Infinite Lotteries, Perfectly Thin Darts and Infinitesimals

One of the problems that Bayesian regularity, the thesis that all contingent propositions should be given probabilities strictly between zero and one, faces is the possibility of random processes that randomly and uniformly choose a number between zero and one. According to classical probability theory, the probability that such a process picks a particular number in the range is zero, but of course any number in the range can indeed be picked. There is a solution to this particular problem on the books: a measure that assigns the same infinitesimal probability to each number between zero and one. I will show that such a measure, while mathematically interesting, is pathological for use in confirmation theory, for the same reason that a measure that assigns an infinitesimal probability to each possible outcome in a countably infinite lottery is pathological. The pathology is that one can force someone to assign a probability within an infinitesimal of one to an unlikely event
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DOI 10.1002/tht3.13
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References found in this work BETA
I. J. Good (1967). On the Principle of Total Evidence. British Journal for the Philosophy of Science 17 (4):319-321.

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Citations of this work BETA
Gordon Belot (forthcoming). Curve-Fitting for Bayesians? British Journal for the Philosophy of Science:axv061.

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Yaroslav Sergeyev (2010). Lagrange Lecture: Methodology of Numerical Computations with Infinities and Infinitesimals. Rendiconti Del Seminario Matematico dell'Università E Del Politecnico di Torino 68 (2):95–113.

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