Skip to main content
SpringerLink
Log in

Many-Valued Logic And Fuzzy Set Theory

  • Chapter
Mathematics of Fuzzy Sets

Part of the book series: The Handbooks of Fuzzy Sets Series ((FSHS,volume 3))

Abstract

Rather early in the (short) history of fuzzy sets it became clear that there is an intimate relationship between fuzzy set theory and many-valued logic. In the early days of fuzzy sets the main connection was given by fuzzy logic — in the understanding of this notion in those days: and this was as switching logic within a multiple-valued setting.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 379.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baudry, L. (1950). La Querelle des Future Contingents: Louvain 1465–1475, Vrin, Paris.

    Google Scholar 

  2. Belluce, L.P. (1964). Further results on infinite valued predicate logic. J. Symbolic Logic 29, 69–78.

    Article  MathSciNet  MATH  Google Scholar 

  3. Belluce, L.P. and Chang, C.C. (1963). A weak completeness theorem for infinite valued first-order logic. J. Symbolic Logic 28, 43–50.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bernays, P. (1926). Axiomatische Untersuchungen des Aussagenkalküls der “Principia Mathematica”. Math. Zeitschr. 25, 305–320.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bočvar, D.A. (1938). Ob odnom trechznačnom isčislenii i ego primenenii k analizu paradoksov klassičeskogo funkcional’nogo isčislenija. Mat. Sbornik 4(46), 287–308.

    MATH  Google Scholar 

  6. Bočvar, D.A. (1943). K voprosu o neprotivorecivosti odnogo trechznačnogo isčislenija. Mat. Sbornik 12(54), 353–369.

    Google Scholar 

  7. Bolc, L. and Borowik, P. (1992). Many-Valued Logics, 1. Theoretical Foundations. Springer, Berlin.

    MATH  Google Scholar 

  8. Börger, E., Grädel, E. and Gurevich, Y. (1997). The Classical Decision Problem. Perspectives in Mathematical Logic, Springer, Berlin.

    Google Scholar 

  9. Butnariu, D. and Klement, E.P. (1993). Triangular Norm-Based Measures and Games with Fuzzy Coalitions. Kluwer Acad. Publ., Dordrecht.

    Book  MATH  Google Scholar 

  10. Chang, C.C. (1958). Algebraic analysis of many valued logics. Trans. Amer. Math. Soc. 88, 467–490.

    Article  MathSciNet  MATH  Google Scholar 

  11. Chang, C.C. (1959). A new proof of the completeness of the Lukasiewicz axioms. Trans. Amer. Math. Soc. 93, 74–80.

    MathSciNet  MATH  Google Scholar 

  12. Chang, C.C. (1965). Infinite valued logic as a basis for set theory. In: Logic, Methodology and Philosophy of Science. Proc. Internat. Congress Jerusalem 1964 (Y. Bar-Hillel et al. eds.), North-Holland Publ. Comp., Amsterdam, 93–100.

    Google Scholar 

  13. Chang, C.C. and Keisler, H.J. (1966). Continuous Model Theory. Ann. of Math. Stud., vol. 58, Princeton Univ. Press, Princeton, NJ.

    Google Scholar 

  14. Cignoli, R. (1970). Moisil Algebras. Notas Lógica Matemática, vol. 27, Universidad Nacional del Sur, Bahía Bianca.

    Google Scholar 

  15. Cignoli, R., d’Ottaviano, I. and Mundici, D. (1994). Álgebras das Lógicas de Lukasiewicz. Coleção CLE, vol. 12, Centro de Lógica, Epistemologica e História da Ciéncia — UNICAMP, Campinas, Brasil. [English translation Algebraic Foundations of Many-Valued Reasoning in preparation (to appear approximately 1998).]

    Google Scholar 

  16. De Baets, B. and Mesiar, R. (1996) Residual implicators of continuous t-norms. In: Fourth European Congress on Intelligent Techniques and Soft Computing, EUFIT ′96, Aachen, Sept. 1993, Proceedings (H.-J. Zimmermann ed.), ELITE Foundation, Aachen, 27–31.

    Google Scholar 

  17. Dummett, M. (1959). A propositional calculus with denumerable matrix. J. Symbolic Logic 24, 97–106.

    Article  MathSciNet  MATH  Google Scholar 

  18. Dunn, M. and Meyer, R.K. (1971). Algebraic completeness results for Dummett’s LC and its extensions. Ztschr. Math. Logik Grundlag. Math. 17, 225–230.

    Article  MathSciNet  MATH  Google Scholar 

  19. Dwinger, Ph. (1977). A survey of the theory of Post algebras and their generalizations. In: Modern Uses of Multiple-Valued Logic (M. Dunn and G. Epstein eds.), Reidel, Dordrecht, 53–75.

    Google Scholar 

  20. Epstein, G. (1960). The lattice of Post algebras. Trans. Amer. Math. Soc. 95, 300–317.

    Article  MathSciNet  MATH  Google Scholar 

  21. Fisch, M. and Turquette, A.R. (1966). Peirce’s triadic logic. Trans. Charles Sanders Peirce Society 2, 71–85.

    MathSciNet  Google Scholar 

  22. Fodor, J. (1993). A new look at fuzzy connectives. Fuzzy Sets and Systems 57, 141–148.

    Article  MathSciNet  MATH  Google Scholar 

  23. Fodor, J. (1993). On contrapositive symmetry of implications in fuzzy logic. In: First European Congress on Fuzzy and Intelligent Technologies, EUFIT ′93, Aachen, Sept. 1993, Proceedings (H.-J. Zimmermann ed.), Verlag Augustinus Buchhandlung, Aachen, 1342–1348.

    Google Scholar 

  24. Fodor, J. and Roubens, M. (1994). Fuzzy Preference Modelling and Multkriteria Decision Support. Kluwer Acad. Publ., Dordrecht.

    Book  Google Scholar 

  25. Frank, H.J. (1979). On the simultaneous associativity of F(x,y) and x + y-F(x,y). Aequationes Math. 19, 194–226.

    Article  MathSciNet  MATH  Google Scholar 

  26. Girard, J.Y. (1987). Linear logic. Theor. Comp. Sci. 50, 1–102.

    Article  MathSciNet  MATH  Google Scholar 

  27. Gluschankof, D. (1992). Prime deductive systems and injective objects in the algebras of Lukasiewicz infinite-valued calculi. Algebra Universalis 29, 354–377.

    Article  MathSciNet  MATH  Google Scholar 

  28. Gödel, K. (1932). Zum intuitionistischen Aussagenkalkül. Anzeiger Akad. Wiss. Wien, math.-naturwiss. Kl., 69, 65–66.

    MATH  Google Scholar 

  29. Gottwald, S. (1979). Set theory for fuzzy sets of higher level. Fuzzy Sets Syst. 2, 25–51.

    Article  MathSciNet  Google Scholar 

  30. Gottwald, S. (1980). Fuzzy uniqueness of fuzzy mappings. Fuzzy Sets Syst. 3, 49–74.

    Article  MathSciNet  MATH  Google Scholar 

  31. Gottwald, S. (1985). A generalized Lukasiewicz-style identity logic. In: Mathematical Logic and Formal Systems (L.P. de Alcantara (ed.), Lecture Notes Pure and Applied Mathematics, vol. 94, Marcel Dekker, New York, 183–195.

    Google Scholar 

  32. Gottwald, S. (1986). Characterizations of the solvability of fuzzy equations. Elektron. Informationsverarb. Kybernet. EIK 22, 67–91.

    MathSciNet  MATH  Google Scholar 

  33. Gottwald, S. (1989). Mehrwertige Logik. Eine Einführung in Theorie und Anwendungen. Akademie-Verlag, Berlin.

    Google Scholar 

  34. Gottwald, S. (1993). Fuzzy Sets and Fuzzy Logic. Vieweg, Wiesbaden.

    Book  MATH  Google Scholar 

  35. Gottwald, S. (1995). An approach to handle partially sound rules of inference. In: Advances in Intelligent Computing — IPMU ′94, Selected Papers (B. Bouchon-Meunier, R.R. Yager, L.A. Zadeh eds.), Lecture Notes Computer Sci., vol. 945, Springer, Berlin, 380–388.

    Google Scholar 

  36. Hähnle, R. (1994). Many-valued logic and mixed integer programming. Annals of Math. and Artif. Intelligence 12, 231–263.

    Article  MATH  Google Scholar 

  37. Hájek, P. (1995). Fuzzy logic and arithmetical hierarchy. Fuzzy Sets and Systems 73, 359–363.

    Article  MathSciNet  MATH  Google Scholar 

  38. Hájek, P. (1997). Fuzzy logic and arithmetical hierarchy II. Studia Logica 58, 129–141.

    Article  MathSciNet  MATH  Google Scholar 

  39. Hájek, P. (1998). Metamathematics of Fuzzy Logic. Kluwer Acad. Publ., Dordrecht.

    Book  MATH  Google Scholar 

  40. Hájek, P., Godo, L. and Esteva, F. (1996). A complete many-valued logic with product-conjunction. Arch. Math. Logic 35, 191–208.

    MathSciNet  MATH  Google Scholar 

  41. Harnacher, H. (1978). Über logische Aggregationen nicht-binär explizierter Entscheidungskriterien. Rita G. Fischer Verlag, Frankfurt/Main.

    Google Scholar 

  42. Hay, L.S. (1963). Axiomatization of the infinite-valued predicate calculus. J. Symbolic Logic 28, 77–86.

    Article  MathSciNet  Google Scholar 

  43. Höhle, U. (1992). M-valued sets and sheaves over integral commutative CL-monoids. In: Applications of Category Theory to Fuzzy Subsets (S.E. Rod-abaugh, E.P. Klement, U. Höhle eds.), Kluwer Acad. Publ., Dordrecht, 33–72.

    Chapter  Google Scholar 

  44. Höhle, U. (1994). Monoidal logic. In: Fuzzy Systems in Computer Science (R. Kruse, J. Gebhardt, R. Palm eds.), Vieweg, Wiesbaden, 233–243.

    Chapter  Google Scholar 

  45. Höhle, U. (1995). Commutative, residuated 1-monoids. In: Non-Classical Logics and Their Applications to Fuzzy Subsets (U. Höhle, E.P. Klement eds.), Kluwer Acad. Publ., Dordrecht, 53–106.

    Chapter  Google Scholar 

  46. Höhle, U. (1995). Presheaves over GL-monoids. In: Non-Classical Logics and Their Applications to Fuzzy Subsets (U. Höhle, E.P. Klement eds.), Kluwer Acad. Publ., Dordrecht, 127–157.

    Chapter  Google Scholar 

  47. Höhle, U. (1996). On the fundamentals of fuzzy set theory. J. Math. Anal. Appl. 201, 786–826.

    Article  MathSciNet  MATH  Google Scholar 

  48. Jaśkowski, S. (1936). Recherches sur le système de la logique intuitioniste. In: Actes du Congrès Internat. de Philosophie Scientifique 1936, vol. 6, Actualitès Scientifique et Industriel, vol. 393, Hermann, Paris, 58–61.

    Google Scholar 

  49. Klawonn, F. and Kruse, R. (1993). Equality relations as a basis for fuzzy control. Fuzzy Sets and Systems 54, 147–156.

    Article  MathSciNet  MATH  Google Scholar 

  50. Kleene, S.C. (1938). On notation for ordinal numbers. J. Symbolic Logic 3, 150–155.

    Article  MATH  Google Scholar 

  51. Kleene, S.C. (1952). Introduction to Metamathematics, North-Holland Publ. Comp./van Nostrand, Amsterdam/New York.

    MATH  Google Scholar 

  52. Klement, E.P. (1997). Some mathematical aspects of fuzzy sets: triangular norms, fuzzy logics, and generalized measures. Fuzzy Sets and Systems 90, 133–140.

    Article  MathSciNet  MATH  Google Scholar 

  53. Klement, E.P. and Mesiar, R. (1997). Triangular Norms. Tatra Mountain Math. Publ. 13, 169–193.

    MathSciNet  MATH  Google Scholar 

  54. Klement, E.P., Mesiar, R. and Pap, E. (1994). Triangular Norms. (book in preparation)

    Google Scholar 

  55. Lewis, C.J. and Langford, C.H. (1932). Symbolic Logic, Dover, New York.

    MATH  Google Scholar 

  56. Ling, C.H. (1965). Representation of associative functions. Publ. Math. Debrecen 12, 182–212.

    Google Scholar 

  57. Lovett, E.O. (1900–01). Mathematics at the International Congress of Philosophy, Paris 1900. Bull. Amer. Math. Soc. 7, 157–183.

    Article  Google Scholar 

  58. Lukasiewicz, J. (1913). Die logischen Grundlagen der Wahrscheinlichkeitsrechnung, Krakow [cf. also his (1970)].

    Google Scholar 

  59. Lukasiewicz, J. (1920). O logice trójwartościowej. Ruch Filozoficzny 5, 170–171.

    Google Scholar 

  60. Lukasiewicz, J. (1930). Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls. C. R. Séances Soc. Sci. et Lettr. Varsovie, cl. III, 23, 51–77.

    Google Scholar 

  61. Lukasiewicz, J. (1935). Zur Geschichte der Aussagenlogik. Erkenntnis 5, 111–131.

    Article  Google Scholar 

  62. Lukasiewicz, J. (1970). Selected Works (L. Borkowski, ed.). North-Holland Publ. Comp., Amsterdam.

    Google Scholar 

  63. Lukasiewicz, J. and Tarski, A. (1930). Untersuchungen über den Aussagenkalkül. C. R. Séances Soc. Sci. et Lettr. Varsovie, cl. III, 23, 30–50.

    MATH  Google Scholar 

  64. Mangani, P. (1973). Su certe algebre connesse con logiche a piú valori. Boll. Unione Math. Italiana (4) 8, 68–78.

    MathSciNet  Google Scholar 

  65. McColl, H. (1897). Symbolic reasoning, II. Mind, N.S., 6, 493–510.

    Article  Google Scholar 

  66. McNaughton, R. (1951). A theorem about infinite-valued sentential logic. J. Symbolic Logic 16, 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  67. Menger, K. (1979). Selected Papers in Logic and Foundations, Didactics, Economics, Reidel, Dordrecht.

    Book  MATH  Google Scholar 

  68. Michalski, K. (1937). Le problème de la volonté à Oxford et à Paris au XIVe siècle. Studia Philosophica 2, 233–365.

    Google Scholar 

  69. Moisil, G.C. (1935). Recherches sur l’algèbre de la logique. Ann. Sci. Univ. Jassy 22, 1–117.

    Google Scholar 

  70. Moisil, G.C. (1940). Recherches sur les logiques non chrysipiennes. Ann. Sci. Univ. Jassy 26, 431–466.

    MathSciNet  Google Scholar 

  71. Moisil, G.C. (1941). Notes sur les logiques non-chrysipiennes. Ann. Sci. Univ. Jassy 27, 86–98.

    MathSciNet  Google Scholar 

  72. Morgan, C.G. (1974). A theory of equality for a class of many-valued predicate calculi, Zeitschr. math. Logik Grundl. Math. 20, 427–432.

    Article  MATH  Google Scholar 

  73. Trakhtenbrot, B.A. (1950). Nevozmožnost’ algorifma dlja problemy razrešimosti na konečnykh klassakh. Doklady Akad. Nauk SSSR 70, 569–572. [English Translation: The impossibility of an algorithm for the decision problem for finite models. AMS Transl. Series 2 (1963), 1-6.]

    MathSciNet  Google Scholar 

  74. Turquette, A.R. (1967). Peirce’s Phi and Psi operators for triadic logic. Trans. Charles Sanders Peirce Society 3, 66–73.

    MathSciNet  Google Scholar 

  75. Turunen, E. (1995). Well-defined fuzzy sentential logic. Math. Logic Quarterly 41, 236–248

    Article  MathSciNet  MATH  Google Scholar 

  76. Turunen, E. (1997). Rules of inference in fuzzy sentential logic. Fuzzy Sets and Systems 85, 63–72.

    Article  MathSciNet  MATH  Google Scholar 

  77. Wajsberg, M. (1977). Axiomatization of the three-valued propositional calculus. In: M. Wajsberg: Logical Works (S.S. Surma ed.), Ossolineum, Wroclaw, 12–29.

    Google Scholar 

  78. Weber, S. (1983). A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms. Fuzzy Sets and Systems 11, 115–134.

    Article  MathSciNet  MATH  Google Scholar 

  79. Yager, R.R. (1980). On a general class of fuzzy connectives. Fuzzy Sets and Systems 4, 235–242.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Authors
  1. S. Gottwald
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Fachbereich Mathematik, Bergische Universität, Wuppertal, Germany

    Ulrich Höhle

  2. Department of Mathematics and Statistics, Youngstown State University, Youngstown, Ohio, USA

    Stephen Ernest Rodabaugh

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer Science+Business Media New York

About this chapter

Cite this chapter

Gottwald, S. (1999). Many-Valued Logic And Fuzzy Set Theory. In: Höhle, U., Rodabaugh, S.E. (eds) Mathematics of Fuzzy Sets. The Handbooks of Fuzzy Sets Series, vol 3. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5079-2_2

Download citation

  • DOI: https://doi.org/10.1007/978-1-4615-5079-2_2

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7310-0

  • Online ISBN: 978-1-4615-5079-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation