Skip to main content
Log in

Actuality in Propositional Modal Logic

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

We show that the actuality operator A is redundant in any propositional modal logic characterized by a class of Kripke models (respectively, neighborhood models). Specifically, we prove that for every formula \({\phi}\) in the propositional modal language with A, there is a formula \({\psi}\) not containing A such that \({\phi}\) and \({\psi}\) are materially equivalent at the actual world in every Kripke model (respectively, neighborhood model). Inspection of the proofs leads to corresponding proof-theoretic results concerning the eliminability of the actuality operator in the actuality extension of any normal propositional modal logic and of any “classical” modal logic. As an application, we provide an alternative proof of a result of Williamson’s to the effect that the compound operator A□ behaves, in any normal logic between T and S5, like the simple necessity operator □ in S5.

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

Access this article

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. Blackburn P., Marx M.: Remarks on Gregory’s “actually” operator. Journal of Philosophical Logic 31, 281–288 (2002)

    Article  Google Scholar 

  2. Chellas B. F.: Modal Logic—An Introduction. Cambridge University Press, Cambridge (1980)

    Book  Google Scholar 

  3. Crossley J. N., Humberstone L.: The logic of “actually”. Reports on Mathematical Logic 8, 11–29 (1977)

    Google Scholar 

  4. Gregory D.: Completeness and decidability results for some propositional modal logics containing “actually” operators. Journal of Philosophical Logic 30, 57–78 (2001)

    Article  Google Scholar 

  5. Hazen A. P.: Expressive completeness in modal language. Journal of Philosophical Logic 5, 25–46 (1976)

    Article  Google Scholar 

  6. Hazen A. P.: The eliminability of the actuality operator in propositional modal logic. Notre Dame Journal of Formal Logic 4, 617–622 (1978)

    Article  Google Scholar 

  7. Kripke S.A.: Semantical analysis of modal logic I: Normal modal propositional calculi. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 9, 67–96 (1963)

    Article  Google Scholar 

  8. Kripke, S. A., Semantical analysis of modal logic II: Non-normal modal propositional calculi, in J. W. Addison, L. Henkin, and A. Tarski (eds.), The Theory of Models, North-Holland, Amsterdam, 1965, pp. 206–220.

  9. Segerberg, K., An Essay in Classical Modal Logic, The Philosophical Society, University of Uppsala, 1971.

  10. Stephanou Y.: Indexed actuality. Journal of Philosophical Logic 30, 355–393 (2001)

    Article  Google Scholar 

  11. Wehmeier, K. F., World travelling and mood swings, in B. Löwe, T. Räsch, and W. Malzkorn (eds.), Foundations of the Formal Sciences II, Kluwer, Dordrecht, 2003, pp. 257–260.

  12. Williamson, T., Iterated attitudes, in T. Smiley (ed.), Philosophical Logic (= Proceedings of the British Academy 95), Oxford University Press, Oxford, 1998, pp. 85–133.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kai F. Wehmeier.

Additional information

Author names are listed alphabetically. Sections 2 and 3 contain joint work by KW and BR, with a crucial suggestion from AH. The remainder of the paper consists of joint work by AH and KW.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hazen, A.P., Rin, B.G. & Wehmeier, K.F. Actuality in Propositional Modal Logic. Stud Logica 101, 487–503 (2013). https://doi.org/10.1007/s11225-012-9395-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-012-9395-x

Keywords

Navigation