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Fuzzy Set Theory: Some Aspects of the Early Development

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Aspects of Vagueness

Part of the book series: Theory and Decision Library ((TDLU,volume 39))

Abstract

In a well-known study on connections of fuzzy set theory with many-valued logic, R. Giles (1976) refers to “a neglected series of papers by Klaua (1965–1970)”, noticing that “Klaua… develops a ‘many-valued set theory’ based on Łukasiewicz logic in a manner similar to (but much more sophisticated than) that adopted here. … His work goes far beyond the present paper in the mathematical development of the subject and deserves much more attention than it appears to have received.”

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References

  • Bellman, R.E. and L.A. Zadeh: 1970, ‘Decision making in a fuzzy environment’, Management Sci. 17, 141–164.

    Article  Google Scholar 

  • Chang, C.C.: 1963, ‘The axiom of comprehension in infinite valued logic’, Mathematica Scand. 13, 9–30.

    Google Scholar 

  • deLuca, A. and S. Termini: 1972, ‘A definition of a non-probabilistic entropy in the setting of fuzzy sets theory’, Inform, and Control 20, 301–312.

    Article  Google Scholar 

  • Giles, R.: 1976, ‘Lukasiewicz logic and fuzzy set theory’, Internat. J. Man-Machine Stud. 8, 313–327.

    Article  Google Scholar 

  • Goguen, J.A.: 1967, ‘L-fuzzy sets’, J. Math. Anal. Appl. 18, 145–174.

    Article  Google Scholar 

  • Gottwald, S.: 1969, Konstruktion von Zahlbereichen und die Grundlagen der Inhaltstheorie in einer mehrwertigen Mengenlehre, Ph.D. Thesis, Leipzig, Karl Marx University.

    Google Scholar 

  • Gottwald, S.: 1971a, ‘Zahlbereichskonstruktionen in einer mehrwertigen Mengenlehre’, Zeitschr. math. Logik Grundlagen Math. 17, 145–188.

    Article  Google Scholar 

  • Gottwald, S.: 1971b, ‘Elementare Inhalts- und Masstheorie in einer mehrwertigen Mengenlehre’, Mathematische Nachr. 50, 27–68.

    Article  Google Scholar 

  • Gottwald, S.: 1972, ‘Verallgemeinerte Peano-Systeme’, Zeitschr. math. Logik Grundlagen Math. 18, 19–30.

    Article  Google Scholar 

  • Gottwald, S.: 1973, ‘Uber Einbettungen in Zahlenbereiche einer mehrwertigen Mengenlehre’, Mathematische Nachr. 56, 43–46.

    Article  Google Scholar 

  • Gottwald, S.: 1974, ‘Mehrwertige Anordnungsrelationen in klassischen Mengen’, ibid. 63, 205–212.

    Google Scholar 

  • Gottwald, S.: 1981, ‘Fuzzy-Mengen und ihre Anwendungen Ein Uberblick’, Elektron. Informationsverarb. Kybernetik EIK 17, 207–235.

    Google Scholar 

  • Jahn, K.U.: 1969, Uber einen Ansatz zur mehrwertigen Mengenlehre unter Zulassung kontinuum-vieler Wahrheitswerte, Master Thesis, Leipzig, Karl Marx University.

    Google Scholar 

  • Jahn, K.U.: 1974, ‘Eine Theorie der Gleichungssysteme mit Intervall-Koeffizienten’, ZAMM 54, 405–412.

    Article  Google Scholar 

  • Jahn, K.U.: 1975a, ‘Eine auf den Intervall-Zahlen fussende 3-wertige lineare Algebra’, Mathematische Nachr. 65, 105–116.

    Article  Google Scholar 

  • Jahn, K.U.: 1975b, ‘Intervall-wertigen Mengen’, ibid. 68, 115–132.

    Google Scholar 

  • Klaua, D.: 1965, ‘Uber einen Ansatz zur mehrwertigen Mengenlehre’, Monatsber. Deut. Akad. Wiss. Berlin 7, 859–876.

    Google Scholar 

  • Klaua, D.: 1966a, ‘Über einen zweiten Ansatz zur mehrwertigen Mengenlehre’, ibid. 8, 161–177.

    Google Scholar 

  • Klaua, D.: 1966b, ‘Grundbegriffe einer mehrwertigen Mengenlehre’, ibid. 8, 782–802.

    Google Scholar 

  • Klaua, D.: 1967a, ‘Ein Ansatz zur mehrwertigen Mengenlehre’, Mathematische Nachr. 33, 273–296.

    Article  Google Scholar 

  • Klaua, D.: 1967b, ‘Einbettung der klassischen Mengenlehre in die mehrwertige’, Monatsber. Deut. Akad. Wiss. Berlin 9, 258–272.

    Google Scholar 

  • Klaua, D.: 1968, ‘Partiell definierte Mengen’, ibid. 10, 571–578.

    Google Scholar 

  • Klaua, D.: 1969a, ‘Partielle Mengen mit mehrwertigen Grundbeziehungen’, ibid. 11, 573–584.

    Google Scholar 

  • Klaua, D.: 1969b, ‘Partielle Mengen und Zahlen’, ibid. 11, 585–599.

    Google Scholar 

  • Klaua, D.: 1970, ‘Stetige Gleichmächtigkeiten kontinuier-lich-wertiger Mengen’, ibid. 12, 749–758.

    Google Scholar 

  • Klaua, D.: 1972, ‘Zum Kardinalzahlbegriff in der mehrwertigen Mengenlehre’, in: G. Asser et al. (eds.), Theory of Sets and Topology, Deutscher Verlag d. Wiss., Berlin, pp. 313–325.

    Google Scholar 

  • Klaua, D.: 1973, ‘Zur Arithmetik mehrwertiger Zahlen’, Mathematische Nachr. 57, 275–306.

    Article  Google Scholar 

  • Negoita, C.V. and D.A. Ralescu: 1975, Applications of Fuzzy Sets to Systems Analysis, Birkhäuser-Verlag, Basel.

    Google Scholar 

  • Schwartz, D.: 1972, ‘Mengenlehre über vorgegebenen algegebraischen Systemen’, Mathematische Nachr. 53, 365–370.

    Article  Google Scholar 

  • Skolem, Th.: 1957, ‘Bemerkungen zum Komprehensionsaxiom’. Zeitschr. math. Logik Grundlagen Math. 3, 1–17.

    Article  Google Scholar 

  • Skolem, Th.: 1960, ‘A set theory based on a certain 3-valued logic’, Mathematica Scand. 8, 127–136.

    Google Scholar 

  • Sugeno, M.: 1974, Theory of Fuzzy Integrals and its Application, Ph.D. Thesis, Tokyo, Tokyo Institute of Technology.

    Google Scholar 

  • Vuckovic, V.: 1959, ‘Partially ordered recursive arithmetics’, Mathematica Scand. 7, 305–320.

    Google Scholar 

  • Wechler, W. and V. Dimitrov: 1974, ‘R-fuzzy automata’, in: J.L. Rosenfeld (ed.), Information Processing 1974 North-Holland Pubi. Comp., Amsterdam, pp. 657–660.

    Google Scholar 

  • Zadeh, L.A.: 1965, ‘Fuzzy sets’. Inform, and Control 8, 338–353.

    Article  Google Scholar 

  • Zadeh, L, A.: 1971, ‘Similarity relations and fuzzy orderings’, Information Sci. 3, 177–200.

    Article  Google Scholar 

  • Zadeh, L.A.: 1975, ‘The concept of a linguistic variable and its application to approximate reasoning – I’, ibid. 8, 199–250.

    Google Scholar 

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© 1984 D. Reidel Publishing Company

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Gottwald, S. (1984). Fuzzy Set Theory: Some Aspects of the Early Development. In: Skala, H.J., Termini, S., Trillas, E. (eds) Aspects of Vagueness. Theory and Decision Library, vol 39. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-6309-2_2

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  • DOI: https://doi.org/10.1007/978-94-009-6309-2_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-009-6311-5

  • Online ISBN: 978-94-009-6309-2

  • eBook Packages: Springer Book Archive

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