Granularity, Multi-valued Logic, Bayes’ Theorem and Rough Sets

  • Zdzisław Pawlak
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 95)


Granularity of knowledge attracted attention of many researchers recently. This paper concerns this issue from the rough set perspective. Granularity is inherently connected with foundation of rough set theory. The concept of the rough set hinges on classification of objects of interest into similarity classes, which form elementary building blocks (atoms, granules) of knowledge. These granules are employed to define basic concepts of the theory. In the paper basic concepts of rough set theory will be defined and their granular structure will be pointed out. Next the consequences of granularity of knowledge for reasoning about imprecise concepts will be discussed. In particular the relationship between some ideas of Lukasiewicz’s multi-valued logic, Bayes’ Theorem and rough sets will be pointed out.


Decision Rule Granular Structure Information Granulation Fuzzy Graph Certainty Factor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Adams E W. (1975) The Logic of Conditionals, an Application of Probability to Deductive Logic. D. Reidel Publishing Company, Dordrecht, BostonGoogle Scholar
  2. 2.
    P. Apostoli, A. Kanda, Parts of the continuum: towards a modern ontology of science, (to appear), 1999Google Scholar
  3. 3.
    Black M. (1952) The Identity of Indiscernibles, Mind, 61Google Scholar
  4. 4.
    Borkowski L. (Ed.) (1970) Jan Lukasiewicz — Selected Works, North Holland Publishing Company, Amsterdam, London, Polish Scientific Publushers, WarszawaGoogle Scholar
  5. 5.
    Cattaneo G. (1993) Fuzzy Quantum Logic: The Logic of Unsharp Quantum Mechanics. Int. Journal of Theoretical Physics 32: 1709–1734MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Cattaneo G. (1996) Mathematical Foundations of Roughness and Fuzziness. In: Tsumoto S. at al (Eds.) The fourth International Workshop on Rough Sets, Fuzzy Sets and Machine Discovery, Proceedings, The University of Tokyo 241247Google Scholar
  7. 7.
    Chattebrjee A. (1994) Understanding vagueness, pragati publications. DehliGoogle Scholar
  8. 8.
    Dubois D., Prade H. (1999) Foreword. In: Pawlak Z. Rough Sets - Theoretical Aspect of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, Boston, LondonGoogle Scholar
  9. 9.
    Forrest P. (1997) Identity of Indiscernibilities. Stanford Encyclopedia of PhilosophyGoogle Scholar
  10. 10.
    French S. (1998) Quantum Physics and the Identity of Indiscernibles. British Journal of the Philosophy of Sciences 39Google Scholar
  11. 11.
    Krawiec K., Slowinski R., Vanderpooten D. (1996) Construction of Rough Classifiers Based on Application of Similarity Relation. In: Proceedings of the Fourth International Workshop on Rough Sets, Fuzzy Sets and Machine Discovery, November 6–8, Tokyo, Japan, 23–30Google Scholar
  12. 12.
    Legniewski S. (1992) Foundations of the General Theory of Sets. In: Surma, Srzednicki, Barnett, Riskkey (Eds) Stanislaw Legniweski Collected Works, Kluwer Academic Publishers, Dordrecht, Boston, London, 128–173Google Scholar
  13. 13.
    Lukasiewicz J. (1913) Die logishen Grundlagen der Wahrscheinichkeitsrechnung, KrakowGoogle Scholar
  14. 14.
    Parker-Rhodes A. F. (1981) The Theory of Indistinguishables. D. Reidel Publishing Company, Dordrecht, Boston, LondonGoogle Scholar
  15. 15.
    Pawlak, Z. (1998) Granurality of Knowledge, Indiscernibility and Rough Sets. In: IEEE International Conference on Granulationary Computing, May 5–9, Anchorage, Alaska, 100–103Google Scholar
  16. 16.
    Polkowski L., Skowron A. (1997) Towards Adaptative Calculus of Granules. ManuscriptGoogle Scholar
  17. 17.
    Polkowski L., Skowron A. (1994) Rough Mereology. In: Proc. of the Symphosium on Methodologies for Intelligent Systems, Charlotte, N.C., Lecture Notes in Artificial Intelligence 869, Springer Verlag, Berlin, 85–94Google Scholar
  18. 18.
    Polkowski L., Skowron A. (1996) Rough Mereology: A new Paradigm for Approximate Reasoning. Journ. of Approximate Reasoning 15 (4): 333–365MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Polkowski L., Skowron A. (Eds.) (1998) Rough Sets in Knowledge Discovery. Physica-Verlag Vol. 1, 2Google Scholar
  20. 20.
    Skowron A., Stepaniuk J. (1997) Information Granulation — a Rough Set Approach. ManuscrriptGoogle Scholar
  21. 21.
    Skowron A., Stepaniuk J. (1996) Tolerance Approximation Spaces. Fundamenta Informaticae 27: 245–253MathSciNetMATHGoogle Scholar
  22. 22.
    Sorensen R. (1997) Vagueness. Stanford Encyclopedia of PhilosophyGoogle Scholar
  23. 23.
    Suppes P. (1972) Some Open Problems in the Philosophy of Space and Time. Synthese 24: 298–316MATHCrossRefGoogle Scholar
  24. 24.
    Tsumoto S. (1998) Modelling Medical Diagnostic Rules Based on Rough Sets. In: Polkowski L., Skowron A. (Eds.) Rough Sets and Current Trends in Computing, Lecture Notes in Artificial Intelligence 1424 Springer, First International Conference, RSCTC’98, Warsaw, Poland, June, Proceedings, 475–482Google Scholar
  25. 25.
    Williamson T. (1990) Identity and Discrimination. BlackwellGoogle Scholar
  26. 26.
    Yao Y.Y., Wong S.K.M. (1995) Generalization of Rough Sets using Relationships between Attribute Values. In: Proceedings of the Second Annual Joint Conference on Information Sciences, Wrightsville Beach, N.C. USA, September 28–October 1, 245–253Google Scholar
  27. 27.
    Zadeh L. (1994) Fuzzy Graphs, Rough Sets and Information Granularity. In: Proc. Third Int. Workshop on Rough Sets and Soft Computing, Nov. 10–12, San JoseGoogle Scholar
  28. 28.
    Zadeh L. (1996) The Key Rules of Information Granulation and Fuzzy Logic in Human Reasoning, Concept Formulation and Computing with Words. In: Proc. FUZZ-96: Fifth IEEE International Conference on Fuzzy Systems, September 8–11, New OrleansGoogle Scholar
  29. 29.
    Zadeh L. (1996) Information Granulation, Fuzzy Logic and Rough Sets. In: Proc. of the Fourth Int. Workshop on Rough Sets, and Machine Discovery, November 6–8, TokyoGoogle Scholar
  30. 30.
    Ziarko W. (1993) Variable Precison Rough Set Model. Journal of Computer and System Sciences 46 /1: 39–59MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Zdzisław Pawlak
    • 1
  1. 1.Institute for Theoretical and Applied InformaticsPolish Academy of SciencesGliwicePoland

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