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Redefinition of the concept of fuzzy set based on vague partition from the perspective of axiomatization

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A Correction to this article was published on 07 February 2018

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Abstract

Based on the in-depth analysis of the essence and key features of vague phenomena, this paper focuses on establishing the axiomatical foundation of membership degree theory using for modeling vague phenomena, presents an axiomatic system to govern membership degrees and their interconnections. The concept of vague partition is introduced, on this basis, the concept of fuzzy set in Zadeh’s sense is redefined based on vague partition from the perspective of axiomatization. The thesis defended in this paper is that the mutual constraint relationship among vague attribute values in a vague partition should be the starting point to recognize and model vague phenomena by the quantitative analysis method.

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  • 07 February 2018

    I recently found that several errors occur in the statement of Definition 5.2 in Section 5 in the paper “Redefinition of the concept of fuzzy set based on vague partition from the perspective of axiomatization”.

References

  • Atanassov KT (1983) Intuitionistic fuzzy sets. VII ITKR’s Session, Sofia (deposited in Central Sci.-Technical Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)

  • Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96

    Article  MATH  Google Scholar 

  • Atanassov KT, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31(3):343–349

    Article  MathSciNet  MATH  Google Scholar 

  • Bede B (2013) Mathematics of fuzzy sets and fuzzy logic. Springer, Berlin

    Book  MATH  Google Scholar 

  • Behounek L(2009) Logical foundations of fuzzy mathematics. Doctoral dissertation, Charles University in Prague, Prague, Czech. http://mujweb.cz/behounek/logic/papers/Behounek-PhD-summary.pdf

  • Cintula P, Hájek P, Noguera C (eds) (2011) Studies in logic: handbook of mathematical fuzzy logic, vol 37 and 38. College Publications, London

    MATH  Google Scholar 

  • Dubois D, Prade H (2015) The legacy of 50 years of fuzzy sets: a discussion. Fuzzy Sets Syst 281:21–31

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois D, Prade H, Klement EP (1999) Fuzzy sets logics and reasoning about knowledge. Kluwer Academic, Dordrecht

    Book  MATH  Google Scholar 

  • Dvořák A, Novák V (2004) Formal theories and linguistic descriptions. Fuzzy Sets Syst 143(1):169–188

    Article  MathSciNet  MATH  Google Scholar 

  • Goguen JA (1967) L-fuzzy sets. Math Appl 18:145–174

    MathSciNet  MATH  Google Scholar 

  • Goguen JA (1969) The logic of inexact concepts. Synthese 19(3–4):325–373

    Article  MATH  Google Scholar 

  • Grattan-Guinness I (1976) Fuzzy membership mapped onto intervals and many-valued quantities. Zeitschr. f. math. Logik und Grundlagen d. Math. Bd 22(1):149–160

    Article  MathSciNet  MATH  Google Scholar 

  • Greenfield S (2016) Uncertainty measurement for the interval type-2 fuzzy set. In: Rutkowski L, Korytkowski M, Scherer R, Tadeusiewicz R, Zadeh L, Zurada J (eds) Artificial intelligence and soft computing (Lecture notes in computer science), vol 9692. ICAISC 2016. Springer, Cham

  • Hájek P (1998) Metamathematics of fuzzy logic. Kluwer, Dordrecht

    Book  MATH  Google Scholar 

  • Hu BQ, Kwong CK (2014) On type-2 fuzzy sets and their t-norm operations. Inf Sci 255(1):58–81

    Article  MathSciNet  MATH  Google Scholar 

  • Hubbard DW (2010) How to measure anything: finding the value of intangibles in business, 2nd edn. Wiley, New York

    Google Scholar 

  • Karnik NN, Mendel JM (2001) Operations on type-2 fuzzy sets. Fuzzy Sets Syst 122(2):327–348

    Article  MathSciNet  MATH  Google Scholar 

  • Keefe BR, Smith P (1996) Vagueness: a reader. MIT Press, Cambridge

    Google Scholar 

  • Klement EP, Mesiar R, Pap E (2000) Triangular norms. Springer, Berlin

    Book  MATH  Google Scholar 

  • Kolmogorov AN (1956) Foundations of the theory of probability (English edition). Chelsea Publishing Company, White River Junction

    MATH  Google Scholar 

  • Mendel JM (2007) Advances in type-2 fuzzy sets and systems. Inf Sci 177(1):84–110

    Article  MathSciNet  MATH  Google Scholar 

  • Novák V (2006) Which logic is the real fuzzy logic? Fuzzy Sets Syst 157(5):635–641

    Article  MathSciNet  MATH  Google Scholar 

  • Novák V (2012) Reasoning about mathematical fuzzy logic and its future. Fuzzy Sets Syst 192:25–44

    Article  MathSciNet  MATH  Google Scholar 

  • Novák V, Perfilieva I, Močkoř J (1999) Mathematical principles of fuzzy logic. Kluwer Academic, Dordrecht

    Book  MATH  Google Scholar 

  • Pan XD, & Xu Y (2016a) Vague partition. In: International conference on intelligent systems and knowledge engineering. IEEE

  • Pan XD, Xu Y (2016b) Vague membership measure: uncertainty modelling in knowledge engineering and decision making. In: Proceedings of the 12th international FLINS conference, pp 527–534

  • Pavelka J (1979) On fuzzy logic i: many-valued rules of inference, ii: enriched residuated lattices and semantics of propositional calculi. iii: semantical completeness of some many-valued propositional calculi. Math Logic Q 25(7–12):45–52, 1191–34, 447–464, 119–134

  • Peirce CS (1902) Vague. In: Baldwin JM (ed) Dictionary of philosophy and psychology. MacMillan, New York, p 748

    Google Scholar 

  • Sadeghian A, Mendel J, Tahayori H (2013) Advances in type-2 fuzzy sets and systems: theory and applications. Springer, New York

    Book  MATH  Google Scholar 

  • Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25:529–539

    MATH  Google Scholar 

  • Ward M, Dilworth RP (1938) Residuated lattices. Proc Nat Acad Sci USA 24(3):162–164

    Article  MATH  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  MATH  Google Scholar 

  • Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning—1. Inf Sci 8:199–249

  • Zadeh LA (1978) Pruf-a meaning representation language for natural languages. Int J Man Mach Stud 10(4):395–460

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh LA (2015) Fuzzy logic—a personal perspective. Fuzzy Sets Syst 281(C):4–20

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to express my warm thanks to Prof. P. Eklund and Prof. D.W. Pei for valuable discussions on some of the problems considered here, and are very grateful to the Editors and the anonymous reviewers for their insightful and constructive comments and suggestions that have led to an improved version of this paper. This work was partially funded by the National Natural Science Foundation of China (Grant Nos. 61305074, 61673320).

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Authors and Affiliations

  1. School of Mathematics, Southwest Jiaotong University, West Section, High-Tech Zone, Chengdu, 611756, Sichuan, People’s Republic of China

    Xiaodong Pan & Yang Xu

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  1. Xiaodong Pan
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  2. Yang Xu
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Correspondence to Xiaodong Pan.

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Communicated by A. Di Nola.

A correction to this article is available online at https://doi.org/10.1007/s00500-018-3061-9.

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Pan, X., Xu, Y. Redefinition of the concept of fuzzy set based on vague partition from the perspective of axiomatization. Soft Comput 22, 1777–1789 (2018). https://doi.org/10.1007/s00500-017-2855-5

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