaults and extensions. Our algebraic treatment of default logics concludes with an algebraic completeness result. To our knowledge, our approach is novel, and it lays the groundwork for studying default logics from a dynamic logic perspective.

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Notes

  1. 1.

    Typically the underlying proof calculi is one for Classical First-Order Logic (\(\mathsf {FOL}\)) (see, e.g., [21]) or for Classical Propositional Logic (\(\mathsf {CPL}\)) (see, e.g., [6, 17, 19, 22]).

  2. 2.

    In the literature on Default Logic well-behaved defaults are called normal. We avoid using this terminology here to avoid any confusion with normality in Modal Logic.

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Acknowledgment

We would like to thank the anonymous reviewers for their helpful comments and suggestions. This work is supported by projects ANPCyT-PICTs-2017-1130 and 2016-0215, Stic-AmSud 20-STIC-03 “DyLo-MPC”, Secyt-UNC, GRFT Mincyt-Cba, and by the Laboratoire International Associé SINFIN.

Author information

Authors and Affiliations

  1. Consejo Nacional de Investigaciones Científicas y Técnicas, Buenos Aires, Argentina

    Valentin Cassano, Raul Fervari, Carlos Areces & Pablo F. Castro

  2. Universidad Nacional de Córdoba, Córdoba, Argentina

    Valentin Cassano, Raul Fervari & Carlos Areces

  3. Universidad Nacional de Río Cuarto, Río Cuarto, Argentina

    Valentin Cassano & Pablo F. Castro

Authors
  1. Valentin Cassano
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  2. Raul Fervari
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  3. Carlos Areces
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  4. Pablo F. Castro
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Correspondence to Valentin Cassano .

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Editors and Affiliations

  1. Department of Mathematics, University of Aveiro, Aveiro, Portugal

    Manuel A. Martins

  2. Institute of Computer Science, Czech Academy of Sciences, Praha 8, Czech Republic

    Igor Sedlár

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Cassano, V., Fervari, R., Areces, C., Castro, P.F. (2020). Default Modal Systems as Algebraic Updates. In: Martins, M.A., Sedlár, I. (eds) Dynamic Logic. New Trends and Applications. DaLi 2020. Lecture Notes in Computer Science(), vol 12569. Springer, Cham. https://doi.org/10.1007/978-3-030-65840-3_7

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  • DOI: https://doi.org/10.1007/978-3-030-65840-3_7

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