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HOW TO EXPRESS SELF-REFERENTIAL PROBABILITY. A KRIPKEAN PROPOSAL

Published online by Cambridge University Press:  30 April 2015

CATRIN CAMPBELL-MOORE*
Affiliation:
Munich Center for Mathematical Philosophy, Ludwig-Maximilians-Universität München
*
*MUNICH CENTER FOR MATHEMATICAL PHILOSOPHY LUDWIG-MAXIMILIANS-UNIVERSITÄT MÜNCHEN E-mail:catrin@ccampbell-moore.com

Abstract

We present a semantics for a language that includes sentences that can talk about their own probabilities. This semantics applies a fixed point construction to possible world style structures. One feature of the construction is that some sentences only have their probability given as a range of values. We develop a corresponding axiomatic theory and show by a canonical model construction that it is complete in the presence of the ω-rule. By considering this semantics we argue that principles such as introspection, which lead to paradoxical contradictions if naively formulated, should be expressed by using a truth predicate to do the job of quotation and disquotation and observe that in the case of introspection the principle is then consistent.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2015 

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References

BIBLIOGRAPHY

Aumann, R. J. (1999). Interactive epistemology II: Probability. International Journal of Game Theory, 28(3), 301314.CrossRefGoogle Scholar
Bacchus, F. (1990). Lp, a logic for representing and reasoning with statistical knowledge. Computational Intelligence, 6(4), 209231.CrossRefGoogle Scholar
Caie, M. (2011). Paradox and Belief. Berkeley: University of California. Unpublished doctoral dissertation.Google Scholar
Caie, M. (2013). Rational probabilistic incoherence. Philosophical Review, 122(4), 527575.Google Scholar
Caie, M. (2014). Calibration and probabilism. Ergo, 1, 1338.Google Scholar
Campbell-Moore, C. (2015). Rational probabilistic incoherence? A reply to Michael Caie. Philosophical Review, 124(3).CrossRefGoogle Scholar
Chang, C., & Keisler, H. (1990). Model Theory. Amsterdam, The Netherlands: Elsevier Science. Retrieved from http://books.google.de/books?id=uiHq0EmaFp0C.Google Scholar
Christiano, P., Yudkowsky, E., Herresho, M., & Barasz, M. (n.d.). Definability of Truth in Probabilistic Logic, early draft. Retrieved from https://intelligence.org/files/DefinabilityTruthDraft.pdf (Accessed June 10, 2013).Google Scholar
Fagin, R., Halpern, J. Y., & Megiddo, N. (1990). A logic for reasoning about probabilities. Information and computation, 87(1), 78128.Google Scholar
Fischer, M., Halbach, V., Kriener, J., & Stern, J. (2015, 2). Axiomatizing semantic theories of truth? The Review of Symbolic Logic, FirstView, 122. Retrieved from http://journals.cambridge.org/article_S1755020314000379 doi: 10.1017/S1755020314000379Google Scholar
Goldblatt, R. (2014). The countable Henkin principle. In Manzano, M., Sain, I., and Alonso, E., editors. The Life and Work of Leon Henkin. Birkhäuser Basel: Springer, pp. 179201.Google Scholar
Halbach, V. (2014). Axiomatic Theories of Truth (revised edition). Cambridge University Press.Google Scholar
Halbach, V., Leitgeb, H., & Welch, P. (2003). Possible-worlds semantics for modal notions conceived as Predicates. Journal of Philosophical Logic, 32, 179222.Google Scholar
Halbach, V., & Welch, P. (2009). Necessities and necessary truths: A prolegomenon to the use of modal logic in the analysis of intensional notions. Mind, 118(469), 71100.Google Scholar
Harsanyi, J. C. (1967). Games with incomplete information played by bayesian players, I-III part I. The basic model. Management Science, 14(3), 159182.CrossRefGoogle Scholar
Heifetz, A., & Mongin, P. (2001). Probability logic for type spaces. Games and economic behavior, 35(1), 3153.Google Scholar
Kripke, S. (1975). Outline of a theory of truth. The journal of philosophy, 72(19), 690716.CrossRefGoogle Scholar
Leitgeb, H. (2008). On the probabilistic convention T. The Review of Symbolic Logic, 1(2), 218224.Google Scholar
Leitgeb, H. (2012). From type-free truth to type-free probability. In Restall, G. and Russel, G., editors. New Waves in Philosophical Logic. New York: Palgrave Macmillan, pp. 8494.CrossRefGoogle Scholar
McGee, V. (1985). How truthlike can a predicate be? A negative result. Journal of Philosophical Logic, 14(4), 399410.Google Scholar
Ognjanović, Z., & Rašković, M. (1996). A logic with higher order probabilities. Publications de l’Institut Mathématique. Nouvelle Série, 60, 14.Google Scholar
Skyrms, B. (1980). Higher order degrees of belief. In Ramsey, F. P. and Melor, D. H., editors. Prospects for Pragmatism. Cambridge, UK: Cambridge University Press, pp. 109137.Google Scholar
Stern, J. (2014a). Modality and axiomatic theories of truth I: Friedman-Sheard. The Review of Symbolic Logic, 7(2), 273298.Google Scholar
Stern, J. (2014b). Modality and axiomatic theories of truth II: Kripke-Feferman. The Review of Symbolic Logic, 7(2), 299318.Google Scholar
Weaver, G. (1992). Unifying some modifications of the Henkin construction. Notre Dame journal of formal logic, 33(3), 450460.Google Scholar
Williams, J. R. G. (in press). Probability and non-classical logic. In Hitchcock, C. and Hájek, A., editors. Oxford Handbook of Probability and Philosophy. Oxford, UK: Oxford University Press.Google Scholar
Zhou, C. (2013). Belief functions on distributive lattices. Artificial Intelligence, 201(0), 131. Retrieved from http://www.sciencedirect.com/science/article/pii/S000437021300043X, doi: http://dx.doi.org/10.1016/j.artint.2013.05.003.Google Scholar