Abstract
This paper explores De Finetti’s generalized versions of Dutch Book and Accuracy Domination theorems. Following proposals due to Jeff Paris, we construe these as underpinning a generalized probabilism appropriate to belief states against either a classical or a non-classical background. Both results are straightforward corollaries of the separating hyperplane theorem; their geometrical relationship is examined. It is shown that each point of Accuracy Domination for b induces a Dutch Book on b; but Dutch Books may need to be ‘scaled’ in order to find a point of Accuracy-Domination. Finally, diachronic Dutch Book defences of conditionalization are examined in the general setting. The formulation and limitations of the generalized conditionalization this delivers are examined.
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References
Border, K. C. (2009). Separating hyperplane theorems. http://www.hss.caltech.edu/ kcb/Notes/SeparatingHyperplane.pdf. Accessed 14 September 2010.
Choquet, G. (1953). Theory of capacities. Annales D’institute Fourier, 5, 131–295.
De Finetti, B. (1974). Theory of probability (Vol. 1). New York: Wiley.
Field, H. H. (2000). Indeterminacy, degree of belief, and excluded middle. Nous, 34, 1–30. Reprinted in field, Truth and the absence of fact (pp. 278–311). Oxford University Press, 2001.
Fine, K. (1975). Vagueness, truth and logic. Synthese, 30, 265–300. Reprinted with corrections in Keefe and Smith (Eds.), Vagueness: A reader (pp. 119–150). Cambridge: MIT Press, 1997.
Greaves, H., & Wallace, D. (2006). Justifying conditionalization: Conditionalization maximizes expected epistemic utility. Mind, 115(459), 607–632.
Hájek, A. (2008). Arguments for—or against—probablism?. British Journal for the Philosophy of Science, 59(4), 783–819.
Hájek, P. (1998). Metamathematics of fuzzy logic. Springer.
Halpern, J. Y. (1995). Reasoning about uncertainty. Revised edn. MIT Press. Revised paperback edition published 2003.
Jaffray, J.-Y. (1989). Coherent bets under partially resolving uncertainty and belief functions. Theory and Decision, 26, 90–105.
Joyce, J. M. (1998). A non-pragmatic vindication of probabilism. Philosophy of Science, 65, 575–603.
Joyce, J. M. (2009). Accuracy and coherence: prospects for an alethic epistemology of partial belief. In F. Huber, & C. Schmidt-Petri (Eds.), Degrees of belief (pp. 263–297). Springer.
Keefe, R. (2000). Theories of vagueness. Cambridge: Cambridge University Press.
Leitgeb, H., & Pettigrew, R. (2010). An objective justification of Bayesianism II: The consequences of minimizing inaccuracy. Philosophy of Science, 77(2), 236–272.
Lewis, D. K. (1999). Why conditionalize? Papers on metaphysics and epistemology (pp. 403–7). Cambridge: Cambridge University Press. Written in 1972; introduction dated 1997. First published in this collection.
Paris, J. B. (2001). A note on the Dutch Book method. In Proceedings of the second international symposium on imprecise probabilities and their applications, Isipta (pp. 301–306). Ithaca: Shaker.
Shafer, G. (1976). A mathematical theory of evidence. Princeton: Princeton University Press.
Skyrms, B. (2006). Diachronic coherence and radical probabilism. Philosophy of Science, 73, 959–96.
Teller, P. (1973). Conditionalization and observation. Synthese, 26(2), 218–258.
van Fraassen, B. (1966). Singular terms, truth-value gaps, and free logic. The journal of Philosophy, 63(17), 481–495.
Williams, J. R. G. (2011). Gradational accuracy and non-classical semantics. http://philpapers.org/rec/WILGAA-2. Accessed 7 May 2011.
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Williams, J.R.G. Generalized Probabilism: Dutch Books and Accuracy Domination. J Philos Logic 41, 811–840 (2012). https://doi.org/10.1007/s10992-011-9192-4
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DOI: https://doi.org/10.1007/s10992-011-9192-4