Abstract
The principle of indifference has fallen from grace in contemporary philosophy, yet some papers have recently sought to vindicate its plausibility. This paper follows suit. In it, I articulate a version of the principle and provide what appears to be a novel argument in favour of it. The argument relies on a thought experiment where, intuitively, an agent’s confidence in any particular outcome being true should decrease with the addition of outcomes to the relevant space of possible outcomes. Put simply: the greater the number of outcomes, the weaker your confidence should be in any one of those outcomes. The argument holds that this intuition favours the principle of indifference. I point out that, in contrast, the intuition is also incompatible with a major alternative to the principle which prescribes imprecise credences: the so-called wide interval view. Consequently, the argument may also be seen as an argument against the wide interval view.
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Notes
For example, White (2010) and Kaplan (1996) liberally use the terms “objective chance” or “objective probability” without any conceptual analysis of them, and perhaps rightly so. In Sect. 3.1, I will discuss issues involved in understanding objective chances and what it means to have evidence about them.
I have chosen the term “stochastic ignorance” for the concept since already established nomenclature for it is lacking and the concept denotes ignorance of the kind of probabilities that concern what Howson and Urbach call a type of “stochastic (random or chance-like) experiment” (Howson and Urbach 2006, 15, emphasis in the original).
Given that the subjective interpretation of probabilities as credences dominates formal epistemology, I sometimes use the terms “probability” and “credence” interchangeably in this paper.
See this terminology in, for instance, Rinard (2013).
Rinard claims that Walley (1991), Joyce (2005), and Weatherson (2007) are also of the opinion that, roughly speaking, “in a case of no evidence, rationality requires credence to be spread over [0, 1]” (Rinard 2013, 157). However, a thorough discussion of their opinions is beyond the scope of this paper.
One might think of the “space of possible outcomes” in terms of a standard sample space which contains all possible combinations of outcomes that are a priori constructible in a given first-order language (including those specifying that the ball is black and the ball is white). While the argument could be made with this notion of a space of possible outcomes, this is not the way I am thinking of a space of possible outcomes. Instead, I use the term to refer to the set of outcomes which are consistent with what the agent knows, and later on, I shall talk of two spaces of possible outcomes: one for the colour of one ball and one for the colour of another.
I believe this argument is probably novel—or at the very least, not widely-known—for several reasons. I could not find an argument of this sort when searching the literature. This might be explained by the fact that the imprecise credence view which it bears on is somewhat of a newcomer to philosophy, perhaps only emerging in the last 25 years. Furthermore, twenty-one academics have given feedback on the argument (all of whom either were reviewers of earlier versions of this paper or were formal epistemologists I spoke with in person), and most of them did not object to its novelty. However, seven academics did have concerns about its originality. Some thought the argument was similar to White (2010) and Rinard’s (2013) arguments. But, as I have argued in various places in this paper, there are specific reasons why their arguments are substantially different to that of this paper. Others instead had qualms about the originality of the argument, not necessarily because it had been discussed in the literature, but because its key points may seem quite simple or obvious to people regardless. For example, one thought that the argument’s key intuition is an immediate consequence of the principle and consequently may have been inexplicitly understood by others as a consideration in favour of the principle. Regardless, it still looks like the literature lacks an explicit discussion of the extent to which this intuition supports the principle of indifference over the imprecise view. This discussion is not found, for example, in White’s (2010) review of arguments for the principle of indifference, nor in his argument against the wide-interval view in that same paper. Yet it seems to me that such a discussion is warranted, at the very least because some accept the value and force of the intuition while others object to it—that is, if the feedback I have received is anything to go by. Consequently, I have tried to rectify this situation by explicitly defending the force of the intuition and by addressing various objections later in the paper. In any case, it may be that the main argument of this paper is not obvious at first, although it may strike one as such after it has been pointed out; after all, it was not obvious to me at first, even after reading the relevant literature. Regardless, it is possible that the literature already contains an argument of this sort, but that is the case with virtually any new argument. So I am inclined to treat the novelty of the argument as innocent until proven guilty.
Of course, this is not a pragmatic argument for the principle of indifference. It is an argument for the principle as an epistemically rational constraint on credences. But the point is that pragmatic considerations (about which choice to make) reflect deeper epistemic intuitions about how confident we should be that one ball is black compared to the other.
This, however, is if the number of outcomes is finite. Since the restricted principle of indifference only concerns finitely many outcomes, I have nothing to say about infinite sets of outcomes.
To give one example of another objection I have not discussed here, one critic claimed that we could try to save the wide interval view by adopting an “overall confidence level” which is a function of the functions in the representor and which somehow assigns a low degree of confidence to the outcomes. I have a response to this objection and others in a longer version of this paper. There, my response to this objection is that having a sharp and low overall confidence level is inconsistent with the motivation for the wide interval view—namely, to have no particularly committal confidence level at all.
See a similar response discussed in White (2010).
This would be a probability assignment in conformity with the principle of the narrowest reference class. For more on the preference for narrower or more specific reference classes, see Thorn (2017).
Of course, one might think that the principal principle (Lewis 1980), or something like it, is one such alternative principle—at least if one defines objective chances so as to include the number of outcomes in a uniquely correct partition when no other evidence favours one outcome over another. Given this definition, the principle would prescribe that we match our credences to the chances qua the proportions of outcomes where a given proposition is true. I have not seen anyone in the literature who would adopt such a definition and consequently think that the principal principle applies as such, but even if they did so, I argued in Sect. 3.1 that this paper’s argument would still achieve its main purpose: showing that sharp and symmetrical credences are warranted in cases where others (notably Kaplan) think they are not. But granted, there are relevant background questions about how to define objective chances and the principle of indifference, although, as mentioned in Sect. 3.1, these are not the focus of this paper.
More specifically, other problems arise from the possibility that the open interval response is ad hoc, less simple and invokes a representor of functions which could not adequately represent any actual credal state since it’s unclear as to what it would mean to have a credal state of confidence that conforms to the view (unlike precise probabilities and the representor of the wide interval view).
In particular, if a probability less than 0.1 is assigned to the proposition that ball2 is some particular colour, then a probability higher than 0.1 must be assigned to ball2 being some other particular colour. Otherwise, the sum of the probabilities for the ten colours would be less than 1, thereby violating the probability calculus. Then, this assignment will be higher than the probability that ball1 is black, thus contradicting the general intuition of the thought experiment: you should be less confident that ball2 is any particular colour of its ten possible colours in comparison to the two possible colours for ball1.
For a concise and insightful discussion of the programme, see O’Hagan et al. (2006, 31–52).
In saying that, I suspect that the restricted principle of indifference is more gradationally accurate than any competing candidate principles, and that there may be evolutionary reasons why humans have intuitions which accord with the principle’s prescriptions. However, I do not have strong evidence to support this suspicion which could also justify trusting the intuitions of the thought experiment.
The reader may notice that these next two objections are general objections to the principle of indifference rather than specifically to this paper’s argument itself. Nevertheless, I discuss these objections here since a few commentators on the argument have raised them, essentially claiming that the argument is insignificant if it is ultimately for a principle that could not plausibly address these obvious challenges.
A critic may attempt to level an objection against this example on the basis of difficulties in characterising and partitioning colour predicates. Nevertheless, we can suppose realistically that in this circumstance, the flowers obviously fall under exactly one of the colour categories such that Nancy would prefer one set of flowers over the other, but Josh lacks evidence to discern what her preference would be.
For example, here’s one candidate: a partition of n propositions \(\left\{ {p_{i} , \ldots ,p_{n} } \right\}\) is uniquely correct iff (1) the most fine grained a posteriori knowledge or evidence one has about the truth of any proposition in \(\left\{ {p_{i} , \ldots ,p_{n} } \right\}\) is that exactly one of the n propositions is true and (2) all other a posteriori knowledge about the truth of any such proposition is implied by this fine grained knowledge. This is similar to Keynes’s proposal that the elements of the partition be “indivisible” in the sense that they cannot be “further split up” into alternative elements (Keynes 2010 [1921], 67–68); this much is captured with the clause that the partition be fine-grained. However, the clause differs in that it accords a privileged role to a posteriori knowledge of the possible outcomes rather than regarding it on par with a priori knowledge of possible outcomes. In any case, I am sure this candidate is not the final word and is probably problematic, although a whole paper could be devoted to exploring whether any other alternative fares better.
As mentioned, Meacham also appeals to the concepts of inverse time and inverse distance to argue against the principle of indifference and its application in two particular cases. Unfortunately, I am not knowledgeable about the relevant physics. However, I have discussed Meacham's paper with Brendon Brewer, a statistician and physicist, and he informs me that most, if not all, physicists would probably find Meacham’s utilisation of the concepts quite unintuitive. This is because they would be inclined to give ontological priority to more familiar concepts of time and space which accord with the way that physicists think of physical symmetry. Nonetheless, I think that the above response to Meacham’s worries suffices irrespective of the physics; and presumably an in-depth analysis of his appeal to physics would take me well out of the domain of formal epistemology.
Thorn (2017) is an example of someone who rejects the principle of indifference which concerns abstract proportions while accepting a frequency-credence principle that concerns concrete proportions.
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Acknowledgements
I would like to thank various people for their helpful comments on the ideas in this paper. They include Brendon Brewer, Ray Briggs, Nick DiBella, Anthony Eagle, Marc Fischer, Alan Hájek, Susanna Rinard, Michael Strevens and some anonymous reviewers.
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Wilcox, J.E. An Argument for the Principle of Indifference and Against the Wide Interval View. J Gen Philos Sci 51, 65–87 (2020). https://doi.org/10.1007/s10838-019-09488-0
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DOI: https://doi.org/10.1007/s10838-019-09488-0