Abstract
A possible event always seems to be more probable than an impossible event. Although this constraint, usually alluded to as regularity, is prima facie very attractive, it cannot hold for standard probabilities. Moreover, in a recent paper Timothy Williamson has challenged even the idea that regularity can be integrated into a comparative conception of probability by showing that the standard comparative axioms conflict with certain cases if regularity is assumed. In this note, we suggest that there is a natural weakening of the standard comparative axioms. It is shown that these axioms are consistent both with the regularity condition and with the essential feature of Williamson’s example.
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Notes
The regularity condition has already been proposed by Finetti (1964: 100).
Generally, we will presuppose a subjective conception of probability and possibility. But most of what is said will apply mutatis mutandis to objective kinds of probability and possibility.
It should be noted that the regularity constraint can be satisfied in a setting with countably many possible outcomes if the outcomes are not equiprobable. Yet as soon as there are uncountably many possible outcomes, the regularity constraint becomes unsatisfiable.
Of course, standard probabilities will remain a valuable tool for modeling various other important phenomena.
Cf. Williamson (2007).
For such multiple runnings of an experiment involving infinitely many tasks to be possible, the tasks need to be performed at an accelerating and converging time scale so that a coin is tossed infinitely many times within a finite amount of time. For instance, the coin may be tossed for the first time at point 0, for the second time at 1/2, for the third time at 2/3, and so on and so forth; the n-th toss of the coin occurring at 1 − 1/n.
Recall that \({{\mathcal{A}}}\) is called a Boolean algebra on \(\Upomega\) if \({{\mathcal{A}}}\) is a set of subsets of \(\Upomega\) which is closed under union and complementation and which contains \(\Upomega. \)
Adopting the nonstrict relation ≤ as basic, the relations ≈ (‘equiprobable’) and < (‘less probable than’) can be defined in the usual way: \(A \approx B \quad :\Leftrightarrow \quad A \leq B \wedge B \leq A\quad \hbox{ and} \quad A<B \quad :\Leftrightarrow \quad A\leq B \wedge B \nleq A.\)
Sometimes transitivity is doubted, but we would like to think of the axioms as stating epistemic norms rather than psychological facts.
In set-theoretical terms: \({\leq^* := \leq \setminus \left( \{(A,\emptyset): A \in {\mathcal{A}} \wedge A \neq \emptyset \} \cup \{(\Upomega,A): A \in {\mathcal{A}} \wedge A \neq \Upomega \} \right).}\)
If ≤ is already regular, we have ≤ * = ≤ .
If one were to require a comparative probability relation to be linear and/or non-trivial, the class of comparative probability relations would still be closed under regularization.
This is obvious, since the regularization agrees with the original relation on the relevant cases.
The operation of regularization can be extended to the standard case, where probability distributions \({P: {\mathcal{A}} \to [0,1]}\) are considered. One would expand the image of P by adding two new elements s < 0 and 1 < t. Then a new function P * could be defined which differs from P only in mapping \(\emptyset \) to s and \(\Upomega\) to t. The induced quasi-odering of P * would then be the regularization of the quasi-ordering induced by P. Similarly to the comparative case, and as one would expect anyway, the new function does not satisfy the numerical version of additivity: if \({A \in {\mathcal{A}}^*, }\) then \(t = P^*(A \cup \overline{A}) \neq P^*(A) + P^*(\overline{A}) = 1\). On the other hand, its restriction to \(\mathcal{A}^*\) coincides with P’s restriction to \({{\mathcal{A}}^*, }\) and, as a standard probability distribution, it maps \(\emptyset \) to the smallest and \(\Upomega\) to the largest value in the linearly ordered value space. To the extent that one takes additivity to be an essential feature of numerical probability, it might be debatable whether one is willing to call P * a measure of probability.
References
Finetti, B. de (1964). Foresight: lts logical laws, its subjective sources. In Kyburg, H., & Smokler, H. (Eds.), Studies in subjective probability (pp. 93–158). Huntington, NY: Krieger.
Fishburn, P.C. (1986). The axioms of subjective probability. Statistical Science, 1, 335–345.
Weintraub, R. (2008). How probable is an infinite sequence of heads? A reply to Williamson. Analysis, 68, 247–50.
Williamson, T. (2007). How probable is an infinite sequence of heads?. Analysis, 67, 173–80.
Acknowledgments
We would like to thank the other members of the Phlox research group, Miguel Hoeltje, Benjamin Schnieder and Alexander Steinberg, for very helpful comments and support. Many thanks are also due to two anonymous referees of this journal and the participants of Phloxshop I, Berlin 2008, at which an earlier version of the present paper was presented. Special thanks to Timothy Williamson whose comments on multiple drafts of this paper have led to significant improvements. Finally, research for this paper has profited from the generous support of the Deutsche Forschungsgemeinschaft (DFG).
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Haverkamp, N., Schulz, M. A Note on Comparative Probability. Erkenn 76, 395–402 (2012). https://doi.org/10.1007/s10670-011-9307-x
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DOI: https://doi.org/10.1007/s10670-011-9307-x