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Generalizing Functional Completeness in Belnap-Dunn Logic

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Abstract

One of the problems we face in many-valued logic is the difficulty of capturing the intuitive meaning of the connectives introduced through truth tables. At the same time, however, some logics have nice ways to capture the intended meaning of connectives easily, such as four-valued logic studied by Belnap and Dunn. Inspired by Dunn’s discovery, we first describe a mechanical procedure, in expansions of Belnap-Dunn logic, to obtain truth conditions in terms of the behavior of the Truth and the False, which gives us intuitive readings of connectives, out of truth tables. Then, we revisit the notion of functional completeness, which is one of the key notions in many-valued logic, in view of Dunn’s idea. More concretely, we introduce a generalized notion of functional completeness which naturally arises in the spirit of Dunn’s idea, and prove some fundamental results corresponding to the classical results proved by Post and Słupecki.

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Authors and Affiliations

  1. The Graduate Center, City University of New York, 365 Fifth Avenue, New York, NY, 10016, USA

    Hitoshi Omori

  2. School of Information Science, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa, 923-1292, Japan

    Katsuhiko Sano

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  1. Hitoshi Omori
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  2. Katsuhiko Sano
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Correspondence to Hitoshi Omori.

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Omori, H., Sano, K. Generalizing Functional Completeness in Belnap-Dunn Logic. Stud Logica 103, 883–917 (2015). https://doi.org/10.1007/s11225-014-9597-5

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  • DOI: https://doi.org/10.1007/s11225-014-9597-5

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