Abstract
There are concepts of belief on different scales of measurement. In particular, it is common practice to ascribe beliefs to a person both in terms of a categorical (all-or-nothing) concept of belief and in terms of a numerical concept of degree of belief; the formal structure of categorical belief being the subject of doxastic logic, the formal structure of degrees of belief being the topic of subjective probability theory. How do these two kinds of belief relate to each other? We derive an answer to this question from one basic norm: rational categorical belief ought to be a simplified version of subjective probability, where the corresponding concept of simplification can be made mathematically precise in terms of minimizing sums of errors of the result of approximating the probability of a proposition by means of belief or disbelief in the proposition. As it turns out, essentially (glossing over a couple of details) the answer to our question is: a rational person’s set of doxastically accessible worlds must have a stably high probability with respect the person’s subjective probability measure.
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Notes
- 1.
In a different paper (H. Leitgeb, unpublished [5]), we approximate subjective probability in terms of plausibility orders of possible worlds, that is, by belief on an ordinal scale; in this way, also the dynamical aspects of belief can be taken into account. (The method of approximating probability by belief in Leitgeb, unpublished, is completely different from the one that will be employed in the next section.) Unfortunately, we will not be able to deal with this in the present chapter—sorry for remaining on the static side here, Johan!
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Leitgeb, H. (2014). Belief as a Simplification of Probability, and What This Entails. In: Baltag, A., Smets, S. (eds) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-06025-5_14
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