Truth, disjunction, and induction

  • Ali EnayatEmail author
  • Fedor Pakhomov
Open Access


By a well-known result of Kotlarski et al. (1981), first-order Peano arithmetic \({{\mathsf {P}}}{{\mathsf {A}}}\) can be conservatively extended to the theory \({{\mathsf {C}}}{{\mathsf {T}}}^{-}\mathsf {[PA]}\) of a truth predicate satisfying compositional axioms, i.e., axioms stating that the truth predicate is correct on atomic formulae and commutes with all the propositional connectives and quantifiers. This result motivates the general question of determining natural axioms concerning the truth predicate that can be added to \({{\mathsf {C}}}{{\mathsf {T}}}^{-}\mathsf {[PA]}\) while maintaining conservativity over \( {{\mathsf {P}}}{{\mathsf {A}}}\). Our main result shows that conservativity fails even for the extension of \({{\mathsf {C}}}{{\mathsf {T}}}^{-}\mathsf {[PA]}\) obtained by the seemingly weak axiom of disjunctive correctness \({{\mathsf {D}}}{{\mathsf {C}}}\) that asserts that the truth predicate commutes with disjunctions of arbitrary finite size. In particular, \({{\mathsf {C}}}{\mathsf {T}}^{-}\mathsf {[PA]}+\mathsf {DC}\) implies \(\mathsf {Con}(\mathsf {PA})\). Our main result states that the theory \({\mathsf {C}}{\mathsf {T}}^{-}\mathsf {[PA]}+\mathsf {DC}\) coincides with the theory \({\mathsf {C}}{\mathsf {T}}_{0}\mathsf {[PA]}\) obtained by adding \( \Delta _{0}\)-induction in the language with the truth predicate. This result strengthens earlier work by Kotlarski (1986) and Cieśliński (2010). For our proof we develop a new general form of Visser’s theorem on non-existence of infinite descending chains of truth definitions and prove it by reduction to (Löb’s version of) Gödel’s second incompleteness theorem, rather than by using the Visser–Yablo paradox, as in Visser’s original proof (1989).


Axiomatic truth Compositional theory of truth Conservativity 

Mathematics Subject Classification




  1. 1.
    Boolos, G.S., Burgess, J.P., Jeffrey, R.C.: Computability and Logic, 5th edn. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  2. 2.
    Buss, S.: Proof theory of arithmetic. In: Buss, S. (ed.) Handbook of Proof Theory, pp. 79–147. Elsevier, Amsterdam (1998)CrossRefGoogle Scholar
  3. 3.
    Cieśliński, C.: Deflationary truth and pathologies. J. Philos. Logic 39, 325–337 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cieśliński, C.: The Epistemic Lightness of Truth. Deflationism and its Logic. Cambridge University Press, Cambridge (2017)CrossRefGoogle Scholar
  5. 5.
    Enayat, A., Visser, A.: New constructions of full satisfaction classes. In: Achourioti, T., Galinon, H., Fujimoto, K., Martínez-Fernández, J. (eds.) Unifying the Philosophy of Truth, pp. 321–325. Springer, Berlin (2015)CrossRefGoogle Scholar
  6. 6.
    Engström, F.: Satisfaction classes in nonstandard models of first-order arithmetic (2002). arXiv:math/0209408
  7. 7.
    Flumini, D., Sato, K.: From hierarchies to well-foundedness. Arch. Math. Log. 54, 855–863 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hájek, P., Pudlák, P.: Metamathematics of First-Order Arithmetic. Springer, Berlin (1993)CrossRefGoogle Scholar
  9. 9.
    Halbach, V.: Axiomatic Theories of Truth, 2nd edn. Cambridge University Press, Cambridge (2015)zbMATHGoogle Scholar
  10. 10.
    Kaye, R.: Models of Peano Arithmetic. Oxford University Press, Oxford (1991)zbMATHGoogle Scholar
  11. 11.
    Kotlarski, H.: Bounded induction and satisfaction classes. Zeitschrift für matematische Logik und Grundlagen der Mathematik 32, 531–544 (1986)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kotlarski, H., Krajewski, S., Lachlan, A.: Construction of satisfaction classes for nonstandard models. Can. Math. Bull. 24, 283–293 (1981)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Krajewski, S.: Nonstandard satisfaction classes. In: Marek, W., et al. Set Theory and Hierarchy Theory: A Memorial Tribute to Andrzej Mostowski. Lecture Notes in Mathematics, vol. 537, pp. 121–144 . Springer, Berlin (1976)Google Scholar
  14. 14.
    Leigh, G.: Conservativity for theories of compositional truth via cut elimination. J. Symb. Log. 80, 845–865 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Łełyk, M.: Axiomatic theories of truth, bounded induction and reflection principles. Ph.D. disseration, University of Warsaw (2017)Google Scholar
  16. 16.
    Łełyk, M., Wcisło, B.: Notes on bounded induction for the compositional truth predicate. Rev. Symb. Logic 10, 455–480 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Löb, M.H.: Solution of a problem of Leon Henkin. J. Symb. Logic 20, 115–118 (1955)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Pakhomov, F., Walsh, J.: Reflection ranks and ordinal analysis (2018). arXiv:1805.02095
  19. 19.
    Pudlák, P.: The lengths of proofs. In: Buss, S. (ed.) Handbook of Proof Theory, pp. 547–637. Elsevier, Amsterdam (1998)CrossRefGoogle Scholar
  20. 20.
    Visser, A.: Semantics and the liar paradox. In: Gabbay, D., Günthner, F. (eds.) Handbook of Philosophical Logic, vol. 4, pp. 149–240. Reidel, Dordrecht (1989)Google Scholar
  21. 21.
    Visser, A.: From Tarski to Gödel. Or, how to derive the second incompleteness theorem from the undefinability of truth without self-reference (2018). arXiv:1803.03937

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Philosophy, Linguistics, and the Theory of ScienceUniversity of GothenburgGothenburgSweden
  2. 2.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

Personalised recommendations