Empirical Negation, Co-negation and Contraposition Rule I: Semantical Investigations

Authors

DOI:

https://doi.org/10.18778/0138-0680.2020.12

Keywords:

empirical negation, co-negation, Beth semantics, Kripke semantics, intuitionism

Abstract

We investigate the relationship between M. De's empirical negation in Kripke and Beth Semantics. It turns out empirical negation, as well as co-negation, corresponds to different logics under different semantics. We then establish the relationship between logics related to these negations under unified syntax and semantics based on R. Sylvan's CCω.

References

[1] L. E. J. Brouwer, Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten. Zweiter Teil, [in:] A. Heyring (ed.), L.E.J. Brouwer Collected Works 1: Philosophy and Foundations of Mathematics, North-Holland (1975), pp. 191–221, DOI: http://dx.doi.org/10.1016/C2013-0-11893-4
Google Scholar

[2] J. L. Castiglioni, R. C. E. Biraben, Strict paraconsistency of truth-degree preserving intuitionistic logic with dual negation, Logic Journal of the IGPL, vol. 22(2) (2014), pp. 268–273, DOI: http://dx.doi.org/10.1093/jigpal/jzt027
Google Scholar

[3] M. De, Empirical Negation, Acta Analytica, vol. 28 (2013), pp. 49–69, DOI: http://dx.doi.org/10.1007/s12136-011-0138-9
Google Scholar

[4] M. De, H. Omori, More on Empirical Negation, [in:] R. Goreé, B. Kooi, A. Kurucz (eds.), Advances in Modal Logic, vol. 10, College Publications (2014), pp. 114–133.
Google Scholar

[5] K. Došen, Negation on the Light of Modal Logic, [in:] D. M. Gabbay, H. Wansing (eds.), What is Negation?, Kluwer Academic Publishing. (1999), DOI: http://dx.doi.org/10.1007/978-94-015-9309-04
Google Scholar

[6] T. M. Ferguson, Extensions of Priest-da Costa Logic, Studia Logica, vol. 102 (2013), pp. 145–174, DOI: http://dx.doi.org/10.1007/s11225-013-9469-4
Google Scholar

[7] A. B. Gordienko, A Paraconsistent Extension of Sylvan's Logic, Algebra and Logic, vol. 46(5) (2007), pp. 289–296, DOI: http://dx.doi.org/10.1007/s10469-007-0029-8
Google Scholar

[8] A. Heyting, Intuitionism: An Introduction, third revised ed., North Holland (1976).
Google Scholar

[9] M. Osorio, J. L. Carballido, C. Zepeda, J. A. Castellanos, Weakening and Extending Z, Logica Universalis, vol. 9(3) (2015), pp. 383–409, DOI: http://dx.doi.org/10.1007/s11787-015-0128-6
Google Scholar

[10] M. Osorio, J. A. C. Joo, Equivalence among RC-type paraconsistent logics, Logic Journal of the IGPL, vol. 25(2) (2017), pp. 239–252, DOI: http://dx.doi.org/10.1093/jigpal/jzw065
Google Scholar

[11] G. Priest, Dualising Intuitionistic Negation, Principia, vol. 13(2) (2009), pp. 165–184, DOI: http://dx.doi.org/10.5007/1808-1711.2009v13n2p165
Google Scholar

[12] C. Rauszer, A formalization of the propositional calculus of H-B logic, Studia Logica, vol. 33(1) (1974), pp. 23–34, DOI: http://dx.doi.org/10.1007/BF02120864
Google Scholar

[13] C. Rauszer, Applications of Kripke models to Heyting-Brouwer logic, Studia Logica, vol. 36(1) (1977), pp. 61–71, DOI: http://dx.doi.org/10.1007/BF02121115
Google Scholar

[14] G. Restall, Extending intuitionistic logic with subtraction (1997), unpublished.
Google Scholar

[15] R. Sylvan, Variations on da Costa C Systems and dual-intuitionistic logics I. Analyses of Cω and CCω, Studia Logica, vol. 49(1) (1990), pp. 47–65, DOI: http://dx.doi.org/10.1007/BF00401553
Google Scholar

[16] A. S. Troelstra, J. R. Moschovakis, A.S. Troelstra, D. van Dalen, Constructivism in Mathematics Corrections, URL: https://www.math.ucla.edu/~joan/ourTvDcorr030818 [accessed 20/Jul/2020].
Google Scholar

[17] A. S. Troelstra, D. van Dalen, Constructivism in Mathematics: An Introduction, vol. I, Elsevier (1988).
Google Scholar

[18] A. S. Troelstra, D. van Dalen, Constructivism in Mathematics: An Introduction, vol. II, Elsevier (1988).
Google Scholar

[19] D. van Dalen, L.E.J. Brouwer: Topologist, Intuitionist, Philosopher, Springer (2013), DOI: http://dx.doi.org/10.1007/978-1-4471-4616-2
Google Scholar

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Published

2020-11-04

How to Cite

Niki, S. (2020). Empirical Negation, Co-negation and Contraposition Rule I: Semantical Investigations. Bulletin of the Section of Logic, 49(3), 231-253. https://doi.org/10.18778/0138-0680.2020.12

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Research Article