Notre Dame Journal of Formal Logic

Substructural Fuzzy-Relevance Logic

Eunsuk Yang

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Abstract

This paper proposes a new topic in substructural logic for use in research joining the fields of relevance and fuzzy logics. For this, we consider old and new relevance principles. We first introduce fuzzy systems satisfying an old relevance principle, that is, Dunn’s weak relevance principle. We present ways to obtain relevant companions of the weakening-free uninorm (based) systems introduced by Metcalfe and Montagna and fuzzy companions of the system R of relevant implication (without distributivity) and its neighbors. The algebraic structures corresponding to the systems are then defined, and completeness results are provided. We next consider fuzzy systems satisfying new relevance principles introduced by Yang. We show that the weakening-free uninorm (based) systems and some extensions and neighbors of R satisfy the new relevance principles.

Article information

Source
Notre Dame J. Formal Logic, Volume 56, Number 3 (2015), 471-491.

Dates
Received: 11 December 2010
Accepted: 13 May 2013
First available in Project Euclid: 22 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1437570857

Digital Object Identifier
doi:10.1215/00294527-3132824

Mathematical Reviews number (MathSciNet)
MR3373615

Zentralblatt MATH identifier
1334.03030

Subjects
Primary: 02C
Secondary: 02J

Keywords
(substructural) fuzzy-relevance logic fuzzy logic relevance logic uninorm (based) logic

Citation

Yang, Eunsuk. Substructural Fuzzy-Relevance Logic. Notre Dame J. Formal Logic 56 (2015), no. 3, 471--491. doi:10.1215/00294527-3132824. https://projecteuclid.org/euclid.ndjfl/1437570857


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References

  • [1] Anderson, A. R., and N. D. Belnap, Jr., “The pure calculus of entailment,” Journal of Symbolic Logic, vol. 27 (1962), pp. 19–52.
  • [2] Anderson, A. R., and N. D. Belnap, Jr., Entailment, I: The Logic of Relevance and Necessity, Princeton University Press, Princeton, 1975.
  • [3] Anderson, A. R., N. D. Belnap, Jr., and J. M. Dunn, Entailment, II: The Logic of Relevance and Necessity, Princeton University Press, Princeton, 1992.
  • [4] Beall, J. C., and G. Restall, Logical Pluralism, Oxford University Press, Oxford, 2006.
  • [5] Cintula, P., “Weakly implicative (fuzzy) logics, I: Basic properties,” Archive for Mathematical Logic, vol. 45 (2006), pp. 673–704.
  • [6] Czelakowski, J., Protoalgebraic Logics, vol. 10 of Trends in Logic—Studia Logica Library, Kluwer Academic, Dordrecht, 2001.
  • [7] Dunn, J. M., “Algebraic completeness for $R$-mingle and its extensions,” Journal of Symbolic Logic, vol. 35 (1970), pp. 1–13.
  • [8] Dunn, J. M., “Relevance logic and entailment,” pp. 117–224 in Handbook of Philosophical Logic, edited by D. Gabbay and F. Guenthner, vol. 166 of Synthese Library, D. Reidel, Dordrecht, 1986.
  • [9] Esteva, F., and L. Godo, “Monoidal t-norm based logic: Towards a logic for left-continuous t-norms,” pp. 271–88 in Fuzzy Logic (Palma, 1999/Liptovský Ján, 2000), vol. 124 of Fuzzy Sets and Systems, North-Holland, Amsterdam, 2001.
  • [10] Esteva, F., L. Godo, P. Hájek, and M. Navara, “Residuated fuzzy logics with an involutive negation,” Archive for Mathematical Logic, vol. 39 (2000), pp. 103–24.
  • [11] Galatos, N., P. Jipsen, T. Kowalski, and H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, vol. 151 of Studies in Logic and the Foundations of Mathematics, Elsevier B. V., Amsterdam, 2007.
  • [12] Hájek, P., Metamathematics of Fuzzy Logic, vol. 4 of Trends in Logic—Studia Logica Library, Kluwer, Amsterdam, 1998.
  • [13] Hájek, P., “Fuzzy logics with noncommutative conjunctions,” Journal of Logic and Computation, vol. 13 (2003), pp. 469–79.
  • [14] Hájek, P., “Observations on non-commutative fuzzy logic,” Soft Computing, vol. 8 (2003), pp. 38–43.
  • [15] Metcalfe, G., “Uninorm based logics,” Proceedings of EUROFUSE, Warsaw, Poland, 2004, 85–90.
  • [16] Metcalfe, G., and F. Montagna, “Substructural fuzzy logics,” Journal of Symbolic Logic, vol. 72 (2007), pp. 834–64.
  • [17] Meyer, R. K., “Intuitionism, entailment, negation,” pp. 168–98 in Truth, Syntax, and Modality (Philadelphia, 1970), edited by H. Leblanc, vol. 68 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1973.
  • [18] Meyer, R. K., J. M. Dunn, and H. Leblanc, “Completeness of relevant quantification theories,” Notre Dame Journal of Formal Logic, vol. 15 (1974), pp. 97–121.
  • [19] Novak, V., “On the syntactico-semantical completeness of first-order fuzzy logic, I: Syntax and semantics,” Kybernetika (Prague), vol. 26 (1990), pp. 47–66; II: “Main results,” pp. 134–54.
  • [20] Restall, G., An Introduction to Substructural Logics, Routledge, New York, 2000.
  • [21] Yager, R. R., and A. Rybalov, “Uninorm aggregation operators,” Fuzzy Sets and Systems, vol. 80 (1996), pp. 111–20.
  • [22] Yang, E., “R and relevance principle revisited,” Journal of Philosophical Logic, vol. 42 (2013), pp. 767–82.