Abstract
The assignment of probabilities is the most established way of measuring uncertainties on a quantitative scale. In the framework of subjective probability, the probabilities are interpreted as someone’s (the agent’s) degrees of belief. Since justified belief amounts to knowledge, the assignment of probabilities, in as much as it can be justified, expresses knowledge. Indeed, knowledge of probabilities, appears to be the basic kind of knowledge that is provided by the experimental sciences today.
A part of this paper has been included in a talk given in a NSF symposium on foundations of probability and causality, organized by W. Harper and B. Skyrms at UC Irvine, July 1985. I wish to thank the organizers for the opportunity to discuss and clarify some of these ideas.
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Notes
- 1.
My Salzburg paper (1983) has been devoted to these questions. The upshot of the analysis there has been that even a “purely subjective” probability implies a kind of factual claim, for one can asses its success in the actual world. Rather than two different kinds, subjective and objective probabilities are better to be regarded as two extremes of a spectrum.
- 2.
It is important to restrict C in Axiom (VI) to an intersection of such events. The removal of this restriction will cause the p x ’s to be two-valued functions, meaning that all facts are known in the maximal knowledge states.
- 3.
I am thankful to my colleagues at the Hebrew University H. Furstenberg, I. Katzenelson and B. Weiss for their help in this item. Needless to say that errors, if any, are my sole responsibility.
- 4.
Actually there are 51 real numbers α n such that the event N = n is the same as PR(H ≤ 1/2, α n ).
- 5.
They are seperable, i.e., for some countably generated field every event in the space differs from a set in the field by a 0-set.
- 6.
The list is far from being complete. Some important papers not mentioned in the abstract are to be found in Ifs, W. Harper ed. Boston Reidel, 1981. Material related in a less direct way: non-probabilistic measures of certainty (e.g., the Dempster-Shafer measure), expert systems involving reasoning with uncertainties, probabilistic protocols and distributed systems, has not been included, but should be included in a fuller exposition.
References
The list is far from being complete. Some important papers not mentioned in the abstract are to be found in Ifs, W. Harper ed. Boston Reidel, 1981. Material related in a less direct way: non-probabilistic measures of certainty (e.g., the Dempster-Shafer measure), expert systems involving reasoning with uncertainties, probabilistic protocols and distributed systems, has not been included, but should be included in a fuller exposition.
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Gaifman, H. (2016). A Theory of Higher Order Probabilities. In: Arló-Costa, H., Hendricks, V., van Benthem, J. (eds) Readings in Formal Epistemology. Springer Graduate Texts in Philosophy, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-20451-2_6
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