Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T08:22:42.942Z Has data issue: false hasContentIssue false
  • English
  • Français

CONSERVATIVITY FOR THEORIES OF COMPOSITIONAL TRUTH VIA CUT ELIMINATION

Published online by Cambridge University Press:  22 July 2015

GRAHAM E. LEIGH
GRAHAM E. LEIGH*
Affiliation:
INSTITUTE OF DISCRETE MATHEMATICS AND GEOMETRY VIENNA UNIVERSITY OF TECHNOLOGY WIEDNER HAUPTSTRAßE 8–10 1040 VIENNA, AUSTRIAE-mail: graham.leigh@tuwien.ac.at
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a cut elimination argument that witnesses the conservativity of the compositional axioms for truth (without the extended induction axiom) over any theory interpreting a weak subsystem of arithmetic. In doing so we also fix a critical error in Halbach’s original presentation. Our methods show that the admission of these axioms determines a hyper-exponential reduction in the size of derivations of truth-free statements.

Keywords

proof theorycut eliminationaxiomatic truthcompositional theory of truth
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 3 , September 2015 , pp. 845 - 865
Copyright
Copyright © The Association for Symbolic Logic 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Barwise, J., Admissible Sets and Structures: An Approach to Definability Theory, Springer–Verlag, Berlin, 1975.CrossRefGoogle Scholar
Enayat, A. and Visser, A., Full satisfaction classes in a general setting (Part 1), unpublished manuscript, 2012.Google Scholar
Enayat, A. and Visser, A., New constructions of satisfaction classes, Unifying the philosophy of truth (Achourioti, T., Galinon, H., Fujimoto, K. and Martínez-Fernández, J., editors), in press.Google Scholar
Feferman, S., Reflecting on incompleteness, this Journal, vol. 56 (1991), pp. 1–49.Google Scholar
Fischer, M., Minimal truth and interpretability. Review of Symbolic Logic, vol. 2 (2009), no. 4, pp. 799–815.CrossRefGoogle Scholar
Friedman, H. and Sheard, M., An Axiomatic Approach to Self-Referential Truth. Annals of Pure and Applied Logic, vol. 33 (1987), pp. 1–21.CrossRefGoogle Scholar
Halbach, V., Conservative theories of classical truth. Studia Logica, vol. 62 (1999), pp. 353–370.CrossRefGoogle Scholar
Halbach, V., Axiomatic theories of truth, Cambridge University Press, Cambridge, 2011.CrossRefGoogle Scholar
Halbach, V., Axiomatic theories of truth, second edition. Cambridge University Press, Cambridge, 2014.CrossRefGoogle Scholar
Halbach, V. and Leigh, G. E., Axiomatic theories of truth (Zalta, E. N., editors), The Stanford Encyclopedia of Philosophy (Summer 2014 Edition), url http://plato.stanford.edu/archives/sum2014/entries/truth-axiomatic/.Google Scholar
Krajewski, S., Nonstandard satisfaction classes, Set Theory and Hierarchy Theory: A Memorial Tribute to Andrzej Mostowski (Marek, W. et al. , editors), Lecture Notes in Mathematics, vol. 537, Springer-Verlag, Berlin, 1976, pp. 121–144.CrossRefGoogle Scholar
Kotlarski, H., Bounded Induction and Satisfaction Classes. Zeitschrift für Mathematische Logik, vol. 32 (1986), pp. 531–44.CrossRefGoogle Scholar
Kotlarski, H., Krajewski, S., and Lachlan, A. H., Construction of satisfaction classes for nonstandard models. Canadian Mathematical Bulletin, vol. 24 (1981), pp. 283–93.CrossRefGoogle Scholar
Leigh, G. E., A proof-theoretic account of classical principles of truth. Annals of Pure and Applied Logic, vol. 164 (2013), pp. 1009–1024.CrossRefGoogle Scholar
Leigh, G. E. and Rathjen, M., An ordinal analysis for theories of self-referential truth. Archive for Mathematical Logic, vol. 49 (2010), no. 2, pp. 213–247.CrossRefGoogle Scholar
Leigh, G. E. and Rathjen, M., The Friedman-Sheard programme in intuitionistic logic, this Journal, vol. 77 (2012), no. 3, pp. 777–806.Google Scholar