Skip to main content
SpringerLink
Log in

The Modal Logic of Gödel Sentences

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

Abstract

The modal logic of Gödel sentences, termed as GS, is introduced to analyze the logical properties of ‘true but unprovable’ sentences in formal arithmetic. The logic GS is, in a sense, dual to Grzegorczyk’s Logic, where modality can be interpreted as ‘true and provable’. As we show, GS and Grzegorczyk’s Logic are, in fact, mutually embeddable. We prove Kripke completeness and arithmetical completeness for GS. GS is also an extended system of the logic of ‘Essence and Accident’ proposed by Marcos (Bull Sect Log 34(1):43–56, 2005). We also clarify the relationships between GS and the provability logic GL and between GS and Intuitionistic Propositional Logic.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Boolos, G. (1993). The logic of provability. Cambridge: Cambridge University Press.

    Google Scholar 

  2. Chagrov, A., & Zakharyaschev, M. (1997). Modal logic. London: Oxford University Press.

    Google Scholar 

  3. Goldblatt, R. (1978). Arithmetical necessity, provability and intuitionistic logic. Theoria, 44, 38–46.

    Article  Google Scholar 

  4. Gödel, K. (1933). Eine interpretation des intuitionistischen Aussagenkalküls. Ergebnisse Eines Mathematischen Kolloquiums, 4, 39–40.

    Google Scholar 

  5. Grzegorczyk, A. (1967). Some relational systems and the associated topological spaces. Fundamenta Mathematicae, 60, 223–231.

    Google Scholar 

  6. Marcos, J. (2005). Logics of essence and accident. Bulletin of the Section of Logic, 34(1), 43–56.

    Google Scholar 

  7. Mckinsey, J. C. C., & Tarski, A. (1948). Some theorems about the sentential calculi of Lewis and Heyting. Journal of Symbolic Logic, 13, 1–15.

    Article  Google Scholar 

  8. Segerberg, K. (1971). An essay in classical modal logic, filosofiska studier. Uppsala Universitet.

  9. Solovay, R. M. (1976). Provability interpretation of modal logic. Israel Journal of Mathematics, 25, 287–304.

    Article  Google Scholar 

  10. Smorỳnski, C. (1985). Self-reference and modal logic. Heidelberg: Springer.

    Google Scholar 

  11. Steinsvold, C. (2008). Completeness for various logics of essence and accident. Bulletin of the Section of Logic, 37(2), 93–102.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Philosophy, Keio University, Tokyo, Japan

    Hirohiko Kushida

Authors
  1. Hirohiko Kushida
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hirohiko Kushida.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kushida, H. The Modal Logic of Gödel Sentences. J Philos Logic 39, 577–590 (2010). https://doi.org/10.1007/s10992-010-9140-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-010-9140-8

Keywords

Search

Navigation