Abstract
In his seminal work, A Subjectivist’s Guide to Objective Chance, David Lewis considered the possibility for a subjectivist to accept the existence of objective chance, and argued how this acceptance would affect the subjectivist’s degrees of belief (credences) about the world: they have to satisfy the Principal Principle. Lewis did not put his proposal into mathematically precise terms. Most importantly, he did not define what kind of mathematical object objective chance was. In this work we pay careful attention to identify what mathematical model the subjectivist agent considers, and through several simple examples we illustrate how objective chance can be incorporated in the subjectivist’s model in a mathematically rigorous manner.
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- 1.
That is, closed under finite union and complementation.
- 2.
Recall that a function \(f: W \rightarrow \mathbb{R}\) is measurable with respect to \(\Sigma \ \sigma\)-algebra on W if and only if for all intervals \(I \subseteq \mathbb{R}\), the set \(f^{-1}(I):=\{ w \in W: f(w) \in I\} \in \Sigma \). In case \(\Sigma \) is atomic (finite spaces are always such), measurable functions are those that are constant on the atoms of \(\Sigma \).
- 3.
Hence we strongly disagree with the treatments of some other authors such as Pettigrew (1986) on page 6.
References
Bana, G. 2016. On the formal consistency of the principal principle. Philosophy of Science 83(5): 988–1001. University of Chicago Press.
Lewis, D. 1980. A subjectivist’s guide to objective chance. In Studies in inductive logic and probability, ed. R.C. Jeffrey, vol. II, 263–293. Berkely: University of California Press.
Lewis, D. 1994. Humean supervenience debugged. Mind 103(412): 473–490. Oxford University Press.
Pettigrew, R. 2013. What chance-credence norms should not be. Noûs 73(3): 177–196. Wiley.
Rédei, M., and Z. Gyenis. 2016. Measure theoretic analysis of consistency of the principal principle. Philosophy of Science 83(5): 972–987. University of Chicago Press.
Acknowledgements
I would like to thank to the organizers of the Budapest-Krakow Workshop on Probability, Causality and Determinism for granting me the opportunity to present my ideas about objective chance and the Principal Principle. I would also like to thank Leszek Wroński, Miklós Rédei, Zalán Gyenis, and Balázs Gyenis for the inspiring discussions and their insightful comments.
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Bana, G. (2017). Models of Objective Chance: An Analysis through Examples. In: Hofer-Szabó, G., Wroński, L. (eds) Making it Formally Explicit. European Studies in Philosophy of Science, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-55486-0_3
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DOI: https://doi.org/10.1007/978-3-319-55486-0_3
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