A Treatise on Rough Sets

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3700)


This article presents some general remarks on rough sets and their place in general picture of research on vagueness and uncertainty – concepts of utmost interest, for many years, for philosophers, mathematicians, logicians and recently also for computer scientists and engineers particularly those working in such areas as AI, computational intelligence, intelligent systems, cognitive science, data mining and machine learning. Thus this article is intended to present some philosophical observations rather than to consider technical details or applications of rough set theory. Therefore we also refrain from presentation of many interesting applications and some generalizations of the theory.


Sets fuzzy sets rough sets antinomies vagueness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Apostoli, P., Kanda, A.: Parts of the Continuum: Towards a Modern Ontology of Sciences. Technical Reports in Philosophical Logic 96,97(1), Revised March, The University of Toronto, Department of Philosophy (1999)Google Scholar
  2. 2.
    Banerjee, M., Chakraborty, M.K.: Rough Consequence and Rough Algebra. In: Ziarko, W.P. (ed.) Rough Sets, Fuzzy Sets and Knowledge Discovery, Proc. Int. Workshop on Rough Sets and Knowledge Discovery (RSKD 1993), Workshops in Computing, pp. 196–207. Springer-Verlag & British Computer Society (1993)Google Scholar
  3. 3.
    Banerjee, M., Chakraborty, M.K.: Algebras from Rough Sets. In: [27], pp. 157–188 (2004)Google Scholar
  4. 4.
    Banerjee, M.: Rough truth, consequence, consistency and belief revision. In: Tsumoto, S., Słowiński, R., Komorowski, J., Grzymała-Busse, J.W., et al. (eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 95–102. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Bazan, J.G., Peters, J.F., Skowron, A.: Behavioral pattern identification through rough set modelling. In: Ślęzak, D., Yao, J., Peters, J.F., Ziarko, W.P., Hu, X. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3642, pp. 688–697. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Chakraborty, M.K., Banerjee, M.: Rough Consequence. Bull. Polish Acad. Sc (Math.) 41(4), 299–304 (1993)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Breiman, L.: Statistical Modeling: The Two Cultures. Statistical Science 16(3), 199–231 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cantor, G.: Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Leipzig (1883)Google Scholar
  9. 9.
    Casati, R., Varzi, A.: Parts and Places. In: The Structures of Spatial Representation. MIT Press, Bradford Books (1999)Google Scholar
  10. 10.
    Doherty, P., Łukaszewicz, W., Skowron, A., Szałas, A.: Knowledge Engineering: A Rough Set Approach. Springer, Heidelberg (2005) (to appear)Google Scholar
  11. 11.
    Dubois, D., Prade, H.: Foreword. In: Pawlak, Z. (ed.) Rough Sets, Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)Google Scholar
  12. 12.
    Frege, G.: Grundlagen der Arithmetik, vol. 2. Verlag von Herman Pohle, Jena (1893)Google Scholar
  13. 13.
    Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.): Handbook of Logic in Aretificial Intelligence and Logic Programming: Nonmonotonic Reasoning and Uncertain Reasoning, vol. 3. Calderon Press, Oxford (1994)Google Scholar
  14. 14.
    Greco, S., Matarazzo, B., Słowiński, R.: Rough Set Theory for Multicriteria Decision Analysis. European Journal of Operational Research 129(1), 1–47 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Grzymała-Busse, J.W.: Managing Uncertainty in Expert Systems. Kluwer Academic Publishers, Norwell (1990)Google Scholar
  16. 16.
    Friedman, J., Hastie, T., Tibshirani, R.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  17. 17.
    Keefe, R.: Theories of Vagueness. Cambridge Studies in Philosophy, Cambridge (2000)Google Scholar
  18. 18.
    Keefe, R., Smith, P. (eds.): Vagueness: A Reader. MIT Press, Massachusetts (1997)Google Scholar
  19. 19.
    Leśniewski, S.: Grungzüge eines neuen Systems der Grundlagen der Mathematik. Fundamenta Matemaicae 14, 1–81 (1929)zbMATHGoogle Scholar
  20. 20.
    Łukasiewicz, J.: Die Logischen grundlagen der Wahrscheinlichkeitsrechnung. Kraków. In: Borkowski, L. (ed.) Jan Łukasiewicz - Selected Works, North Holland Publishing Company, Polish Scientific Publishers (1913/1970)Google Scholar
  21. 21.
    Marcus, S.: The paradox of the heap of grains in respect to roughness, fuzziness and negligibility. In: Polkowski, L., Skowron, A. (eds.) RSCTC 1998. LNCS (LNAI), vol. 1424, pp. 19–23. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  22. 22.
    Nguyen, S.H., Bazan, J.G., Skowron, A., Nguyen, H.S.: Layered learning for concept synthesis. In: Peters, J.F., Skowron, A., Grzymała-Busse, J.W., Kostek, B.z., Świniarski, R.W., Szczuka, M.S. (eds.) Transactions on Rough Sets I. LNCS, vol. 3100, pp. 187–208. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  23. 23.
    Orłowska, E.: Semantics of Vague Conepts. In: Dorn, G., Weingartner, P. (eds.) Foundation of Logic and Linguistics, pp. 465–482. Plenum Press, New York (1984)Google Scholar
  24. 24.
    Orłowska, E.: Reasoning about Vague Concepts. Bull. Polish Acad. Sci. Math. 35, 643–652 (1987)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Pal, S.K., Skowron, A.: Rough Fuzzy Hybridization: A New Trend in Decision-Making. Springer, Singapore (1999)zbMATHGoogle Scholar
  26. 26.
    Pal, S.K., Mitra, P.: Pattern Recognition Algorithms for Data Mining. CRC Press, Boca Raton (2004)zbMATHCrossRefGoogle Scholar
  27. 27.
    Pal, S.K., Polkowski, L., Skowron, A. (eds.): Rough-Neural Computing: Techniques for Computing with Words. Springer, Heidelberg (2004)zbMATHGoogle Scholar
  28. 28.
    Pawlak, Z.: Rough Sets. Int. J. of Information and Computer Sciences 11(5), 341–356 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Pawlak, Z.: Rough Logic. Bull. Polish. Acad. Sci. Tech. 35(5-6), 253–258 (1987)zbMATHMathSciNetGoogle Scholar
  30. 30.
    Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. In: System Theory, Knowledge Engineering and Problem Solving, vol. 9. Kluwer Academic Publishers, Dordrecht (1991)Google Scholar
  31. 31.
    Pawlak, Z., Skowron, A.: Rough Membership Functions. In: Yager, R.R., Fedrizzi, M., Kacprzyk, J. (eds.) Advances in the Dempster-Schafer Theory of Evidence, pp. 251–271. John Wiley and Sons, New York (1994)Google Scholar
  32. 32.
    Peters, J.F., Skowron, A., Synak, P., Ramanna, S.: Rough Sets and Information Granulation. In: De Baets, B., Kaynak, O., Bilgiç, T. (eds.) IFSA 2003. LNCS, vol. 2715, pp. 370–377. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  33. 33.
    Polkowski, L.: Rough Sets: Mathematical Foundations. Physica, Heidelberg (2002)zbMATHGoogle Scholar
  34. 34.
    Polkowski, L., Skowron, A.: Rough Mereology: A New Paradigm for Approximate Reasoning. International Journal of Approximate Reasoning 15, 333–365 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Polkowski, L., Skowron, A.: Rough Mereological Calculi of Granules: A Rough Set Approach to Computation. Computational Intelligence 17, 472–492 (2001)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Read, S.: Thinking about Logic. An Introduction to the Philosophy of Logic. Oxford University Press, Oxford (1995)Google Scholar
  37. 37.
    Russell, B.: The Principles of Mathematics, 1st edn. George Allen & Unwin Ltd., London (1903) (2nd edn. in 1937)Google Scholar
  38. 38.
    Russell, B.: An Inquiry into Meaning and Truth. George Allen and Unwin, London (1940)Google Scholar
  39. 39.
    Skowron, A.: Rough Sets in KDD (plenary lecture). In: Shi, Z., Faltings, B., Musen, M. (eds.) 16-th World Computer Congress (IFIP 2000): Proceedings of Conference on Intelligent Information Processing (IIP 2000), pp. 1–17. Publishing House of Electronic Industry, Beijing (2000)Google Scholar
  40. 40.
    Skowron, A.: Approximate Reasoning in Distributed Environments. In: Zhong, N., Liu, J. (eds.) Intelligent Technologies for Information Analysis, pp. 433–474. Springer, Heidelberg (2004)Google Scholar
  41. 41.
    Skowron, A.: Rough Sets and Vague Concepts. Fundamenta Informaticae 64(1-4), 417–431 (2005)zbMATHMathSciNetGoogle Scholar
  42. 42.
    Skowron, A., Stepaniuk, J.: Tolerance Approximation Spaces. Fundamenta Informaticae 27(2-3), 245–253 (1996)zbMATHMathSciNetGoogle Scholar
  43. 43.
    Skowron, A., Peters, J.: Rough Sets: Trends and Challenges (plenary talk). In: Wang, G., Liu, Q., Yao, Y.Y., Skowron, A. (eds.) RSFDGrC 2003. LNCS (LNAI), vol. 2639, pp. 25–34. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  44. 44.
    Skowron, A., Świniarski, R.W., Synak, P.: Approximation spaces and information granulation. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets III. LNCS, vol. 3400, pp. 175–189. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  45. 45.
    Słowiński, R., Vanderpooten, D.: Similarity Relation as a Basis for Rough Approximations. In: Wang, P. (ed.) Advances in Machine Intelligence and Soft Computing, vol. 4, pp. 17–33. Duke University Press (1997)Google Scholar
  46. 46.
    Swift, J.: Gulliver’s Travels into Several Remote Nations of the World. London, M, DCC, XXVI (1726)Google Scholar
  47. 47.
    Vapnik, V.: Statistical Learning Theory. John Wiley & Sons, New York (1998)zbMATHGoogle Scholar
  48. 48.
    Vitória, A.: A Framework for Reasoning with Rough Sets. Licentiate Thesis, Linköping University, Transactions on Rough Sets IV: Journal Subline, LNCS. Springer, Heidelberg (2005) (to appear)Google Scholar
  49. 49.
    Vopenka, P.: Mathematics in the Alternative Set Theory, Teubner, Leipzig (1979)Google Scholar
  50. 50.
    Ziarko, W.: Variable Precision Rough Set Model. Journal of Computer and System Sciences 46, 39–59 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Zadeh, L.A.: Fuzzy Sets. Information and Control 8, 338–353 (1965)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Institute for Theoretical and Applied InformaticsPolish Academy of SciencesGliwicePoland
  2. 2.Warsaw School of Information TechnologyWarsawPoland

Personalised recommendations