×

zbMATH — the first resource for mathematics

Mathematical fuzzy logics. (English) Zbl 1144.03023
This paper brings a nice historical survey of the development of mathematical fuzzy logics and a state-of-the-art overview, including a rich bibliography with 120 items. The paper concentrates on the propositional fuzzy logics. The first-order systems are discussed in less detail. In 13 sections the author treats algebraic properties of the sets of truth degrees with the corresponding connectives, fuzzy propositional logics, the logic of continuous t-norms (BL-logic of P. Hájek), the logic of left-continuous t-norms (MTL-logic of F. Esteva and L. Godo), extensions to first-order logics and some generalizations, extensions with graded notions of inference, complexity issues, and an implication-based approach to fuzzy logic. Not discussed topics are mentioned in concluding remarks giving the relevant sources (compactness, interpolation, etc.).

MSC:
03B52 Fuzzy logic; logic of vagueness
03B50 Many-valued logic
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03-03 History of mathematical logic and foundations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1002/(SICI)1521-3870(200001)46:1<77::AID-MALQ77>3.0.CO;2-X · Zbl 0957.68112
[2] DOI: 10.1007/s00153-005-0303-1 · Zbl 1099.03016
[3] DOI: 10.1007/s00153-005-0287-x · Zbl 1095.03014
[4] DOI: 10.2307/2586560 · Zbl 0971.03025
[5] Fuzzy relational systems: Foundations and principles 20 (2002) · Zbl 1067.03059
[6] DOI: 10.1080/0308107031000152522 · Zbl 1044.03017
[7] DOI: 10.1006/jmaa.2000.7456 · Zbl 0989.54006
[8] Beyond two: theory and applications of multiple-valued logic pp 273– (2003)
[9] DOI: 10.1016/j.fss.2005.10.005 · Zbl 1106.03019
[10] DOI: 10.1007/BF01268618 · Zbl 0848.03005
[11] DOI: 10.1016/j.fss.2004.12.010 · Zbl 1086.03043
[12] DOI: 10.2178/jsl/1154698581 · Zbl 1111.03030
[13] DOI: 10.1007/s001530050151 · Zbl 0936.03026
[14] Fuzzy logic and related structures (2005)
[15] DOI: 10.1007/s00153-005-0284-0 · Zbl 1096.03064
[16] Multiple-Valued Logic and Soft Computing 11 pp 137– (2005)
[17] DOI: 10.1007/s00500-004-0448-6 · Zbl 1093.03012
[18] First-order logic revised pp 107– (2004)
[19] DOI: 10.1002/malq.19790252510 · Zbl 0446.03016
[20] Trends in logic. 50 years of Studia Logica 21 pp 193– (2003) · Zbl 1029.00014
[21] Mathematical principles of fuzzy logic (1999) · Zbl 0940.03028
[22] Fuzzy Sets and Systems 159 (2008)
[23] DOI: 10.1093/logcom/13.4.469 · Zbl 1036.03018
[24] DOI: 10.1023/A:1008311022292 · Zbl 0951.03024
[25] DOI: 10.1016/S0165-0114(01)00098-7 · Zbl 0994.03017
[26] Mathware & Soft Computing 6 pp 219– (1999)
[27] DOI: 10.1016/j.fss.2006.11.010 · Zbl 1117.03030
[28] New trends in quantum structures 516 (2000) · Zbl 0987.81005
[29] Algebraic methods in philosophical logic 41 (2001)
[30] DOI: 10.1016/S0888-613X(96)00137-5 · Zbl 0935.03038
[31] DOI: 10.1023/A:1011958407631 · Zbl 0985.03014
[32] Proceedings of the 11th International Symposium on Multiple-Valued Logic, Norman/Oklahoma, 1981 pp 232– (1981)
[33] DOI: 10.1023/A:1016500922708 · Zbl 1013.03021
[34] Journal of Multiple-Valued Logic 8 pp 671– (2002)
[35] DOI: 10.1145/1071596.1071600 · Zbl 1407.03037
[36] Universal algebra and applications in theoretical computer science (2002) · Zbl 0993.08001
[37] Logic Journal of the Interest Group in Pure and Applied Logics 13 pp 561– (2005)
[38] DOI: 10.1016/j.fss.2003.06.002 · Zbl 1040.03019
[39] Neural Network World 13 pp 549– (2003)
[40] DOI: 10.1016/j.apal.2007.09.002 · Zbl 1140.03010
[41] DOI: 10.2178/jsl/1191333844 · Zbl 1139.03017
[42] DOI: 10.1007/s00153-007-0047-1 · Zbl 1128.03015
[43] Mathematica Bohemica 128 pp 199– (2003)
[44] Fuzzy logic and related structures (2005)
[45] Triangular norms (2000)
[46] Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalences (2007)
[47] DOI: 10.1007/s00153-006-0011-5 · Zbl 1101.03015
[48] DOI: 10.1007/s00153-002-0152-0 · Zbl 1026.03017
[49] DOI: 10.1016/S0019-9958(65)90241-X · Zbl 0139.24606
[50] DOI: 10.1016/S0165-0114(01)00099-9 · Zbl 0994.03015
[51] Neural Network World 124 pp 561– (2001)
[52] DOI: 10.1002/mana.19670330503 · Zbl 0154.25904
[53] DOI: 10.1007/s005000000044 · Zbl 02181428
[54] Monatsberichte der Deutschen Akademie der Wissenschaften Berlin 8 pp 161– (1966)
[55] DOI: 10.1007/978-3-540-32275-7_33
[56] Monatsberichte der Deutschen Akademie der Wissenschaften Berlin 7 pp 859– (1965)
[57] Fuzzy logic in knowledge systems, decision, and control pp 247– (1988)
[58] Neural Network World 13 pp 481– (2003)
[59] DOI: 10.1080/11663081.1994.10510820 · Zbl 0794.03032
[60] DOI: 10.1023/A:1015122331293 · Zbl 0997.03027
[61] DOI: 10.1016/0165-0114(94)00172-4 · Zbl 0844.03011
[62] DOI: 10.1007/978-94-011-0215-5_5
[63] Multiple-Valued Logic and Soft Computing 9 pp 343– (2003)
[64] Neural Network World 12 pp 453– (2002)
[65] DOI: 10.1016/S0165-0114(01)00101-4 · Zbl 0994.03018
[66] DOI: 10.1002/malq.19950410209 · Zbl 0829.03011
[67] Management decision support systems using fuzzy sets and possibility theory pp 198– (1985)
[68] DOI: 10.1002/malq.19580040704 · Zbl 0088.24702
[69] DOI: 10.1016/0888-613X(87)90023-5 · Zbl 0643.03018
[70] The fuzzification of systems 216 (2007)
[71] Probabilistic metric spaces (1983) · Zbl 0546.60010
[72] DOI: 10.1016/S0888-613X(03)00016-1 · Zbl 1045.68137
[73] A course in universal algebra (1981) · Zbl 0478.08001
[74] An algebraic approach to non-classical logics (1974) · Zbl 0299.02069
[75] DOI: 10.1016/S0888-613X(98)00018-8 · Zbl 0947.68142
[76] DOI: 10.1007/BF02023019 · Zbl 0533.03007
[77] Tatra Mountains Mathematical Publications 27 pp 125– (2003)
[78] DOI: 10.1007/s11225-007-9078-1 · Zbl 1127.03049
[79] DOI: 10.1007/s00500-002-0195-5 · Zbl 1018.03021
[80] DOI: 10.1023/A:1011906423560 · Zbl 0988.03042
[81] Metamathematics of fuzzy logic 4 (1998) · Zbl 0937.03030
[82] DOI: 10.1007/s005000050043 · Zbl 05469956
[83] DOI: 10.1016/S0165-0114(01)00100-2 · Zbl 0998.03024
[84] Logical, algebraic, analytic, and probabilistic aspects of triangular norms pp 275– (2005)
[85] Fundamenta Informaticae 81 pp 123– (2007)
[86] DOI: 10.1016/j.fss.2005.10.010 · Zbl 1100.03013
[87] DOI: 10.1016/j.fss.2004.03.027 · Zbl 1068.03019
[88] Fuzzy logic and related structures (2005)
[89] Annals of Pure and Applied Logic 113 pp 3– (2002)
[90] DOI: 10.1016/j.fss.2006.03.004 · Zbl 1111.03033
[91] Fundamenta Informaticae 59 pp 315– (2004)
[92] DOI: 10.1016/j.fss.2005.10.018 · Zbl 1106.03020
[93] Gödel ’96 pp 23– (1996)
[94] DOI: 10.1007/s00153-004-0231-5 · Zbl 1070.03013
[95] DOI: 10.1007/BF01531058 · Zbl 0865.03042
[96] Triangular norms and copulas (2006) · Zbl 1100.39023
[97] Neural Network World 12 pp 407– (2002)
[98] DOI: 10.1007/s00500-002-0245-z
[99] DOI: 10.1007/s001530100118 · Zbl 1032.03017
[100] DOI: 10.1007/s11225-006-9001-1 · Zbl 1124.03027
[101] DOI: 10.1007/s11225-006-7197-8 · Zbl 1111.03047
[102] DOI: 10.1016/j.ins.2005.02.004 · Zbl 1079.03014
[103] A treatise on many-valued logics 9 (2001)
[104] Fuzzy sets and fuzzy logic. The foundations of application–from a mathematical point of view (1993) · Zbl 0782.94025
[105] Mathematical logic and formal systems pp 183– (1985)
[106] DOI: 10.1016/0165-0114(80)90053-6 · Zbl 0426.03030
[107] Synthese 19 pp 325– (1968)
[108] DOI: 10.1080/11663081.1999.10510957 · Zbl 1033.03018
[109] DOI: 10.1016/S0020-7373(76)80003-X · Zbl 0335.02037
[110] Fuzzy logic. Mathematical tools for approximate reasoning 11 (2001) · Zbl 0976.03026
[111] Residuated lattices: An algebraic glimpse at substructural logics 151 (2007) · Zbl 1171.03001
[112] DOI: 10.1007/s005000100137 · Zbl 0995.03048
[113] TABLEAUX 2003 2796 pp 48– (2003)
[114] DOI: 10.1023/B:STUD.0000032084.12744.e3 · Zbl 1045.03048
[115] DOI: 10.1007/s001530050173 · Zbl 0966.03022
[116] DOI: 10.1007/s001530050006 · Zbl 0965.03035
[117] Journal of Logic and Computation 13 pp 531– (2003)
[118] Beyond two: theory and applications of multiple-valued logic 114 pp 251– (2003) · Zbl 1015.00007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.