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Classical Natural Deduction for S4 Modal Logic

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Abstract

This paper proposes a natural deduction system CNDS4 for classical S4 modal logic with necessity and possibility modalities. This new system is an extension of Parigot’s Classical Natural Deduction with dualcontext to formulate S4 modal logic. The modal λμ-calculus is also introduced as a computational extraction of CNDS4. It is an extension of both the λμ-calculus and the modal λ-calculus. Subject reduction, confluency, and strong normalization of the modal λμ-calculus are shown. Finally, the computational interpretation of the modal λμ-calculus, especially the computational meaning of the modal possibility operator, is discussed.

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Authors and Affiliations

  1. National Institute of Informatics, 2-1-2 Hitotsubashi, Tokyo, 101-8430, Japan

    Daisuke Kimura

  2. Department of Information Science, University of Tokyo, 7-3-1 Hongo, Tokyo, 113-8656, Japan

    Yoshihiko Kakutani

Authors
  1. Daisuke Kimura
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  2. Yoshihiko Kakutani
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Correspondence to Daisuke Kimura.

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Kimura, D., Kakutani, Y. Classical Natural Deduction for S4 Modal Logic. New Gener. Comput. 29, 61–86 (2011). https://doi.org/10.1007/s00354-010-0099-3

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  • DOI: https://doi.org/10.1007/s00354-010-0099-3

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