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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 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.
The name derives from the algebraic operation of residuation.
References
Albrycht, J., Wiśniewski, H. (eds.): Proceedings of the Polish Symposium on Interval & Fuzzy Mathematics, Poznań. Wydawn Politech Pozn (1983)
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)
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)
Gottwald, S.: Set theory for fuzzy sets of higher level. Fuzzy Sets Syst. 2, 125–151 (1979)
Gottwald, S.: Fuzzy uniqueness of fuzzy mappings. Fuzzy Sets Syst. 3, 49–74 (1980)
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)
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)
Gottwald, S.: Generalized solvability criteria for fuzzy equations. Fuzzy Sets Syst. 17, 285–296 (1985)
Gottwald, S.: Characterizations of the solvability of fuzzy equations. Elektron. Informationsverarbeitung Kybernetik 22, 67–91 (1986)
Hájek, P.: Metamathematics of Fuzzy Logic. In: Trends in Logic, vol. 4. Kluwer Acad. Publ, Dordrecht (1998)
Gottwald, S.: A Treatise on Many-Valued Logics. In: Studies in Logic and Computation, vol. 9. Research Studies Press, Baldock (2001)
Butnariu, D., Klement, E.P., Zafrany, S.: On triangular norm-based propositional fuzzy logics. Fuzzy Sets Syst. 69, 241–255 (1995)
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)
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)
Goguen, J.A.: The logic of inexact concepts. Synthese 19, 325–373 (1968–69)
Gottwald, S.: Mehrwertige Logik. Logica Nova. Akademie-Verlag, Berlin (1989)
Chang, C.C.: Algebraic analysis of many valued logics. Trans. Am. Math. Soc. 88, 476–490 (1958)
Hájek, P., Lluís, G., Francesc, E.: A complete many-valued logic with product-conjunction. Arch. Math. Log. 35, 191–208 (1996)
Pavelka, J.: On fuzzy logic. I–III. Zeitschr. math. Logik Grundl. Math. 25, 45–52, 119–134, 447–464 (1979)
Jayaram, B.: On the continuity of residuals of triangular norms. Nonlinear Anal. 72, 1010–1018 (2010)
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)
Novák, V.: A formal theory of intermediate quantifiers. Fuzzy Sets Syst. 159(10), 1229–1246 (2008)
Gerla, G.: Fuzzy logic. Mathematical Tools for Approximate Reasoning. In: Trends in Logic, vol. 11. Kluwer Academic Publishers (2001)
Rodabaugh, S.E., Klement, E.P., Höhle, U. (eds.): Applications of Category Theory to Fuzzy Subsets. Kluwer Acad. Publ, Dordrecht (1992)
Höhle, U., Klement, E.P. (eds.): Fuzzy Sets and Systems, vol. 256 (2014)
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)
Höhle, U.: Many valued logic and sheaf theory. Sci. Math. Japon. 68(3), 417–433 (2008)
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)
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)
Thiele, H.: Theorie der endlichwertigen Łukasiewiczschen Prädikatenkalküle der ersten Stufe. Zeitschr. math. Logik Grundl. Math 4, 108–142 (1958)
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)
Gottwald, S.: Fuzzy Sets and Fuzzy Logic. Artificial Intelligence. Verlag Vieweg, Wiesbaden, and Tecnea, Toulouse (1993)
Hájek, P.: On equality and natural numbers in Cantor-Łukasiewicz set theory. Log. J. IGPL 21(3), 91–100 (2013)
Skolem, Th.: Bemerkungen zum Komprehensionsaxiom. Zeitschr. math. Logik Grundl. Math. 3, 1–17 (1957)
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)
Arnon, A.: Hypersequents, logical consequence and intermediate logics for concurrency. Ann. Math. Log. AI 4, 225–248 (1991)
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)
Metcalfe, G., Olivetti, N., Gabbay, D.: Proof theory for product logics. Neural Netw. World 13, 549–558 (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-28808-6_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-28807-9
Online ISBN: 978-3-319-28808-6
eBook Packages: EngineeringEngineering (R0)