Abstract
In this chapter, we embark on a more specific granulation theory, stemming from the mereological theory of things. This mechanism provides us with tolerance and weak tolerance relations forming a graded similarity in the sense of Chap. 1 and with resulting therefrom granules.
Ex pluribus unum
[Saint Augustine. Confessions]
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aristotle. Lawson-Tancred, H. (transl.): Metaphysics. Penguin Classics. The Penguin Group, London, Book Delta 1203 b (2004)
Casati, R., Varzi, A.C.: Parts and Places. The Structures of Spatial Representations. MIT Press, Cambridge (1999)
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)
Iwanuś, B.: On Leśniewski’s elementary ontology. Stud. Log. XXXI, 73–119 (1973)
Leśniewski, S.: Podstawy Ogólnej Teoryi Mnogości, I (Foundations of General Set Theory, I, in Polish). Prace Polskiego Koła Naukowego w Moskwie, Sekcya Matematyczno-przyrodnicza, No. 2, Moscow (1916)
Leśniewski, S.: O podstawach matematyki (On foundations of mathematics, in Polish). Przegla̧d Filozoficzny, XXX, pp. 164–206; Przegla̧d Filozoficzny, XXXI, pp. 261–291; Przegla̧d Filozoficzny, XXXII, pp. 60–101; Przegla̧d Filozoficzny, XXXIII, pp. 77–105; Przegla̧d Filozoficzny, XXXIV, pp. 142–170 (1927–1931)
Leśniewski, S.: Über die Grundlagen der Ontologie. Comptes rendus Social Science Letters de Varsovie Cl. III, 23 Anneé, pp. 111–132 (1930)
Leśniewski, S.: On the foundations of mathematics. Topoi 2, pp. 7–52 [[5] transl. E. Luschei] (1982)
Leśniewski, S., Srzednicki, J., Surma, S.J., Barnett, D., Rickey, V.F. (eds.): Collected Works of Stanisław Leśniewski. Kluwer, Dordrecht (1992)
Ling, C.-H.: Representations of associative functions. Publ. Math. Debr. 12, 189–212 (1965)
Luschei, E.C.: The Logical Systems of Leśniewski. North Holland, Amsterdam (1962)
Łukasiewicz, J.: Die Logischen Grundlagen der Wahrscheinlichkeitsrechnung. Cracow (1913); cf., [Engl. transl.] In: Borkowski L. (ed.) Jan Łukasiewicz. Selected Works. North Holland, pp. 16–63. Polish Scientific Publishers, Amsterdam-Warsaw (1970)
Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer, Dordrecht (1991)
Polkowski, L.: Rough Sets: Mathematical Foundations. Physica Verlag, Heidelberg (2002)
Polkowski, L.: A rough set paradigm for unifying rough set theory and fuzzy set theory. Fundamenta Informaticae 54, 67–88. In: Proceedings RSFDGrC03, Chongqing, China, 2003. Lecture Notes in Artificial Intelligence, vol. 2639, pp. 70–78. Springer, Berlin (2003)
Polkowski, L.: Toward rough set foundations. Mereological approach. In: Proceedings RSCTC04, Uppsala, Sweden. Lecture Notes in Artificial Intelligence, vol. 3066, pp. 8–25. Springer, Berlin (2004)
Polkowski, L.: Formal granular calculi based on rough inclusions. In: Proceedings of IEEE 2005 Conference on Granular Computing GrC05, pp. 57–62. IEEE Press, Beijing (2005)
Polkowski, L.: Rough-fuzzy-neurocomputing based on rough mereological calculus of granules. Int. J. Hybrid Intell. Syst. 2, 91–108 (2005)
Polkowski, L.: A model of granular computing with applications. In: Proceedings of IEEE 2006 Conference on Granular Computing GrC06, pp. 9–16. IEEE Press, Atlanta (2006)
Polkowski L.: Granulation of knowledge in decision systems: the approach based on rough inclusions. The method and its applications. In: Proceedings RSEISP 07, Warsaw, Poland, June 2007. Lecture Notes in Artificial Intelligence, vol. 4585, pp. 271–279. Springer, Berlin (2007)
Polkowski, L.: A unified approach to granulation of knowledge and granular computing based on rough mereology: a survey. In: Pedrycz W., Skowron A., Kreinovich V. (eds.) Handbook of Granular Computing, pp. 375–400. Wiley, Chichester (2008)
Polkowski, L.: Granulation of knowledge: similarity based approach in information and decision systems. In: Meyers, R.A. (ed.) Springer Encyclopedia of Complexity and System Sciences. Springer, Berlin, article 00 788 (2009)
Polkowski, L.: Data-mining and knowledge discovery: case based reasoning, nearest neighbor and rough sets. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and System Sciences. Springer, Berlin, Article 00 391 (2009)
Polkowski, L.: Approximate Reasoning by Parts. An Introduction to Rough Mereology. Springer, Berlin (2011)
Polkowski, L., Skowron, A.: Rough mereology. In: Proceedings of ISMIS’94. Lecture Notes in Artificial Intelligence, vol. 869, pp. 85–94. Springer, Berlin (1994)
Polkowski, L., Skowron, A.: Rough mereology: a new paradigm for approximate reasoning. Int. J. Approx. Reason. 15(4), 333–365 (1997)
Simons, P.: Parts: A Study in Ontology, 2nd edn. Clarendon Press, Oxford (2003)
Słupecki, J.: S. Leśniewski’s calculus of names. Stud. Log. III, 7–72 (1955)
Sobociński, B.: L’analyse de l’antinomie Russellienne par Leśniewski. Methodos I, pp. 94–107, 220–228, 308–316; Methodos II, pp. 237–257 (1949–1950)
Sobociński, B.: Studies in Leśniewski’s mereology. Yearbook for 1954–55 of the Polish Society of Art and Sciences Abroad, vol. V, pp. 34–43. London (1955)
Stanfill, C., Waltz, D.: Toward memory-based reasoning. Commun. ACM 29, 1213–1228 (1986)
Tarski, A.: Les fondements de la géométrie des corps. Supplement to Annales de la Société Polonaise de Mathématique 7, 29–33 (1929)
Tarski, A.: Zur Grundlegung der Booleschen Algebra. I. Fundamenta Mathematicae 24, 177–198 (1935)
Tarski, A.: Appendix E. In: Woodger, J.H. (ed.) The Axiomatic Method in Biology, p. 160. Cambridge University Press, Cambridge (1937)
Wilson, D.R., Martinez, T.R.: Improved heterogeneous distance functions. J. Artif. Intell. Res. 6, 1–34 (1997)
Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Polkowski, L., Artiemjew, P. (2015). Mereology and Rough Mereology: Rough Mereological Granulation. In: Granular Computing in Decision Approximation. Intelligent Systems Reference Library, vol 77. Springer, Cham. https://doi.org/10.1007/978-3-319-12880-1_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-12880-1_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12879-5
Online ISBN: 978-3-319-12880-1
eBook Packages: EngineeringEngineering (R0)