Skip to main content
SpringerLink
Log in

Truth in Applicative Theories

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

We give a survey on truth theories for applicative theories. It comprises Frege structures, universes for Frege structures, and a theory of supervaluation. We present the proof-theoretic results for these theories and show their syntactical expressive power. In particular, we present as a novelty a syntactical interpretation of ID1 in a applicative truth theory based on supervaluation.

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. P. Aczel, 'The strength of Martin Lóf's intuitionistic type theory with one universe', in S. Miettinen and J. Väänäanen, editors, Proceedings of the Symposiums on Mathematical Logic in Oulo 1974 and in Helsinki 1975, Report No. 2 from the Dept. of Philosophy, University of Helsinki, 1977.

  2. P. Aczel, 'Frege structures and the notion of proposition, truth and set', in J. Barwise, H. Keisler, and K. Kunen, editors, The Kleene Symposium, pages 31-59, North-Holland, 1980.

  3. M. Beeson, Foundations of Constructive Mathematics, Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, Bd. 6, Springer, Berlin, 1985.

  4. W. Buchholz, S. Feferman, W. Pohlers, and W. Sieg, Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical studies, volume 897 of Lecture Notes in Mathematics, Springer, 1981.

  5. A. Cantini, 'Notes on formal theories of truth', Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 35:97-130, 1989.

    Google Scholar 

  6. A. Cantini, 'A theory of formal truth arithmetically equivalent to ID1', Journal of Symbolic Logic 55(1):244-259, 1990.

    Google Scholar 

  7. A. Cantini, 'A logic of abstraction related to finite constructive number classes', Archive for Mathematical Logic, 31:69-83, 1991.

    Google Scholar 

  8. A. Cantini, 'Extending the first-order theory of combinators with selfreferential truth', Journal of Symbolic Logic 58(2), 1993.

  9. A. Cantini, 'Levels of truth', Notre Dame Journal of Formal Logic, 36:185-213, 1995.

    Google Scholar 

  10. A. Cantini, Logical Frameworks for Truth and Abstraction, volume 135 of Studies in Logic and the Foundations of Mathematics, North-Holland, 1996.

  11. S. Feferman, 'A language and axioms for explicit mathematics', in J. Crossley, editor, Algebra and Logic, volume 450 of Lecture Notes in Mathematics, pages 87-139, Springer, 1975.

  12. S. Feferman, 'Constructive theories of functions and classes', in M. Boffa, D. van Dalen, and K. McAloon, editors, Logic Colloquium 78, pages 159-224, North-Holland, Amsterdam, 1979.

    Google Scholar 

  13. S. Feferman, 'Iterated inductive fixed-point theories: Application to Hancock's conjecture', in G. Metakides, editor, Patras Logic Symposion, pages 171-196, North-Holland, Amsterdam, 1982.

    Google Scholar 

  14. S. Feferman, 'Hilbert's program relativized: Proof-theoretical and foundational reductions', Journal of Symbolic Logic 53(2):364-384, 1988.

    Google Scholar 

  15. S. Feferman, 'Reflection on imcompleteness', Journal of Symbolic Logic 57(1):1-49, 1991.

    Google Scholar 

  16. S. Feferman, 'Does reductive proof theory have a viable rationale?', Erkenntnis 53(1-2):63-96, 2000.

    Google Scholar 

  17. S. Feferman, and G. Jäger, 'Systems of explicit mathematics with nonconstructive µ-operator. Part I', Annals of Pure and Applied Logic 65(3):243-263, December 1993.

    Google Scholar 

  18. S. Feferman, and G. Jäger, 'Systems of explicit mathematics with nonconstructive µ-operator. Part II,' Annals of Pure and Applied Logic 79:37-52, 1996.

    Google Scholar 

  19. R. Flagg and J. Myhill, 'An extension of Frege structures', in D. Kueker, E. Lopez-Escobar, and C. Smith, editors, Mathematical Logic and Theoretical Computer Science, pages 197-217, Dekker, 1987.

  20. R. Flagg and J. Myhill, 'Implication and analysis in classical Frege structures', Annals of Pure and Applied Logic 34:33-85, 1987.

    Google Scholar 

  21. H. Friedman and M. Sheard, 'An axiomatic approach to self-referential truth', Annals of Pure and Applied Logic 33:1-21, 1982.

    Google Scholar 

  22. V. Halbach, Axiomatische Wahrheitstheorien, Akademie-Verlag, Berlin, 1996.

    Google Scholar 

  23. F. Hamm, Modelltheoretische Untersuchungen zur Semantik von Nominalisierungen, Habilitionsschrift, Seminar für Sprachwissenschaften, Universität Tübingen, 1999.

  24. S. Hayashi and S. Kobayashi, 'A new formulation of Feferman's system of functions and classes and its relation to Frege structures', International Journal of Foundations of Computer Science 6(3):187-202, 1995.

    Google Scholar 

  25. G. Jãger, 'Zur Beweistheorie der Kripke-Platek-Mengenlehre über den natürlichen Zahlen', Archiv für mathematische Logik und Grundlagenforschung 22:121-139, 1982.

    Google Scholar 

  26. G. Jäger, R. Kahle, and TH. Strahm, 'On applicative theories', in A. Cantini, E. Casari, and P. Minari, editors, Logic and Foundation of Mathematics, pages 88-92, Kluwer, 1999.

  27. G. Jäger, R. Kahle, and TH. Studer,'Universes in explicit mathematics', Annals of Pure and Applied Logic 109(3):141-162, 2001.

    Google Scholar 

  28. G. Jäger, R. Kahle, A. Setzer, and TH. Strahm, 'The proof-theoretic analysis of transfinitely iterated fixed point theories', Journal of Symbolic Logic 64(1):53-67, March 1999.

    Google Scholar 

  29. G. Jäger and TH. Strahm, 'Totality in applicative theories', Annals of Pure and Applied Logic 74:105-120, 1995.

    Google Scholar 

  30. G. Jäger and TH. Strahm, 'Some theories with positive induction of ordinal strength ϕω0', Journal of Symbolic Logic, 61(3):818-842, September 1996.

    Google Scholar 

  31. R. Kahle, Applikative Theorien und Frege-Strukturen, PhD thesis, Institut für Informatik und angewandte Mathematik, Universität Bern, 1997.

  32. R. Kahle, 'Uniform limit in explicit mathematics with universes', Technical Report IAM-97-002, IAM, Universität Bern, März 1997.

  33. R. Kahle, 'Frege structures for partial applicative theories', Journal of Logic and Computation 9(5):683-700, October 1999.

    Google Scholar 

  34. R. Kahle, 'Mathematical proof theory in the light of ordinal analysis', Synthese, 200x, (to appear).

  35. R. Kahle, 'Universes over Frege structures', 200x (submitted).

  36. R. Kahle and TH. Studer, 'A theory of explicit mathematics equivalent with ID1', in P. Clote and H. Schwichtenberg, editors, CSL 2000, volume 1862 of LNCS, pages 356-370, Springer, 2000.

  37. R. Kahle and TH. Studer, 'Formalizing non-termination of recursive programs', Journal of Logic and Algebraic Programming, 200x (to appear).

  38. S. Kripke, 'Outline of a theory of truth', Journal of Philosophy 72:690-716, 1975.

    Google Scholar 

  39. P. Martin-lóf, Intuitionistic Type Theory, Bibliopolis, 1984.

  40. M. Marzetta, 'Universes in the theory of types and names', in E. Bórger et al., editor, Computer Science Logic '92, volume 702 of Lecture Notes in Computer Science, pages 340-351, Springer, 1993.

  41. M. Marzetta, Predicative Theories of Types and Names, PhD thesis, Universität Bern, Institut f¨ur Informatik und angewandte Mathematik, 1994.

  42. W. Pohlers, 'Subsystems of set theory and second order number theory', in S. Buss, editor, Handbook of Proof Theory, chapter IV, pages 209-335, North-Holland, 1998.

  43. K. Schütte, Proof Theory, volume 225 of Grundlehren der mathematischen Wissenschaften, Springer, Berlin, 1977.

    Google Scholar 

  44. D. Scott, 'Combinators and classes', in C. Bóhm, editor, λ-Calculus and Computer Science Theory, volume 37 of Lecture Notes in Computer Science, pages 1-26, Springer, 1975.

  45. J. Smith, On the relation between a type theoretic and a logical formulation of the theory of constructions, PhD thesis, Dept. of Mathematics, University of Góteborg, 1978.

  46. J. Smith, 'An interpretation of Martin-Lóf's type theory in a type-free theory of proposition', Journal of Symbolic Logic 49(3):730-753, 1984.

    Google Scholar 

  47. TH. Strahm, On the proof theory of applicative theories, PhD thesis, Universität Bern, Institut für Informatik und angewandte Mathematik, 1996.

  48. TH. Strahm, 'First steps into metapredicativity in explicit mathematics', in S. Cooper and J. Truss, editors, Sets and Proofs, volume 258 of London Mathematical Society Lecture Note Series, pages 383-402, Cambridge University Press, 1999.

  49. A. Tarski, 'Der Wahrheitsbegriff in den formalisierten Sprachen', Studia Philosophica 1:261-405, 1935. Reprinted in: A. Tarski, Collected Papers, Volume 2, pp. 51-198, Birkhäuser, Basel, 1986. Original version of [Tar56].

  50. A. Tarski, 'The concept of truth in formalized languages', in Logic, Semantics, Metamathematics, pages 152-278, Clarendon Press, Oxford, 1956. Revised and supplemented English translation of [Tar35].

    Google Scholar 

  51. B. van Fraassen, 'Presupposition, implication and self-reference', Journal of Philosophy 65:135-152, 1968.

    Google Scholar 

  52. B. van Fraassen, 'Inference and self-reference', Synthese, 21:425-438, 1970.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. WSI, Universität Tübingen, Sand 13, D-72076, Tübingen, Germany

    Reinhard Kahle

Authors
  1. Reinhard Kahle
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kahle, R. Truth in Applicative Theories. Studia Logica 68, 103–128 (2001). https://doi.org/10.1023/A:1011954206722

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011954206722

Search

Navigation