Abstract

The lottery paradox can be solved if epistemic justification is assumed to be a species of permissibility. Given this assumption, the starting point of the paradox can be formulated as the claim that, for each lottery ticket, I am permitted to believe that it will lose. This claim is ambiguous between two readings, depending on the scope of ‘permitted’. On one reading, the claim is false; on another, it is true, but, owing to the general failure of permissibility to agglomerate, does not generate the paradox. The solution generalizes to formulations of the paradox in terms of rational acceptability and doxastic rationality.

The lottery paradox, epistemic justification and permissibility

Suppose that epistemic justification is a species of permissibility. Suppose, that is, that my being justified in believing such-and-such consists in my being epistemically permitted to believe such-and-such.1 As it stands, this characterization of epistemic justification might not sound terribly informative. It does, however, afford a novel solution to the lottery paradox.

Here is a standard presentation of the lottery paradox.2 It seems plausible that, for each ticket in a large and fair lottery, I am justified in believing that the ticket will lose. It seems plausible that it follows from this that I am justified in believing that all tickets will lose. That, however, does not seem plausible at all.

The premiss of the paradox is If justification is a species of permissibility, premiss (1-J) is equivalent to The paradoxical conclusion that seems to follow from (1-J) is Put in terms of permissibility, the conclusion (2-J) is equivalent to The solution to the paradox is that (1-PE) is ambiguous. On one reading, it is false; on another, it is true but does not entail (2-PE).

  • (1-J)

    For each ticket, I am justified in believing that it will lose.

  • (1-PE)

    For each ticket, I am permitted to believe that it will lose.

  • (2-J)

    I am justified in believing that all tickets will lose.

  • (2-PE)

    I am permitted to believe that all tickets will lose.

The ambiguity of (1-PE) can be made perspicuous by formalization. Assume, purely for simplicity, that the lottery involves merely three tickets. Let p, q and r be, respectively, the sentences ‘Ticket #1 will lose’, ‘Ticket #2 will lose’ and ‘Ticket #3 will lose’. Let be the sentence ‘I believe that φ’. Let PEψ be the sentence ‘It is permissible for me that ψ’. Then (1-PE) can be formalized in two ways, depending on the scope of ‘permitted’. First, we can assign narrow scope to ‘permitted’, yielding the claim that I am separately permitted to believe that Ticket #1 will lose, to believe that Ticket #2 will lose and to believe that Ticket #3 will lose: The second reading assigns wide scope to ‘permitted’, yielding the claim that I am permitted to have the three beliefs at once:

  • (NARROW)

    PEBp & PEBq & PEBr

  • (WIDE)

    PE[Bp & Bq & Br]

Claim (WIDE) has paradoxical consequences. For it seems plausible that being permitted to at once hold several beliefs entails being permitted to have a single belief whose content is the conjunction of their contents. This can be expressed formally by the following closure principle: The consequent of (CLOSURE) is the highly implausible claim (2-PE), according to which I am permitted to believe that all tickets will lose.3 So if (1-PE) is read as (WIDE), it generates the paradox together with the plausible principle (CLOSURE). This is a good reason for thinking that (WIDE) is false.

  • (CLOSURE)

     If PE[Bp & Bq & Br], then PEB[p & q & r].

Claim (NARROW), by contrast, does not have these paradoxical consequences. For (NARROW) does not entail (WIDE). It is a common phenomenon that someone is permitted to do this, permitted to do that, etc. without being permitted to do all of these things. For instance, I might be permitted to eat this piece of the cake, permitted to eat that piece of the cake, etc. without being permitted to eat the whole cake. Technically speaking, permissibility does not agglomerate. Given this feature of permissibility in general, it should not come as a surprise that epistemic permissibility does not agglomerate either. That is, it should not come as a surprise that I am permitted to believe that Ticket #1 will lose, permitted to believe that Ticket #2 will lose and permitted to believe that Ticket #3 will lose without being permitted to have the three beliefs at once.

I have formulated the lottery paradox in terms of justification. Alternatively, it may be formulated in terms of rational acceptability or doxastic rationality. The premiss then reads, respectively: The same strategy can be applied to solve the paradox formulated in these terms if rational acceptability and doxastic rationality are species of permissibility too. Thus, assume that such-and-such is rationally acceptable for me if and only if it is rationally permissible for me to believe such-and-such, and likewise that it is rational for me to believe such-and-such if and only if it is rationally permissible for me to believe such-and-such. Then (1-RA) and (1-RB) are each equivalent to (1-PE), and the solution can proceed as above. It is not my aim to defend the assumption that rational acceptability and doxastic rationality are species of permissibility here. Since it is similar in spirit to the assumption that justification is a species of permissibility, however, it enjoys at least a similar prima facie plausibility.

  • (1-RA)

    For each ticket, it is rationally acceptable for me that it will lose.

  • (1-RB)

    For each ticket, it is rational for me to believe that it will lose.

Abstracting from particular formulations of the paradox, the underlying strategy of the solution is to make the high probability of a proposition sufficient not for believing that proposition, but for the permissibility of believing it (by ‘believing’ I always mean ‘fully believing’ or ‘outright believing’). Since each lottery ticket has a high probability of losing, I am permitted to believe that Ticket #1 will lose, permitted to believe that Ticket #2 will lose, etc. But since permissibility does not agglomerate, it does not follow that I am permitted to have all these beliefs at once. Neither does it follow that I am permitted to have a single belief whose content is the conjunction of the contents of these beliefs – even if we uphold the closure principle.

Suppose, then, that I come to believe that Ticket #3 (say) will lose and abstain from judgement about the other tickets. Given what I have said, I would be epistemically permitted to so believe. But wouldn't this be arbitrary? Yes and no. There is no arbitrariness in the claim (NARROW) that licenses my belief, since it treats all ticket-propositions on a par. What is arbitrary is that I come to believe of this specific ticket that it will lose. It is hard to see what is supposed to be bad about this sort of arbitrariness, however – especially if one compares the situation with solutions to the paradox that forbid me to form any such beliefs (Douven 2008: 214–17). Analogously, in the cake example it might be arbitrary which piece I choose, but nobody would take this to show that I was not permitted to have any cake in the first place.4

1 For recent discussions of this claim and of deontic conceptions of epistemic justification in general, see Steup 2001, Chuard and Southwood 2009 and Nelson 2010.
2 The paradox is due to Kyburg (1961: 197). Wheeler 2007 is a recent review of the debate.
3 I do not distinguish between believing that all tickets will lose and believing that [p & q & r], which is harmless here.
4 Thanks to Franz Huber, Erasmus Mayr, Moritz Schulz, Ralph Wedgwood, and Timothy Williamson for helpful comments and suggestions.

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