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Mathematical aspects of fuzzy sets and fuzzy logic. Some reflections after 40 years. (English) Zbl 1088.03024

This survey paper considers some of the main trends of the development of mathematical fuzzy logic and of formalized fuzzy set theory. In four decades of fuzzy set theory and fuzzy logic, the last one has seen a tremendous wealth of new and important approaches and results which paved the way for the future success. Especiall BL-algebras, introduced by P. Hájek, brought a qualitative jump in the fuzzy logic theory and they have initiated several new streams of research. Some important open problems for further research are mentioned in the concluding remarks of this interesting paper that is worth reading for any scientist interested in logic, artificial intelligence or soft computing.

MSC:

03B52 Fuzzy logic; logic of vagueness
03E72 Theory of fuzzy sets, etc.
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03-03 History of mathematical logic and foundations
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[1] L. Běhounek, P. Cintula, Fuzzy class theory, Fuzzy Sets and Systems, in press, doi:10.1016/j.fss.2004.12.010.; L. Běhounek, P. Cintula, Fuzzy class theory, Fuzzy Sets and Systems, in press, doi:10.1016/j.fss.2004.12.010.
[2] Bell, J. L., Boolean-Valued Models and Independence Proofs in Set Theory (1977), Clarendon Press: Clarendon Press Oxford · Zbl 0371.02028
[3] Botta, O.; Delorme, M., An \(f\)-sets universe \(\widetilde{V} \), (Gupta, M. M.; Sanchez, E., Fuzzy Information and Decision Processes (1982), North-Holland: North-Holland Amsterdam), 133-141 · Zbl 0524.03046
[4] U. Cerruti, U. Höhle. Categorical foundations of fuzzy set theory with applications to algebra and topology, in: The Mathematics of Fuzzy Systems, Verlag TÜV Köln, 1986, pp. 51-86.; U. Cerruti, U. Höhle. Categorical foundations of fuzzy set theory with applications to algebra and topology, in: The Mathematics of Fuzzy Systems, Verlag TÜV Köln, 1986, pp. 51-86. · Zbl 0586.18002
[5] Cignoli, R.; Esteva, F.; Godo, L.; Torrens, A., Basic fuzzy logic is the logic of continuous t-norms and their residua, Soft Computing, 4, 106-112 (2000)
[6] Esteva, F.; Godo, L., Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems, 124, 271-288 (2001) · Zbl 0994.03017
[7] Esteva, F.; Godo, L.; Montagna, F., Equational characterization of the subvarieties of BL generated by t-norm algebras, Studia Logica, 76, 161-200 (2004) · Zbl 1045.03048
[8] Eytan, M., Fuzzy setsa topos-logical point of view, Fuzzy Sets and Systems, 5, 47-67 (1981) · Zbl 0453.03059
[9] Fourman, M. P., The logic of topoi, (Barwise, J., Handbook of Mathematical Logic (1977), North-Holland: North-Holland Amsterdam), 1053-1090
[10] M.P. Fourman, D.S. Scott. Sheaves and logic, in: M.P. Fourman, C.J. Mulvey, D.S. Scott (Eds.), Applications of sheaves, Lecture Notes in Mathematics, vol. 753, Springer, Berlin, 1979, pp. 302-401.; M.P. Fourman, D.S. Scott. Sheaves and logic, in: M.P. Fourman, C.J. Mulvey, D.S. Scott (Eds.), Applications of sheaves, Lecture Notes in Mathematics, vol. 753, Springer, Berlin, 1979, pp. 302-401. · Zbl 0415.03053
[11] Giles, R., Łukasiewicz logic and fuzzy set theory, Internat. J. Man-Machine Studies, 8, 313-327 (1976) · Zbl 0335.02037
[12] Giles, R., A formal system for fuzzy reasoning, Fuzzy Sets and Systems, 2, 233-257 (1979) · Zbl 0411.03018
[13] K. Gödel, Zum intuitionistischen Aussagenkalkül, Anzeiger Akademie der Wissenschaften Wien, Math.-Naturwiss. Klasse 69 (1932) 65-66; also in: Ergebnisse eines mathematischen Kolloquiums 4 (1933) 40.; K. Gödel, Zum intuitionistischen Aussagenkalkül, Anzeiger Akademie der Wissenschaften Wien, Math.-Naturwiss. Klasse 69 (1932) 65-66; also in: Ergebnisse eines mathematischen Kolloquiums 4 (1933) 40. · JFM 58.1001.03
[14] Goguen, J. A., \(L\)-fuzzy sets, J. Math. Anal. Appl., 18, 145-174 (1967) · Zbl 0145.24404
[15] Goguen, J. A., Concept representation in natural and artificial languagesaxioms, extensions and applications for fuzzy sets, Int. J. Man-Machine Studies, 6, 513-561 (1974) · Zbl 0321.68055
[16] S. Gottwald, A cumulative system of fuzzy sets, in: A. Zarach (Ed.), Set theory hierarchy theory, Memorial Tribute A. Mostowski, Bierutowice 1975, Lecture Notes in Mathematics, vol. 537, Springer, Berlin, 1976, pp.109-119.; S. Gottwald, A cumulative system of fuzzy sets, in: A. Zarach (Ed.), Set theory hierarchy theory, Memorial Tribute A. Mostowski, Bierutowice 1975, Lecture Notes in Mathematics, vol. 537, Springer, Berlin, 1976, pp.109-119. · Zbl 0339.02055
[17] Gottwald, S., Set theory for fuzzy sets of higher level, Fuzzy Sets and Systems, 2, 125-151 (1979) · Zbl 0408.03042
[18] S. Gottwald, Fuzzy set theory: some aspects of the early development, in: H.J. Skala, S. Termini, E. Trillas (Eds.), Aspects of vagueness, Theory and Decision Library, vol. 39, Reidel, Dordrecht, 1984, pp. 13-29.; S. Gottwald, Fuzzy set theory: some aspects of the early development, in: H.J. Skala, S. Termini, E. Trillas (Eds.), Aspects of vagueness, Theory and Decision Library, vol. 39, Reidel, Dordrecht, 1984, pp. 13-29. · Zbl 0543.03035
[19] S. Gottwald, Fuzzy sets and fuzzy Logic, The Foundations of Application—From a Mathematical Point of View, Vieweg, Braunschweig, Wiesbaden, Teknea, Toulouse, 1993.; S. Gottwald, Fuzzy sets and fuzzy Logic, The Foundations of Application—From a Mathematical Point of View, Vieweg, Braunschweig, Wiesbaden, Teknea, Toulouse, 1993. · Zbl 0782.94025
[20] S. Gottwald, Universes of fuzzy sets—a short survey, in: Proc. 33rd Internat. Symp. Multiple-Valued Logic, Tokyo 2003, IEEE Computer Soc. Press, Los Alamitos, 2003, pp. 71-76.; S. Gottwald, Universes of fuzzy sets—a short survey, in: Proc. 33rd Internat. Symp. Multiple-Valued Logic, Tokyo 2003, IEEE Computer Soc. Press, Los Alamitos, 2003, pp. 71-76.
[21] S. Gottwald, Mathematical fuzzy logic as a tool for the treatment of vague information, Inform. Sci. 172 (2005) 41-71.; S. Gottwald, Mathematical fuzzy logic as a tool for the treatment of vague information, Inform. Sci. 172 (2005) 41-71. · Zbl 1079.03014
[22] Gottwald, S.; Hájek, P., T-norm based mathematical fuzzy logics, (Klement, E. P.; Mesiar, R., Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms (2005), Elsevier: Elsevier Dordrecht), 275-299 · Zbl 1078.03020
[23] P. Hájek, Metamathematics of fuzzy logic, Trends in Logic, vol. 4, Kluwer Academic Publishers, Dordrecht, 1998.; P. Hájek, Metamathematics of fuzzy logic, Trends in Logic, vol. 4, Kluwer Academic Publishers, Dordrecht, 1998. · Zbl 0937.03030
[24] Hájek, P.; Godo, L.; Esteva, F., A complete many-valued logic with product-conjunction, Arch. Math. Logic, 35, 191-208 (1996) · Zbl 0848.03005
[25] P. Hájek, Z. Haniková, A set theory within fuzzy logic, in: Proc.; P. Hájek, Z. Haniková, A set theory within fuzzy logic, in: Proc.
[26] D. Higgs, A category approach to Boolean-valued set theory, Preprint, University of Waterloo, 1973.; D. Higgs, A category approach to Boolean-valued set theory, Preprint, University of Waterloo, 1973.
[27] Höhle, U., \(M\)-valued sets and sheaves over integral commutative CL-monoids, (Rodabaugh, S. E.; Klement, E. P.; Höhle, U., Applications of Category Theory to Fuzzy Subsets (1992), Reidel: Reidel Dordrecht), 33-72 · Zbl 0766.03037
[28] Höhle, U., Commutative, residuated l-monoids, (Höhle, U.; Klement, E. P., Non-Classical Logics and Their Application to Fuzzy Subsets (1995), Reidel: Reidel Dordrecht), 53-106 · Zbl 0838.06012
[29] Höhle, U., Presheaves over GL-monoids, (Höhle, U.; Klement, E. P., Non-Classical Logics and Their Application to Fuzzy Subsets (1995), Reidel: Reidel Dordrecht), 127-157 · Zbl 0838.06013
[30] Höhle, U., Classification of subsheaves over GL-Algebras, (Buss, R.; Hájek, P.; Pudlak, P., Logic Colloquium ’98 (2000), A.K. Peters: A.K. Peters Natick, MA.), 238-261 · Zbl 0946.03080
[31] Höhle, U.; Stout, L. N., Foundations of fuzzy sets, Fuzzy Sets and Systems, 40, 257-296 (1991) · Zbl 0725.03031
[32] Jenei, S.; Montagna, F., A proof of standard completeness for Esteva and Godo’s logic MTL, Studia Logica, 70, 183-192 (2002) · Zbl 0997.03027
[33] Klaua, D., Über einen Ansatz zur mehrwertigen Mengenlehre, Monatsber. Deutsch. Akad. Wiss. Berlin, 7, 859-867 (1965) · Zbl 0154.26001
[34] Klaua, D., Über einen zweiten Ansatz zur mehrwertigen Mengenlehre, Monatsber. Deutsch. Akad. Wiss. Berlin, 8, 161-177 (1966) · Zbl 0154.26002
[35] Klaua, D., Grundbegriffe einer mehrwertigen Mengenlehre, Monatsber. Deutsch. Akad. Wiss. Berlin, 8, 781-802 (1966) · Zbl 0168.00802
[36] Lawvere, F. W., An elementary theory of the category of sets, Proc. Nat. Acad. Sci. USA, 52, 1506-1511 (1964) · Zbl 0141.00603
[37] Łukasiewicz, J., Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls, Comptes Rendus Séances Société des Sciences et Lettres Varsovie, cl. III, 23, 51-77 (1930) · JFM 57.1319.02
[38] A. Pultr, Closed categories of \(L\); A. Pultr, Closed categories of \(L\)
[39] A. Pultr, Fuzziness and fuzzy equality, in: H.J. Skala, S. Termini, E. Trillas (Eds.), Aspects of vagueness, Theory and Decision Library, vol. 39, Reidel, Dordrecht, 1984, pp. 119-135.; A. Pultr, Fuzziness and fuzzy equality, in: H.J. Skala, S. Termini, E. Trillas (Eds.), Aspects of vagueness, Theory and Decision Library, vol. 39, Reidel, Dordrecht, 1984, pp. 119-135. · Zbl 0541.04001
[40] D.S. Scott. Lectures on Boolean-valued models for set theory, Mimeographed Typescript of Lectures Symposium on Axiomatic Set Theory, American Mathematical Society, Berkeley, 1967, unpublished.; D.S. Scott. Lectures on Boolean-valued models for set theory, Mimeographed Typescript of Lectures Symposium on Axiomatic Set Theory, American Mathematical Society, Berkeley, 1967, unpublished.
[41] D.S. Scott, Identity and existence in intuitionistic logic, in: M.P. Fourman, C.J. Mulvey, D.S. Scott (Eds.), Applications of sheaves, Lecture Notes in Mathematics, vol. 753, Springer, New York, 1979, pp. 660-696.; D.S. Scott, Identity and existence in intuitionistic logic, in: M.P. Fourman, C.J. Mulvey, D.S. Scott (Eds.), Applications of sheaves, Lecture Notes in Mathematics, vol. 753, Springer, New York, 1979, pp. 660-696. · Zbl 0418.03016
[42] Stout, L. N., Topoi and categories of fuzzy sets, Fuzzy Sets and Systems, 12, 169-184 (1984) · Zbl 0557.03045
[43] Stout, L. N., Categories of fuzzy sets with values in a quantale or projectale, (Höhle, U.; Klement, E. P., Non-Classical Logics and Their Application to Fuzzy Subsets (1995), Reidel: Reidel Dordrecht), 219-234 · Zbl 0827.03038
[44] Takeuti, G.; Titani, S., Intuitionistic fuzzy logic and intuitionistic fuzzy set theory, J. Symbolic Logic, 49, 851-866 (1984) · Zbl 0575.03015
[45] G. Takeuti, S. Titani, Global intuitionistic fuzzy set theory, in: A. di Nola, A.G.S. Ventre (Eds.), The mathematics of fuzzy systems, Interdisciplinary Systems Research, vol. 88, Verlag TÜV Rheinland, Cologne, 1986, pp. 291-301.; G. Takeuti, S. Titani, Global intuitionistic fuzzy set theory, in: A. di Nola, A.G.S. Ventre (Eds.), The mathematics of fuzzy systems, Interdisciplinary Systems Research, vol. 88, Verlag TÜV Rheinland, Cologne, 1986, pp. 291-301. · Zbl 0593.03031
[46] Takeuti, G.; Titani, S., Fuzzy logic and fuzzy set theory, Arch. Math. Logic, 32, 1-32 (1992) · Zbl 0786.03039
[47] O. Wyler, Lecture Notes on Topoi and Quasitopoi, Singapore, 1991.; O. Wyler, Lecture Notes on Topoi and Quasitopoi, Singapore, 1991. · Zbl 0727.18001
[48] Wyler, O., Fuzzy logic and categories of fuzzy sets, (Höhle, U.; Klement, E. P., Non-Classical Logics and Their Application to Fuzzy Subsets (1995), Reidel: Reidel Dordrecht), 235-268 · Zbl 0827.03039
[49] Zadeh, L. A., Fuzzy sets, Inform. and Control, 8, 338-353 (1965) · Zbl 0139.24606
[50] Zhang, J. W., A unified treatment of fuzzy set theory and Boolean-valued set theory—fuzzy set structures and normal fuzzy set structures, J. Math. Anal. Appl., 76, 297-301 (1980) · Zbl 0452.03043
[51] Zhang, J. W., Between fuzzy set theory and Boolean valued set theory, (Gupta, M. M.; Sanchez, E., Fuzzy Information and Decision Processes (1982), North-Holland: North-Holland Amsterdam), 143-147 · Zbl 0529.03033
[52] J.W. Zhang, Fuzzy set structure with strong implication, in: P.P. Wang (Ed.), Advances in Fuzzy Sets, Possibility Theory, and Applications, Plenum Press, New York, 1983, pp. 107-136.; J.W. Zhang, Fuzzy set structure with strong implication, in: P.P. Wang (Ed.), Advances in Fuzzy Sets, Possibility Theory, and Applications, Plenum Press, New York, 1983, pp. 107-136. · Zbl 0608.94015
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