Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-31T14:28:20.268Z Has data issue: false hasContentIssue false
  • English
  • Français

PROVING UNPROVABILITY

Published online by Cambridge University Press:  21 November 2016

BRUNO WHITTLE
BRUNO WHITTLE*
Affiliation:
Department of Philosophy, University of Glasgow
*
*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF GLASGOW GLASGOW G12 8QQ, UK E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper addresses the question: given some theory T that we accept, is there some natural, generally applicable way of extending T to a theory S that can prove a range of things about what it itself (i.e., S) can prove, including a range of things about what it cannot prove, such as claims to the effect that it cannot prove certain particular sentences (e.g., 0 = 1), or the claim that it is consistent? Typical characterizations of Gödel’s second incompleteness theorem, and its significance, would lead us to believe that the answer is ‘no’. But the present paper explores a positive answer. The general approach is to follow the lead of recent (and not so recent) approaches to truth and the Liar paradox.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 10 , Issue 1 , March 2017 , pp. 92 - 115
Copyright
Copyright © Association for Symbolic Logic 2016 

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

Artemov, S. N. & Beklemishev, L. D. (2005). Provability logic. In Gabbay, D. M. and Guenthner, F., editors. Handbook of Philosophical Logic, Second Edition, Vol. 13. Dordrecht: Springer, pp. 189–360.Google Scholar
des Rivières, J. & Levesque, H. J. (1988). The consistency of syntactical treatments of knowledge (how to compile quantificational modal logics into classical FOL). Computational Intelligence, 4, 31–41.CrossRefGoogle Scholar
Detlefsen, M. (1986). Hilbert’s Program: An Essay on Mathematical Instrumentalism. Dordrecht: D. Reidel.Google Scholar
Feferman, S. (1990). Introductory note to Gödel 1972. In Gödel 1990, pp. 281–287.Google Scholar
Gaifman, H. (2000a). Pointers to propositions. In Chapuis, A. and Gupta, A., editors. Circularity, Definition, and Truth. New Delhi: Indian Council of Philosophical Research, pp. 79–121.Google Scholar
Gaifman, H. (2000b). What Gödel’s incompleteness result does and does not show. Journal of Philosophy, 97, 46–70.Google Scholar
Gödel, K. (1972). Some remarks on the undecidability results. In Gödel 1990, pp. 305–306.Google Scholar
Gödel, K. (1990). Collected Works: Volume II, Publications 1938–1972. Feferman, S., Dawson, J. W. Jr., Kleene, S. C., Moore, G. H., Solovay, R. M., and van Heijenoort, J., editors. Oxford: Oxford University Press.Google Scholar
Gupta, A. (1982). Truth and paradox. Journal of Philosophical Logic, 11, 1–60.Google Scholar
Gupta, A. & Belnap, N. (1993). The Revision Theory of Truth. Cambridge, MA: MIT Press.Google Scholar
Halbach, V., Leitgeb, H., & Welch, P. (2003). Possible-worlds semantics for modal notions conceived as predicates. Journal of Philosophical Logic, 32, 179–223.Google Scholar
Herzberger, H. G. (1982). Notes on naive semantics. Journal of Philosophical Logic, 11, 61–102.Google Scholar
Hughes, G. E. & Cresswell, M. J. (1996). A New Introduction to Modal Logic. London: Routledge.Google Scholar
Japaridze, G. & de Jongh, D. (1998). The logic of provability. In Buss, S. R., editor. Handbook of Proof Theory. Amsterdam: Elsevier, pp. 475–546.CrossRefGoogle Scholar
Kaplan, D. & Montague, R. (1960). A paradox regained. Notre Dame Journal of Formal Logic, 1, 79–90.Google Scholar
Kreisel, G. (1953). On a problem of Henkin’s. Indagationes Mathematicae (Proceedings), 56, 405–406.Google Scholar
Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72, 690–716.Google Scholar
Lewis, D. (1983). New work for a theory of universals. Australasian Journal of Philosophy, 61, 343–377.Google Scholar
Maudlin, T. (2004). Truth and Paradox: Solving the Riddles. Oxford: Clarendon Press.CrossRefGoogle Scholar
Raatikainen, P. (2015). Gödel’s incompleteness theorems. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Spring 2015 Edition). Available at: http://plato.stanford.edu/archives/spr2015/entries/goedel-incompleteness/.Google Scholar
Skyrms, B. (1984). Intensional aspects of self-reference. In Martin, R. L., editor. Recent Essays on Truth and the Liar Paradox. Oxford: Clarendon Press, pp. 119–131.Google Scholar
Smith, P. (2013). An Introduction to Gödel’s Theorems (second edition). Cambridge: Cambridge University Press.Google Scholar
Visser, A. (1989). Peano’s smart children: A provability logical study of systems with built-in consistency. Notre Dame Journal of Formal Logic, 30, 161–196.Google Scholar