Abstract
A genuine fuzzy approach to fuzzy mathematics consists in constructing axiomatic theories over suitable systems of formal fuzzy logic. The features of formal fuzzy logics (esp. the invalidity of the law of contraction) entail certain differences in form between theories axiomatized in fuzzy logic and usual theories known from classical mathematics. This paper summarizes the most important differences and presents guidelines for constructing new theories, defining new notions, and proving new theorems in formal fuzzy mathematics.
Keywords
- Formal fuzzy logic
- axiomatic theories
- the law of contraction
- fuzzy mathematics
- graded properties
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References
Běhounek, L., Cintula, P.: From fuzzy logic to fuzzy mathematics: A methodological manifesto. Fuzzy Sets and Systems 157(5), 642–646 (2006)
Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980)
Dubois, D., Prade, H. (eds.): Fundamentals of Fuzzy Sets. The Handbooks of Fuzzy Sets, vol. 7. Kluwer Academic Publishers, Boston (2000)
Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice-Hall, Englewood Cliffs (1995)
Běhounek, L., Bodenhofer, U., Cintula, P.: Relations in Fuzzy Class Theory: Initial steps. Submitted to Fuzzy Sets and Systems (2006)
Kroupa, T.: Filters in fuzzy class theory. Submitted (2006)
Běhounek, L.: Extensionality in graded properties of fuzzy relations. In: Proceedings of 11th IPMU Conference, pp. 1604–1611. Edition EDK, Paris (2006)
Běhounek, L.: On the difference between traditional and formal fuzzy logic. Submitted (2007)
Esteva, F., Godo, L.: Monoidal t-norm based logic: Towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124(3), 271–288 (2001)
Hájek, P., Cintula, P.: Triangular norm predicate fuzzy logics. To appear in Proceedings of Linz Seminar 2005 (2006)
Běhounek, L., Cintula, P.: Fuzzy class theory. Fuzzy Sets and Systems 154(1), 34–55 (2005)
Novák, V.: On fuzzy type theory. Fuzzy Sets and Systems 149(2), 235–273 (2004)
Běhounek, L., Cintula, P.: Fuzzy Class Theory: A primer v1.0. Technical Report V-939, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague (2006), Available at http://www.cs.cas.cz/research/library/reports_900.shtml
Hájek, P.: Metamathematics of Fuzzy Logic. Trends in Logic, vol. 4. Kluwer, Dordrecht (1998)
Gottwald, S.: Fuzzy Sets and Fuzzy Logic: Foundations of Application—from a Mathematical Point of View. Vieweg, Wiesbaden (1993)
Gottwald, S.: A Treatise on Many-Valued Logics. Studies in Logic and Computation, vol. 9. Research Studies Press, Baldock (2001)
Bělohlávek, R.: Fuzzy Relational Systems: Foundations and Principles. IFSR International Series on Systems Science and Engineering, vol. 20. Plenum Press, New York (2002)
Höhle, U.: Many Valued Topology and Its Applications. Kluwer, Boston (2001)
Höhle, U.: Fuzzy real numbers as Dedekind cuts with respect to a multiple-valued logic. Fuzzy Sets and Systems 24(3), 263–278 (1987)
Cintula, P.: Weakly implicative (fuzzy) logics I: Basic properties. Archive for Mathematical Logic 45(6), 673–704 (2006)
Ying, M.: Fuzzy topology based on residuated lattice-valued logic. Acta Mathematica Sinica (English Series) 17, 89–102 (2001)
Valverde, L.: On the structure of F-indistinguishability operators. Fuzzy Sets and Systems 17(3), 313–328 (1985)
Bandler, W., Kohout, L.J.: On the universality of the triangle superproduct and the square product of relations. International Journal of General Systems 25, 399–403 (1997)
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Běhounek, L., Cintula, P. (2007). Features of Mathematical Theories in Formal Fuzzy Logic. In: Melin, P., Castillo, O., Aguilar, L.T., Kacprzyk, J., Pedrycz, W. (eds) Foundations of Fuzzy Logic and Soft Computing. IFSA 2007. Lecture Notes in Computer Science(), vol 4529. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72950-1_52
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DOI: https://doi.org/10.1007/978-3-540-72950-1_52
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