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A Granularity Approach to Vague Quantification

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Integrated Uncertainty in Knowledge Modelling and Decision Making (IUKM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10758))

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Abstract

Motivated by the observation that vagueness often involves a shift of underlying granularity levels, we introduce models of quantifiers like ‘about one half’, ‘roughly 10%’, etc., that refer to granules of proportionality levels. Corresponding (partial) truth functions are extracted from rough set based models, which may then be systematically mapped into corresponding fuzzy quantifier models.

C. G. Fermüller — This work is supported by Austrian Science Fund (FWF) grant I1827-N25.

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Notes

  1. 1.

    Of course, linguists and philosophers recognize the importance of emphasis, hedging, uncertainty, etc., as triggered by vagueness. However these phenomena are modeled beneath or in addition to the outer level of communication via declarative statements.

  2. 2.

    Recall from Sect. 2 that we identify domain elements with constant symbols.

  3. 3.

    Not every such function is admitted as meaning of a logical quantifier. We focus on proportional quantifiers here, which are definitely to be classified as logical.

  4. 4.

    Modifiers like \(\textsf {approximately}\) (\(\approx \)) are usually left implicit when communicating by asserting sentences like 10% of the population lives in poverty.

  5. 5.

    There are obvious connections to epistemic theories of vagueness [15, 18] here. Moreover, a similar setup is also discussed in [17].

References

  1. Delgado, M., Ruiz, M.D., Sánchez, D., Vila, M.A.: Fuzzy quantification: a state of the art. Fuzzy Sets Syst. 242, 1–30 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Fermüller, C.G.: Combining fuzziness and context sensitivity in game based models of vague quantification. In: Huynh, V.-N., Inuiguchi, M., Denoeux, T. (eds.) IUKM 2015. LNCS (LNAI), vol. 9376, pp. 19–31. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25135-6_4

    Chapter  Google Scholar 

  3. Fermüller, C.G., Hofer, M., Ortiz, M.: Querying with vague quantifiers using probabilistic semantics. In: Christiansen, H., Jaudoin, H., Chountas, P., Andreasen, T., Legind Larsen, H. (eds.) FQAS 2017. LNCS (LNAI), vol. 10333, pp. 15–27. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59692-1_2

    Chapter  Google Scholar 

  4. Fermüller, C.G., Roschger, C.: Randomized game semantics for semi-fuzzy quantifiers. Logic J. IGPL 223(3), 413–439 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fine, K.: Vagueness, truth and logic. Synthese 30(3), 265–300 (1975)

    Article  MATH  Google Scholar 

  6. Giles, R.: A non-classical logic for physics. Stud. Logica 33(4), 397–415 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  7. Glöckner, I.: Fuzzy Quantifiers: A computational Theory. Studies in Fuzziness and Soft Computing, vol. 193. Springer, Heidelberg (2006). https://doi.org/10.1007/3-540-32503-4

    MATH  Google Scholar 

  8. Krifka, M.: Approximate interpretation of number words: A case for strategic communication. In: Hinrichs, E., Nerbonne, J. (eds.) Theory and Evidence in Semantics, pp. 109–132. CSLI Publications, Stanford (2009)

    Google Scholar 

  9. Lawry, J.: A voting mechanism for fuzzy logic. Int. J. Approximate Reasoning 19(3–4), 315–333 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu, Y., Kerre, E.E.: An overview of fuzzy quantifiers (I) Interpretations. Fuzzy Sets Syst. 95(1), 1–21 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Pawlak, Z.: Vagueness and uncertainty: a rough set perspective. Comput. Intell. 11(2), 227–232 (1995)

    Article  MathSciNet  Google Scholar 

  12. Pawlak, Z., Grzymala-Busse, J., Slowinski, R., Ziarko, W.: Rough sets. Commun. ACM 38(11), 88–95 (1995)

    Article  Google Scholar 

  13. Shapiro, S.: Vagueness in Context. Oxford University Press, Oxford (2006)

    Book  Google Scholar 

  14. Smith, N.J.J.: Vagueness and Degrees of Truth. Oxford University Press, Oxford (2008)

    Book  Google Scholar 

  15. Sorensen, R.: Vagueness and Contradiction. Clarendon Press, Oxford (2001)

    MATH  Google Scholar 

  16. Vetterlein, T.: Vagueness: a mathematicians perspective. In: Cintula, P., et al. (eds.) Understanding Vagueness - Logical, Philosophical and Linguistic Perspectives, pp. 67–86. College Publications (2011)

    Google Scholar 

  17. Vetterlein, T.: Logic of prototypes and counterexamples: possibilities and limits (2015). https://www.atlantis-press.com/proceedings/ifsa-eusflat-15/

  18. Williamson, T.: Vagueness. Routledge, London (2002)

    Google Scholar 

  19. Yao, Y.Y.: A comparative study of fuzzy sets and rough sets. Inf. Sci. 109(1–4), 227–242 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yao, Y.Y.: Information granulation and rough set approximation. Int. J. Intell. Syst. 16(1), 87–104 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zadeh, L.A.: A computational approach to fuzzy quantifiers in natural languages. Comput. Math. Appl. 9(1), 149–184 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Vienna University of Technology, Vienna, Austria

    Christian G. Fermüller

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  1. Christian G. Fermüller
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Correspondence to Christian G. Fermüller .

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Editors and Affiliations

  1. Japan Advanced Institute of Science and Technology, Nomi, Japan

    Van-Nam Huynh

  2. Osaka University, Osaka, Japan

    Masahiro Inuiguchi

  3. Hanoi National University of Education, Hanoi, Vietnam

    Dang Hung Tran

  4. Université de Technologie de Compiègne, Compiègne, France

    Thierry Denoeux

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Fermüller, C.G. (2018). A Granularity Approach to Vague Quantification. In: Huynh, VN., Inuiguchi, M., Tran, D., Denoeux, T. (eds) Integrated Uncertainty in Knowledge Modelling and Decision Making. IUKM 2018. Lecture Notes in Computer Science(), vol 10758. Springer, Cham. https://doi.org/10.1007/978-3-319-75429-1_1

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  • DOI: https://doi.org/10.1007/978-3-319-75429-1_1

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