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A Study of Subminimal Logics of Negation and Their Modal Companions

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Language, Logic, and Computation (TbiLLC 2018)

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Abstract

We study propositional logical systems arising from the language of Johansson’s minimal logic and obtained by weakening the requirements for the negation operator. We present their semantics as a variant of neighbourhood semantics. We use duality and completeness results to show that there are uncountably many subminimal logics. We also give model-theoretic and algebraic definitions of filtration for minimal logic and show that they are dual to each other. These constructions ensure that the propositional minimal logic has the finite model property. Finally, we define and investigate bi-modal companions with non-normal modal operators for some relevant subminimal systems, and give infinite axiomatizations for these bi-modal companions.

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Notes

  1. 1.

    This property is equivalent to \(w \in \mathrm{N}(X) \Longleftrightarrow w \in \mathrm{N}(X \cap R(w))\) for every \(w \in W\) and \(X \in \mathcal {U}(W,\le )\); see, e.g., [12, Lemma 4.3.1].

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Acknowledgments

We would like to thank the referees for their careful reading, which has helped us to clarify a number of issues and improve the paper.

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Authors and Affiliations

  1. Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherlands

    Nick Bezhanishvili & Dick de Jongh

  2. Mathematical Institute, University of Bern, Bern, Switzerland

    Almudena Colacito

Authors
  1. Nick Bezhanishvili
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  2. Almudena Colacito
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  3. Dick de Jongh
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Corresponding author

Correspondence to Almudena Colacito .

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Editors and Affiliations

  1. Department of Computer Science, University College London, London, UK

    Alexandra Silva

  2. Department of Computer Science, University of Oxford, Oxford, UK

    Sam Staton

  3. Department of Linguistics, Heinrich Heine University Düsseldorf, Düsseldorf, Germany

    Peter Sutton

  4. Leibniz-Zentrum Allgemeine Sprachwissenschaft, Berlin, Germany

    Carla Umbach

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Bezhanishvili, N., Colacito, A., Jongh, D.d. (2019). A Study of Subminimal Logics of Negation and Their Modal Companions. In: Silva, A., Staton, S., Sutton, P., Umbach, C. (eds) Language, Logic, and Computation. TbiLLC 2018. Lecture Notes in Computer Science(), vol 11456. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59565-7_2

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  • DOI: https://doi.org/10.1007/978-3-662-59565-7_2

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