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Logics and Their Galaxies

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The Road to Universal Logic

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

Abstract

This chapter introduces some concepts that help exploring the ontological import of universal logic. It studies the notions of an antilogic and counterlogic associated with each logic and shows some of their properties. It presents the notion of galaxy, as the class of possible worlds compatible with a given logic. We explore some consequences of these developments.

Work supported by a grant of the National Council for Scientific and Technological Development – CNPq – Brazil (486635/2013-9).

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Notes

  1. 1.

    Tarski proposed an operator of logical consequence with conditions on it (see [20]). We use a consequence relation without restrictions in the sense of [2].

  2. 2.

    The concept of antilogic has been developed by Łukasiewicz in [12], and studied in [17] as rejected propositions or in [18] under the label of refutation systems. For anticlassical propositional logic, there are results in [3], [6], and  [21].

  3. 3.

    Henceforth, whenever nonambiguous we use simply ‘‘counterlogic’’.

  4. 4.

    The term is used here in a way that is reminiscent but not equal to that used by Priest (see [15]). He takes a dialethea to be a part of reality which harbours inconsistencies, and true contradictions. We take it to be simply an inconsistent world. It is also worth noticing that if L is such that it is not the case that \(\varnothing\vdash_{L}\varphi\wedge\neg\varphi\) then \(\varnothing\vdash_{\bar{L}}\varphi\wedge\neg\varphi\). Therefore \(\bar{L}\) could contain dialetheas (true contradictions) also in Priest’s sense.

  5. 5.

    Kit Fine [7], for example, considers three irreducible varieties of necessity: metaphysical necessity, natural necessity and normative necessity.

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Acknowledgment

Thanks to Arnold Koslow and Graham Priest for discussions concerning ideas of this paper.

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Authors and Affiliations

  1. Campus Universitário Darcy Ribeiro, Instituto de Ciências Humanas, Departamento de Filosofia, ICC Ala Norte, Universidade de Brasília, Brasilia, Brazil

    Hilan Bensusan & Alexandre Costa-Leite

  2. Departamento de Filosofia, Universidade de São Paulo, São Paulo, Brazil

    Edelcio Gonçalves de Souza

Authors
  1. Hilan Bensusan
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  2. Alexandre Costa-Leite
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  3. Edelcio Gonçalves de Souza
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Corresponding author

Correspondence to Alexandre Costa-Leite .

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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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Bensusan, H., Costa-Leite, A., Gonçalves de Souza, E. (2015). Logics and Their Galaxies. 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_10

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