Skip to main content

Mathematical Fuzzy Logic: A Good Theory for Practice

  • Chapter
35 Years of Fuzzy Set Theory

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 261))

  • 1037 Accesses

Abstract

We discuss the present state of mathematical fuzzy logic in narrow sense, its extension - fuzzy logic in broader sense (FLb) as a logic of natural human reasoning and also some related theories, e.g., the fuzzy transform. We argue that these are good theories with potential to be very practical.

The research was supported by the project MSM 6198898701 of the MŠMT ČR.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
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. Běhounek, L., Cintula, P.: Fuzzy class theory. Fuzzy Sets and Systems 154, 34–55 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Běhounek, L., Cintula, P.: From fuzzy logic to fuzzy mathematics: A methodological manifesto. Fuzzy Sets and Systems 157(5), 642–646 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cignoli, R.L.O., D’ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht (2000)

    MATH  Google Scholar 

  4. Cintula, P., Hájek, P., Horčík, R.: Formal systems of fuzzy logic and their fragments. Annals of Pure and Applied Logic 150, 40–65 (2007)

    MATH  MathSciNet  Google Scholar 

  5. Davis, E., Morgenstern, L.: Introduction:progress in formal commonsense reasoning. Artifical Intelligence 153, 1–12 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. De Cooman, G., Kerre, E., Cappelle, B., Ruan, D., Vanmassenhove, F.: On the extension of classical propositional logic by means of a triangular norm. Int. J. of Intelligent Systems 5, 307–322 (1990)

    Article  MATH  Google Scholar 

  7. Di Martino, F., Loia, V., Perfilieva, I., Sessa, S.: An image coding/decoding method based on direct and inverse fuzzy transforms. Int. Journ. of Appr. Reasoning 48, 110–131 (2008)

    Article  MATH  Google Scholar 

  8. Dubois, D., Prade, H.: What are fuzzy rules and how to use them. Fuzzy Sets and Systems 84, 169–185 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dvořák, A., Novák, V.: Fuzzy logic deduction with crisp observations. Soft Computing 8, 256–263 (2004)

    MATH  Google Scholar 

  10. Dvořák, A., Holčapek, M.: L-fuzzy quantifiers of the type 〈1〉 determined by measures. Fuzzy Sets and Systems 160, 3425–3452 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dvořák, A., Novák, V.: Formal theories and linguistic descriptions. Fuzzy Sets and Systems 143, 169–188 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dvořák, A., Novák, V.: Towards automatic modeling of economic texts. Mathware & Soft Computing XIV(3), 217–231 (2007)

    Google Scholar 

  13. 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)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fitting, M.: Intensional logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (2006)

    Google Scholar 

  15. Flaminio, T., Montagna, F.: MV-algebras with internal states and probabilistic fuzzy logics. International Journal of Approximate Reasoning 50, 138–152 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices. In: An Algebraic Glimpse At Substructural Logics. Elsevier, Amsterdam (2007)

    Google Scholar 

  17. Goguen, J.A.: The logic of inexact concepts. Synthese 19, 325–373 (1968-1969)

    Google Scholar 

  18. Gottwald, S.: A Treatise on Many-Valued Logics. Research Studies Press Ltd., Baldock (2001)

    MATH  Google Scholar 

  19. Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)

    MATH  Google Scholar 

  20. Hájek, P., Cintula, P.: On theories and models in fuzzy predicate logics. Journal of Symbolic Logic 71(3), 863–880 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Holčapek, M.: Monadic L-fuzzy quantifiers of the type 〈1n, 1〉. Fuzzy Sets and Systems 159, 1811–1835 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Jenei, S., Kerre, E.: Convergence of residuated operators and connective stability. Fuzzy Sets and Systems 114, 411–415 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kennedy, C.: Vagueness and grammar: The semantics of relative and absolute gradable adjectives. Linguistics and Philosophy 30(3), 1–45 (2007)

    Article  Google Scholar 

  24. Kerre, E., De Cock, M.: A comparative study of the behaviour of some popular fuzzy implication operators on the generalized modus ponens. In: Martinez, J. (ed.) Fuzzy Logic for the Management of Uncertainty, pp. 281–296. John Wiley and Sons, New York (1992)

    Google Scholar 

  25. Kerre, E., De Cock, M.: Linguistic modifiers: an overview. In: Martinez, J. (ed.) Fuzzy Logic and Soft Computing, pp. 69–86. Kluwer Academic, Boston (1999)

    Google Scholar 

  26. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Dordrecht (2000)

    MATH  Google Scholar 

  27. Lewin, K.: Field theory in social science: Selected theoretical papers. Harper & Row, New York (1951)

    Google Scholar 

  28. Mamdani, E., Assilian, S.: An experiment in linguistic synthesis with a fuzzy logic controller. Int. J. of Man-Machine Studies 7, 1–13 (1975)

    Article  MATH  Google Scholar 

  29. Materna, P.: Concepts and Objects, Acta Philosophica Fennica 63, Helsinki (1998)

    Google Scholar 

  30. Materna, P.: Conceptual Systems. Logos Verlag, Berlin (2004)

    MATH  Google Scholar 

  31. McCarthy, J.: Programs with common sense. In: Proc. of the Teddington Conf. on Mechanization of Thought Processes, Her Majesty’s Stationary Office, London (1959)

    Google Scholar 

  32. Murinová, P., Novák, V.: A formal theory of generalized intermediate syllogisms. Fuzzy Sets and Systems

    Google Scholar 

  33. Novák, V.: On the syntactico-semantical completeness of first-order fuzzy logic I, II. Kybernetika 26, 47–66, 134–154 (1990)

    MATH  MathSciNet  Google Scholar 

  34. Novák, V.: Towards formalized integrated theory of fuzzy logic. In: Bien, Z., Min, K. (eds.) Fuzzy Logic and Its Applications to Engineering, Information Sciences, and Intelligent Systems, pp. 353–363. Kluwer, Dordrecht (1995)

    Google Scholar 

  35. Novák, V.: Fuzzy relation equations with words. In: Nikravesh, M., Zadeh, L., Korotkikh, V. (eds.) Fuzzy Partial Differential Equations and Relational Equations, pp. 167–185. Springer, Berlin (2004)

    Google Scholar 

  36. Novák, V.: On fuzzy type theory. Fuzzy Sets and Systems 149, 235–273 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  37. Novák, V.: Perception-based logical deduction. In: Reusch, B. (ed.) Computational Intelligence, Theory and Applications, pp. 237–250. Springer, Berlin (2005)

    Chapter  Google Scholar 

  38. Novák, V.: Perception-based logical deduction as alternative approximate reasoning method. In: Proc. Int. Conf. FUZZ-IEEE 2005, Reno, USA (May 2005)

    Google Scholar 

  39. Novák, V.: Fuzzy logic theory of evaluating expressions and comparative quantifiers. In: Proc. 11th Int. Conf. IPMU, Paris, Éditions EDK, Les Cordeliers, Paris, vol. 2 (July 2006)

    Google Scholar 

  40. Novák, V.: Which logic is the real fuzzy logic? Fuzzy Sets and Systems 157, 635–641 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  41. Novák, V.: Mathematical fuzzy logic in modeling of natural language semantics. In: Wang, P., Ruan, D., Kerre, E. (eds.) Fuzzy Logic – A Spectrum of Theoretical & Practical Issues, pp. 145–182. Elsevier, Berlin (2007)

    Google Scholar 

  42. Novák, V.: A comprehensive theory of trichotomous evaluative linguistic expressions. Fuzzy Sets and Systems 159(22), 2939–2969 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  43. Novák, V.: A formal theory of intermediate quantifiers. Fuzzy Sets and Systems 159(10), 1229–1246 (2008), doi:10.1016/j.fss.2007.12.008

    MATH  MathSciNet  Google Scholar 

  44. Novák, V.: EQ-algebra-based fuzzy type theory and its extensions. Logic Journal of the IGPL (to appear, 2010)

    Google Scholar 

  45. Novák, V., Lehmke, S.: Logical structure of fuzzy IF-THEN rules. Fuzzy Sets and Systems 157, 2003–2029 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  46. Novák, V., Perfilieva, I. (eds.): Discovering the World With Fuzzy Logic. Studies in Fuzziness and Soft Computing, vol. 57. Springer, Heidelberg (2000)

    MATH  Google Scholar 

  47. Novák, V., Perfilieva, I.: On the semantics of perception-based fuzzy logic deduction. International Journal of Intelligent Systems 19, 1007–1031 (2004)

    Article  MATH  Google Scholar 

  48. Novák, V., Perfilieva, I., Močkoř, J.: Mathematical Principles of Fuzzy Logic. Kluwer, Boston (1999)

    MATH  Google Scholar 

  49. Novák, V., Štěpnička, M.U., Dvořák, A., Perfilieva, I., Pavliska, V., Vavříčková, L.: Analysis of seasonal time series using fuzzy approach. Int. Journal of General Systems 39, 305–328 (2010)

    MATH  Google Scholar 

  50. Pavelka, J.: On fuzzy logic I, II, III, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 25, 45–52, 119–134, 447–464 (1979)

    Google Scholar 

  51. Perfilieva, I.: Fuzzy logic normal forms for control law representation. In: Verbruggen, H., Zimmermann, H.-J., Babusha, R. (eds.) Fuzzy Algorithms for Control, pp. 111–125. Kluwer, Boston (1999)

    Google Scholar 

  52. Perfilieva, I.: Logical approximation. Soft Computing 7(2), 73–78 (2002)

    Article  MATH  Google Scholar 

  53. Perfilieva, I.: Fuzzy transform: Application to reef growth problem. In: Demicco, R.B., Klir, G.J. (eds.) Fuzzy Logic in Geology, pp. 275–300. Academic Press, Amsterdam (2003)

    Google Scholar 

  54. Perfilieva, I.: Fuzzy function as an approximate solution to a system of fuzzy relation equations. Fuzzy Sets and Systems 147, 363–383 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  55. Perfilieva, I.: Normal forms in BL-algebra of functions and their contribution to universal approximation. Fuzzy Sets and Systems 143, 111–127 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  56. Perfilieva, I.: Fuzzy transforms: theory and applications. Fuzzy Sets and Systems 157, 993–1023 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  57. Perfilieva, I.: Fuzzy transforms: A challenge to conventional transforms. In: Hawkes, P.W. (ed.) Advances in Images and Electron Physics, vol. 147, pp. 137–196. Elsevier Academic Press, San Diego (2007)

    Google Scholar 

  58. Perfilieva, I., De Meyer, H., De Baets, B., Plskova, D.: Cauchy problem with fuzzy initial condition and its approximate solution with the help of fuzzy transform. In: Proc. of WCCI 2008, IEEE Int. Conf. on Fuzzy Systems, Hong Kong (2008)

    Google Scholar 

  59. Perfilieva, I., Lemhke, S.: Correct models of fuzzy if–then rules are continuous. Fuzzy Sets and Systems 157, 3188–3197 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  60. Perfilieva, I., Nosková, L.: System of fuzzy relation equations with \(\inf\rightarrow\) composition: complete set of solutions. Fuzzy Sets and Systems 159, 2256–2271 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  61. Perfilieva, I., Novák, V.: System of fuzzy relation equations as a continuous model of IF-THEN rules. Information Sciences 177, 3218–3227 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  62. Perfilieva, I., Novák, V., Dvořák, A.: Fuzzy transform in the analysis of data. Int. Journ. of Appr. Reasoning 48, 36–46 (2008)

    Article  MATH  Google Scholar 

  63. Peterson, P.: Intermediate Quantifiers. Logic, linguistics, and Aristotelian semantics, Ashgate, Aldershot (2000)

    Google Scholar 

  64. Plšková, D.: Fuzzy transform in geological applications. Journal of Electrical Engineering 57, 43–46 (2006)

    MATH  Google Scholar 

  65. Sanchez, E.: Resolution of composite fuzzy relation equations. Information and Control 30, 38–48 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  66. Sgall, P., Hajičová, E., Panevová, J.: The Meaning of the Sentence in Its Syntactic and Pragmatic Aspects. D. Reidel, Dordrecht (1986)

    Google Scholar 

  67. Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  68. Zadeh, L.A.: Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. on Systems, Man, and Cybernetics SMC 3, 28–44 (1973)

    MATH  MathSciNet  Google Scholar 

  69. Zadeh, L.A.: A rationale for fuzzy control, Trans. ASME, Ser. G, J. Dynamic. Systems, Measurement and Control 94, 3–4 (1974)

    Google Scholar 

  70. Zadeh, L.A.: Precisiated natural language. AI Magazine 25, 74–91 (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Novák, V., Perfilieva, I. (2010). Mathematical Fuzzy Logic: A Good Theory for Practice. In: Cornelis, C., Deschrijver, G., Nachtegael, M., Schockaert, S., Shi, Y. (eds) 35 Years of Fuzzy Set Theory. Studies in Fuzziness and Soft Computing, vol 261. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16629-7_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-16629-7_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-16628-0

  • Online ISBN: 978-3-642-16629-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics