Abstract
The Principle of Indifference (POI) was once regarded as a linchpin of probabilistic reasoning, but has now fallen into disrepute as a result of the so-called problem of multiple of partitions. In ‘Evidential symmetry and mushy credence’ Roger White suggests that we have been too quick to jettison this principle and argues that the problem of multiple partitions rests on a mistake. In this paper I will criticise White’s attempt to revive POI. In so doing, I will argue that what underlies the problem of multiple partitions is a fundamental tension between POI and the very idea of evidential incomparability.
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Notes
Novack (2010) objects to the transitivity of evidential symmetry by appealing to the apparent failure of the transitivity of indistinguishability. Suppose I’m shown a sequence of colour samples ranging from, say, red to blue, such that adjacent samples are pairwise indistinguishable and one of the samples matches the colour of a getaway car that I saw speeding away from a robbery. If I’m shown any two adjacent samples, my evidence, plausibly, no more supports the proposition that the first matches the colour of the car than the proposition that the second matches the colour of the car, since I can’t tell them apart. But if we consider the proposition that the first (red) colour in the sequence matches the colour of the car and the proposition that the last (blue) colour in the sequence matches the colour of the car, my evidence could well support one of these more strongly than the other. Novack attributes this example to Branden Fitelson. White also briefly considers this kind of objection to Transitivity and attributes it both to Fitelson and to Elliot Sober. Whatever we make of this example, though, it seems to be of limited value in responding to White’s argument. After all, White does not require that Transitivity hold universally—merely that it hold in the square factory and in other cases like it. And the present considerations don’t give us any obvious reason to question that.
A representor is often required to be convex—that is, closed under the operation of taking weighted averages of probability functions. More formally, a representor Γ is convex just in case for every Prx ∈ Γ and Pry ∈ Γ, (αPrx + (1 − α)Pry) ∈ Γ, for all real numbers α, 0 ≤ α ≤ 1. The weighted average of two probability functions is a function that assigns to each proposition the weighted average of the two values assigned by the two functions. As can be clearly seen, the values assigned to a proposition by the functions in a convex representor must themselves be closed under the taking of weighted averages and, thus, form a real interval.
Orthodox Bayesians famously demand that the only way to rationally update one’s credence function is by conditionalising on new evidence—if one receives evidence E and no further evidence and Cr and CrE are one’s credence functions before and after the receipt of this evidence then, provided Cr(E) > 0 and one is rational, CrE(P) = Cr(P | E) = Cr(P ∧ E)/Cr(E) for any P. This rule has an obvious analogue within the mushy credence model that we might adopt—the only way to rationally update one’s representor is by conditionalising every function therein on new evidence. If, however, one is in a state of total agnosticism about a proposition P then, given certain assumptions, conditionalising on a proposition E that is assigned a positive value by every function in the representor will leave the distribution of values assigned to P untouched—that is, it will leave one in a state of total agnosticism about P. If one is in a state of total agnosticism about P then there is a function in one’s representor that assigns P a probability of 0. This function will continue to assign P a probability of 0 even after being conditionalised on E. If one is in a state of total agnosticism about P then there is a function in one’s representor that assigns P a probability of 1. This function will continue to assign P a probability of 1 even after being conditionalised on E. The convexity constraint will then ensure that the functions in one’s representor continue to assign all real values in the unit interval to P. This would appear to make agnosticism into a very unappealing prospect—a state that, once entered into, may never be rationally escaped. For this reason, those who adopt the mushy credence model—particularly those who adopt it as a way of doing justice to the attitude of agnosticism—often use alternative, more permissive, update rules (see for instance Weatherson 2007).
If one wishes to impose a convexity constraint upon representors, then we could simply consider the closure of {Pr1, Pr2} under the taking of weighted averages. As can be easily checked, moving to this representor will alter none of the above verdicts.
What if my representor included some functions that were indifferent with respect to side length and some functions that were indifferent with respect to area and no functions of any other kind? In this case I wouldn’t count as being totally agnostic about the side length and area of the plate. My credence in L1 for instance would be {0.25, 0.5} while my credence in L2 would be {0.5, 0.75}. Could I rationally adopt such a credal state in the square factory case? One concern about this representor is that it would violate the convexity constraint mentioned in footnote 3. But even if we put aside such worries, it’s not clear that such a representor would be a legitimate response to the evidence I have available. After all, I have no reason to think that the squares produced by this factory must be evenly distributed with respect to side length or with respect to area or, indeed, with respect to any other parameter that might strike me as being natural. As far as my evidence is concerned, the squares produced by the factory could have any distribution whatever, provided none of them have side lengths in excess of two feet. I won’t explore this further here.
This argument doesn’t force one to admit the existence of evidential incomparability. One might, for instance, insist that the proposition that I’ll draw a black marble is more strongly supported than the proposition that I’ll draw a white-7 marble, on the grounds that white-7 is a more precise, circumscribed colour than black—almost as though there’s some objective ‘colour space’ in which black occupies a larger region that white-7. This is not an incoherent line of thought—but it does strike me as difficult to maintain. Even if we can make sense of an objective colour space, the fact remains that I have no reason at all to think that the distribution of marbles in the urn should, in any way, conform to it and, thus, no reason at all to think that it has any bearing on the question of what colour marble I’ll draw.
If counterexamples to the transitivity of indistinguishability do serve as counterexamples to the transitivity of evidential symmetry, then they will presumably also serve as counterexamples to the transitivity of equal evidential support. If the proposition that the first of two adjacent colour samples matches the colour of the car is evidentially symmetric with the proposition that the second of two adjacent colour samples matches the colour of the car, then this would have to be because they are equally strongly supported—they don’t look to be evidentially incomparable. The transitivity of equal evidential support could perhaps be preserved, in the face of such cases, if we are prepared to give up on the idea that discrepancies in evidential support, no matter how small, must always be distinguishable. I won’t pursue this further here.
References
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Joyce, J. (2005). How probabilities reflect evidence. Philosophical Perspectives, 19, 153–178.
Novack, G. (2010). A defense of the principle of indifference. Journal of Philosophical Logic, 39, 655–678.
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Van Fraassen, B. (1989). Laws and symmetry. Oxford: Oxford University Press.
Weatherson, B. (2007). The Bayesian and the dogmatist. Proceedings of the Aristotelian Society, 107, 169–185.
White, R. (2010). Evidential symmetry and mushy credence. In T. Szabo Gendler & J. Hawthorne (Eds.), Oxford studies in epistemology (Vol. 3). New York: Oxford University Press.
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Smith, M. Evidential Incomparability and the Principle of Indifference. Erkenn 80, 605–616 (2015). https://doi.org/10.1007/s10670-014-9665-2
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DOI: https://doi.org/10.1007/s10670-014-9665-2