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On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory pp 1–11Cite as

Fuzzy Logic and the Linz Seminar: Themes and Some Personal Reminiscences

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Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 336))

Abstract

The paper discusses concisely the main developments in the field of mathematically oriented fuzzy logics and how they found their representation over the years in the Linz Seminars on Fuzzy Set Theory.

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Notes

  1. 1.

    This comes from the fact that in early phases of these considerations t-conorms also had been discussed under the name “S-norms”.

  2. 2.

    The name derives from the algebraic operation of residuation.

References

  1. Albrycht, J., Wiśniewski, H. (eds.): Proceedings of the Polish Symposium on Interval & Fuzzy Mathematics, Poznań. Wydawn Politech Pozn (1983)

    Google Scholar 

  2. Bocklisch, S., Orlovski, S., Peschel, M., Nishiwaki, Y. (eds.): Fuzzy sets applications, methodological approaches, and results. In: Mathematische Forschung, vol. 30. Akademie-Verlag, Berlin (1986)

    Google Scholar 

  3. Gottwald, S.: A cumulative system of fuzzy sets. In: Marek, W., Srebrny, M., Zarach, A. (eds.) Set Theory Hierarchy Theory, Mem. Tribute A. Mostowski, Bierutowice (1975) Lecture Notes In Mathematics, vol. 537, pp. 109–119. Springer, Berlin (1976)

    Google Scholar 

  4. Gottwald, S.: Set theory for fuzzy sets of higher level. Fuzzy Sets Syst. 2, 125–151 (1979)

    Google Scholar 

  5. Gottwald, S.: Fuzzy uniqueness of fuzzy mappings. Fuzzy Sets Syst. 3, 49–74 (1980)

    Google Scholar 

  6. Gottwald, S.: \({T}\)-Normen und \(\varphi \)-Operatoren als Wahrheitswertfunktionen mehrwertiger Junktoren. In: Wechsung, G. (ed.) Frege Conference 1984 (Schwerin, 1984). Mathematical Research, vol. 20, pp. 121–128. Akademie-Verlag, Berlin (1984)

    Google Scholar 

  7. Gottwald, S.: Fuzzy set theory with \(t\)-norms and \(\varphi \)-operators. In: Di Nola, A., Ventre, A.G.S. (eds.) The Mathematics of Fuzzy Systems. Interdisciplinary Systems Research, vol. 88, pp. 143–195. TÜV Rheinland, Köln (Cologne) (1986)

    Google Scholar 

  8. Gottwald, S.: Generalized solvability criteria for fuzzy equations. Fuzzy Sets Syst. 17, 285–296 (1985)

    Google Scholar 

  9. Gottwald, S.: Characterizations of the solvability of fuzzy equations. Elektron. Informationsverarbeitung Kybernetik 22, 67–91 (1986)

    Google Scholar 

  10. Hájek, P.: Metamathematics of Fuzzy Logic. In: Trends in Logic, vol. 4. Kluwer Acad. Publ, Dordrecht (1998)

    Google Scholar 

  11. Gottwald, S.: A Treatise on Many-Valued Logics. In: Studies in Logic and Computation, vol. 9. Research Studies Press, Baldock (2001)

    Google Scholar 

  12. Butnariu, D., Klement, E.P., Zafrany, S.: On triangular norm-based propositional fuzzy logics. Fuzzy Sets Syst. 69, 241–255 (1995)

    Google Scholar 

  13. Höhle, U.: Monoidal logic. In: Kruse, R., Gebhard, J., Palm, R. (eds.) Fuzzy Systems in Computer Science. Artificial Intelligence, pp. 233–243. Verlag Vieweg, Wiesbaden (1994)

    Google Scholar 

  14. Höhle, U.: Commutative, residuated \(l\)-monoids. In: Höhle, U., Klement, E.P. (eds.) Non-Classical Logics and Their Applications to Fuzzy Subsets. Theory and Decision Library Series B, vol. 32, pp. 53–106. Kluwer Acad. Publ., Dordrecht (1995)

    Google Scholar 

  15. Goguen, J.A.: The logic of inexact concepts. Synthese 19, 325–373 (1968–69)

    Google Scholar 

  16. Gottwald, S.: Mehrwertige Logik. Logica Nova. Akademie-Verlag, Berlin (1989)

    Google Scholar 

  17. Chang, C.C.: Algebraic analysis of many valued logics. Trans. Am. Math. Soc. 88, 476–490 (1958)

    Google Scholar 

  18. Hájek, P., Lluís, G., Francesc, E.: A complete many-valued logic with product-conjunction. Arch. Math. Log. 35, 191–208 (1996)

    Google Scholar 

  19. Pavelka, J.: On fuzzy logic. I–III. Zeitschr. math. Logik Grundl. Math. 25, 45–52, 119–134, 447–464 (1979)

    Google Scholar 

  20. Jayaram, B.: On the continuity of residuals of triangular norms. Nonlinear Anal. 72, 1010–1018 (2010)

    Google Scholar 

  21. Novák, V.: On the syntactico-semantical completeness of first-order fuzzy logic. I: Syntax and semantics. II: Main results. Kybernetika 26, 47–66, 134–154 (1990)

    Google Scholar 

  22. Novák, V.: A formal theory of intermediate quantifiers. Fuzzy Sets Syst. 159(10), 1229–1246 (2008)

    Google Scholar 

  23. Gerla, G.: Fuzzy logic. Mathematical Tools for Approximate Reasoning. In: Trends in Logic, vol. 11. Kluwer Academic Publishers (2001)

    Google Scholar 

  24. Rodabaugh, S.E., Klement, E.P., Höhle, U. (eds.): Applications of Category Theory to Fuzzy Subsets. Kluwer Acad. Publ, Dordrecht (1992)

    Google Scholar 

  25. Höhle, U., Klement, E.P. (eds.): Fuzzy Sets and Systems, vol. 256 (2014)

    Google Scholar 

  26. Höhle, U.: \(M\)-valued sets and sheaves over integral commutative \(CL\)-monoids. In: Rodabaugh, S.E., et al. (eds.) Applications of Category Theory to Fuzzy Subsets, Theory and Decision Library Series B, vol. 14, pp. 34–72. Kluwer Acad. Publ., Dordrecht (1992)

    Google Scholar 

  27. Höhle, U.: Many valued logic and sheaf theory. Sci. Math. Japon. 68(3), 417–433 (2008)

    Google Scholar 

  28. Höhle, U.: Many-valued equalities and their representations. In: Klement, E.P., Mesiar, R. (eds.) Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, pp. 301–319. Elsevier, Dordrecht (2005)

    Google Scholar 

  29. Scott, D.S.: Identity and existence in intuitionistic logic. In: Fourman, M.P., Mulvey, C.J., Scott, D.S. (eds.) Applications of Sheaves. Lecture Notes in Mathematics, vol. 753, pp. 660–696. Springer, New York (1979)

    Google Scholar 

  30. Thiele, H.: Theorie der endlichwertigen Łukasiewiczschen Prädikatenkalküle der ersten Stufe. Zeitschr. math. Logik Grundl. Math 4, 108–142 (1958)

    Google Scholar 

  31. Gottwald, S.: A generalized Łukasiewicz-style identity logic. In: de Alcantara, L.P. (ed.) Mathematical Logic and Formal Systems. Lecture Notes Pure Applied Mathematics, vol. 94, pp. 183–195. Marcel Dekker, New York (1985)

    Google Scholar 

  32. Gottwald, S.: Fuzzy Sets and Fuzzy Logic. Artificial Intelligence. Verlag Vieweg, Wiesbaden, and Tecnea, Toulouse (1993)

    Google Scholar 

  33. Hájek, P.: On equality and natural numbers in Cantor-Łukasiewicz set theory. Log. J. IGPL 21(3), 91–100 (2013)

    Google Scholar 

  34. Skolem, Th.: Bemerkungen zum Komprehensionsaxiom. Zeitschr. math. Logik Grundl. Math. 3, 1–17 (1957)

    Google Scholar 

  35. Běhounek, L., Haniková, Z.: Set theory and arithmetic in fuzzy logic. In: Montagna, F. (ed.) Petr Hájek on Mathematical Fuzzy Logic. Outstanding Contributions to Logic, vol. 6, pp. 63–89. Springer, Switzerland (2015)

    Google Scholar 

  36. Arnon, A.: Hypersequents, logical consequence and intermediate logics for concurrency. Ann. Math. Log. AI 4, 225–248 (1991)

    Google Scholar 

  37. Baaz, M., Ciabattoni, A., Fermüller, C., Veith, H. (eds.): Proof theory of fuzzy logics: urquhart’s C and related logics. In: Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 1450, pp. 203–212. Springer, Berlin (1998)

    Google Scholar 

  38. Metcalfe, G., Olivetti, N., Gabbay, D.: Proof theory for product logics. Neural Netw. World 13, 549–558 (2003)

    Google Scholar 

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Authors and Affiliations

  1. Universität Leipzig, Abt. Logik Am Institut Für Philosophie, Beethovenstr. 15, 04107, Leipzig, Germany

    Siegfried Gottwald

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  1. Siegfried Gottwald
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Editors and Affiliations

  1. Dept of Knowledge-Based Math. Systems, Johannes Kepler University Linz, Linz, Austria

    Susanne Saminger-Platz

  2. Dept. of Mathematics & Descriptive Geo., Slovak University of Technology, BRATISLAVA, Slovakia

    Radko Mesiar

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Gottwald, S. (2016). Fuzzy Logic and the Linz Seminar: Themes and Some Personal Reminiscences. In: Saminger-Platz, S., Mesiar, R. (eds) On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory. Studies in Fuzziness and Soft Computing, vol 336. Springer, Cham. https://doi.org/10.1007/978-3-319-28808-6_1

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