Journal of Philosophical Logic

, Volume 45, Issue 1, pp 89–119

A Note on Typed Truth and Consistency Assertions

Article

Abstract

In the paper we investigate typed (mainly compositional) axiomatizations of the truth predicate in which the axioms of truth come with a built-in, minimal and self-sufficient machinery to talk about syntactic aspects of an arbitrary base theory. Expanding previous works of the author and building on recent works of Albert Visser and Richard Heck, we give a precise characterization of these systems by investigating the strict relationships occurring between them, arithmetized model constructions in weak arithmetical systems and suitable set existence axioms. The framework considered will give rise to some methodological remarks on the construction of truth theories and provide us with a privileged point of view to analyze the notion of truth arising from compositional principles in a typed setting.

Keywords

Axiomatic theories of truth Subsystems of first-order arithmetic Truth-theoretic deflationism 

References

  1. 1.
    Bar-On, D., & Simmons, K. (2007). The use of force against deflationism: Assertion and truth’. In Greimann, D., & Siegwart, G. (Eds.), Truth and Speech Acts: Studies in the Philosophy of Language. (pp. 61–89). New York: Routledge.Google Scholar
  2. 2.
    Bennet, J.H. (1962). On Spectra. PhD Thesis, Princeton.Google Scholar
  3. 3.
    Buss, S. (1986). Bounded Arithmetic. Naples: Bibliopolis.Google Scholar
  4. 4.
    Buss, S. (ed.) (1998). Handbook of Proof Theory. Elsevier.Google Scholar
  5. 5.
    Cantini, A. (1989). Notes on formal theories of truth. Archives for Mathematical Logic, 35, 97–139.CrossRefGoogle Scholar
  6. 6.
    Cieśliński, C. (2010). Truth, Conservativeness, Provability. Mind, 119(474).Google Scholar
  7. 7.
    Craig, W., & Vaught, W. (1958). Finite axiomatizability using additional predicates. The Journal of Symbolic Logic, 23, 289–308.CrossRefGoogle Scholar
  8. 8.
    Enayat, A., & Visser, A. (2013). New constructions of satisfaction classes. Logic Preprints Series: University of Utrecht.Google Scholar
  9. 9.
    Enderton, H. (2001). A mathematical introduction to logic Harcourt Press.Google Scholar
  10. 10.
    Feferman, S. (1960). Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae, 49, 35–91.Google Scholar
  11. 11.
    Feferman, S. (1991). Reflecting on incompleteness. Journal of Symbolic Logic, 56, 1–49.CrossRefGoogle Scholar
  12. 12.
    Ferreira, F. (1988). Polynomial Time Computable Arithmetic and Conservative Extensions. Ph.D.thesis, The Pennsylvania State University, State College.Google Scholar
  13. 13.
    Field, H. (1999). Deflating the conservativeness argument. Journal of Philosophy, 96(10), 533–540.CrossRefGoogle Scholar
  14. 14.
    Field, H. (2005). Compositional principles vs. schematic reasoning. The Monist, 89, 9–27.CrossRefGoogle Scholar
  15. 15.
    Fischer, M. (2009). Minimal Truth and Interpretability. The Review of symbolic logic, 2, 799–815.CrossRefGoogle Scholar
  16. 16.
    Fischer, M., Halbach, V., Speck, J., & Stern, J. Axiomatizing Semantic Theories of Truth?. Unpublished Manuscript.Google Scholar
  17. 17.
    Fujimoto, K. (2012). Classes and truths in set theory. Annals of Pure and Applied Logic, 164(11), 1484–1523.CrossRefGoogle Scholar
  18. 18.
    Gödel, K. (1931). Über formal unentscheidbare sätze der principia mathematica und verwandter systeme, I. Monatshefte für Mathematik und Physik, 38, 173–98.CrossRefGoogle Scholar
  19. 19.
    Hájek, P., & Pudlák, P. (1993). Metamathematics of first-order arithmetic. Berlin: Springer.CrossRefGoogle Scholar
  20. 20.
    Halbach, V. (1994). A system of complete and consistent truth. Notre Dame Journal of Formal Logic, 35, 311–327.CrossRefGoogle Scholar
  21. 21.
    Halbach, V. (2000). Truth and reduction. Erkenntnis, 53, 197–126.CrossRefGoogle Scholar
  22. 22.
    Halbach, V. (2001a). How innocent is deflationism?. Synthese, 126, 167–194.CrossRefGoogle Scholar
  23. 23.
    Halbach, V. (2014). Axiomatic theories of truth. Revised Edition: Cambridge University Press.CrossRefGoogle Scholar
  24. 24.
    Heck, R. Consistency and the Theory of Truth. forthcoming in The Review of Symbolic Logic.Google Scholar
  25. 25.
    Horwich, P. (1998). Truth, 2nd ed. Blackwell.Google Scholar
  26. 26.
    Horsten, L. (2011). The Tarskian Turn. Deflationism and Axiomatic Truth. Princeton University Press.Google Scholar
  27. 27.
    Joosten, J., & Visser, A. (2004). Visser (2004), ‘Characterizations of Interpretability’, in Logic Preprint Group Series Utrecht.Google Scholar
  28. 28.
    Kaye, R. (1991). Models of peano arithmetic. Oxford University Press, 1991.Google Scholar
  29. 29.
    Ketland, J. (1999). Deflationism and Tarski’s paradise. Mind, 108(429), 69–94.CrossRefGoogle Scholar
  30. 30.
    Kikuchi, M., & Tanaka, K. (1994). On formalizations of model-theoretic proofs of Gödel’s theorems’. Notre Dame Journal of Formal Logic, 35(3), 403–412.CrossRefGoogle Scholar
  31. 31.
    Lachlan, A. (1981). Full satisfaction classes and recursive saturation. Canadian Mathematical Bulletin, 24, 295–297.Google Scholar
  32. 32.
    Lavine, S. (1999). Skolem was wrong, unpublished manuscript.Google Scholar
  33. 33.
    Leigh, G., & Nicolai, C. (2013). Axiomatic truth, syntax and metatheoretic reasoning. The Review of Symbolic Logic, 6(4), 613–636.CrossRefGoogle Scholar
  34. 34.
    McGee, V. (2006). In praise of a free lunch: why disquotationalists should embrace compositional semantics In: Bolander et alii (eds.), Self Reference, CSL Publications.Google Scholar
  35. 35.
    Nicolai, C. (2014). Truth, deflationism and the ontology of expressions An Axiomatic Study. Oxford: DPhil Thesis.Google Scholar
  36. 36.
    Nicolai, C. (2014). Deflationary truth and the ontology of expressions. To appear in Synthese.Google Scholar
  37. 37.
    Paris, J.B., & Wilkie, A. (1987). On the scheme of induction for bounded formulas. Annals of Pure and Applied Logic, 35, 261–302.CrossRefGoogle Scholar
  38. 38.
    Pudlák, P. (1985). Cuts, consistency statements and interpretations. The Journal of Symbolic Logic, 50, 423–441.CrossRefGoogle Scholar
  39. 39.
    Shapiro, S. (1998). Truth and Proof: through thick and thin. Journal of Philosophy.Google Scholar
  40. 40.
    Simpson, S. (2009). Subsystems of Second-Order Arithmetic. Second edition: Cambridge University press–ASL.CrossRefGoogle Scholar
  41. 41.
    Tarski, A. (1956a). The concept of truth in formalized languages. In Woodger, H.J. (Ed.), Logic, Semantic, Metamathematics: papers of Alfred Tarski from, 1922-1938. (pp. 152–278). Oxford: Clarendon Press.Google Scholar
  42. 42.
    Tennant, N. (2002). Deflationism and the Gödel phenomena. Mind, 111, 551–82.CrossRefGoogle Scholar
  43. 43.
    Vaught, R.A. (1967). Axiomatizability by a schema. The Journal of Symbolic Logic, 32, 473–479.CrossRefGoogle Scholar
  44. 44.
    Visser, A. (1991). The formalization of interpretability. Studia Logica, 50(1), 81–106.CrossRefGoogle Scholar
  45. 45.
    Visser, A. (2009). Can we make the second incompleteness theorem coordinate free?. Journal of Logic and Computation, 21(4), 543–560.CrossRefGoogle Scholar
  46. 46.
    Visser, A. (2009). The predicative frege hierarchy. Annals of Pure and Applied Logic 3c, 160, 129–153.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Faculty of PhilosophyUniversity of OxfordOxfordEngland

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