Abstract
This chapter covers the epistemic or information-based interpretations of probability: logical, subjective, objective Bayesian, and group level. It explains how these differ from aleatory or world-based interpretations of probability, presents each in detail, and then discusses its strengths and weaknesses.
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Notes
- 1.
Shafer and Vovk [44] argue that we should not begin by understanding probability in a measure-theoretic way, but instead in a game-theoretic way. As such, their interpretative strategy is different from those considered here (although the Dutch Book argument, which we will cover in due course, is game-theoretic in nature).
- 2.
I call the two kinds of probability ‘world-based’ and ‘information-based’ in Rowbottom [35]; I think this is better terminology, but it’s non-standard.
- 3.
There is also a ‘classical interpretation’, which predates the logical one. On the classical view, probabilities are defined (roughly) in terms of the ratio of favourable outcomes to possible outcomes. The problem with this view is that it appears to require that each outcome be equipossible. So it could not handle a biased coin; e.g. in calculating the probability of heads on one flip, we’d always arrive at one half. (And someone might very well take the coin landing on its edge to be possible, and thereby be forced to conclude that the probability of heads was a third.) For more on the classical interpretation, see Gillies [13] and Rowbottom [35].
- 4.
How to connect reasons for belief and entailment is much more complicated than it may first appear. See, for example, Streumer [45].
- 5.
In the words of De Finetti ([7], p. 23): ‘For any proposed interpretation of Probability, a proper operational definition must be worked out: that is, a device apt to measure it must be constructed.’ One could argue with this, of course, but it seems odd to want to posit a kind of probability that isn’t generally measurable! What purpose would it serve?
- 6.
Keynes also believed in non-numerical probabilities, which complicates matters further, but we can put this to one side for present purposes.
- 7.
This was earlier called ‘the principle of non-sufficient reason’, and goes back (although not under the same name) to Bernoulli [2].
- 8.
- 9.
In the words of Ramsey ([27], pp. 72–73): ‘the kind of measurement of belief with which probability is concerned is…of belief qua basis of action…with beliefs like my belief that the earth is round…which would guide my action in any case to which it was relevant.’ Ramsey’s example seems somewhat odd, however, because it seems that all conceivable beliefs can be relevant to action in appropriate cases. For example, I could be asked what I believe about some obscure philosophical issue and desire to express the truth. An asseveration I made in response would be guided by that belief.
- 10.
The following statement, for example, is misleading: ‘only subjective probabilities exist—i.e. the degree of belief in the occurrence of an event attributed by a given person at a given instant with a given set of information.’ (De Finetti [8], pp. 3–4)
- 11.
A good place to read more about Dutch Books is Hájek [17].
- 12.
Note, in particular, the comment about ascribing numbers to intensities of feeling. The idea that people can have precise degrees of belief corresponding to any rational number between 0 and 1—or perhaps beyond, if we’re discussing degrees of belief that don’t satisfy the probability calculus—is clearly an idealisation. It’s more realistic to think that they lie in particular intervals. There is a related literature on imprecise probabilities, and in fact the idea of working with intervals was discussed at considerable length by Keynes [21]. For more on the notion, which is growing in popularity, see http://www.sipta.org and Bradley [4].
- 13.
For example, I argue that Keynes did hold that observed frequencies should constrain our degrees of belief, or at least that his interpretation could easily accommodate this idea. I also dispute the view that the principle of indifference is not as well motivated as the maximum entropy principle.
- 14.
Williamson would presumably insist that the sample space in this case should be the latter. However, when the sample spaces are continuous, e.g. in the paradox of Bertrand [3], he thinks that it is allowable to equivocate on the basis of different sample spaces. In short, the idea is that the sample space to use is not clearly specified in the way the problem is set up.
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Acknowledgements
Work on this chapter was supported by a General Research Fund grant, ‘Computational Social Epistemology and Scientific Method’ (#341413), from Hong Kong’s Research Grants Council. Thanks to Teddy Seidenfeld, Glenn Shafer, and the editors for comments on earlier versions.
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Rowbottom, D.P. (2018). Probability Theory. In: Hansson, S., Hendricks, V. (eds) Introduction to Formal Philosophy. Springer Undergraduate Texts in Philosophy. Springer, Cham. https://doi.org/10.1007/978-3-319-77434-3_21
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