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Published online by Cambridge University Press: 02 January 2020
A starting point for this paper is that there is at least one concept of probability, call it epistemic probability, which can be identified with belief or some sort of idealised belief (e.g., rational belief). If this identification is to be of any significance, then it needs to be shown that epistemic probability is a ‘true’ probability concept and is subject to those restrictions and requirements which relate and govern probabilities, which we call the probability calculus.
The most rehearsed argument to establish the probability calculus for epistemic probabilities is the Dutch Book Argument (DBA). There are two intuitions behind the DBA. The first is that if we can find some fine-grained behavioural measure of epistemic probability, then we may be able to show that epistemic probabilities obey the probability calculus by showing that the behaviour is of a kind which is, as a matter of necessity, subject to certain limitations and restrictions.
1 The form of the DBA presented is basically the same as that used by Jackson, F. and Pargetter, R. in ‘A Modified Dutch Book Argument,’ Philosophical Studies 29 (1976) 403–7.CrossRefGoogle Scholar
2 It is not essential to the argument that the bets involved concern propositions. If, like David Lewis for example, we reject the view that propositions are the uniform objects of belief and hold instead that all belief involves the ascription of some property to oneself (or the self-ascription of some property), then the DBA would involve bets for a person, X, concerning whether X himself has some property. Such a change would not affect the soundness of our argument. Lewis’ view is found in ‘Attitudes De Dicto and De Se,’ The Philosophical Review 88 (1979) 513-43.
3 Baillie, Patricia ‘Confirmation and the Dutch Book Argument,’ British Journal for the Philosophy of Science 24 (1973) 396.CrossRefGoogle Scholar
4 See, for instance, B. de Finetti ‘Foresight: Its Logical Laws, Its Subjective Sources’ in Kyburg, Henry E. Jr. and Smokier, Howard E. eds., Studies in Subjective Probability, Wiley, 1964, 102.Google Scholar
5 See Baillie, 395
6 Jackson and Pargetter use this example.
7 This is claimed, for instance, in de Finetti.
8 This is a slightly modified definition to that given by Jackson and Pargetter, but is essentially in the same spirit as their definition.
9 Len O'Neill and John Bigelow showed us other examples of essentially the same kind which made us modify our suggested additional condition for a CBS. We are grateful to them for this.
10 Kennedy, Ralph and Chihara, Charles (‘The Dutch Book Argument: Its Logical Flaws, Its Subjective Sources,’ Philosophical Studies, 36 (1979) 19–33)CrossRefGoogle Scholar give another example of circumstances in which, they claim, it would be rational for a person to set his FBQs different from his degrees of belief. Suppose X is in a CBS and is required to set FBQs for propositions p and rvp; and X knows that either p or its negation is a logical truth but he is not sure which of these it is. Thus X's degree of belief concerning p (p) ≠ 1 or 0. However unless he specifies 0 for one of these bets and 1 for the other he can have a Dutch Book made against him. This difficulty too is met by the further restriction, (v), on a CBS. The problem only arises if Y has knowledge which X lacks at the time the direction of the bet is specified, namely, whether the proposition is logically true or logically false. Without such knowledge at that time, Y cannot ensure that X loses.
11 Kennedy and Chihara, 26
12 This statement of the universalisability principle used by Jackson and Pargetter is taken from Kennedy and Chihara, 29.
13 Ellis, Brian ‘The Logic of Subjective Probability,’ British Journal for the Philosophy of Science 24 (1973) 125–52,CrossRefGoogle Scholar takes such cases to present a problem for the DBA as presented here.
14 This strategy is adopted by Jackson and Pargetter.
15 Ellis, in ‘The Logic of Subjective Probability,’ claims that such problems show that, for the DBA to succeed, it must be founded on a notion of rationality on which it is irrational to undertake a course of action which might involve a nett loss and cannot involve a nett gain.
16 We are indebted to many invaluable comments and suggestions by Frank Jackson.
