Abstract
A probability distribution is regular if it does not assign probability zero to any possible event. Williamson (Analysis, 67, 173–180, 2007) argued that we should not require probabilities to be regular, for if we do, certain “isomorphic” physical events (infinite sequences of coin flip outcomes) must have different probabilities, which is implausible. His remarks suggest an assumption that chances are determined by intrinsic, qualitative circumstances. Weintraub (Analysis, 68, 247–250, 2008) responds that Williamson’s coin flip events differ in their inclusion relations to each other, or the inclusion relations between their times, and this can account for their differences in probability. Haverkamp and Schulz (Erkenntnis, 76, 395–402, 2012) rebut Weintraub, but their rebuttal fails because the events in their example are even less symmetric than Williamson’s. However, Weintraub’s argument also fails, for it ignores the distinction between intrinsic, qualitative differences and relations of time and bare identity. Weintraub could rescue her argument by claiming that the events differ in duration, under a non-standard and problematic conception of duration. However, we can modify Williamson’s example with Special Relativity so that there is no absolute inclusion relation between the times, and neither event has longer duration except relative to certain reference frames. Hence, Weintraub’s responses do not apply unless chance is observer-relative, which is also problematic. Finally, another symmetry argument defeats even the appeal to frame-dependent durations, for there the events have the same finite duration and are entirely disjoint, as are their respective times and places.
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Notes
In general, a probability function assigns probabilities to all sets in some algebra of subsets of a sample space, not all subsets of the sample space. Hájek (Staying regular? http://hplms.berkeley.edu/HajekStayingRegular.pdf, unpublished) takes regularity to imply that all sets of possible outcomes are assigned non-zero probability, rather than non-zero or none at all. We could call Hájek’s regularity strong regularity, and the weaker condition that every non-empty set of outcomes is assigned either non-zero probability or none at all, weak regularity. Hájek argues against strong regularity, but here we are mainly concerned with weak regularity, for, as we will see, the latter is already problematic, provided that a few very simple events do have probabilities.
Proof: Suppose 0 < ε ∈ R. Choose n > log1/2 ε. Write H(m, m + 1,…, n) for the event that flips m, m + 1,…, n all come up heads. Since the flips are fair and independent, Prob(H(1, 2, 3,…, n)) = (½)n < ε. By independence, Prob(H(1, 2, 3,…)) = Prob(H(1, 2, 3,…, n)) × Prob(H(n + 1, n + 2, n + 3,…)). By the normality axiom, Prob(H(n + 1, n + 2, n + 3, …)) < 1. Hence Prob(H(1, 2, 3, …)) < ε × 1 = ε.
Howson continued his critique of Williamson in 2019a and 2019b. His 2019a does not address the arguments of Parker (2019). Howson (2019b) acknowledges a related argument, based on the claim that spacetime invariance is “a fundamental feature of nature”, but Howson then downplays the infinitesimal asymmetries implied by regularity as lying within an empirically “more-than-acceptable margin of error”. We will not pursue this dispute at length here, but a brief remark is warranted: If we are willing to accept an infinitesimal margin of error, then there is no need to introduce infinitesimals. Regularists could instead just accept the classical real-valued theory of probability as empirically close enough. On the contrary, it seems that what regularists about chances want is a better or more accurate theory of chance, at a level of detail that exceeds the empirical discernibility of individual chance values. For that purpose, we must decide whether the principle of spacetime invariance and other symmetry considerations outweigh the arguments for regularity, which even Howson (ibid.) regarded as weak.
Of course, the probability of one being blinded by sunlight is lower at night than during the day, as one referee pointed out, but that is due to a difference in circumstances, not the mere times of the proposed events.
I am speaking loosely here of “the same event” occurring at different times or places or involving different coins. We can speak this way if we regard an event as a class of possible configurations in space and time (including any motions) of matter, energy, fields, or whatever physical entities exist, identified by the intrinsic structure of the configuration rather than the absolute place or time at which it hypothetically occurs or the bare identities of the entities involved. Alternatively we might understand an event as a class of spacetime configurations in a particular place or time, with specific samples of matter or what have you, so that it makes no sense to speak of the same event occurring in a different place or time or with a different coin. But in that case we can instead speak of qualitatively similar events at different times and places. The present point is just that, on Williamson’s view, perfectly similar events in different times and places or involving numerically distinct but perfectly similar coins should have the same chance. For further clarification of Williamson’s notion of an event, see Parker, 2019.
In particular, this does not assume countable additivity. Regularists often introduce infinitesimal probabilities in order to obtain regular, uniform distributions over infinite sample spaces, and these infinitesimal probabilities are not countably additive. Since regularists are already willing to sacrifice countable additivity, Parker, 2019 does not take it for granted, and nor will we here.
This is essentially the argument for IP′ in Parker, 2019. Of course, it is conceivable that the laws of nature might depend on haecceitistic properties or lack the relevant spacetime invariance, but it is at least plausible that they need not. To accept such capricious “laws” as fundamental would amount to saying that the way things behave, including chances, varies for no underlying reason, and it would be hard to accept such a system as the best possible. In any case, we have had considerable scientific success with spacetime invariant, qualitative laws so far and can plausibly continue to do so.
Note also that Arntzenius and Hall (2003) argue, contra Lewis, that uniform dependence on qualitative properties is a compelling requirement of chance and shows that the Principal Principle is not all we know about it. According to Schaffer (2007, footnote 17), Lewis himself considered accepting this argument.
The fact that Lewis himself favoured regular, hyperreal probabilities is no counter-argument, since Lewis was not aware of Williamson’s later argument and, to the best of my knowledge, never commented on any symmetry arguments against regularity (but see the preceding note regarding his response to Arntzenius & Hall, 2003). Lewis apparently thought that the best system would involve regular, hyperreal chances, but that is a point that Williamson’s argument calls into question.
Thanks to an anonymous referee for raising this point.
Bottazzi and Katz (2020, 2021) have argued against such arbitrariness claims in the context of Robinson- or Nelson-style non-standard analysis. They do not appear to have resolved all arbitrariness worries, and some of their arguments seem to be countered already in Barrett, 2010, but we cannot take that up here.
Weintraub made this remark several years ago in an informal context. She should not be held accountable for it, but the suggestion it makes is important to consider here.
Actually, we could say this about Williamson’s original story as well. H(2…) is after all an outcome of a second run of qualitatively the same experiment with the very same coin as that in H(1…). It just happens also that the experiment of H(1…) properly includes that of H(2…), which gives Weintraub her foothold to argue that the events are physically different. Nonetheless, it is still a repeat of (qualitatively) the same experiment with (numerically) the same device.
Pruss (2021b) gives a similar mathematical example (p. 9) and an enlightening general theorem (p. 8), but does not take up questions of inclusion, intersection, or physical examples.
We can think of this interval either as an abstract sample space, or as a physical interval in space or time. In the applications mentioned below, we can use the abstract mathematical interval [0, 1) to represent the physical one, or the physical interval itself can serve as the relevant sample space, provided space and time are continuous.
More formally, let T be a translation Tx = x + c. Suppose A, TA ⊆ [0, 1) and Prob(A) ≠ Prob(TA). Choose n ∈ N so that 1/n < c. For each whole number i < n, let Ai = A ∩ [i/n, (i + 1)/n). Then Ai and TAi are disjoint, and by finite additivity, ∑i ∈ {0, 1,..., n – 1}Prob(Ai) = Prob(A) ≠ Prob(TA) = ∑i ∈ {0, 1,..., n – 1}Prob(TAi). So for at least one i, Prob(Ai) ≠ Prob(TAi). Hence there is a set Ai and a disjoint translation of Ai that differ in probability.
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Acknowledgements
I would like to thank Dimitrios Athanasiou, Mark Baker, Kenny Easwaran, Craig Fox, Marie Gueguen, Niels Linnemann, Yichen Luo, Chris Smeenk, the audience at PSA2018, and three anonymous referees for helpful comments and discussion. This work was partly supported by the John Templeton Foundation under grant #61048.
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Partly supported by the John Templeton Foundation under grant #61048.
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This paper is dedicated to the memory of Colin Howson, a brilliant and cheerful scholar.
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Parker, M.W. Weintraub’s response to Williamson’s coin flip argument. Euro Jnl Phil Sci 11, 71 (2021). https://doi.org/10.1007/s13194-021-00389-y
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DOI: https://doi.org/10.1007/s13194-021-00389-y