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Negative Modalities in the Light of Paraconsistency

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The Road to Universal Logic

Part of the book series: Studies in Universal Logic ((SUL))

Abstract

Modality and non-classical negation have some interesting connections. One of the most famous connections is the relation between S4-modality and intuitionistic negation. In this chapter, we focus on the negative modalities in the perspective of paraconsistency. The basic idea here is to consider the negative modality defined as ‘not necessarily’ or equivalently ‘possibly not’ where ‘not’ is classical negation, and ‘necessarily’ and ‘possibly’ are modalities in modal logics. This chapter offers a solution to the problem of axiomatizing systems of modal logic such as D and S4 in terms of negative modalities. One of the upshots of this solution is that we may consider the semantics of paraconsistency with the help of various considerations known in the literature of modal logics related to D and S4.

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Notes

  1. 1.

    We only note that as Michael De is pointing out in [7, Chap. 2], a semantic approach rather than a syntactic approach seems to be promising in this context.

  2. 2.

    Strictly speaking, this will exclude the connexive negation (cf. [28]), but our purpose here is to clarify our usage of the word ‘negation’, not to claim something about negation in general. Thus we do not necessarily claim that connexive negation is not a negation.

  3. 3.

    For the purpose of emphasizing that negation in the language \({\mathcal{L}}_{\mathbf{CL}}\) is not necessarily intended to be classical negation whereas \({\mathcal{L}}_{\mathbf{ML}}\) is intended to be classical negation, we distinguished the negations by writing \({\sim}\) and \(\neg\), respectively.

  4. 4.

    These results are proved in [22].

  5. 5.

    This is also mentioned in [9, p. 69, Exercise 3.13].

  6. 6.

    Where ‘n’ is for ‘negative’.

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Acknowledgment

Hitoshi Omori is a JSPS Postdoctoral Fellow for Research Abroad. An earlier version of the chapter was presented at a special session ‘Negation’ of UNILOG III in Lisbon. We would like to thank Michael De for his careful reading and suggestions to improve both content and English of the chapter. We would also like to thank Katsuhiko Sano and Makoto Kikuchi for their helpful discussions on the chapter.

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Authors and Affiliations

  1. The Graduate Center, City University of New York, New York, USA

    Hitoshi Omori

  2. Graduate School of Decision Science and Technology, Tokyo Institute of Technology, Tokyo, Japan

    Toshiharu Waragai

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  1. Hitoshi Omori
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  2. Toshiharu Waragai
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Correspondence to Hitoshi Omori .

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Editors and Affiliations

  1. City University of New York, New York, New York, USA

    Arnold Koslow

  2. Federal University of Santa Catarina, Florianópolis, Santa Catarina, Brazil

    Arthur Buchsbaum

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Omori, H., Waragai, T. (2015). Negative Modalities in the Light of Paraconsistency. In: Koslow, A., Buchsbaum, A. (eds) The Road to Universal Logic. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-15368-1_23

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