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On the difference between traditional and deductive fuzzy logic. (English) Zbl 1175.03012
Author’s abstract: “In three case studies on notions of fuzzy logic and fuzzy set theory (Dubois-Prade’s gradual elements, the entropy of a fuzzy set, and aggregation operators), the paper exemplifies methodological differences between traditional and deductive fuzzy logic. While traditional fuzzy logic admits various interpretations of membership degrees, deductive fuzzy logic always interprets them as degrees of truth preserved under inference. The latter fact imposes several constraints on systems of deductive fuzzy logic, which need not be followed by mainstream fuzzy logic. That makes deductive fuzzy logic a specific area of research that can be characterized both methodologically (by constraints on meaningful definitions) and formally (as a specific class of logical systems). An analysis of the relationship between deductive and traditional fuzzy logic is offered.”

03B52 Fuzzy logic; logic of vagueness
03E72 Theory of fuzzy sets, etc.
68T37 Reasoning under uncertainty in the context of artificial intelligence
Full Text: DOI
[1] Atanassov, K.T., Intuitionistic fuzzy sets, Fuzzy sets and systems, 20, 87-96, (1986) · Zbl 0631.03040
[2] L. Běhounek, Towards a formal theory of fuzzy Dedekind reals, in: Proc. Joint EUSFLAT-LFA Conf. Barcelona, 2005, pp. 946-954.
[3] L. Běhounek, Two notions of fuzzy lattice completion, in: V. Novák, M. Štěpnička (Eds.), Proc. LSC IV & 4th Workshop of the ERCIM Working Group on Soft Computing, Ostrava, 2005, pp. 22-24, Research Report No. 77, Institute for Research and Application of Fuzzy Modeling, University of Ostrava.
[4] Běhounek, L.; Cintula, P., Fuzzy class theory, Fuzzy sets and systems, 154, 1, 34-55, (2005) · Zbl 1086.03043
[5] L. Běhounek, P. Cintula, Fuzzy class theory as foundations for fuzzy mathematics, in: Fuzzy Logic, Soft Computing and Computational Intelligence: 11th IFSA World Congr., Vol. 2, Beijing, 2005, Tsinghua University Press, Springer, pp. 1233-1238.
[6] Běhounek, L.; Cintula, P., From fuzzy logic to fuzzy mathematics: a methodological manifesto, Fuzzy sets and systems, 157, 5, 642-646, (2006) · Zbl 1108.03027
[7] L. Běhounek, P. Cintula, 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 \(\langle\)www.cs.cas.cz/research/library/reports_900.shtml⟩.
[8] Běhounek, L.; Cintula, P., Fuzzy logics as the logics of chains, Fuzzy sets and systems, 157, 5, 604-610, (2006) · Zbl 1106.03019
[9] Běhounek, L.; Cintula, P., Features of mathematical theories in formal fuzzy logic, (), 523-532 · Zbl 1202.03034
[10] L. Běhounek, M. Daňková, Relational compositions in Fuzzy Class Theory, Fuzzy Sets and Systems (2007), submitted for publication.
[11] R. Bělohlávek, Fuzzy relational systems: foundations and principles, IFSR International Series on Systems Science and Engineering, Vol. 20, Kluwer Academic, Plenum Press, New York, 2002. · Zbl 1067.03059
[12] L. Běhounek, Dubois and Prade’s fuzzy elements: a challenge for formal fuzzy logic, in: S. Gottwald, P. Hájek, M. Ojeda-Aciego (Eds.), Proc. LSC IV & 4th Workshop of the ERCIM Working Group on Soft Computing, Málaga, 2006, pp. 90-95.
[13] Běhounek, L., An alternative justification of the axioms of fuzzy logics, Bull. symb. logic, 13, 267, (2007)
[14] Cintula, P., Weakly implicative (fuzzy) logics I: basic properties, Arch. math. logic, 45, 6, 673-704, (2006) · Zbl 1101.03015
[15] de Luca, A.; Termini, S., A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory, Inform. control, 20, 301-312, (1972) · Zbl 0239.94028
[16] Dubois, D.; Godo, L.; Prade, H.; Esteva, F., An information-based discussion of vagueness, (), 892-913, (Chapter 40)
[17] Dubois, D.; Prade, H., The three semantics of fuzzy sets, Fuzzy sets and systems, 90, 141-150, (1997) · Zbl 0919.04006
[18] D. Dubois, H. Prade, Fuzzy elements in a fuzzy set, in: Y. Liu, G. Chen, M. Ying (Eds.), Fuzzy Logic, Soft Computing and Computational Intelligence: 11th IFSA World Congr., Vol. 1, Beijing, 2005, Tsinghua University Press, Springer, pp. 55-60.
[19] D. Dubois, H. Prade, Fuzzy intervals versus fuzzy numbers: is there a missing concept in fuzzy set theory? in: S. Gottwald, P. Hájek, U. Höhle, E.P. Klement (Eds.), Fuzzy Logics and Related Structures: Abstracts of 26th Linz Seminar on Fuzzy Set Theory, Linz, 2005, pp. 45-46.
[20] 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) · Zbl 0994.03017
[21] Esteva, F.; Godo, L.; Montagna, F., The ł\(\operatorname{\Pi}\) and ł\(\operatorname{\Pi} \frac{1}{2}\) logics: two complete fuzzy systems joining łukasiewicz and product logics, Arch. math. logic, 40, 1, 39-67, (2001) · Zbl 0966.03022
[22] Hájek, P., Metamathematics of fuzzy logic, trends in logic, Vol. 4, (1998), Kluwer Dordercht · Zbl 0937.03030
[23] Hájek, P., What is mathematical fuzzy logic, Fuzzy sets and systems, 157, 5, 597-603, (2006) · Zbl 1108.03028
[24] P. Hájek, L. Godo, F. Esteva, Fuzzy logic and probability, in: Proc. 11th UAI Conf., Montreal, 1995, pp. 237-244.
[25] Hájek, P.; Haniková, Z., A development of set theory in fuzzy logic, (), 273-285 · Zbl 1040.03041
[26] Hájek, P.; Novák, V., The sorites paradox and fuzzy logic, Internat. J. gen. systems, 32, 373-383, (2003) · Zbl 1044.03017
[27] Höhle, U., Fuzzy real numbers as Dedekind cuts with respect to a multiple-valued logic, Fuzzy sets and systems, 24, 3, 263-278, (1987) · Zbl 0638.03051
[28] Jenei, S.; Montagna, F., A proof of standard completeness for esteva and Godo’s logic MTL, Studia logica, 70, 2, 183-192, (2002) · Zbl 0997.03027
[29] Kaufmann, A., Introduction to the theory of fuzzy subsets, Vol. I, (1975), Academic Press New York · Zbl 0332.02063
[30] Klir, G.J.; Yuan, B., Fuzzy sets and fuzzy logic: theory and applications, (1995), Prentice-Hall Englewood Cliffs, NJ · Zbl 0915.03001
[31] Kosko, B., Fuzzy entropy and conditioning, Inform. sci., 40, 165-174, (1986) · Zbl 0623.94005
[32] Kreisel, G., Informal rigour and completeness proofs, (), 138-157
[33] Metcalfe, G.; Montagna, F., Substructural fuzzy logics, J. symb. logic, 72, 3, 834-864, (2007) · Zbl 1139.03017
[34] Montagna, F.; Panti, G., Adding structure to MV-algebras, J. pure appl. algebra, 164, 3, 365-387, (2001) · Zbl 0992.06012
[35] Novák, V., On fuzzy type theory, Fuzzy sets and systems, 149, 2, 235-273, (2004) · Zbl 1068.03019
[36] H. Ono, Substructural logics and residuated lattices—an introduction, in: V.F. Hendricks, J. Malinowski (Eds.), 50 Years of Studia Logica, Trends in Logic, Vol. 21, Kluwer, Dordrecht, 2003, pp. 193-228. · Zbl 1048.03018
[37] Rasiowa, H., An algebraic approach to non-classical logics, (1974), North-Holland Amsterdam · Zbl 0299.02069
[38] Takeuti, G.; Titani, S., Fuzzy logic and fuzzy set theory, Arch. math. logic, 32, 1-32, (1992) · Zbl 0786.03039
[39] Yager, R.R., On the measure of fuzziness and negation part I: membership in the unit interval, Internat. J. gen. systems, 5, 221-229, (1979) · Zbl 0429.04007
[40] Zadeh, L.A., Concept of a linguistic variable and its application to approximate reasoning—I, Inform. sci., 8, 199-249, (1975) · Zbl 0397.68071
[41] Zadeh, L.A., Fuzzy logic and approximate reasoning, Synthese, 30, 407-428, (1975) · Zbl 0319.02016
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