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Logics with disjunction and proof by cases

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Abstract

This paper is a contribution to the general study of consequence relations which contain (definable) connective of “disjunction”. Our work is centered around the “proof by cases property”, we present several of its equivalent definitions, and show some interesting applications, namely in constructing axiomatic systems for intersections of logics and recognizing weakly implicative fuzzy logics among the weakly implicative ones.

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References

  1. Běhounek L., Cintula P.: Fuzzy logics as the logics of chains. Fuzzy Sets Syst. 157(5), 604–610 (2006)

    Article  MATH  Google Scholar 

  2. Cintula P.: Weakly implicative (fuzzy) logics I: Basic properties. Arch. Math. Logic 45(6), 673–704 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Czelakowski J.: Protoalgebraic Logics, Trends in Logic, vol. 10. Kluwer, Dordrecht (2001)

    Google Scholar 

  4. Dunn J.M., Hardegree G.M.: Algebraic Methods in Philosophical Logic, Oxford Logic Guides, vol. 41. Oxford University Press, Oxford (2001)

    Google Scholar 

  5. Esteva F., Godo L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst. 124(3), 271–288 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Font J.M., Jansana R., Pigozzi D.: A survey of Abstract Algebraic Logic. Studia Logica 74(1–2), 13–97 (Special Issue on Abstract Algebraic Logic II) (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Galatos N.: Equational bases for joins of residuated-lattice varieties. Studia Logica 76(2), 227–240 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hájek P.: Metamathematics of fuzzy logic, Trends in Logic, vol. 4. Kluwer, Dordercht (1998)

    Google Scholar 

  9. Horčík R., Cintula P.: Product Łukasiewicz logic. Arch. Math. Logic 43(4), 477–503 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Jansana R.: Selfextensional logics with implication. In: Béziau, J.(eds) Logica Universalis, pp. 65–88. Birkhäuser, Basel (2005)

    Chapter  Google Scholar 

  11. Jansana R.: Selfextensional logics with conjunction. Studia Logica 84(1), 63–104 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ono H.: Substructural logics and residuated lattices—an introduction. In: Hendricks, V.F., Malinowski, J.(eds) 50 Years of Studia Logica, Trends in Logic, vol. 21, pp. 193–228. Kluwer, Dordrecht (2003)

    Google Scholar 

  13. Rasiowa H.: An Algebraic Approach to Non-Classical Logics. North-Holland, Amsterdam (1974)

    MATH  Google Scholar 

  14. Restall G.: An Introduction to Substructural Logics. Routledge, New York (2000)

    Google Scholar 

  15. Wang S.M., Lu Z.J.: How to construct formal systems for fuzzy logics. In: Castillo, O., Melin, P., Ross, O.M., Cruz, R.S., Pedrycz, W., Kacprzyk, J.(eds) Theoretical Advances and Applications of Fuzzy Logic and Soft Computing, Advances in Soft Computing, vol. 42/2007, pp. 593–601. Springer, Berlin (2007)

    Chapter  Google Scholar 

  16. Wang S.M., Wang M.Y.: Disjunctive elimination rule and its application in MTL. Fuzzy Sets Syst. 157(24), 3169–3176 (2006)

    Article  MATH  Google Scholar 

  17. Wójcicki R.: Theory of Logical Calculi, Synthese Library, vol. 199. Kluwer, Dordrecht (1988)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Nanchang University, 330031, Nanchang, People’s Republic of China

    San-min Wang

  2. Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou věží 2, 182 07, Prague 8, Czech Republic

    Petr Cintula

Authors
  1. San-min Wang
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  2. Petr Cintula
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Corresponding author

Correspondence to Petr Cintula.

Additional information

The work of the first author was supported by the National Foundation of Natural Sciences of China (Grant no. 60663002) and by the Grant Project of science and technology of The Education Department of Jiangxi Province under Grant no. 200618. The work of the second author was supported by grant A100300503 of the Grant Agency of the Academy of Sciences of the Czech Republic and by Institutional Research Plan AVOZ10300504.

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Wang, Sm., Cintula, P. Logics with disjunction and proof by cases. Arch. Math. Logic 47, 435–446 (2008). https://doi.org/10.1007/s00153-008-0088-0

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  • DOI: https://doi.org/10.1007/s00153-008-0088-0

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