Hostname: page-component-6bf8c574d5-b4m5d Total loading time: 0 Render date: 2025-03-01T12:27:04.489Z Has data issue: false hasContentIssue false
  • English
  • Français

MODELS OF POSITIVE TRUTH

Published online by Cambridge University Press:  26 December 2018

MATEUSZ ŁEŁYK  and
BARTOSZ WCISŁO
MATEUSZ ŁEŁYK*
Affiliation:
Institute of Philosophy, University of Warsaw
BARTOSZ WCISŁO*
Affiliation:
Institute of Mathematics, University of Warsaw
*
*INSTITUTE OF PHILOSOPHY UNIVERSITY OF WARSAW WARSAW 00–927, POLAND E-mail: mlelyk@uw.edu.pl
†INSTITUTE OF MATHEMATICS UNIVERSITY OF WARSAW WARSAW 00–927, POLAND E-mail: b.wcislo@mimuw.edu.pl
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is a follow-up to [4], in which a mistake in [6] (which spread also to [9]) was corrected. We give a strenghtening of the main result on the semantical nonconservativity of the theory of PT− with internal induction for total formulae ${(\rm{P}}{{\rm{T}}^ - } + {\rm{INT}}\left( {{\rm{tot}}} \right)$, denoted by PT− in [9]). We show that if to PT− the axiom of internal induction for all arithmetical formulae is added (giving ${\rm{P}}{{\rm{T}}^ - } + {\rm{INT}}$), then this theory is semantically stronger than ${\rm{P}}{{\rm{T}}^ - } + {\rm{INT}}\left( {{\rm{tot}}} \right)$. In particular the latter is not relatively truth definable (in the sense of [11]) in the former. Last but not least, we provide an axiomatic theory of truth which meets the requirements put forward by Fischer and Horsten in [9]. The truth theory we define is based on Weak Kleene Logic instead of the Strong one.

Keywords

03H15axiomatic theories of truthconservativitynonstandard models of arithmeticfinite axiomatizabilityspeed-up
Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 1 , March 2019 , pp. 144 - 172
Copyright
Copyright © Association for Symbolic Logic 2018 

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

BIBLIOGRAPHY

Cantini, A. (1990). A theory of formal truth as strong as ID1. The Journal of Symbolic Logic, 55(1), 244–259.CrossRefGoogle Scholar
Cieliski, C. (2015). The innocence of truth. Dialectica, 69(1), 61–85.Google Scholar
Cieśliński, C. (2015). Typed and untyped disquotational truth. In Achourioti, T., Galinon, H., Fernández, J. M., and Fujimoto, K., editors. Unifying the Philosophy of Truth. Dordrecht: Springer-Verlag, pp. 307–320.CrossRefGoogle Scholar
Cieśliński, C., Lelyk, M., & Wcislo, B. (2017). Models of PT− with internal induction for total formulae. The Review of Symbolic Logic, 10(1), 187–202.CrossRefGoogle Scholar
Feferman, S. (1991). Reflecting on incompleteness. Journal of Symbolic Logic, 56(1), 1–49.CrossRefGoogle Scholar
Fischer, M. (2009). Minimal truth and interpretability. Review of Symbolic Logic, 2, 799–815.CrossRefGoogle Scholar
Fischer, M. (2014). Truth and speed-up. Review of Symbolic Logic, 7, 319–340.CrossRefGoogle Scholar
Fischer, M. (2015). Deflationism and instrumentalism. In Fujimoto, K., Fernández, J. M., Galinon, H., and Achourioti, T., editors. Unifying the Philosophy of Truth. Dordrecht: Springer, pp. 293–306.CrossRefGoogle Scholar
Fischer, M. & Horsten, L. (2015). The expressive power of truth. Review of Symbolic Logic, 8, 345–369.CrossRefGoogle Scholar
Fitting, M. (1994). Kleene’s three valued logics and their children. Fundamenta Informaticae, 20(1–3), 113–131.Google Scholar
Fujimoto, K. (2010). Relative truth definability in axiomatic truth theories. Bulletin of Symbolic Logic, 16, 305–344.CrossRefGoogle Scholar
Halbach, V. (2011). Axiomatic Theories of Truth. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Halbach, V. & Horsten, L. (2006). Axiomatizing kripke’s theory of truth. Journal of Symbolic Logic, 71(2), 677–712.CrossRefGoogle Scholar
Halbach, V. & Leigh, G. E. (2014). Axiomatic theories of truth. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Summer 2014 edition). Available at https://plato.stanford.edu/entries/truth-axiomatic/.Google Scholar
Halbach, V. & Nicolai, C. (2018). On the costs of nonclassical logic. Journal of Philosophical Logic, 47(2), 227–257.CrossRefGoogle Scholar
Horsten, L. (1995). The semantical paradoxes, the neutrality of truth and the neutrality of the minimalist theory of truth. In Cartois, P., editor. The Many Problems of Realism (Studies in the General Philosophy of Science: Volume 3). Tilburg: Tilburg University Press, pp. 173–187.Google Scholar
Kaufmann, M. (1977). A rather classless model. Proceedings of American Mathematical Society, 62(2), 330–333.CrossRefGoogle Scholar
Kaye, R. (1991). Models of Peano Arithmetic. Oxford, UK: Clarendon Press.Google Scholar
Ketland, J. (1999). Deflationism and tarski’s paradise. Mind, 108, 69–94.CrossRefGoogle Scholar
Kripke, S. (1975). Outline of a theory of truth. The Journal of Philosophy, 72(19), 690–716.CrossRefGoogle Scholar
Łełyk, M. & Wcisło, B. (2017). Models of weak theories of truth. Archive for Mathematical Logic, 56(5), 1–26. doi:10.1007/s00153-017-0531-1.CrossRefGoogle Scholar
Mostowski, A. (1950). Some impredicative definitions in the axiomatic set theory. Fundamenta Mathematicae, 37(1), 111–124.CrossRefGoogle Scholar
Schindler, R. ACA0 and ${\rm{\Pi }}_1^1 - {\rm{C}}{{\rm{A}}_0}$ and the semantics of arithmetic and BG and ,${\rm{BG}} + {\rm{\Sigma }}_1^1 - {\rm{Ind}}$, and the semantics of set theory. Unpublished manuscript. Available at www.math.uni-muenster.de/logik/Personen/rds/.Google Scholar
Shapiro, S. (1998). Proof and truth: Through thick and thin. Jornal of Philosophy, 95(10), 493–521.Google Scholar
Shelah, S. (1978). Models with second order properties II. Trees with no undefined branches. Annals of Mathematical Logic, 14, 73–87.CrossRefGoogle Scholar
Smoryński, C. (1981). Recursively saturated nonstandard models of arithmetic. The Journal of Symbolic Logic, 46(2), 259–286.CrossRefGoogle Scholar