Skip to main content
SpringerLink
Log in

Global Neighbourhood Completeness of the Gödel-Löb Provability Logic

  • Conference paper
  • First Online:
Logic, Language, Information, and Computation (WoLLIC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10388))

  • 459 Accesses

Abstract

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.

This work was supported by the Russian Science Foundation (grant no. 14-50-00005).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aguilera, J.P., Fernández-Duque, D.: Strong completeness of provability logic for ordinal spaces (2015). arXiv:1511.05882v1

  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. 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. Hakli, R., Negri, S.: Does the deduction theorem fail for modal logic? Synthese 187(3), 849–867 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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. Litak, T.: An algebraic approach to incompleteness in modal logic. Ph.D. thesis. Japan Advanced Institute of Science and Technology (2005)

    Google Scholar 

  7. Montague, R.: Universal grammar. Theoria 36(3), 373–398 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  8. Scott, D.: Advice in modal logic. In: Lambert, K. (ed.) Philosophical Problems in Logic. Reidel, Kufstein (1970)

    Google Scholar 

  9. Segerberg, K.: An Essay in Classical Modal Logic. Filosofiska Studier, vol. 13. Uppsala University (1971)

    Google Scholar 

  10. Shamkanov, D.: Circular proofs for the Gödel-Löb provability logic. Math. Not. 96(3), 575–585 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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. Simmons, H.: Topological aspects of suitable theories. Proc. Edinb. Math. Soc. 19(4), 383–391 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  13. Solovay, R.: Provability interpretations of modal logic. Isr. J. Math. 25, 287–304 (1976)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The development of main ideas of this paper took place during my stay in Tash-Bulak village, the Kyrgyz Republic, in 2015. I heartily thank my uncle Kengebek Shamkanov and his family for their hospitality. In addition, I am grateful to Tadeusz Litak, whose constructive comments have helped me to improve the manuscript.

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina str. 8, 119991, Moscow, Russia

    Daniyar Shamkanov

Authors
  1. Daniyar Shamkanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniyar Shamkanov .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland

    Juliette Kennedy

  2. Centro de Informática, Recife, Pernambuco, Brazil

    Ruy J.G.B. de Queiroz

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer-Verlag GmbH Germany

About this paper

Cite this paper

Shamkanov, D. (2017). Global Neighbourhood Completeness of the Gödel-Löb Provability Logic. In: Kennedy, J., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2017. Lecture Notes in Computer Science(), vol 10388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55386-2_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-55386-2_26

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-55385-5

  • Online ISBN: 978-3-662-55386-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation