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Graded Generalized Hexagon in Fuzzy Natural Logic

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Book cover Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2016)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 611))

Abstract

In our previous papers, we formally analyzed the generalized Aristotle’s square of opposition using tools of fuzzy natural logic. Namely, we introduced general definitions of selected intermediate quantifiers, constructed a generalized square of opposition consisting of them and syntactically analyzed the emerged properties. The main goal of this paper is to extend the generalized square of opposition to graded generalized hexagon.

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Notes

  1. 1.

    Blanché in [7] introduced \(\mathbf {Y}\) at first, before completing it with \(\mathbf {U}\) in [8].

  2. 2.

    In some papers, the term “generalized Aristotle square” is replaced by “graded on”.

  3. 3.

    Let \(\mathcal {M}\models T^{\text {IQ}}\). Then we denote \(\mathcal {M}(\top )=1\) and \(\mathcal {M}(\bot )=0\).

References

  1. Andrews, P.: An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof. Kluwer, Dordrecht (2002)

    Book  MATH  Google Scholar 

  2. Béziau, J.: New light on the square of oppositions and its nameless corner. Log. Investig. 10, 218–233 (2003)

    MathSciNet  MATH  Google Scholar 

  3. Béziau, J.: The power of the hexagon. Logica Universalis 6(1–2), 1–43 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Béziau, J., Gan-Krzywoszyńska, K.: Handbook of Abstracts of the 2nd World Congress on the Square of Opposition, Corte, Corsica, 17–20 June 2010

    Google Scholar 

  5. Béziau, J., Gan-Krzywoszyńska, K.: Handbook of Abstracts of the 3rd World Congress on the Square of Opposition, Beirut, Lebanon, 26–30 June 2010

    Google Scholar 

  6. Béziau, J., Gan-Krzywoszyńska, K.: Handbook of Abstracts of the 4th World Congress on the Square of Opposition, Roma, Vatican, pp. 26–30, 5–9 May 2014

    Google Scholar 

  7. Blanché, R.: Quantitiy, modality, and other kindred systems of categories. Mind LXI(243), 369–375 (1952)

    Article  Google Scholar 

  8. Blanché, R.: Sur l’oppostion des concepts. Theoria 19, 89–130 (1953)

    Article  Google Scholar 

  9. Cignoli, R.L.O., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer, Dordrecht (2000)

    Book  MATH  Google Scholar 

  10. Ciucci, D., Dubois, D., Prade, H.: Oppositions in rough set theory. In: Li, T., Nguyen, H.S., Wang, G., Grzymala-Busse, J., Janicki, R., Hassanien, A.E., Yu, H. (eds.) RSKT 2012. LNCS, vol. 7414, pp. 504–513. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  11. Ciucci, D., Dubois, D., Prade, H.: The structure of oppositions in rough set theory and formal concept analysis - toward a new bridge between the two settings. In: Beierle, C., Meghini, C. (eds.) FoIKS 2014. LNCS, vol. 8367, pp. 154–173. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  12. Ciucci, D., Dubois, D., Prade, H.: Structures of opposition induced by relations: the boolean and the gradual cases. Ann. Math. Artif. Intell. 76(3–4), 351–373 (2015). doi:10.1007/s10472-015-9480-8

    MathSciNet  MATH  Google Scholar 

  13. Dubois, D., Prade, H.: From blanche’s hexagonal organization of concepts to formal concepts analysis and possibility theory. Logica Universalis 6(1–2), 149–169 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dubois, D., Prade, H.: Gradual structures of oppositions. In: Esteva, F., Magdalena, L., Verdegay, J.L. (eds.) Enric Trillas: Passion for Fuzzy Sets. Studies in Fuzziness and Soft Computing, vol. 322, pp. 79–91. Springer, Switzerland (2015)

    Google Scholar 

  15. Dubois, D.: P.H.: from blanches hexagonal organization of concepts to formal concept analysis and possibility theory. Logica Universalis 6, 149–169 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dvořák, A., Holčapek, M.: L-fuzzy quantifiers of the type \(\langle {1}\rangle \) determined by measures. Fuzzy Sets Syst. 160, 3425–3452 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Moretti, A.: The geometry of logical oppositions and the opposition of logic to it. In: Bianchi, I., Savaradi, U. (eds.) The Perception and Cognition of Contraries (2009)

    Google Scholar 

  18. Murinová, P., Novák, V.: A formal theory of generalized intermediate syllogisms. Fuzzy Sets Syst. 186, 47–80 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Murinová, P., Novák, V.: Analysis of generalized square of opposition with intermediate quantifiers. Fuzzy Sets Syst. 242, 89–113 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Murinová, P., Novák, V.: The structure of generalized intermediate syllogisms. Fuzzy Sets Syst. 247, 18–37 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Murinová, P., Novák, V.: Analysis of the intermediate quantifier “many” in fuzzy natural logic. In: Proceedings of International Conference IFSA-EUSFLAT 2015, pp. 1147–1153 (2015)

    Google Scholar 

  22. Murinová, P., Novák, V.: On properties of the intermediate quantifier “many”. Fuzzy Sets Syst. (submitted)

    Google Scholar 

  23. Novák, V.: On fuzzy type theory. Fuzzy Sets Syst. 149, 235–273 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Novák, V.: Perception-based logical deduction. In: Reusch, B. (ed.) Computational Intelligence, Theory and Applications. Advances in Soft Computing, vol. 33, pp. 237–250. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  25. Novák, V.: A comprehensive theory of trichotomous evaluative linguistic expressions. Fuzzy Sets Syst. 159(22), 2939–2969 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Novák, V.: A formal theory of intermediate quantifiers. Fuzzy Sets Syst. 159(10), 1229–1246 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Novák, V., Lehmke, S.: Logical structure of fuzzy IF-THEN rules. Fuzzy Sets Syst. 157, 2003–2029 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Novák, V., Perfilieva, I., Močkoř, J.: Mathematical Principles of Fuzzy Logic. Kluwer, Boston (1999)

    Book  MATH  Google Scholar 

  29. Pellissier, R.: Setting n-opposition. Logica Universalis 2(2), 235–263 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Peters, S., Westerståhl, D.: Quantifiers in Language and Logic. Claredon Press, Oxford (2006)

    Google Scholar 

  31. Peterson, P.: Intermediate Quantifiers. Logic, linguistics, and Aristotelian Semantics. Ashgate, Aldershot (2000)

    Google Scholar 

  32. Smesaert, H.: The classical aristotelian hexagon versus the modern duality hexagon. Logica Universalis 6, 171–199 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Smessaert, H.: On the 3D-visualisationof logical relations. Logica Universalis 3(2), 303–332 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Westerståhl, D.: The traditional square of opposition and generalized quantifiers. Stud. Log. 2, 1–18 (2008)

    Google Scholar 

  35. Wikipedia (2004). http://en.wikipedia.org/wiki/aristotle

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Authors and Affiliations

  1. Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03, Ostrava 1, Czech Republic

    Petra Murinová & Vilém Novák

Authors
  1. Petra Murinová
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  2. Vilém Novák
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Corresponding author

Correspondence to Petra Murinová .

Editor information

Editors and Affiliations

  1. INESC-ID,Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Joao Paulo Carvalho

  2. LIP 6, Université Pierre et Marie Curie, Paris, France

    Marie-Jeanne Lesot

  3. School of Industrial Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

    Uzay Kaymak

  4. IDMEC,Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Susana Vieira

  5. LIP6, Université Pierre et Marie Curie, CNRS, Paris, France

    Bernadette Bouchon-Meunier

  6. Iona College, Machine Intelligence Institute, New Rochelle, New York, USA

    Ronald R. Yager

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Murinová, P., Novák, V. (2016). Graded Generalized Hexagon in Fuzzy Natural Logic. In: Carvalho, J., Lesot, MJ., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2016. Communications in Computer and Information Science, vol 611. Springer, Cham. https://doi.org/10.1007/978-3-319-40581-0_4

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  • DOI: https://doi.org/10.1007/978-3-319-40581-0_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-40580-3

  • Online ISBN: 978-3-319-40581-0

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