Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-16T16:57:28.677Z Has data issue: false hasContentIssue false
  • English
  • Français

Mathematical Fuzzy Logics

Published online by Cambridge University Press:  15 January 2014

Siegfried Gottwald
Siegfried Gottwald*
Affiliation:
Universität LeipzigInstitut Für Logik und Wissenschaftstheorie, Beethovenstr. 15, 04107 Leipzig, GermanyE-mail: [email protected]: http://www.uni-leipzig.de/~logik/gottwald/
Get access Rights & Permissions [Opens in a new window]

Abstract

The last decade has seen an enormous development in infinite-valued systems and in particular in such systems which have become known as mathematical fuzzy logics.

The paper discusses the mathematical background for the interest in such systems of mathematical fuzzy logics, as well as the most important ones of them. It concentrates on the propositional cases, and mentions the first-order systems more superficially. The main ideas, however, become clear already in this restricted setting.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 14 , Issue 2 , June 2008 , pp. 210 - 239
Copyright
Copyright © Association for Symbolic Logic 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Aglianó, Paolo, Ferreirim, Isabel M. A., and Montagna, Franco, Basic hoops: an algebraic study of continuous t-norms, Studia Logica, vol. 87 (2007), pp. 73–98.CrossRefGoogle Scholar
[2] Aguzzoli, Stefano and Ciabattoni, Agata, Finiteness in infinite-valued Łukasiewicz logic, Journal of Logic, Language and Information, vol. 9 (2000), pp. 5–29.CrossRefGoogle Scholar
[3] Aguzzoli, Stefano and Gerla, Brunella, Finite-valued reductions of infinite-valued logics, Archive for Mathematical Logic, vol. 41 (2002), pp. 361–399.CrossRefGoogle Scholar
[4] Aguzzoli, Stefano and Gerla, Brunella, On countermodels in basic logic, Neural Network World, vol. 12 (2002), pp. 407–420.Google Scholar
[5] Alsina, Ciaudi, Maurice J., Frank, and Schweizer, Berthold, Associative functions. Triangular norms and copulas, World Scientific Publishing Co., Hackensack, NJ, 2006.CrossRefGoogle Scholar
[6] Avron, Arnon, Hypersequents, logical consequence and intermediate logics for concurrency, Annals of Mathematics and Artificial Intelligence, vol. 4 (1991), pp. 225–248.CrossRefGoogle Scholar
[7] Baaz, Matthias, Infinite-valued Gödel logics with 0-1 projections and relativizations, Gödel '96 (Hajek, P., editor), Lecture Notes in Logic 6, Springer, Berlin, 1996, pp. 23–33.Google Scholar
[8] Baaz, Matthias, Ciabattoni, Agata, and Montagna, Franco, Analytic calculi for monoidal t-norm based logic, Fundamenta Informaticae, vol. 59 (2004), pp. 315–332.Google Scholar
[9] Baaz, Matthias, Hájek, Petr, Montagna, Franco, and Veith, Helmut, Complexity of t-tautologies, Annals of Pure and Applied Logic, vol. 113 (2002), pp. 3–11.CrossRefGoogle Scholar
[10] Baaz, Matthias and Veith, Helmut, Interpolation in fuzzy logic, Archive for Mathematical Logic, vol. 38 (1999), pp. 461–489.CrossRefGoogle Scholar
[11] Běhounek, Libor and Cintula, Petr, Fuzzy class theory, Fuzzy Sets and Systems, vol. 154 (2005), pp. 34–55.CrossRefGoogle Scholar
[12] Běhounek, Libor and Cintula, Petr, Fuzzy logics as the logics of chains, Fuzzy Sets and Systems, vol. 157 (2006), pp. 604–610.CrossRefGoogle Scholar
[13] Bělohlávek, Radim, Fuzzy closure operators, Journal of Mathematical Analysis and Applications, vol. 262 (2001), pp. 473–489.CrossRefGoogle Scholar
[14] Bělohlávek, Radim Fuzzy relational systems: Foundations and principles, IFSR International Series on Systems Science and Engineering, vol. 20, Kluwer Academic Publishers, New York, 2002.CrossRefGoogle Scholar
[15] Bělohlávek, Radim and Vychodil, Vilém, Fuzzy Horn logic. I. Proof theory, Archive for Mathematical Logic, vol. 45 (2006), pp. 3–51.CrossRefGoogle Scholar
[16] Bělohlávek, Radim, Vychodil, Vilém, Fuzzy Horn logic. II. Implicationally defined classes, Archive for Mathematical Logic, vol. 45 (2006), pp. 149–177.CrossRefGoogle Scholar
[17] Biacino, Loredana, Gerla, Giangiacomo, and Sheng, Ying Ming, Approximate reasoning based on similarity, Mathematical Logic Quarterly, vol. 46 (2000), pp. 77–86.3.0.CO;2-X>CrossRefGoogle Scholar
[18] Boixader, Dionis and Jacas, Juan, Extensionality based approximate reasoning, International Journal of Approximate Reasoning, vol. 19 (1998), pp. 221–230.CrossRefGoogle Scholar
[19] Burris, Stanley and Sankappanavar, H.P., A course in universal algebra, Springer, New York, 1981.CrossRefGoogle Scholar
[20] Butnariu, Dan, Klement, Erich Peter, and Zafrany, Samy, On triangular norm-based propositional fuzzy logics, Fuzzy Sets and Systems, vol. 69 (1995), pp. 241–255.CrossRefGoogle Scholar
[21] Castro, J. Luis, Trillas, Enric, and Cubillo, Susana, On consequence in approximate reasoning, Journal of Applied Non-Classical Logics, vol. 4 (1994), pp. 91–103.CrossRefGoogle Scholar
[22] Chakraborty, Mihir K., Use of fuzzy set theory in introducing graded consequence in multiple valued logic, Fuzzy logic in knowledge systems, decision, and control (Gupta, M. M. and Yamakawa, T., editors), North-Holland Publishing Company, Amsterdam, 1988, pp. 247–257.Google Scholar
[23] Ciabattoni, Agata, Fermüller, Christian, and Metcalfe, George, Uniform rules and dialogue games for fuzzy logics, Logic for programming, artificial intelligence, and reasoning, Lecture Notes in Computer Science, vol. 3452, Springer, Berlin, 2005, pp. 496–510.CrossRefGoogle Scholar
[24] Cignoli, Roberto, Esteva, Francesc, Godo, Lluís, and Torrens, Antoni, Basic fuzzy logic is the logic of continuous t-norms and their residua, Soft Computing, vol. 4 (2000), pp. 106–112.CrossRefGoogle Scholar
[25] Cintula, Petr, An alternative approach to the ŁΠ logic, Neural Network World, vol. 124 (2001), pp. 561–575.Google Scholar
[26] Cintula, Petr The ŁΠ and ŁΠ½ propositional and predicate logics, Fuzzy Sets and Systems, vol. 124 (2001), pp. 289–302.CrossRefGoogle Scholar
[27] Cintula, Petr Advances in ŁΠ and ŁΠ½ logics, Archive for Mathematical Logic, vol. 42 (2003), pp. 449–468.CrossRefGoogle Scholar
[28] Cintula, Petr Weakly implicative [fuzzy] logics I: Basic properties, Archive for Mathematical Logic, vol. 45 (2006), pp. 673–704.CrossRefGoogle Scholar
[29] Cintula, Petr, Esteva, Francesc, Gispert, Joan, Godo, Lluís, Montagna, Franco, and Noguera, Carles, Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalences, submitted, 2007.Google Scholar
[30] Cintula, Petr, Háiek, Petr, and Horčík, Rostislav, Formal systems of fuzzy logic and their fragments, Annals of Pure and Applied Logic, vol. 150 (2007), pp. 40–65.CrossRefGoogle Scholar
[31] Cintula, Petr and Navara, Mirko, Compactness of fuzzy logics, Fuzzy Sets and Systems, vol. 143 (2004), pp. 59–73.CrossRefGoogle Scholar
[32] Denecke, Klaus and Wismath, Shelly L., Universal algebra and applications in theoretical computer science, Chapman & Hall/CRC, Boca Raton/FL, 2002.Google Scholar
[33] di Nola, Antonio, Georgescu, George, and Iorgulescu, Afrodita, Pseudo-BL algebras. I and II, Journal of Multiple-Valued Logic, vol. 8 (2002), pp. 671–750.Google Scholar
[34] Domingo, Xavier, Trillas, Enric, and Valverde, Llorenc, Pushing Łukasiewicz-Tarski implication a little farther, Proceedings of the 11th International Symposium on Multiple-Valued Logic, Norman/Oklahoma, 1981, IEEE Computer Society, New York, 1981, pp. 232–243.Google Scholar
[35] Dubois, Didier, Esteva, Francesc, Garcia, Pere, Godo, Lluís, and Prade, Henri, A logical approach to interpolation based on similarity relations, International Journal of Approximate Reasoning, vol. 17 (1997), pp. 1–36.CrossRefGoogle Scholar
[36] Dunn, J. Michael and Hardegree, Gary M., Algebraic methods in philosophical logic, Oxford Logic Guides, vol. 41, Oxford University Press, Oxford, 2001.CrossRefGoogle Scholar
[37] Dvurečenskij, Anatolij and Pulmannová, Sylvia, New trends in quantum structures, Mathematics and Its Applications, vol. 516, Kluwer Academic Publishers, Dordrecht, 2000.CrossRefGoogle Scholar
[38] Esteva, Francesc, Gispert, Joan, Godo, Lluís, and Noguera, Carles, Adding truth-constants to logics of continuous t-norms: axiomatization and completeness results, Fuzzy Sets and Systems, vol. 158 (2007), pp. 597–618.CrossRefGoogle Scholar
[39] Esteva, Francesc and Godo, Lluís, Putting together Łukasiewicz and product logics, Mathware & Soft Computing, vol. 6 (1999), pp. 219–234.Google Scholar
[40] Esteva, Francesc, Godo, Lluís, Monoidal t-norm based logic: toward a logic for left-continuous t-norms, Fuzzy Sets and Systems, vol. 124 (2001), pp. 271–288.CrossRefGoogle Scholar
[41] Esteva, Francesc, Godo, Lluís, and García-Cerdaña, Àngel, On the hierarchy of t-norm based residuated fuzzy logics, Beyond two: theory and applications of multiple-valued logic (Fitting, M. and Orłowska, E., editors), Studies in Fuzziness and Soft Computing, vol. 114, Physica, Heidelberg, 2003, pp. 251–272.CrossRefGoogle Scholar
[42] Esteva, Francesc, Godo, Lluís, Hájek, Petr, and Montagna, Franco, Hoops and fuzzy logic, Journal of Logic and Computation, vol. 13 (2003), pp. 531–555.CrossRefGoogle Scholar
[43] Esteva, Francesc, Godo, Lluís, Hájek, Petr, and Navara, Mirko, Residuated fuzzy logic with an involutive negation, Archive for Mathematical Logic, vol. 39 (2000), pp. 103–124.CrossRefGoogle Scholar
[44] Esteva, Francesc, Godo, Lluís, and Montagna, Franco, The łΠ and ŁΠ½ logics: two complete fuzzy systems joining łukasiewicz and product logics, Archive for Mathematical Logic, vol. 40 (2001), pp. 39–67.CrossRefGoogle Scholar
[45] Esteva, Francesc, Godo, Lluís, Equational characterization of the subvarieties of BL generated by t-norm algebras, Studia Logica, vol. 76 (2004), pp. 161–200.CrossRefGoogle Scholar
[46] Fermüller, Christian, Parallel dialogue games and hypersequents for intermediate logics, TABLEAUX 2003 (Mayer, M. C. and Pirri, F., editors), Lecture Notes in Computer Science, vol. 2796, Springer, Rome, 2003, pp. 48–64.Google Scholar
[47] Flondor, Paul, Georgescu, George, and Iorgulescu, Afrodita, Pseudo t-norms andpseudo-BL algebras, Soft Computing, vol. 5 (2001), pp. 355–371.CrossRefGoogle Scholar
[48] Galatos, Nikolaos, Jipsen, Peter, Kowalski, Tomasz, and Ono, Hiroakira, Residuated lattices: An algebraic glimpse at substructural logics, Studies in Logic and the Foundations of Mathematics, vol. 151, North-Holland Publishing Company, Amsterdam, 2007.Google Scholar
[49] Gerla, Giangiacomo, Fuzzy logic. Mathematical tools for approximate reasoning, Trends in Logic, Studia Logica Library, vol. 11, Kluwer Academic Publishers, Dordrecht, 2001.CrossRefGoogle Scholar
[50] Giles, Robin, Łukasiewicz logic and fuzzy set theory, International Journal of Man-Machine Studies, vol. 8 (1976), pp. 313–327.CrossRefGoogle Scholar
[51] Godo, Lluís and Háiek, Petr, Fuzzy inference as deduction, Journal of Applied Non-Classical Logics, vol. 9 (1999), no. 1, pp. 37–60.CrossRefGoogle Scholar
[52] Goguen, Joseph A., The logic of inexact concepts, Synthese, vol. 19 (1968–1969), pp. 325–373.CrossRefGoogle Scholar
[53] Gottwald, Siegfried, Fuzzy propositional logics, Fuzzy Sets and Systems, vol. 3 (1980), pp. 181–192.CrossRefGoogle Scholar
[54] Gottwald, Siegfried A generalized Łukasiewicz-style identity logic, Mathematical logic and formal systems (de Alcantara, L. P., editor), Lecture Notes in Pure and Applied Mathematics 94, Marcel Dekker, New York, 1985, pp. 183–195.Google Scholar
[55] Gottwald, Siegfried Fuzzy sets and fuzzy logic. The foundations of application—from a mathematical point of view, Artificial Intelligence, Vieweg, Braunschweig/Wiesbaden, 1993.Google Scholar
[56] Gottwald, Siegfried A treatise on many-valued logics, Studies in Logic and Computation, vol. 9, Research Studies Press, Baldock, 2001.Google Scholar
[57] Gottwald, Siegfried Mathematical fuzzy logic as a tool for the treatment of vague information, Information Sciences, vol. 172 (2005), pp. 41–71.CrossRefGoogle Scholar
[58] Gottwald, Siegfried Universes of fuzzy sets and axiomatizations of fuzzy set theory. Part I. Model-based and axiomatic approaches, Studia Logica, vol. 82 (2006), pp. 211–244.CrossRefGoogle Scholar
[59] Gottwald, Siegfried Universes of fuzzy sets and axiomatizations of fuzzy set theory. Part II. Category theoretic approaches, Studia Logica, vol. 84 (2006), pp. 23–50.CrossRefGoogle Scholar
[60] Gottwald, Siegfried Mathematical fuzzy logic: An invitation to interesting research areas, Fundamenta Informaticae, vol. 81 (2007), pp. 123–137.Google Scholar
[61] Gottwald, Siegfried and Háiek, Petr, T-norm based mathematical fuzzy logics, Logical, algebraic, analytic, and probabilistic aspects of triangular norms (Klement, E. P. and Mesiar, R., editors), Elsevier, Dordrecht, 2005, pp. 275–299.CrossRefGoogle Scholar
[62] Gottwald, Siegfried and Jenei, Sándor, A new axiomatization for involutive monoidal t-norm based logic, Fuzzy Sets and Systems, vol. 124 (2001), pp. 303–307.CrossRefGoogle Scholar
[63] Hájek, Petr, Basic fuzzy logic and BL-algebras, Soft Computing, vol. 2 (1998), pp. 124–128.Google Scholar
[64] Hájek, Petr Metamathematics of fuzzy logic, Trends in Logic, Studia Logica Library, vol. 4, Kluwer Academic Publishers, Dordrecht, 1998.CrossRefGoogle Scholar
[65] Hájek, Petr Fuzzy logic and arithmetical hierarchy III, Studia Logica, vol. 68 (2001), pp. 129–142.CrossRefGoogle Scholar
[66] Hájek, Petr Basic fuzzy logic and BL-algebras II, Soft Computing, vol. 7 (2003), pp. 179–183.CrossRefGoogle Scholar
[67] Hájek, Petr Embedding standard BL-algebras into non-commutativepseudo-BL-algebras, Tatra Mountains Mathematical Publications, vol. 27 (2003), pp. 125–130.Google Scholar
[68] Hájek, Petr Fuzzy logics with non-commutative conjunctions, Journal of Logic and Computation, vol. 13 (2003), pp. 469–479.CrossRefGoogle Scholar
[69] Hájek, Petr Observations on non-commutative fuzzy logics, Soft Computing, vol. 8 (2003), pp. 28–43.CrossRefGoogle Scholar
[70] Hájek, Petr Fuzzy logic and arithmetical hierarchy IV, First-order logic revised (Hendricks, V., Neuhaus, F., Pedersen, S. A., Scheffler, U., and Wansing, H., editors), Logos Verlag, Berlin, 2004, pp. 107–115.Google Scholar
[71] Hájek, Petr Arithmetical complexity of fuzzy predicate logics—a survey, Soft Computing, vol. 9 (2005), no. 12, pp. 935–941.CrossRefGoogle Scholar
[72] Hájek, Petr Fleas and fuzzy logic, Multiple-Valued Logic and Soft Computing, vol. 11 (2005), pp. 137–152.Google Scholar
[73] Hájek, Petr On arithmetic in Cantor-Łukasiewicz fuzzy set theory, Archive for Mathematical Logic, vol. 44 (2005), pp. 763–782.CrossRefGoogle Scholar
[74] Hájek, Petr and Cintula, Petr, Triangular norm based predicate fuzzy logics, Fuzzy logic and related structures (Klement, E. P. et al., editors), Proceedings of the Linz Seminar on Fuzzy Set Theory 2005, in print.Google Scholar
[75] Hájek, Petr and Cintula, Petr On theories and models in fuzzy predicate logics, The Jounal of Symbolic Logic, vol. 71 (2006), pp. 863–880.CrossRefGoogle Scholar
[76] Hájek, Petr, Godo, Lluís, and Esteva, Francesc, A complete many-valued logic with product-conjunction, Archive for Mathematical Logic, vol. 35 (1996), pp. 191–208.CrossRefGoogle Scholar
[77] Hájek, Petr and Haniková, Zuzana, A development of set theory in fuzzy logic, Beyond two: theory and applications of multiple-valued logic (Fitting, M. and Orlowska, E., editors), Studies in Fuzziness and Soft Computing, Physica-Verlag, Heidelberg, 2003, pp. 273–285.CrossRefGoogle Scholar
[78] Hájek, Petr and Novák, Vilém, The sorites paradox and fuzzy logic, International Journal of General Systems, vol. 32 (2003), no. 4, pp. 373–383.CrossRefGoogle Scholar
[79] Hájek, Petr, Paris, Jeff, and Shepherdson, John C., Rational Pavelka predicate logic is a conservative extension of Łukasiewicz predicate logic, The Journal of Symbolic Logic, vol. 65 (2000), pp. 669–682.CrossRefGoogle Scholar
[80] Haniková, Zuzana, Standard algebras for fuzzy propositional calculi, Fuzzy Sets and Systems, vol. 124 (2001), pp. 309–320.CrossRefGoogle Scholar
[81] Haniková, Zuzana A note on the complexity of propositional logics of individual t-algebras, Neural Network World, vol. 12 (2002), pp. 453–460.Google Scholar
[82] Haniková, Zuzana Mathematical and metamathematical properties of fuzzy logic, Ph.D. thesis, Charles University, Prague, 2003.Google Scholar
[83] Hekrdla, Josef, Klement, Erich Peter, and Navara, Mirko, Two approaches to fuzzy propositional logics, Multiple-Valued Logic and Soft Computing, vol. 9 (2003), pp. 343–360.Google Scholar
[84] Höhle, Ulrich, Commutative, residuated l-monoids, Non-classical logics and their applications to fuzzy subsets (Hohle, U. and Klement, E. P., editors), Theory and Decision Library, Series B, 32, Kluwer Academic Publ., Dordrecht, 1995, pp. 53–106.CrossRefGoogle Scholar
[85] Jenei, Sándor and Montagna, Franco, A proof of standard completeness for Esteva and Godo's logic MTL, Studia Logica, vol. 70 (2002), pp. 183–192.CrossRefGoogle Scholar
[86] Jenei, Sándor A proof of standard completeness for non-commutative monoidal t-norm logic, Neural Network World, vol. 13 (2003), pp. 481–488.Google Scholar
[87] Klaua, Dieter, Über einen Ansatz zur mehrwertigen Mengenlehre, Monatsberichte der Deutschen Akademie der Wissenschaften Berlin, vol. 7 (1965), pp. 859–867.Google Scholar
[88] Klaua, Dieter Über einen zweiten Ansatz zur mehrwertigen Mengenlehre, Monatsberichte der Deutschen Akademie der Wissenschaften Berlin, vol. 8 (1966), pp. 161–177.Google Scholar
[89] Klaua, Dieter Ein Ansatz zur mehrwertigen Mengenlehre, Mathematische Nachrichten, vol. 33 (1967), pp. 273–296.CrossRefGoogle Scholar
[90] Klement, Erich Peter, Mesiar, Radko, and Pap, Endre, Triangular norms, Kluwer Academic Publishers, Dordrecht, 2000.CrossRefGoogle Scholar
[91] Kowalski, Tomasz and Ono, Hiroakira, Fuzzy logics from a substructural perspective, Fuzzy logic and related structures (Klement, E. P. et al., editors), Proceedings of the Linz Seminar on Fuzzy Set Theory 2005, in print.Google Scholar
[92] Kühr, Jan, Pseudo BL-algebras and DRl-monoids, Mathematica Bohemica, vol. 128 (2003), pp. 199–208.CrossRefGoogle Scholar
[93] Metcalfe, George and Montagna, Franco, Fuzzy logics based on [0,1)-continuous uninorms, Archive for Mathematical Logic, vol. 46 (2007), pp. 425–469.Google Scholar
[94] Metcalfe, George and Montagna, Franco Montagna, Franco, Substructural fuzzy logics, The Journal of Symbolic Logic, vol. 72 (2007), pp. 834–864.CrossRefGoogle Scholar
[95] Metcalfe, George, Olivetti, Nicola, and Gabbay, Dov, Proof theory for product logics, Neural Network World, vol. 13 (2003), pp. 549–558.Google Scholar
[96] Metcalfe, George, Olivetti, Nicola, and Gabbay, Dov, Łukasiewicz logic: from proof systems to logic programming, Logic Journal of the Interest Group in Pure and Applied Logics, vol. 13 (2005), pp. 561–585.Google Scholar
[97] Metcalfe, George Olivetti, Nicola, and Gabbay, Dov, Sequent and hypersequent calculi for abelian and Łukasiewicz logics, ACM Transactions on Computational Logic, vol. 6 (2005), pp. 578–613.CrossRefGoogle Scholar
[98] Montagna, Franco and Ono, Hiroakira, Kripke semantics, undecidability and standard completeness for Esteva and Godo's logic MTL ∀, Studia Logica, vol. 71 (2002), pp. 227–245.CrossRefGoogle Scholar
[99] Montanga, Franco, Three complexity problems in quantifiedfuzzy logic, Studia Logica, vol. 68 (2001), pp. 143–152.Google Scholar
[100] Montanga, Franco On the predicate logics of continuous t-norm BL-algebras, Archive for Mathematical Logic, vol. 44 (2005), pp. 97–114.Google Scholar
[101] Morsi, Nehad N., Lotfallah, Wafik, and El-Zekey, M. S., The logic of tied implications. I. Properties, applications and representation, Fuzzy Sets and Systems, vol. 157 (2006), pp. 647–669.CrossRefGoogle Scholar
[102] Morsi, Nehad N. The logic of tied implications. II. Syntax, Fuzzy Sets and Systems, vol. 157 (2006), pp. 2030–2057.CrossRefGoogle Scholar
[103] Novák, Vilém, A comprehensive theory of trichotomous evaluative linguistic expressions, Fuzzy logic and related structures (Klement, E. P. et al., editors), Proceedings of the Linz Seminar on Fuzzy Set Theory 2005, in print.Google Scholar
[104] Novák, Vilém On fuzzy type theory, Fuzzy Sets and Systems, vol. 149 (2005), pp. 235–273.CrossRefGoogle Scholar
[105] Novák, Vilém Which logic is the real fuzzy logic?, Fuzzy Sets and Systems, vol. 157 (2006), pp. 635–641.CrossRefGoogle Scholar
[106] Novák, Vilém A formal theory of intermediate quantifiers, Fuzzy Sets and Systems, vol. 159 (2008), in print.CrossRefGoogle Scholar
[107] Novák, Vilém, Perfilieva, Irina, and Močkoč, Jiři, Mathematical principles of fuzzy logic, Kluwer Academic Publishers, Boston/Dordrecht/London, 1999.CrossRefGoogle Scholar
[108] Ono, Hiroakira, Substructural logics and residuated lattices—an introduction, Trends in logic. 50 years of Studia Logica (Hendricks, V. F. and Malinowski, J., editors), Trends in Logic, Studia Logica Library, vol. 21, Kluwer Academic Publishers, Dordrecht, 2003, pp. 193–228.CrossRefGoogle Scholar
[109] Pavelka, Jan, On fuzzy logic. I–III, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 25 (1979), pp. 45–52; 119–134; 447–464.CrossRefGoogle Scholar
[110] Ragaz, Matthias, Arithmetische Klassifikation von Formelmengen der unendlich-wertigen Logik, Ph.D. thesis, ETH, Zürich, 1981.Google Scholar
[111] Ragaz, Matthias Die Unentscheidbarkeit der einstelligen unendlichwertigen Prädikatenlogik, Archiv für mathematische Logik und Grundlagenforschung, vol. 23 (1983), pp. 129–139.CrossRefGoogle Scholar
[112] Rasiowa, Helena, An algebraic approach to non-classical logics, North-Holland Publishing Company/PWN, Amsterdam/Warsaw, 1974.Google Scholar
[113] Rodríguez, Ricardo, Esteva, Francesc, Garcia, Pere, and Godo, Lluís, On implicative closure operators in approximate reasoning, International Journal of Approximate Reasoning, vol. 33 (2003), pp. 159–184.CrossRefGoogle Scholar
[114] Schweizer, Berthold and Sklar, Abe, Probabilistic metric spaces, North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing Company, New York, 1983.Google Scholar
[115] Seising, Rudolf, The fuzzification of systems, Studies in Fuzziness and Soft Computing, vol. 216, Springer, Heidelberg, 2007.CrossRefGoogle Scholar
[116] Smets, Philippe and Magrez, Paul, Implication in fuzzy logic, International Journal of Approximate Reasoning, vol. 1 (1987), pp. 327–347.CrossRefGoogle Scholar
[117] Thiele, Helmut, Theorie der endlichwertigen Łukasiewiczschen Prädikatenkalküle der ersten Stufe, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 4 (1958), pp. 108–142.CrossRefGoogle Scholar
[118] Trillas, Enric and Valverde, Llorenc, On implication and indistinguishability in the setting of fuzzy logic, Management decision support systems using fuzzy sets and possibility theory (Kacprzyk, J. and Yager, R. R., editors), TÜV Rheinland, Cologne, 1985, pp. 198–212.Google Scholar
[119] Turunen, Esko, Well-defined fuzzy sentential logic, Mathematical Logic Quarterly, vol. 41 (1995), pp. 236–248.CrossRefGoogle Scholar
[120] Zadeh, Lotfi A., Fuzzy sets, Information and Control, vol. 8 (1965), pp. 338–353.CrossRefGoogle Scholar