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AXIOMS FOR DETERMINATENESS AND TRUTH

Published online by Cambridge University Press:  01 August 2008

SOLOMON FEFERMAN*
Affiliation:
Stanford University
*
*DEPARTMENT OF MATHEMATICS STANFORD UNIVERSITY STANFORD, CA 94305-2125 E-mail:sf@csli.stanford.edu

Abstract

A new formal theory DT of truth extending PA is introduced, whose language is that of PA together with one new unary predicate symbol T (x), for truth applied to Gödel numbers of suitable sentences in the extended language. Falsity of x, F(x), is defined as truth of the negation of x; then, the formula D(x) expressing that x is the number of a determinate meaningful sentence is defined as the disjunction of T(x) and F(x). The axioms of DT are those of PA extended by (I) full induction, (II) strong compositionality axioms for D, and (III) the recursive defining axioms for T relative to D. By (II) is meant that a sentence satisfies D if and only if all its parts satisfy D; this holds in a slightly modified form for conditional sentences. The main result is that DT has a standard model. As an improvement over earlier systems developed by the author, DT meets a number of leading criteria for formal theories of truth that have been proposed in the recent literature and comes closer to realizing the informal view that the domain of the truth predicate consists exactly of the determinate meaningful sentences.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2008

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References

BIBLIOGRAPHY

Aczel, P. (1980). Frege structures and the notions of proposition, truth and set. In Barwise, J., Keisler, H. J., and Kunen, K., editors. The Kleene Symposium. Amsterdam, The Netherlands: North-Holland, pp. 3159.Google Scholar
Beeson, M. J. (1985). Foundations of Constructive Mathematics. Berlin: Springer-Verlag.Google Scholar
Cantini, A. (1989). Notes on formal theories of truth. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 35, 97130.Google Scholar
Feferman, S. (1984). Toward useful type-free theories I. Journal of Symbolic Logic, 49, 75111.Google Scholar
Feferman, S. (1991). Reflecting on incompleteness. Journal of Symbolic Logic, 56, 149.Google Scholar
Feferman, S. (1996). Gödel's program for new axioms: why, where, how and what? In Hájek, P., editor. Gödel ’96. Lecture Notes in Logic, 6, 322.Google Scholar
Feferman, S. (2004). Typical ambiguity: trying to have your cake and eat it too. In Link, G., editor. One Hundred Years of Russell's Paradox. Berlin: Walter de Gruyter, pp. 131151.Google Scholar
Feferman, S. (2006). Enriched stratified systems for the foundations of category theory. In Sica, G., editor. What is Category Theory? Milano, Italy: Polimetrica, pp. 185203.Google Scholar
Feferman, S., & Strahm, T. (2000). The unfolding of non-finitist arithmetic. Annals of Pure and Applied Logic, 104, 7596.Google Scholar
Field, H. (2003). A revenge-immune solution to the semantic paradoxes. Journal of Philosophical Logic, 32, 139177.Google Scholar
Gödel, K. (1944). Russell's mathematical logic. In Schilpp, P. A., editor. The Philosophy of Bertrand Russell, Library of Living Philosophers. Evanston, IL: Northwestern, pp. 123–153. Reprinted in Gödel [1990, pp. 119141].Google Scholar
Gödel, K. (1990). Collected Works, Vol. II. Publications 1938–1974 (Feferman, S., et al. , editors). New York, NY: Oxford University Press.Google Scholar
Halbach, V. (2007). Axiomatic theories of truth. In Zalta, E. N., editor. Stanford Encyclopedia of Philosophy. Available from: http://plato.stanford.edu/entries/truth-axiomatic.Google Scholar
Halbach, V., & Horsten, L. (2006). Axiomatizing Kripke's theory of truth. Journal of Symbolic Logic, 71, 677712.Google Scholar
Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72, 690716.Google Scholar
Leitgeb, H. (2007). What theories of truth should be like (but cannot be). Philosophy Compass, 2, 276290.Google Scholar
McDonald, B. E. (2000). On meaningfulness and truth. Journal of Philosophical Logic, 29, 433482.Google Scholar
McGee, V. (1991). Truth, Vagueness, and Paradox: An Essay on the Logic of Truth. Indianapolis, IN: Hackett.Google Scholar
Priest, G. (2002). Paraconsistent logic. In Gabbay, D., and Guenthner, F., editors. Handbook of Philosophical Logic (second edition), Vol. 6. Dordrecht, The Netherlands: Kluwer, pp. 287393.Google Scholar
Reinhardt, W. N. (1985). Remarks on significance and meaningful applicability. In Paulo de Alcantara, L., editor. Mathematical Logic and Formal Systems. Lecture Notes in Pure and Applied Mathematics, 94, 227242.Google Scholar
Reinhardt, W. N. (1986). Some remarks on extending and interpreting theories, with a partial predicate for truth. Journal of Philosophical Logic, 15, 219251.Google Scholar
Russell, B. (1908). Mathematical logic as based on the theory of types. American Journal of Mathematics, 30, 222262. Reprinted in From Frege to Gödel. A Source Book in Mathematical Logic, 1879–1931 (van Heijenoort, J., editor) [1967]. Cambridge, MA: Harvard University Press, pp. 150182.Google Scholar
Sheard, M. (1994). A guide to truth predicates in the modern era. Journal of Symbolic Logic, 59, 10321054.Google Scholar
Sheard, M. (2002). Truth, provability, and naive criteria. In Halbach, V., and Horsten, L., editors. Principles of Truth. Frankfurt a.M.: Dr. Hänsel-Hohenhausen, pp. 169181.Google Scholar