Skip to main content
SpringerLink
Log in

Provably True Sentences Across Axiomatizations of Kripke’s Theory of Truth

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

We study the relationships between two clusters of axiomatizations of Kripke’s fixed-point models for languages containing a self-applicable truth predicate. The first cluster is represented by what we will call ‘\(\fancyscript{PKF}\)-like’ theories, originating in recent work by Halbach and Horsten, whose axioms and rules (in Basic De Morgan Logic) are all valid in fixed-point models; the second by ‘\(\fancyscript{KF}\)-like’ theories first introduced by Solomon Feferman, that lose this property but reflect the classicality of the metatheory in which Kripke’s construction is carried out. We show that to any natural system in one cluster—corresponding to natural variations on induction schemata—there is a corresponding system in the other proving the same sentences true, addressing a problem left open by Halbach and Horsten and accomplishing a suitably modified version of the project sketched by Reinhardt aiming at an instrumental reading of classical theories of self-applicable truth.

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. Belnap, N., A useful four-valued logic, in J. M. Dunn and G. Epstein, (eds.), Modern Uses of Multiple-Valued Logic, D. Reidel, Dordrecht, 1977.

  2. Blamey, S., Partial logic, in D. M. Gabbay and F. Guenther (eds.), Handbook of Philosophical Logic, 2 ed., vol. 5, Kluwer, Dordrecht, 2002, pp. 261–353.

    Chapter  Google Scholar 

  3. Cantini, A., Notes on formal theories of truth, Zeitschrift für Logik un Grundlagen der Mathematik 35:97–130, 1989.

    Article  Google Scholar 

  4. Feferman, S., Systems of predicative analysis, Journal of Symbolic Logic 29:1–30, 1964.

    Article  Google Scholar 

  5. Feferman, S., Towards useful type-free theories. I, Journal of Symbolic Logic 49(1):75–111, 1984.

    Article  Google Scholar 

  6. Feferman, S., Reflecting on incompleteness, Journal of Symbolic Logic 56:1–49, 1991.

    Article  Google Scholar 

  7. Field, H., Saving Truth from Paradox, Oxford University Press, Oxford, 2008.

    Book  Google Scholar 

  8. Fischer, M., V. Halbach, J. Kriener, and J. Stern, Axiomatizing semantic theories of truth? The Review of Symbolic Logic  8(2):257–278, 2015.

  9. Halbach, V., Axiomatic Theories of Truth. Revised Edition, Cambridge University Press, Cambridge, 2014.

    Book  Google Scholar 

  10. Halbach, V., and L. Horsten, Axiomatizing Kripke’s theory of truth in partial logic, Journal of Symbolic Logic 71:677–712, 2006.

    Article  Google Scholar 

  11. Hilbert, D., Über das Unendliche, in J. Van Heijenoort (ed.), From Frege to Gödel. A Source Book in Mathematical Logic, Harvard University Press, Cambridge, 1967.

    Google Scholar 

  12. Horsten, L., The Tarskian Turn, MIT University Press, Oxford, 2012.

    Google Scholar 

  13. Kremer, M., Kripke and the logic of truth, Journal of Philosophical Logic 17:225–278, 1988.

    Article  Google Scholar 

  14. Kripke, S., Outline of a theory of truth, Journal of Philosophy 72:690–712, 1975.

    Article  Google Scholar 

  15. McGee, V., Truth, Vagueness, and Paradox, MIT University Press, Cambridge, 1991.

    Google Scholar 

  16. Nicolai, C., M. Fischer, and L. Horsten, Iterated reflection over full disquotational truth (Submitted). https://arxiv.org/pdf/1703.02301.pdf.

  17. Nicolai, C., and V. Halbach. On the costs of nonclassical logic, Journal of Philosophical Logic. doi:10.1007/s10992-017-9424-3.

  18. Pohlers, W., Proof Theory. A First Step into Impredicativity, Springer, Berlin, 2009.

    Google Scholar 

  19. Reinhardt, W., Some remarks on extending and interpreting theories with a partial predicate for truth, Journal of Philosophical Logic 15:219–251, 1986.

    Article  Google Scholar 

  20. Schmerl, U. R., Iterated reflection principle and the \(\omega \)-rule, Journal of Symbolic Logic 4:721–733, 1982.

    Article  Google Scholar 

  21. Schwichtenberg, H., Proof theory: some applications of cut-elimination, in J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977.

    Google Scholar 

  22. Tarski, A., Der Wahrhetisbegriff in den formalisierten Sprachen, in J. H. Woodger (ed.), Logic, Semantics, Metamathematics, Clarendon Press, Oxford, 1956, pp. 152–278.

  23. Williamson, T., Semantic paradoxes and abductive methodology, in B. Armour-Garb (ed.), The Relevance of the Liar, Oxford University Press, Oxford, 2017.

Download references

Acknowledgements

This work was supported by the European Commission (Grant No. 658285 FOREMOTIONS). I thank Martin Fischer, Leon Horsten, Johannes Stern, and the anonymous referees for their comments.

Author information

Authors and Affiliations

  1. Ludwig-Maximilians-Universität München, Fakultät für Philosophie, Wissenschaftstheorie und Religionswissenschaft Munich Center for Mathematical Philosophy, Geschwister-Scholl Platz 1, 80539, München, Germany

    Carlo Nicolai

Authors
  1. Carlo Nicolai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carlo Nicolai.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nicolai, C. Provably True Sentences Across Axiomatizations of Kripke’s Theory of Truth. Stud Logica 106, 101–130 (2018). https://doi.org/10.1007/s11225-017-9727-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-017-9727-y

Keywords

Search

Navigation