Cantorian Sets, Fuzzy Sets, Rough Sets and Fregean Sets

Apostoli, P.; Kanda, A.

Artykuł - szczegóły

Tytuł artykułu

Cantorian Sets, Fuzzy Sets, Rough Sets and Fregean Sets

Autorzy

Apostoli P. , Kanda A.

Wybrane pełne teksty z tego czasopisma

http://journals.pan.pl/dlibra/journal/95347

Identyfikatory

Warianty tytułu

Języki publikacji

Abstrakty

Fuzzy set theory [101] and rough set theory [68] deal with the concept of vagueness in the setting of set theory in seemingly different ways. The connection between these two theories of vague sets has been studied in [71], [94], [99], [100], where the probabilistic membership function of fuzzy set theory is derived within approximation spaces. This note initiates the study how "vague power sets" and "vague Cartesian products" may be constructed from given vague sets, in the settings of both fuzzy and rough sets. The introduction of vague sets by the principle of comprehension is also studied in both fuzzy and rough set theory. A marked difference between these two vague set theories with respect to these set constructions is revealed.

Słowa kluczowe

Cantorian set fuzzy set rough set Fregean set

Wydawca

Polska Akademia Nauk, Wydział IV Nauk Technicznych

Czasopismo

Bulletin of the Polish Academy of Sciences. Technical Sciences

Rocznik

2002

Tom

Vol. 50, Nr 3

Strony

247--276

Opis fizyczny

Bibliogr. 104 poz.

Twórcy

autor

Apostoli P.

apostoli@cs.toronto.edu

Department of Philosophy, The University of Toronto, Toronto, Ontario, Canada

autor

Kanda A.

akanda@cs.toronto.edu

Department of Philosophy, The University of Toronto, Toronto, Ontario, Canada

Bibliografia

[1] P. Aczel, Frege structure and the notions of proposition, truth and set, in J. Barwise, H. Keisler and K. Kunen, Eds., The Kleene Symposium, (North-Holland 1980) 31-59.
[2] P. Aczel, Non-well-founded sets, CSLI Lecture Notes, 14 (Stanford, California 1988).
[3] K. Akiba, Vagueness as a modality, Philosophical Quarterly, 50 (2000) 359-70.
[4] K. Akiba, Indefiniteness of mathematical objects, Philoscphia Mathematica, 8 (2000) 26-46. :
[5] P. Apostoli, A. Kanda, Approximation spaces of type-free sets, Proceedings of the Second Annual Conference on Rough Sets and Current Trends in Computing (RSCTC 2000, October 16-19), eds. W. Zairko, Y. Y. Yao. Lecture Notes in Artificial Intelligence, v. 2005. Eds. J. G. Carbonell, J. Sickmann, Springer-Verlag, (2000) 98-105.
[6] P. Apostoli, A. Kanda, Parts of continuum: towards a modern ontology of science, The Poznan Studies in the Philosophy of Science and Humanities (Ed. Leszek Nowak), to be published.
[7] P. Apostoli, A. Kanda, The rough set foundations of arithmetic, ms. in preparation.
[8] P. Apostoli, A. Kanda, 5S. Shapiro, ms. in preparation.
[9] K. J. Barwise, L. Moss, Vicious circles – on the mathematics of non-well-founded phenomena, CSLI Lecture Notes, Stanford, California, 60 (1996).
[10] M. Beeson, Foundations of constructive mathematics, Ergebnisse der Mathematik Grenzgebiete, 3. Folge, Bd. 6 Springer 1985.
[11] S. Blamey, Partial logic, handbook of philosophical logic, eds. D. Gabbay and F. D. Guenthner, Reidel Publishing Company, III (1986) 1-70.
[12] J.L. Bell, Type reducing correspondences and well-orderings: Frege’s and Zermelo’s constructions re-examined, J. of Symbolic Logic, 60 (1995).
[13] J. L. Bell, Set and classes as many, J. of Philosophical Logic, 29, 6, (2000) 595-681.
[14] R. Bunn, Developments in the foundation of mathematics 1870-1910, (Chapter 6 of [44]) 220-255.
[15] C. Burali-Forti, Una questione sui numeri transfiniti, Rend. Circ. Mat. Palermo, 11 (1897) 154-164.
[16] G. Cantor, Uber die ausdehnung eins satzes aus der theorie der trigonometrischen reihen, Mathematische Annalen, 5 (1872) 123-132.
[17] G. Cantor, Beittrage zur Bertindung der transfiniten Mengenehre,I, II, Mathematische Annalen, 46 (1895) 481-512; 49, 207-46. In [18, pp. 282-56].
[18] G. Cantor, Gesammelte Abhandlungen Mathenatischen und Philosopischen Inhalts, Ed. by E. Zermelo, Berlin: Springer; reprinted, Hildesheim: Olms(1962).
[19] S. Chanas, D. Kuchta, Further remarks on the relation between rough and fuzzy sets, Fuzzy sets and Systems, 47 (1992) 391-394.
[20] B. F. Chellas, An introduction to modal logic, Cambridge University Press, Cambridge 1980.
[21] N. Cocchiarella, Whither Russell’s paradox of predication?, in Logic and Ontology, of Studies in Contemporary Philosophy, edited by K. Muntz, New York University Press, 2 (1972a).
[22] N. Cocchiarella, Properties as individuals in formal ontology, Nots, VI, 2, (May 1972b) 165-188.
[23] K. Devlin, The joy of sets, fundamentals of contemporary set theory, Second Edition, Springer-Verlag, Berlin 1993.
[24] D. Dubios, H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems, 17 (1990) 191-209.
[25] H.B. Enderton, Elements of set theory, Academic Press, Orlando 1977.
[26] S. Feferman, Towards useful type-free theories I, Journal of symbolic Logic, 9 (1984) 75-111.
[27] A.A. Fraenkel, Zu den grundlagen der Cantor-Zermeloschen mengenlehre, Mathematische Annalen, 86 (1922) 230-237.
[28] G. Frege, Die grundlagen der arithmetik, Hildesheim 1884.
[29] G. Frege, Grundgesetze der arithmetik, 1, 2 (1893, 1903) Hildesheim.
[30] Frege, Gottlob, Letter to Russell, in van Heijenoort, Jean, From Frege to Gédel, Cambridge: Harvard University Press (1967) 126-128.
[31] G. Frege, The Russell Paradoz, in Frege, Gottlob 1903, The basic laws of arithmetic, Berkeley: University of California Press (1964) 127-143. Abridged and reprinted in Irvine, A.D., Bertrand Russell: critical assessments, 2 (1999) 1-3, London: Routledge.
[32] G. Frege, Concerning Zermelo’s independent discovery of Russell’s paradox, in Frege Against Formalists (1903).
[33] M. Furth, Editors introduction, in G. Frege, The basic laws of arithmetic, M. Furth (translator and editor), Berkeley University of California Press, 1967.
[34] P. T. Geach, Class and concept, The Philosophical Review, LXTV (1955) 561-570.
[35] P. T. Geach, On Frege’s way out, Mind, 65 (1965) 408-409.
[36] P. T. Geach, Identity, Review of Metaphysics, XXI, 1, (1967) 3-12.
[37] P. C. Gilmore, Partial set theory, Los Angeles, July 10-August 4, 1967.
[38] P.C. Gilmore, The consistency of partial set theory without extensionality, IBM Research Report RC, 1973.
[39] P. C. Gilmore, Attributes, sets, partial sets and identity, Essays dedicated to A. Heyting, on the occasion of his 70th birthday, North Holland, Amsterdam 1968.
[40] P.C. Gilmore, A consistent naive set theory; foundations for a formal theory of computation, IBM Research Reports RC 3413 (June 22) 1971.
[41] P. C. Gilmore, Combining unrestricted abstraction with universal quantification, in (87, Seldin and Hindely, 1980).
[42] P.C. Gilmore, Natural deduction based set theories: a new resolution of the old paradoxes, Journal of Symbolic Logic, 51 (1986) 394-411.
[43] K. Gédel, What is Cantor’s continuum problem?, American Mathematical Monthly, 54 (1947) 515-525.
[44] I. Grattan-Guiness, From the calculus to set theory, 163-1910: an introductory history, Gerald Duckworth & Company, London 1980.
[45] M. Hallett, Cantorian set theory and limitation of size, Clarendon Press, Oxford 1984.
[46] A. D. Irvine, Bertrand Russell: critical assessments, 2 (1999) 1-3, London: Routledge.
[47] A. D. Irvine, Russell’s Paradox, The Stanford Encyclopedia of Philosophy (Spring 2002 Edition), Edward N. Zalta (ed.).
[48] T. Jech, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics 13 (1974), part II. American Mathematics Society.
[49] R. E. Jennings, Private Communication, 1977.
[50] B. Jénsson,, A. Tarski, Boolean algebras with operators I, Zeithschrift fur Mathematische Logik und Grundlagen der Mathematik, 9 (1963) 67-96, American J. of Mathematics, 74 (1952) 891-939.
[51] R. Kraut, Indiscernibility and Ontology, Synthese, 44 (1980) 113-135.
[52] S. Kripke, Semantical analysis of modal logic I, Normal Modal Propositional. Calculi, Zeitschrift f. Math. Logik und Grundlagen d. Math., 9 (1963).
[53] S. Kripke, Identity and necessity, in Identity and Individuation, ed., M. Muntz, New York University press (1971) 135-164.
[54] S. Kripke, Outline of a theory of truth, Journal of Philosophy, 72 (1975) 690-716.
[55] G. J. Klir, Multi-valued versus modallogic: alternative frameworks for uncertaintly modelling, (1997) 132-155.
[56] S. Lavine, Understanding the infinite, Harvard Univ. Press, Cambridge, Massachusetts.
[57] G. W. Leibniz, New essays on human understanding, ed. P. Remnant and J. Bennett, Cambridge University Press, Cambridge 1982.
[58] D. Lewis, Counterpart theory and quantified modal logic, J. of Philosophy, 65 (1968) 113-26.
[59] D. K. Lewis, Onthe plurality of worlds, Oxford: Basil Blackwell, 1986.
[60] D. K. Lewis, Parts of classes, Oxford: Basil Blackwell, 1991.
[61] T. Y. Lin, Topological and fuzzy sets, in R. Slowinski (Ed.) Intelligent decision and support: handbook of applications and advances of the rough sets, Kluwer Academic Publishers, Boston (1992) 287-304.
[62] T. Y. Lin, Neighborhood systems: a qualitative theory for fuzzy and rough sets, Advances in information science, ed. Paul Wang, also in: Procceeding of the Second Annual Joint Conference on Information Science: The Fourth Annual International Conference on Fuzzy Theory and Technology, Wrightsville Beach, N. C., (1995) 255-258.
[63] T. Y. Lin, A set theory for soft computing, A unified view of fuzzy sets via neighborhoods, Proceedings of 1996 IEEE International Conference on Fuzzy Systems, New Orleans, Louisiana (September 8-11, 1996) 1140-1146.
[64] Robert L. Martin, Recent essays on truth and the liar paradox, Oxford 1976.
[65] D. Mirimanoff, Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theorie des ensembles, L’enseignment Mathematique, 19 (1917) 37-52.
[66] E. Orlowska, Semantics of vague concepts, in: Foundations of logic and linguistics, Problems and Their Solutions, eds. G. Dorn and P. Weingartner, Plenum Press, New York 1985.
[67] C. Parsons, Whatis the iterative conception of set? in Mathematics in Philosophy, Cornell University Press, Ithaca, New York 1983.
[68] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences, 11.(1982) 341-350.
[69] Z. Pawlak, Rough sets and fuzzy sets, Fuzzy sets and systems, 17 (1985) 99-102.
[70] Z. Pawlak, Rough set elements, Rough sets in knowledge discovery: methodology and applications, eds. L. Polkowski, A. Skowron. Studies in Fuzziness and Soft Computing. Ed.: J. Kacprzyk, 18, Springer (1998).
[71] Z. Pawlak, S.K.M. Wong, W. Zairko, Rough sets: probabilistic versus determinisitic approach, International Journal of Man-Machine Studies, 29 (1988) 81-95.
[72] G. Peano, Arithmetices principia nova methodo exposita, Rome. Translated as “Principles of arithmetic, presented by a new method”in [91].
[73] G. Plotkin, A power domain construction, SIAM Journal on Computing, 5 (1976) 452-487.
[74] W. V.O. Quine, Set theory and its logic, Cambridge, Mass.: Harvard University Press, 1969.
[75] W. V.O. Quine, On the individuation of attributes, in The logical enterprise, eds. A. R. Anderson, R. B. Marcus and R. M. Martin, .Yale Univesity Press, (1975) 3-14.
[76] W. V.O. Quine, Frege’s way out, in Quine’s selected logic papers, Harvard 1995.
[77] M.D. Resnik, Frege and the philosophy of mathematics, Ithica: Cornell University Press 1980.
[78] B. A.W. Russell, Recent work on the principles of mathematics, Int. Monthly, 4 (1901).
[79] B. A.W. Russell, Letter to Frege, (1902). In [91, 124--125].
[80] B. A.W. Russell, Appendix B: The Doctrine of types, in Russell, Bertrand, Principles of Mathematics, Cambridge: Cambridge University Press, (1903) 523-528.
[81] B. A.W. Russell, On some difficulties in the theory of transfinite numbers and order types, Proc. London Math. Soc., 2, 2, (1906-07) 29-35; [84, pp. 135-164].
[82] B. A.W. Russell, Mathematicallogic as based on the theory of types, American Journal of Mathematics, 30 (1908) 222-262.
[83] B. A. W. Russell, The principles of mathematics, 2nd edition, London: George Allen & Unwin, 1937.
[84] B. A.W. Russell, £ssays in analysis, Ed. D. Lackey, London 1973.
[85] D. Scott, Combinators and classes, Lecture Notes in Computer Science, 37 (1975) 1-26.
[86] D. Scott, Data types as lattices, SIAM Journal on Computing, 5 (1976) 522-587.
[87] J. P. Seldin-Hindely, R. J. Hindely, Essays in combinatory logic, lambda calculus and formalism. Academic Press, 1980.
[88] T. Skolem, Some remarks on axiomatic set theory, in From frege to Godel, A source book in mathematical logic, 1879-1981, (1967) 290-301.
[89] M. Smyth, Power domains, Journal of Computer and Systems Science, 16 (1978) 23-36.
[90] M. Smyth-Plotkin, G. Plotkin, The categorical solutions of recursive domain equations, Proc. of the 18th FOCS Conference (1977).
[91] J. van Heijenoort, From Frege to Godel: A source book in matheamtical logic, 1879-1931, Cambridge Mass.: Harvard University Press, 1967.
[92] H. Wang, The concept of set, in From Mathematics to Philosophy, Routledge & Kegan Paul, London (1974) 181-223.
[93] A. Wiweger, On topological rough sets, Bull. Pol. Ac.: Math., 37, 16, (1989).
[94] S.K. M. Wong, W. Zairko, Comparison of the probabilistic approximate classification and the fuzzy set model, Fuzzy Sets and Systems, 21 (1987) 357-362.
[95] M. Wygralak, Rough sets and fuzzy sets - some remarks on interrelations, Fuzzy Sets and Systems, 29 (1998) 241-243.
[96] Y. Y. Yao, Constructive and algebraic methods of the theory of rough sets, Journal of Information Sciences, 109 (1998a) 21-47.
[97] Y. Y. Yao, A comparative study of fuzzy and rough sets, Journalof Information Sciences, 109 (1998b) 227-242.
[98] Y.Y. Yao, On generalizing Pawlak approximation operators, eds. L. Polkowski and A. Skowron, RSCTC’98, LNAI 1414, (1998c) 289-307.
[99] Y. Y. Yao, S.K.M. Wong, A decision theoretic framework for approximating concepts, International Journal of Man-Machine Studies, 37 (1992) 793-809.
[100] Y. Y. Yao, S.K.M. Wong, P. Lingras, A decision theoretic rough set model, in: Methodologies for intelligent systems, New York: North-Holland (1990) 17-24.
[101] L. A. Zadeh, Fuzzy sets, Information and Control, 8, 3, (1965) 338-353.
[102] L. A. Zadeh, Incomplete information: rough set analysis, Physica Verlag, Heidelberg (1998).
[103] E. Zermelo, Untersuchungen tiber die grundlagen der mengenlehre I, Matematische Annalen, 65 (1908) 261-281.
[104] E. Zermelo, Uber grenzzahlen und mengenbereiche, Fundamenta Matematicae, 16 (1930) 29-47.

Typ dokumentu

Bibliografia

Identyfikator YADDA

bwmeta1.element.baztech-article-BPG1-0010-0051