Global Neighbourhood Completeness of the Gödel-Löb Provability Logic

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10388)


The Gödel-Löb provability logic \(\mathsf {GL}\) is strongly neighbourhood complete in the case of the so-called local semantic consequence relation. In the given paper, we consider Hilbert-style non-well-founded derivations in \(\mathsf {GL}\) and establish that \(\mathsf {GL}\) with the obtained derivability relation is strongly neighbourhood complete in the case of the global semantic consequence relation.


Provability logic Neighbourhood semantics Global consequence relations Non-well-founded proofs 


  1. 1.
    Aguilera, J.P., Fernández-Duque, D.: Strong completeness of provability logic for ordinal spaces (2015). arXiv:1511.05882v1
  2. 2.
    Beklemishev, L., Gabelaia, D.: Topological interpretations of provability logic. In: Bezhanishvili, G. (ed.) Leo Esakia on Duality in Modal and Intuitionistic Logics. OCL, vol. 4, pp. 257–290. Springer, Dordrecht (2014). doi:10.1007/978-94-017-8860-1_10 Google Scholar
  3. 3.
    Esakia, L.: Diagonal constructions, Löb’s formula and Cantor’s scattered space. Stud. Logic Semant. 132(3), 128–143 (1981). (in Russian)Google Scholar
  4. 4.
    Hakli, R., Negri, S.: Does the deduction theorem fail for modal logic? Synthese 187(3), 849–867 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Iemhoff, R.: Reasoning in circles. In: van Eijck, J., et al. (eds.) Liber Amicorum Alberti. A Tribute to Albert Visser, pp. 165–178. College Publications, London (2016)Google Scholar
  6. 6.
    Litak, T.: An algebraic approach to incompleteness in modal logic. Ph.D. thesis. Japan Advanced Institute of Science and Technology (2005)Google Scholar
  7. 7.
    Montague, R.: Universal grammar. Theoria 36(3), 373–398 (1970)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Scott, D.: Advice in modal logic. In: Lambert, K. (ed.) Philosophical Problems in Logic. Reidel, Kufstein (1970)Google Scholar
  9. 9.
    Segerberg, K.: An Essay in Classical Modal Logic. Filosofiska Studier, vol. 13. Uppsala University (1971)Google Scholar
  10. 10.
    Shamkanov, D.: Circular proofs for the Gödel-Löb provability logic. Math. Not. 96(3), 575–585 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Shehtman, V.: On neighbourhood semantics thirty years later. In: Artemov, S., et al. (eds.) We Will Show Them! Essays in Honour of Dov Gabbay, vol. 2, pp. 663–692. College Publications, London (2005)Google Scholar
  12. 12.
    Simmons, H.: Topological aspects of suitable theories. Proc. Edinb. Math. Soc. 19(4), 383–391 (1975)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Solovay, R.: Provability interpretations of modal logic. Isr. J. Math. 25, 287–304 (1976)MathSciNetCrossRefMATHGoogle Scholar

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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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