Two Consistent Many-Valued Logics for Paraconsistent Phenomena

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 152)


In this reviewing paper, we recall the main results of our papers [24, 31] where we introduced two paraconsistent semantics for Pavelka style fuzzy logic. Each logic formula \(\alpha \) is associated with a \(2 \times 2\) matrix called evidence matrix. The two semantics are consistent if they are seen from ‘outside’; the structure of the set of the evidence matrices \({{\textit{M}}}\) is an MV-algebra and there is nothing paraconsistent there. However, seen from ‘inside,’ that is, in the construction of a single evidence matrix paraconsistency comes in, truth and falsehood are not each others complements and there is also contradiction and lack of information (unknown) involved. Moreover, we discuss the possible applications of the two logics in real-world phenomena.


Mathematical fuzzy logic Paraconsistent logic MV-algebra 

Mathematics Subject Classification (2000)

03-02 03620 06D35 


  1. 1.
    Arieli, O., Avron, A.: Reasoning with logical bilattices. J. Logic Lang. Inform. 5, 25–63 (1996)Google Scholar
  2. 2.
    Barbon, J.S.; Guido, R.C.; Vieira, L.S.: A neural-network approach for speech features classication based on paraconsistent logic. In: Proceedings of the 11th IEEE conference International Symposium Multimedia, pp. 567–570. San Dego, CA, 14–16 Dec. 2009Google Scholar
  3. 3.
    Belnap, N.D.: A useful four-valued logic. In: Epstein, G., Dumme, J. (eds.) Modern Uses of Multiple Valued Logics, pp. 8–37. D. Reidel, Dordrecht (1977)Google Scholar
  4. 4.
    Chang, C.C.: Algebraic analysis of many-valued logics. Trans. Am. Math. Soc. 88, 476–490 (1958)CrossRefGoogle Scholar
  5. 5.
    Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-valued Reasoning, Trends in Logic, vol. 7. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  6. 6.
    Cornelis, C., Deschrijver, G., Kerre, E.E.: Advances and challenges in interval-valued fuzzy logic. Fuzzy Sets Syst. 157(5), 622–627 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    da Silva Lopes, H.F., Abe, J.M., Anghinah, R.: Application of Paraconsistent artificial neural networks as a method of aid in the diagnosis of alzheimer disease. J. Med. Syst. 34, 1073–81 (2010)Google Scholar
  8. 8.
    Dubois, D.: On ignorance and contradiction considered as truth-values. Logic J. IGPL 16(2), 195–216 (2008)CrossRefzbMATHGoogle Scholar
  9. 9.
    Ginsberg, M.K.: Multivalued logics: a uniform approach to inference in artificial intelligence. Comput. Intell. 4, 265–316 (1988)CrossRefGoogle Scholar
  10. 10.
    Gluschankof, D.: Prime deductive systems and injective objects in the algebras of Łukasiewicz infinite-valued calculi. Algebra Univers. 29, 354–377 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer (1998)Google Scholar
  12. 12.
    Jun, M., Zhang, G., Lu, J.: A state-based knowledge representation approach for information logical inconsistency detection in warning systems. J. Knowl.-Based Syst. 23, 125–131 (2010)CrossRefGoogle Scholar
  13. 13.
    Kukkurainen, P., Turunen, E.: Many-valued similarity reasoning. An axiomatic approach. Int. J. Multiple Valued Logic 8, 751–760 (2002)Google Scholar
  14. 14.
    Nakamatsu, K., Suito, H., Abe, J.M, Suzuki, A.: Paraconsistent logic program based safety verification for air traffic control. IEEE Int. Conf. Syst. Man Cybern. 6, 5 (2002)Google Scholar
  15. 15.
    Novák, V., Perfilieva, I., Mockor, J.: Mathematical Principles of Fuzzy Logic. Kluwer, Boston (1999)CrossRefzbMATHGoogle Scholar
  16. 16.
    Odintsov, S.P.: On axiomatizing Shramko-Wansing’s logic. Stud. Logica 93, 407–428 (2009)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Öztürk, M., Tsoukiás, A.: Modeling uncertain positive and negative reasons in decision aiding. Decis. Support Syst. 43, 1512–1526 (2007)CrossRefGoogle Scholar
  18. 18.
    Pavelka, J.: On fuzzy logic I, II, III. Zeitsch. f. Math. Logik 25, 45–52, 119–134, 447–464 (1979)Google Scholar
  19. 19.
    Perny, P., Tsoukiás, A.: On the continuous extensions of four valued logic for preference modeling. In: Proceedings of the IPMU Conference, pp. 302–309 (1998)Google Scholar
  20. 20.
    Priest, G., Tanaka, K., Weber, Z.: ‘Paraconsistent Logic’, The Stanford Encyclopedia of Philosophy (Fall 2013 Edition), Edward N. Zalta (ed.), URL = \(\langle \) Google Scholar
  21. 21.
    Rivieccio, U.: Neutrosophic logics: prospects and problems. Fuzzy Sets Syst. 159, 1860–1868 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Rodríguez, J.T., Vitoriano, B., Montero, J.: Classification of disasters and emergencies under bipolar knowledge representation. In: Vitoriano, B., Montero, J., Ruan, D. (eds.) Decision Aid Models for Disaster Management and Emergencies. Atlantis Computational Intelligence Systems, vol. 7, pp. 209–232. Atlantis Press, Paris (2013)Google Scholar
  23. 23.
    Rodríguez, J.T., Vitoriano, B., Montero, J.: Fuzzy Dissimilarity–based classification for disaster initial assessment. In: Proceedings of the 8th Conference of the European Society for Fuzzy Logic and Technologies, pp. 448–455 (2013)Google Scholar
  24. 24.
    Rodríguez, J.T., Turunen, E., Ruan, D., Montero, J.: Another paraconsistent algebraic semantics for Łukasiewicz-Pavelka logic. Another paraconsistent algebraic semantics for Lukasiewicz-Pavelka logic. Fuzzy Sets Syst. 242, 132–147 (2014)CrossRefGoogle Scholar
  25. 25.
    Shramko, Y., Wansing, H.: Hypercontradictions, generalized truth values, and logics of truth and falsehood. J. Logic Lang. Inform. 15, 403–424 (2006)Google Scholar
  26. 26.
    Shramko, Y., Wansing, H.: Some useful 16-valued logics: how a computer network should think. J. Philos. Logic 34, 121–153 (2005)Google Scholar
  27. 27.
    Tsoukiás, A.: A first order, four valued, weakly Paraconsistent logic and its relation to rough sets semantics. Foundations Comput. Dec. Sci. 12, 85–108 (2002)zbMATHGoogle Scholar
  28. 28.
    Turunen, E.: Well-defined fuzzy sentential logic. Math. Logic Q. 41, 236–248 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Turunen, E.: Mathematics behind Fuzzy Logic. Springer (1999)Google Scholar
  30. 30.
    Turunen, E.: Interpreting GUHA data mining logic in paraconsistent fuzzy logic framework. In: Tsoukiás, A., Rossi, F. (eds.) Algorithmic Decision Theory, LNCS, vol. 5783. Springer, Berlin/Heidelberg. pp. 284–293 (2009)Google Scholar
  31. 31.
    Turunen, E., Öztürk, M., Tsoukiás, A.: Paraconsistent semantics for Pavelka style fuzzy sentential logic. Fuzzy Sets Syst. 161, 1926–1940 (2010)CrossRefzbMATHGoogle Scholar
  32. 32.
    Turunen, E.: Complete MV–algebra Valued Pavelka Logic. (submitted)Google Scholar
  33. 33.
    Van Gasse, B., Cornelis, C., Deschrijver, G., Kerre, E.E.: Triangle algebras: a formal logic approach to interval-valued residuated lattices. Fuzzy Sets Syst. 159(9), 1042–1060 (2008)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer India 2015

Authors and Affiliations

  1. 1.University of Technology, ViennaWienAustria
  2. 2.Faculty of MathematicsComplutense UniversityMadridSpain

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