Skip to main content
×
Home
    • Aa
    • Aa

Axiomatizing Kripke's theory of truth

  • Volker Halbach (a1) and Leon Horsten (a2)
Abstract
Abstract

We investigate axiomatizations of Kripke's theory of truth based on the Strong Kleene evaluation scheme for treating sentences lacking a truth value. Feferman's axiomatization KF formulated in classical logic is an indirect approach, because it is not sound with respect to Kripke's semantics in the straightforward sense: only the sentences that can be proved to be true in KF are valid in Kripke's partial models. Reinhardt proposed to focus just on the sentences that can be proved to be true in KF and conjectured that the detour through classical logic in KF is dispensable. We refute Reinhardt's Conjecture, and provide a direct axiomatization PKF of Kripke's theory in partial logic. We argue that any natural axiomatization of Kripke's theory in Strong Kleene logic has the same proof-theoretic strength as PKF. namely the strength of the system ramified analysis or a system of Tarskian ramified truth up to ωω. Thus any such axiomatization is much weaker than Feferman's axiomatization KF in classical logic, which is equivalent to the system of ramified analysis up to ε0.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[5] Andrea Cantini , Notes on formal theories of truth, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 35 (1989), pp. 97130.

[8] John Cleave , The notion of logical consequence in the logic of inexact predicates, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 20 (1974), pp. 307324.

[13] Harvey Friedman and Michael Sheard , An axiomatic approach to self-referential truth, Annals of Pure and Applied Logic, vol. 33 (1987), pp. 121.

[14] Harvey Friedman and Michael Sheard , The disjunction and existence properties for axiomatic systems of truth, Annals of Pure and Applied Logic, vol. 40 (1988), pp. 110.

[18] Volker Halbach , Truth and reduction, Erkenntnis, vol. 53 (2000), pp. 97126.

[22] John Kearns , The strong completeness of a system for Kleene's three-valued logic, Zeitschrift für mathematische Logik und Grundlagen der Mathematik. vol. 25 (1979), pp. 6168.

[23] Michael Kremer , Kripke and the logic of truth, Journal of Philosophical Logic, vol. 17 (1988), pp. 225278.

[24] Saul Kripke , Outline of a theory of truth, Journal of Philosophy, vol. 72 (1975), pp. 690712. reprinted in [25].

[26] Robert L. Martin and Peter W. Woodruff , On representing ‘true-in-L’ in L, Philosophia, vol. 5 (1975), pp. 213217. Reprinted in [25].

[27] Tim Maudlin . Truth and paradox, Solving the riddles, Clarendon Press, Oxford. 2004.

[36] William Reinhardt , Some remarks on extending and interpreting theories with a partial predicate for truth. Journal of Philosophical Logic, vol. 15 (1986), pp. 219251.

[37] Helmut Schwichtenberg , Proof theory: Some applications of cut-elimination. Handbook of mathematical logic ( Jon Barwise . editor), Studies in Logic and the Foundations of Mathematics, no. 90. North-Holland, Amsterdam, London, New York and Tokyo, 1977, pp. 867895.

[38] Dana Scott , Combinators and classes, λ-calculus and computer science ( Corrado Böhm . editor), Lecture Notes in Computer Science, Springer, Berlin, 1975, pp. 126.

[40] Scott Soames , Understanding truth, Oxford University Press, New York and Oxford, 1999.

[44] Hao Wang , The calculus of partial predicates and its extension to set theory, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 7 (1961), pp. 283288.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 9 *
Loading metrics...

Abstract views

Total abstract views: 119 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 26th September 2017. This data will be updated every 24 hours.