Notre Dame Journal of Formal Logic

Revisiting Z

Mauricio Osorio, José Luis Carballido, and Claudia Zepeda

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Béziau developed the paraconsistent logic Z, which is definitionally equivalent to the modal logic S5, and gave an axiomatization of the logic Z: the system HZ. Omori and Waragai proved that some axioms of HZ are not independent and then proposed another axiomatization for Z that includes two inference rules and helps to understand the relation between S5 and classical propositional logic. In the present paper, we analyze logic Z in detail; in the process we also construct a family of paraconsistent logics that are characterized by different properties that are relevant in the study of logics.

Article information

Source
Notre Dame J. Formal Logic, Volume 55, Number 1 (2014), 129-155.

Dates
First available in Project Euclid: 20 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1390246443

Digital Object Identifier
doi:10.1215/00294527-2377905

Mathematical Reviews number (MathSciNet)
MR3161417

Zentralblatt MATH identifier
1326.03033

Subjects
Primary: X001 Y002
Secondary: Z003

Keywords
paraconsistent logic logic $\mathbb{Z}$ modal logic nonmonotonic reasoning

Citation

Osorio, Mauricio; Carballido, José Luis; Zepeda, Claudia. Revisiting $\mathbb{Z}$. Notre Dame J. Formal Logic 55 (2014), no. 1, 129--155. doi:10.1215/00294527-2377905. https://projecteuclid.org/euclid.ndjfl/1390246443


Export citation

References

  • [Arieli et al.(2011)Arieli, Avron, and Zamansky] Arieli, O., A. Avron, and A. Zamansky, “Maximal and premaximal paraconsistency in the framework of three-valued semantics,” Studia Logica, vol. 97 (2011), pp. 31–60.
  • [Avron et al.(2010)Avron, Arieli, and Zamansky] Avron, A., O. Arieli, and A. Zamansky, “On strong maximality of paraconsistent finite-valued logics,” pp. 304–13 in 25th Annual IEEE Symposium on Logic in Computer Science LICS 2010, IEEE Computer Society, Los Alamitos, Calif., 2010.
  • [Baral(2003)] Baral, C., Knowledge Representation, Reasoning and Declarative Problem Solving, Cambridge University Press, Cambridge, 2003.
  • [Béziau(2000)] Béziau, J.-Y., “What is paraconsistent logic?” pp. 95–112 in Frontiers of Paraconsistent Logics (Ghent, 1997), vol. 8 of Studies in Logic and Computation, Research Studies Press, Baldock, England, 2000.
  • [Béziau()] Béziau, J.-Y., “Adventures in the paraconsistent jungle,” CLE e-Prints, vol. 4 (2004), no. 1.
  • [Béziau(2005)] Béziau, J.-Y., “Paraconsistent logic from a modal viewpoint,” Journal of Applied Logic, vol. 3 (2005), pp. 7–14.
  • [Béziau(2006)] Béziau, J., “The paraconsistent logic $\mathbf{Z}$: A possible solution to Jaśkowski’s problem,” Logic and Logical Philosophy, vol. 15 (2006), pp. 99–111.
  • [Blackburn et al.(2006)Blackburn, van Benthem, and Wolter] Blackburn, P., J. van Benthem, and F. Wolter, Handbook of Modal Logic, vol. 3 of Studies in Logic and Practical Reasoning, Elsevier, Amsterdam, 2006.
  • [Carnielli et al.(2007)Carnielli, Coniglio, and Marcos] Carnielli, W. A., M. E. Coniglio, and J. Marcos, “Logics of Formal Inconsistency,” pp. 1–93 in Handbook of Philosophical Logic, Vol. 14, 2nd edition, Springer, Dordrecht, 2007.
  • [Carnielli and Marcos(1999)] Carnielli, W. A., and J. Marcos, “Limits for paraconsistent calculi,” Notre Dame Journal of Formal Logic, vol. 40 (1999), pp. 375–90.
  • [Carnielli and Marcos(2002)] Carnielli, W. A., and J. Marcos, “A taxonomy of C-Systems,” pp. 1–94 in Paraconsistency: The Logical Way to the Inconsistent (São Sebastião, Brazil, 2000), vol. 228 of Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 2002.
  • [Carnielli et al.(2000)Carnielli, Marcos, and Amo] Carnielli, W. A., J. Marcos, and S. de Amo, “Formal inconsistency and evolutionary databases,” pp. 115–52 in Parainconsistency, Part II (Toruń, 1998), vol. 8 of Logic and Logical Philosophy, Nicolaus Copernicus University Press, Toruń, Poland, 2000.
  • [da Costa(1963)] da Costa, N. C. A., “On the theory of inconsistent formal systems” (in Portuguese), Ph.D. dissertation, Universidade Federal do Paraná, Brazil, 1963.
  • [da Costa et al.(1995)da Costa, Béziau, and Bueno] da Costa, N. C. A., J.-Y. Béziau, and O. Bueno, “Aspects of paraconsistent logic,” pp. 597–614 in Workshop on Logic, Language, Information and Computation’94 (Recife, Brazil, 1994), vol. 3 of Bulletin of the IGPL, 1995.
  • [Donini et al.(1997)Donini, Nardi, and Rosati] Donini, F. M., D. Nardi, and R. Rosati, “Ground nonmonotonic modal logics,” Journal of Logic and Computation, vol. 7 (1997), pp. 523–48.
  • [Gelfond and Lifschitz(1988)] Gelfond, M., and V. Lifschitz, “The Stable Model Semantics for Logic Programming,” pp. 1070–80 in Proceedings of the Fifth International Conference on Logic Programming, MIT Press, Cambridge, Mass., 1988.
  • [Goldblatt(1992)] Goldblatt, R., Logics of Time and Computation, 2nd edition, vol. 7 of CSLI Lecture Notes, CSLI Publications, Stanford, 1992.
  • [Marcos(2001)] Marcos, J., “On a problem of da Costa,” CLE ePrints, vol. 1 (2001), pp. 39–55.
  • [McDermott(1982)] McDermott, D., “Nonmonotonic logic, II: Nonmonotonic modal theories,” Journal of the ACM, vol. 29 (1982), pp. 33–57.
  • [McDermott and Doyle(1980)] McDermott, D., and J. Doyle, “Nonmonotonic logic, I,” Artificial Intelligence, vol. 13 (1980), pp. 41–72.
  • [Mendelson(1987)] Mendelson, E., Introduction to Mathematical Logic, 3rd edition, Wadsworth and Brooks/Cole, Monterey, Calif., 1987.
  • [Minsky(1975)] Minsky, M., “A framework for representing knowledge,” pp. 211–77 in The Psychology of Computer Vision, McGraw-Hill, New York, 1975.
  • [Omori and Waragai(2008)] Omori, H., and T. Waragai, “On Béziau’s logic Z,” Logic and Logical Philosophy, vol. 17 (2008), pp. 305–20.
  • [Osorio(2007)] Osorio, M., “GLukG logic and its application for non-monotonic reasoning,” in Latin-American Workshop on Non-Monotonic Reasoning (Puebla, Mexico, 2007), University of Skövde, Sweden, 2007, http://www.ceur-ws.org/Vol-286/LANMR07_08.pdf.
  • [Osorio et al.(2008)Osorio, Arrazola, and Carballido] Osorio, M., J. R. Arrazola, and J. L. Carballido, “Logical weak completions of paraconsistent logics,” Journal of Logic and Computation, vol. 18 (2008), pp. 913–40.
  • [Osorio and Carballido(2008)] Osorio, M., and J. L. Carballido, “Brief study of $\mathrm{G}'_{3}$ logic,” Journal of Applied Non-Classical Logics, vol. 18 (2008), pp. 475–99.
  • [Osorio et al.(2011)Osorio, Carballido, and Zepeda] Osorio, M., J. L. Carballido, and C. Zepeda, “An application of clasp in the study of logics,” pp. 278–83 in Logic Programming and Nonmonotonic Reasoning, vol. 6645 of Lecture Notes in Computer Science, Springer, Heidelberg, 2011.
  • [Osorio and Navarro(2003)] Osorio, M., and J. A. Navarro, “Modal logic S5$_{2}$ and PFOUR” (abstract), in Proceedings of the 2003 Annual Meeting of the Association for Symbolic Logic, Chicago, June 2003.
  • [Osorio et al.(2005)Osorio, Navarro, Arrazola, and Borja] Osorio, M., J. A. Navarro, J. R. Arrazola, and V. Borja, “Ground nonmonotonic modal logic S5: New results,” Journal of Logic and Computation, vol. 15 (2005), pp. 787–813.
  • [Osorio et al.(2006)Osorio, Navarro, Arrazola, and Borja] Osorio, M., J. A. Navarro, J. R. Arrazola, and V. Borja, “Logics with common weak completions,” Journal of Logic and Computation, vol. 16 (2006), pp. 867–90.
  • [Osorio et al.(2009)Osorio, Zepeda, Nieves, and Carballido] Osorio, M., C. Zepeda, J. C. Nieves, and J. L. Carballido, “$\mathrm{G}'_{3}$-stable semantics and inconsistency,” Computación y Sistemas, vol. 13 (2009), pp. 75–86.
  • [Rutherford(1965)] Rutherford, D. E., Introduction to Lattice Theory, Hafner, New York, 1965.
  • [Sette(1973)] Sette, A. M., “On the propositional calculus P$^{1}$,” Mathematica Japonicae, vol. 18 (1973), pp. 173–80.
  • [van Dalen(1980)] van Dalen, D., Logic and Structure, 2nd edition, Springer, Berlin, 1980.