A Co-inductive Language and Truth Degrees

  • Shunsuke Yatabe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6878)


We investigate what happens if \({\mathbf{PA}\text{{\bf \L}}\mathbf{Tr}_2}\), a co-inductive language, formalizes itself. We analyze the truth concept in fuzzy logics by formalizing truth degree theory in the framework of truth theories in fuzzy logics. Hájek-Paris-Shepherdson’s paradox [HPS00] involves that so called truth degrees do not represent the degrees of truthhood (defined by the truth predicate) correctly in Łukasiewicz infinite-valued predicate logic \(\forall\text{{\bf \L}}\), therefore truth degree theory fails there.


Fuzzy Logic Classical Logic Object Language Recursive Function Predicate Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [A91]
    Avron, A.: Hypersequents, logical consequence and intermediate logics for concurrency. Annals of Mathematics and Artificial Intelligence 4, 225–248 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [BE87]
    Barwise, J., Etchemendy, J.: The Liar: An Essay in Truth and Circularity. Oxford University Press, Oxford (1987)zbMATHGoogle Scholar
  3. [BM96]
    Barwise, J., Moss, L.: Vicious Circles. CSLI publications, Stanford (1996)zbMATHGoogle Scholar
  4. [BP83]
    Barwise, J., Perry, J.: Situations and Attitudes. MIT Press, Cambridge (1983)zbMATHGoogle Scholar
  5. [B08]
    Beall, J.: Spandrels of Truth. Oxford University Press, Oxford (2008)Google Scholar
  6. [CEGT00]
    Cignoli, R., Esteva, F., Godo, L., Torrens, A.: Basic Fuzzy Logic is the logic of continuous t-norms and their residua. Soft Computing 4, 106–112 (2000)CrossRefGoogle Scholar
  7. [Co07]
    Cook, R.T.: Embracing Revenge: On the Indefinite Extendibility of Language. In: Revenge of the Liar: New Essays on the Paradox. Oxford University Press, Oxford (2007)Google Scholar
  8. [Cq93]
    Coquand, T.: Infinite Objects in Type Theory. In: Barendregt, H., Nipkow, T. (eds.) TYPES 1993. LNCS, vol. 806, pp. 62–78. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  9. [D59]
    Dummett, M.: Truth. Proceedings of the Aristotelian Society 59, 141–162 (1959)CrossRefGoogle Scholar
  10. [EG01]
    Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems (2001)Google Scholar
  11. [EGGM02]
    Esteva, F., Gispert, J., Godo, L., Montagna, F.: On the standard and rational completeness of some axiomatic estensions of to monoidal t-norm logic. Studia Logica 71 (2002)Google Scholar
  12. [Fl08]
    Field, H.: Saving Truth From Paradox. Oxford University Press, Oxford (2008)CrossRefzbMATHGoogle Scholar
  13. [Fn09]
    Font, J.M.: Taking degrees of truth seriously. Studia Logica 91, 383–406 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [GB93]
    Gupta, A., Belnap, N.: The revision theory of truth. MIT Press, Cambridge (1993)zbMATHGoogle Scholar
  15. [Hj01]
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (2001)zbMATHGoogle Scholar
  16. [Hj99]
    Hájek, P.: Ten questions and one problem on fuzzy logic. Annals of Pure and Applied Logic 96, 157–165 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [HPS00]
    Hájek, P., Paris, J.B., Shepherdson, J.C.: The Liar Paradox and Fuzzy Logic. Journal of Symbolic Logic 65(1), 339–346 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [Hl96]
    Halbach, V.: Axiomatische Wahrheitstheorien. Wiley-VCH, Weinheim (1996)CrossRefGoogle Scholar
  19. [HH05]
    Halbach, V., Horsten, L.: The Deflationist’s Axioms for Truth. In: Deflationism and Paradox, pp. 203–217. Oxford University Press, Oxford (2005)Google Scholar
  20. [Hr05]
    Horwich, P.: A minimalist clitique of Tarski on Truth. In: Deflationism and Paradox (2005)Google Scholar
  21. [L04]
    Leitgeb, H.: Circular languages. Journal of Logic, Language and Information 13, 341–371 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [M76]
    Machina, K.F.: Truth, belief and vagueness. Journal of Philosohical Logic 5, 47–78 (1976)MathSciNetzbMATHGoogle Scholar
  23. [MOG08]
    Metcalfe, G., Olivetti, N., Gabbay, D.: Proof Theory for Fuzzy Logics. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  24. [Ml88]
    Martin-Lof, P.: Mathematics of infinity. In: Conference on Computer Logic, pp.146–197 (1988)Google Scholar
  25. [MT91]
    Milner, R., Tofte, M.: Co-induction in relational semantics. Theoretical Computer Science 87, 209–220 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [Msk54]
    Shaw-Kwei, M.: Logical paradoxes for many-valued systems. Journal of Symbolic Logic 19(1), 37–40 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [Pl03]
    Paoli, F.: A really fuzzy approach to the sorites paradox. Synthese 134, 363–387 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Pr06]
    Priest, G.: In Contradiction: A Study of the Transconsistent. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  29. [Rt00]
    Rutten, J.J.M.M.: Universal co-algebra. Theoretical Computer Science 249, 3–80 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  30. [R93]
    Restall, G.: Arithmetic and Truth in Łukasiewicz’s Infinitely Valued Logic. Logique et Analyse 36, 25–38 (1993)MathSciNetGoogle Scholar
  31. [Sh06]
    Shapiro, S.: Vagueness in Context, Oxford (2006)Google Scholar
  32. [Su77]
    Suszko, R.: The Fregean axiom and Polish mathematical logic in the 1920s. Studia Logica 36, 377–380 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  33. [T33]
    Tarski, A.: The conception of truth in the languages of the deductive sciences. In: Corcoran, J. (ed.) Prace Towarzystwa Naukowego Warszawskiego. Warsaw; r expanded English translation in “Logic, Semantics, Metamathematics, papers from 1923 to 1938”, vol. 34, pp. 152–278. Hackett Publishing Company, Indianapolis (1933) (Polish)Google Scholar
  34. [Y09]
    Yatabe, S.: The revenge of the modest liar. Non-Classical Mathematics (2009)Google Scholar
  35. [Y11]
    Yatabe, S.: Yablo-like paradoxes and co-induction. In: Bekki, D. (ed.) JSAI-isAI 2010. LNCS, vol. 6797, pp. 90–103. Springer, Heidelberg (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Shunsuke Yatabe
    • 1
  1. 1.Collaborate Research Team for VerificationNational Institute of Advanced Industrial Science and TechnologyJapan

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