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AXIOMATIC TRUTH, SYNTAX AND METATHEORETIC REASONING

Published online by Cambridge University Press:  28 August 2013

GRAHAM E. LEIGH  and
CARLO NICOLAI
GRAHAM E. LEIGH*
Affiliation:
Faculty of Philosophy, University of Oxford
CARLO NICOLAI*
Affiliation:
Somerville College, University of Oxford
*
*FACULTY OF PHILOSOPHY OXFORD, OX2 6GG, UK E-mail: [email protected]
†FACULTY OF PHILOSOPHY OXFORD, OX2 6GG, UK E-mail: [email protected]
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Abstract

Following recent developments in the literature on axiomatic theories of truth, we investigate an alternative to the widespread habit of formalizing the syntax of the object-language into the object-language itself. We first argue for the proposed revision, elaborating philosophical evidences in favor of it. Secondly, we present a general framework for axiomatic theories of truth with ‘disentangled’ theories of syntax. Different choices of the object theory O will be considered. Moreover, some strengthenings of these theories will be introduced: we will consider extending the theories by the addition of coding axioms or by extending the schemas of O, if present, to the entire vocabulary of our theory of truth. Finally, we touch on the philosophical consequences that the theories described can have on the debate about the metaphysical status of the truth predicate and on the formalization of our informal metatheoretic reasoning.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 6 , Issue 4 , December 2013 , pp. 613 - 636
Copyright
Copyright © Association for Symbolic Logic 2013 

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References

BIBLIOGRAPHY

Barwise, J. (1975). Admissible Sets and Structures. Berlin: SpringerGoogle Scholar
Burgess, J., & Rosen, G. (1997). A Subject With No Object. Strategies for a Nominalistic Interpretation of Mathematics. Oxford: Clarendon Press.Google Scholar
Craig, W., & Vaught, W. (1958). Finite axiomatisability using additional predicates. The Journal of Symbolic Logic, 23, 289–308.Google Scholar
Enderton, H. (2001). A Mathematical Introduction to Logic (second edition). New York: Harcourt/Academic Press.Google Scholar
Feferman, S. (1960). Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae, 49, 35–91.CrossRefGoogle Scholar
Feferman, S. (1982). Inductively presented systems and the formalization of metamathematics. In Van Dalen, D., Lascar, D., and Smiley, T. J., editors. Logic Colloquium ’80. Amsterdam: North-Holland, pp. 95–128.Google Scholar
Feferman, S. (1989). Finitary inductively presented logics. In Ferro, R., Bonotto, C., Valentini, S., and Zanardo, A., editors. Logic Colloquium 1988. Amsterdam: North-Holland, pp. 191–220.Google Scholar
Feferman, S. (1991). Reflecting on incompleteness. Journal of Symbolic Logic, 56, 1–49.Google Scholar
Field, H. (1999). Deflating the conservativeness argument. Journal of Philosophy, 96(10), 533–540.Google Scholar
Ketland, J. (1999). Deflationism and Tarski’s Paradise. Mind, 108(429), 69–94.Google Scholar
Fujimoto, K. (2012). Classes and truths in set theory. Annals of Pure and Applied Logic 164 (11), 1484–1523.CrossRefGoogle Scholar
Hájek, P., & Pudlák, P. (1993). Metamathematics of First-Order Arithmetic. Berlin: Springer.Google Scholar
Halbach, V. (2001). How innocent is deflationism? Synthese,126(1/2), 167–194.Google Scholar
Halbach, V. (2009). Axiomatic Theories of Truth. In Zalta, Edward N., editor. The Stanford Encyclopedia of Philosophy (Winter 2009 Edition). Available from: http://plato.stanford.edu/archives/win2009/entries/truth-axiomatic/.Google Scholar
Halbach, V. (2011). Axiomatic Theories of Truth. Cambridge: Cambridge University Press.Google Scholar
Heck, R. (2009). The Strength of Truth Theories. Unpublished manuscript. Available from: http://rgheck.frege.org/pdf/unpublished/TruthTheories.pdf.Google Scholar
Hofweber, T. (2000). Proof-theoretic reduction as a philosopher’s tool. Erkenntnis, 53, 127–146.Google Scholar
Horsten, L. (1995). The semantical paradoxes, the neutrality of truth and the neutrality of the minimalist theory of truth. In Cortois, 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
Horsten, L. (2011). The Tarskian Turn: Deflationism and Axiomatic Truth. Cambridge, MA: MIT Press.Google Scholar
Lavine, S. (1999). Skolem was wrong. Unpublished manuscript.Google Scholar
Mancosu, P. (1991). Generalizing classical and effective model theory in theories of operations and classes. Annals of Pure and Applied Logic 52, 249–308.Google Scholar
Manzano, M. (1996). Extensions of First-Order Logic. Cambridge: Cambridge University Press.Google Scholar
McGee, V. (1997). How we learn mathematical language The Philosophical Review 106, (1), 35–68.Google Scholar
Nicolai, C. (forthcoming). Axiomatic Truth, Syntax and Deflationism. Dphil Thesis, Oxford.Google Scholar
Niebergall, K. G. (2011). Mereology. In Horsten, L. & Pettigrew, R., editors. Continuum Companion to Philosophical Logic. London: Continuum.Google Scholar
Quine, W. V. (1946). Concatenation as a basis for arithmetic. Journal of Symbolic Logic,11(4), 105–114.CrossRefGoogle Scholar
Shapiro, S. (1998). Truth and proof: through thick and thin. The Journal of Philosophy, 95, 493–521.Google Scholar
Shapiro, S. (2002). Deflation and conservation. In Halbach, V. & Horsten, L.Principles of Truth. Frankfurt: Dr. Hänsel-Hohenhausen.Google Scholar
Simpson, S. (2009). Subsystems of Second Order Arithmetic (second edition). Cambridge: Cambridge University Press.Google Scholar
Smoryński, C. (1977). The incompleteness theorems. In Barwise, J., editor. Handbook of Mathematical Logic. Amsterdam: North-Holland.Google Scholar
Tarski, A. (1936). The concept of truth in formalized languages. In Woodger, H. J. (translator and editor), Logic, Semantic, Metamathematics:Papers of Alfred Tarski From 1922-1938. Oxford: Clarendon Press, 1956, pp. 152–278.Google Scholar
Tarski, A. (1944). The semantic conception of truth. Philosophical and Phenomenological Research 4, 341–375.CrossRefGoogle Scholar
Troelstra, A. S., & Schwichtenberg, H. (2000). Basic Proof Theory. Cambridge tracts intheoretical computer science, no. 43 (second edition). Cambridge: Cambridge University Press.Google Scholar
Visser, A. (2009). Growing commas. A Study of sequentiality and concatenation, Notre Dame Journal of Formal Logic,50(1), 61–85.Google Scholar