Journal of Philosophical Logic

, Volume 45, Issue 4, pp 381–398 | Cite as

Liar-type Paradoxes and the Incompleteness Phenomena

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Abstract

We define a liar-type paradox as a consistent proposition in propositional modal logic which is obtained by attaching boxes to several subformulas of an inconsistent proposition in classical propositional logic, and show several famous paradoxes are liar-type. Then we show that we can generate a liar-type paradox from any inconsistent proposition in classical propositional logic and that undecidable sentences in arithmetic can be obtained from the existence of a liar-type paradox. We extend these results to predicate logic and discuss Yablo’s Paradox in this framework. Furthermore, we define explicit and implicit self-reference in paradoxes in the incompleteness phenomena.

Keywords

Gödel’s incompleteness theorem The Liar Paradox Modal logic Arithmetical realization 

References

  1. 1.
    Artemov, S.N. (1985). Nonarithmeticity of truth predicate logics of provability (in Russian). Doklady Akademii Nauk SSSR, 284(2), 270–271. English translation in Soviet Mathematics Doklady, vol. 32 (1985), no. 2, 403-405.Google Scholar
  2. 2.
    Artemov, S.N., & Japaridze, G.K. (1990). Finite Kripke models and predicate logics of provability. The Journal of Symbolic Logic, 55(3), 1090–1098.CrossRefGoogle Scholar
  3. 3.
    Boolos, G. (1989). A new proof of the Gödel incompleteness theorem. Notices of the American Mathematical Society, 36, 388–390.Google Scholar
  4. 4.
    Boolos, G. (1993). The logic of provability. Cambridge: Cambridge University Press.Google Scholar
  5. 5.
    Chaitin, G.J. (1974). Information-theoretic limitations of formal systems. Journal of the Association for Computing Machinery, 21, 403–424.CrossRefGoogle Scholar
  6. 6.
    Cieśliński, C. (2002). Heterologicality and incompleteness. Mathematical Logic Quarterly, 48(1), 105–110.CrossRefGoogle Scholar
  7. 7.
    Cieśliński, C., & Urbaniak, R. (2013). Gödelizing the Yablo sequence. Journal of Philosophical Logic, 42(5), 679–695.CrossRefGoogle Scholar
  8. 8.
    Fitch, F.B. (1964). A goedelized formulation of the prediction paradox. American Philosophical Quarterly, 1, 161–164.Google Scholar
  9. 9.
    Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. (in German). Monatshefte für Mathematik und Physik, 38(1), 173–198. English translation in Kurt Gödel, Collected Works, Vol. 1 (pp. 145–195).CrossRefGoogle Scholar
  10. 10.
    Japaridze, G., & de Jongh, D. (1998). The logic of provability. In Handbook of proof theory, Studies in Logic and the Foundations of Mathematics, vol. 137, pp. 475546. North- Holland, Amsterdam.Google Scholar
  11. 11.
    Kikuchi, M. (1994). A note on Boolos’ proof of the incompleteness theorem. Mathematical Logic Quarterly, 40(4), 528–532.CrossRefGoogle Scholar
  12. 12.
    Kikuchi, M., & Kurahashi, T. (2011). Three short stories around Gödel’s incompleteness theorems (in Japanese). Journal of the Japan Association for Philosophy of Science, 38(2), 27–32.Google Scholar
  13. 13.
    Kikuchi, M., Kurahashi, T., & Sakai, H. (2012). On proofs of the incompleteness theorems based on Berry’s paradox by Vopėnka, Chaitin, and Boolos. Mathematical Logic Quarterly, 58(4-5), 307–316.CrossRefGoogle Scholar
  14. 14.
    Kikuchi, M., & Tanaka, K. (1994). On formalization of model-theoretic proofs of Gödel’s theorems. Notre Dame Journal of Formal Logic, 35(3), 403–412.CrossRefGoogle Scholar
  15. 15.
    Kritchman, S., & Raz, R. (2010). The surprise examination paradox and the second incompleteness theorem. Notices of the American Mathematical Society, 57 (11), 1454–1458.Google Scholar
  16. 16.
    Kurahashi, T. (2014). Rosser-type undecidable sentences based on Yablo’s paradox. Journal of Philosophical Logic, 43(5), 999–1017.CrossRefGoogle Scholar
  17. 17.
    Montagna, F. (1984). The predicate modal logic of provability. Notre Dame Journal of Formal Logic, 25(2), 179–189.CrossRefGoogle Scholar
  18. 18.
    Priest, G. (1997). Yablo’s paradox. Analysis, 57(4), 236–242.CrossRefGoogle Scholar
  19. 19.
    Segerberg, K. (1971). An essay in classical modal logic. Filosofiska Föreningen och Filosofiska Institutionen vid Uppsala Universitet.Google Scholar
  20. 20.
    Solovay, R.M. (1976). Provability interpretations of modal logic. Israel Journal of Mathematics, 25(3-4), 287–304.CrossRefGoogle Scholar
  21. 21.
    Vopėnka, P. (1966). A new proof of the Gödel’s result on non-provability of consistency. Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques. Astronomiques et Physiques, 14, 111–116.Google Scholar
  22. 22.
    Yablo, S. (1993). Paradox without self-reference. Analysis, 53(4), 251–252.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Graduate School of System InformaticsKobe UniversityKobeJapan
  2. 2.Department of Natural SciencesNational Institute of Technology, Kisarazu CollegeKisarazuJapan

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