Transactions on Rough Sets IX pp 1-13

Part of the Lecture Notes in Computer Science book series (LNCS, volume 5390) | Cite as

Vagueness and Roughness

  • Zbigniew Bonikowski
  • Urszula Wybraniec-Skardowska

Abstract

The paper proposes a new formal approach to vagueness and vague sets taking inspirations from Pawlak’s rough set theory. Following a brief introduction to the problem of vagueness, an approach to conceptualization and representation of vague knowledge is presented from a number of different perspectives: those of logic, set theory, algebra, and computer science. The central notion of the vague set, in relation to the rough set, is defined as a family of sets approximated by the so called lower and upper limits. The family is simultaneously considered as a family of all denotations of sharp terms representing a suitable vague term, from the agent’s point of view. Some algebraic operations on vague sets and their properties are defined. Some important conditions concerning the membership relation for vague sets, in connection to Blizard’s multisets and Zadeh’s fuzzy sets, are established as well. A classical outlook on a logic of vague sentences (vague logic) based on vague sets is also discussed.

Keywords

vagueness roughness vague sets rough sets knowledge vague knowledge membership relation vague logic 

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References

  1. 1.
    Blizard, W.D.: Multiset Theory. Notre Dame J. Formal Logic 30(1), 36–66 (1989)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bonikowski, Z.: A Certain Conception of the Calculus of Rough Sets. Notre Dame J. Formal Logic 33, 412–421 (1992)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bonikowski, Z.: Sets Approximated by Representations (in Polish, the doctoral dissertation prepared under the supervision of Prof. U.Wybraniec-Skardowska), Warszawa (1996)Google Scholar
  4. 4.
    Bonikowski, Z., Wybraniec-Skardowska, U.: Rough Sets and Vague Sets. In: Kryszkiewicz, M., Peters, J.F., Rybinski, H., Skowron, A. (eds.) RSEISP 2007. LNCS, vol. 4585, pp. 122–132. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Bonissone, P., Tong, R.: Editorial: reasoning with uncertainty in expert systems. Int. J. Man–Machine Studies 22, 241–250 (1985)CrossRefGoogle Scholar
  6. 6.
    Codd, E.F.: A Relational Model of Data for Large Shared Data Banks. Comm. ACM 13, 377–387 (1970)CrossRefMATHGoogle Scholar
  7. 7.
    Cresswell, M.J.: Logics and Languages. Methuen, London (1973)MATHGoogle Scholar
  8. 8.
    Demri, S., Orłowska, E.: Incomplete Information: Structure, Inference, Complexity. Springer, Heidelberg (2002)CrossRefMATHGoogle Scholar
  9. 9.
    Fine, K.: Vagueness, Truth and Logic. Synthese 30, 265–300 (1975)CrossRefMATHGoogle Scholar
  10. 10.
    Iwiński, T.: Algebraic Approach to Rough Sets. Bull. Pol. Acad. Sci. Math. 35, 673–683 (1987)MathSciNetMATHGoogle Scholar
  11. 11.
    Malinowski, G.: Many-Valued Logics. Oxford University Press, Oxford (1993)MATHGoogle Scholar
  12. 12.
    Marcus, S.: A Typology of Imprecision. In: Brainstorming Workshop on Uncertainty in Membrane Computing Proceedings, Palma de Mallorca, pp. 169–191 (2004)Google Scholar
  13. 13.
    Marek, W., Pawlak, Z.: Rough Sets and Information Systems, ICS PAS Report 441 (1981)Google Scholar
  14. 14.
    Pagliani, P.: Rough Set Theory and Logic-Algebraic Structures. In: Orłowska, E. (ed.) Incomplete Information: Rough Set Analysis, pp. 109–190. Physica Verlag, Heidelberg (1998)CrossRefGoogle Scholar
  15. 15.
    Parsons, S.: Current approaches to handling imperfect information in data and knowledge bases. IEEE Trans. Knowl. Data Eng. 8(3), 353–372 (1996)CrossRefGoogle Scholar
  16. 16.
    Pawlak, Z.: Information Systems, ICS PAS Report 338 (1979)Google Scholar
  17. 17.
    Pawlak, Z.: Information Systems – Theoretical Foundations (in Polish). PWN – Polish Scientific Publishers, Warsaw (1981)MATHGoogle Scholar
  18. 18.
    Pawlak, Z.: Information Systems – Theoretical Foundations. Information Systems 6, 205–218 (1981)CrossRefMATHGoogle Scholar
  19. 19.
    Pawlak, Z.: Rough Sets. Intern. J. Comp. Inform. Sci. 11, 341–356 (1982)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Pawlak, Z.: Rough Sets. Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)MATHGoogle Scholar
  21. 21.
    Pawlak, Z.: Vagueness and uncertainty: A rough set perspective. Computat. Intelligence 11(2), 227–232 (1995)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Pawlak, Z.: Orthodox and Non-orthodox Sets - some Philosophical Remarks. Found. Comput. Decision Sci. 30(2), 133–140 (2005)Google Scholar
  23. 23.
    Pomykała, J., Pomykała, J.A.: The Stone Algebra of Rough Sets. Bull. Pol. Acad. Sci. Math. 36, 495–508 (1988)MathSciNetMATHGoogle Scholar
  24. 24.
    Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)MATHGoogle Scholar
  25. 25.
    Skowron, A., Komorowski, J., Pawlak, Z., Polkowski, L.: Rough Sets Perspective on Data and Knowledge. In: Klösgen, W., Żytkow, J.M. (eds.) Handbook of Data Mining and Knowlewdge Discovery, pp. 134–149. Oxford University Press, Oxford (2002)Google Scholar
  26. 26.
    Słowiński, R., Stefanowski, J.: Rough-Set Reasoning about Uncertain Data. Fund. Inform. 23(2–3), 229–244 (1996)MathSciNetMATHGoogle Scholar
  27. 27.
    Wybraniec-Skardowska, U.: Knowledge, Vagueness and Logic. Int. J. Appl. Math. Comput. Sci. 11, 719–737 (2001)MathSciNetMATHGoogle Scholar
  28. 28.
    Zadeh, L.A.: Fuzzy Sets. Information and Control 8, 338–353 (1965)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Zadeh, L.A.: PRUF: A meaning representation language for natural languages. Int. J. Man–Machine Studies 10, 395–460 (1978)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Zbigniew Bonikowski
    • 1
  • Urszula Wybraniec-Skardowska
    • 2
  1. 1.Institute of Mathematics and InformaticsUniversity of OpoleOpolePoland
  2. 2.Autonomous Section of Applied Logic, Poznań School of Banking, Faculty in ChorzówPoland

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