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1. Williamson, J. In defence of objective Bayesianism; Oxford University Press: Oxford, UK, 2010.
2. There are several alternatives to the objective Bayesian account of strength of belief, including subjective Bayesianism, imprecise probability, the theory of DempsterShafer belief functions and related theories. Here we only have the space to motivate objective Bayesianism, not to assess these other views.
3. Taking the convex hull may mean that a calibrated belief function does not satisfy the known constraints on physical probability. For example, if is known to be a statement about the past then it is known that its physical probability is 0 or 1; bel is not constrained to be 0 or 1, however, unless it is also known whether or not is true. Similarly, it may be known that two propositions are probabilistically independent with respect to physical probability; this need not imply that they are probabilistically independent with respect to epistemic probability. See Williamson [1] (pp. 4445) for further discussion of this point.
4. Landes, J.; Williamson, J. Objective Bayesianism and the Maximum Entropy Principle. Entropy 2013, 15, 35283591.
5. Gaifman, H. Concerning Measures in First Order Calculi. Isr. J. Math. 1964, 2, 118.
6. Williamson, J. Lectures on Inductive Logic; Oxford University Press: Oxford, UK, 2015.
7. Paris, J.B. The Uncertain Reasoner's Companion; Cambridge University Press: Cambridge, UK, 1994.
8. Williamson, J. Probability logic. In Handbook of the Logic of Argument and Inference: the Turn toward the Practical; Gabbay, D., Johnson, R., Ohlbach, H.J., Woods, J., Eds.; Elsevier: Amsterdam, The Netherlands, 2002; pp. 397424.
9. Haenni, R.; Romeijn, J.W.; Wheeler, G.; Williamson, J. Probabilistic Logics and Probabilistic Networks; Synthese Library, Springer: Dordrecht, The Netherlands, 2011.
10. Cover, T.M.; Thomas, J.A. Elements of Information Theory; John Wiley and Sons: New York, NY, USA, 1991.