Advertisement

Journal of Philosophical Logic

, Volume 47, Issue 5, pp 817–850 | Cite as

Supervaluation-Style Truth Without Supervaluations

  • Johannes SternEmail author
Article
  • 184 Downloads

Abstract

Kripke’s theory of truth is arguably the most influential approach to self-referential truth and the semantic paradoxes. The use of a partial evaluation scheme is crucial to the theory and the most prominent schemes that are adopted are the strong Kleene and the supervaluation scheme. The strong Kleene scheme is attractive because it ensures the compositionality of the notion of truth. But under the strong Kleene scheme classical tautologies do not, in general, turn out to be true and, as a consequence, classical reasoning is no longer admissible once the notion of truth is involved. The supervaluation scheme adheres to classical reasoning but violates compositionality. Moreover, it turns Kripke’s theory into a rather complicated affair: to check whether a sentence is true we have to look at all admissible precisification of the interpretation of the truth predicate we are presented with. One consequence of this complicated evaluation condition is that under the supervaluation scheme a more proof-theoretic characterization of Kripke’s theory becomes inherently difficult, if not impossible. In this paper we explore the middle ground between the strong Kleene and the supervaluation scheme and provide an evaluation scheme that adheres to classical reasoning but retains many of the attractive features of the strong Kleene scheme. We supplement our semantic investigation with a novel axiomatic theory of truth that matches the semantic theory we have put forth.

Keywords

Truth Supervaluation Grounded truth Kripke’s theory of truth Axiomatic theories of truth Strong Kleene 

Notes

Acknowledgments

This work was supported by the DFG-funded research project Syntactical Treatments of Interacting Modalities, the German BA, and by the European Commission through a Marie Sklodowska-Curie Individual Fellowship (TREPISTEME, Grant No. 703529). I wish to thank Catrin Campbell-Moore, Martin Fischer, Carlo Nicolai, Philip Welch and two anonymous referees for helpful comments on and discussion of my paper. Early versions of this paper were presented at Bristol, Bath, and Munich. I thank the audiences for their feedback.

References

  1. 1.
    Burgess, J. P. (1986). The truth is never simple. The Journal of Symbolic Logic, 51(3), 663–681.CrossRefGoogle Scholar
  2. 2.
    Burgess, J. P. (1988). Addendum to the truth is never simple. The Journal of Symbolic Logic, 53(2), 390–392.Google Scholar
  3. 3.
    Cantini, A. (1990). A theory of formal truth arithmetically equivalent to ID1. The Journal of Symbolic Logic, 55, 244–259.CrossRefGoogle Scholar
  4. 4.
    Cantini, A. (1996). Logical frameworks for truth and abstraction. Florenz: Elsevier Science Publisher.Google Scholar
  5. 5.
    Field, H. (2008). Saving truth from paradox. Oxford: Oxford University Press.CrossRefGoogle Scholar
  6. 6.
    Fine, K. (1975). Vagueness, truth and logic. Synthese, 30, 265–300.CrossRefGoogle Scholar
  7. 7.
    Fischer, M., Halbach, V., Kriener, J., & Stern, J. (2015). Axiomatizing semantic theories of truth? Review of Symbolic Logic, 8(2), 257–278.  https://doi.org/10.1017/S1755020314000379.CrossRefGoogle Scholar
  8. 8.
    Friedman, H., & Sheard, M. (1987). An axiomatic approach to self-referential truth. Annals of Pure and Applied Logic, 33, 1–21, 305–344.CrossRefGoogle Scholar
  9. 9.
    Fujimoto, K. (2010). Relative truth definability of axiomatic truth theories. Bulletin of symbolic logic, 16(3).Google Scholar
  10. 10.
    Halbach, V. (2009). Reducing compositional to disquotational truth. The Review of Symbolic Logic, 2, 786–798.CrossRefGoogle Scholar
  11. 11.
    Halbach, V. (2011). Axiomatic theories of truth. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  12. 12.
    Herzberger, H. G. (1970). Paradoxes of grounding in semantics. The Journal of Philosophy, 67, 145–167.CrossRefGoogle Scholar
  13. 13.
    Jäger, G. (1982). Zur Beweistheorie der Kripke-Platek Mengenlehre. Archiv für Mathematische Logik und Grundlagenforschung, 22, 121–139.CrossRefGoogle Scholar
  14. 14.
    Kremer, P., & Kremer, M. (2003). Some supervaluation-based consequence relations. Journal of Philosophical Logic, 32, 225–244.CrossRefGoogle Scholar
  15. 15.
    Kremer, P., & Urquhart, A. (2008). Supervaluation fixed-point logics of truth. Journal of Philosophical Logic, 37, 407–440.CrossRefGoogle Scholar
  16. 16.
    Kripke, S. (1975). Outline of a theory of truth. The Journal of Philosophy, 72, 690–716.CrossRefGoogle Scholar
  17. 17.
    Leitgeb, H. (2005). What truth depends on. Journal of Philosophical Logic, 34, 155–192.CrossRefGoogle Scholar
  18. 18.
    Martin, R. L., & Woodruff, P. W. (1975). On representing true-in-L in L. Philosophia. Philosophical Quarterly of Israel, 5, 213–217.Google Scholar
  19. 19.
    McGee, V. (1991). Truth, vagueness and paradox. Indianapolis: Hackett Publishing Company.Google Scholar
  20. 20.
    van Fraassen, B. C. (1968). Presupposition, implication, and self-reference. The Journal of Philosophy, 65(5), 136–152.CrossRefGoogle Scholar
  21. 21.
    Van Fraassen, B. C. (1969). Presuppositions, supervaluations and free logic. In Lambert, K. (Ed.) The Logical Way of Doing Things. New Haven: Yale University Press.Google Scholar
  22. 22.
    Visser, A. (1984). Semantics and the liar paradox. In Gabbay, D. (Ed.) Handbook of Philosophical Logic, 617–706. Dordrecht.Google Scholar
  23. 23.
    Welch, P. (2015). The complexity of the dependence operator. Journal of Philosophical Logic, 44(3), 337–340.CrossRefGoogle Scholar
  24. 24.
    Yablo, S. (1982). Grounding, dependence, and paradox. Journal of Philosophical Logic, 11, 117–137.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2018

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of BristolBristolEngland

Personalised recommendations