Philosophical Studies

, Volume 161, Issue 1, pp 37–46

Reconsidering the lessons of the lottery for knowledge and belief



DOI: 10.1007/s11098-012-9939-y

Cite this article as:
Ross, G. Philos Stud (2012) 161: 37. doi:10.1007/s11098-012-9939-y


In this paper, I propose that one can have reason to choose a few tickets in a very large lottery and arbitrarily believe of them that they will lose. Such a view fits nicely within portions of Lehrer’s theory of rational acceptance. Nonetheless, the reasonability of believing a lottery ticket will lose should not be taken to constitute the kind of justification required in an analysis of knowledge. Moreover, one should not accept what one takes to have a low chance of being true. Accordingly, one should take care not to believe of too many tickets that they are to lose. Finally, while arbitrariness is no absolute barrier to epistemic reasonability, one may not be able to believe that one’s lottery ticket will lose if one cannot regard oneself as knowing it will lose.


KnowledgeLotteryParadoxReasonable belief

1 Introduction

In a very large lottery, with only one winning ticket, there are epistemic reasons for believing of an arbitrary ticket that it will lose. Accordingly, one can have epistemic reasons for being epistemologically arbitrary. As I argued in (Ross 2003), such a view can fit nicely within Lehrer’s theory of personal justification. I wish now to consider some of Lehrer’s replies to my view, compare it to a recent defense of epistemic arbitrariness, and then to reconsider the matter anew. I propose that the problem with believing or accepting of one’s lottery ticket that it will lose is not with the arbitrariness of one’s reasons for believing that one’s lottery ticket will lose, but with the nature of such a belief itself.

2 A case for credulous consistency

In (Ross 2003), I argued that the lottery paradox for reasonable belief could be seen as a dilemma. I agreed with Lehrer and many others that one horn of the dilemma, an inconsistency resolution—believing of each ticket that it will lose while believing that exactly one ticket will win—was wholly unacceptable. I resisted, however, following Lehrer and many others in swallowing the skeptical horn of the dilemma: believing of no ticket that it will lose. Instead, I endorsed a bold attempt to slip through the horns of the dilemma.

In a fair lottery with a million tickets and only one winning ticket, I should be very confident that my ticket is to lose. Prima facie, my high confidence that my ticket will lose makes it epistemically permissible for me to accept or believe that my ticket will lose. Yet, since my reasons for accepting that my ticket will lose are qualitatively identical to my reasons for accepting the same of any other ticket, it seems wholly arbitrary to accept that my ticket will lose and not accept the same of any other ticket, recognizing, as I do, that the statistical grounds for any other ticket losing are equally strong. To refuse to accept that my ticket will lose seems immoderately skeptical, and to accept that it will lose seems attractively less cautious. Yet, it seems arbitrary to accept that one ticket will lose and not to accept that the other tickets will lose. Nonetheless, to accept that all the tickets will lose is to be self-recognizably inconsistent, since I also believe that one ticket will win.

Lehrer (2000) and many others adopt the extremely cautious skeptical resolution and insist that considerations of consistency demand that one should uniformly withhold judgment on each proposition that any particular lottery ticket will lose. Foley (1993) and Klein (1985) adopt the extremely credulous recommendation that we accept of each ticket that it will lose. They maintain that we have reason to be recognizably inconsistent.

Harman (1986) raised the possibility of a third type of solution. He pointed out that one could make an inference on statistical grounds that any particular ticket will lose, yet that such inferences need not lead to inconsistency:

To say one can infer this of any ticket is not to say one can infer it of all. Given that one has inferred ticket number 1 will not win, then one must suppose the odds against ticket number 2 are no longer 999,999 to 1, but only 999,998 to 1. And after one infers ticket number 2 won’t win, one must change the odds on ticket number 3 to 999,997 to 1, and so on. If one could get to ticket number 999,999, one would have to suppose the odds were even, 1 to 1, so at that point the hypothesis that this ticket will not win would be no better than the hypothesis that it will win, and one could infer no further. (Presumably one would have to have stopped before this point.) But the order of inference really matters here, since one could have inferred that ticket 999,999 won’t win if only one had made this last inference early enough (Harman 1986, p. 71).

Though Harman raises the possibility of this alternative, he does not explicitly endorse it. Indeed, Harman even expresses a reason to doubt that such reasoning can yield epistemically justified belief, as he claims that “in theoretical reasoning one would not be justified in making an arbitrary choice of what to believe among competing hypotheses at the same level” (Harman 1986, p. 68). Since the order of inference in which one considers which tickets will lose is arbitrary, it would seem that Harman endorses, as a principle of epistemic (though not practical) reasoning, a principle of symmetry or non-arbitrariness, viz, that if one has equally good reason to accept one proposition as another, one’s attitude toward each should be the same: either one should accept both or accept neither.

To endorse anything like Harman’s proposal requires rejection of a principle of symmetry or non-arbitrariness applied to the case of the lottery: if one has equally good reason to accept of any ticket that it will lose as one has to accept that any other ticket will lose, one should adopt the same attitude uniformly. In the lottery, symmetry requires that we swallow a horn of the dilemma: either accepting of each ticket that it will lose (the inconsistency horn) or not accepting of any ticket that it will lose (the skeptical horn). Since Harman seems to accept the symmetry principle for theoretical reasoning, it is not clear that Harman means to endorse an epistemically credulous solution. Perhaps he only meant to propose it as a previously ignored alternative that is not inconsistent or merely to endorse such credulity as a pragmatically justified alternative. Moreover, if Harman’s proposal involves a chain of reasoning that extends to many of the tickets, then one will be able to reasonably infer a highly improbable proposition that the winning ticket is among a relatively small group of tickets. I proposed instead that we slip through the horns by adopting a modestly credulous, but consistent approach: accepting that one’s own ticket will lose, while not accepting that of many of the others, even though one is no less confident that the other tickets will lose.1

Turning to Lehrer’s theory of rational acceptance, I examined Lehrer’s argument that we are not justified in accepting that a particular ticket will lose. Lehrer takes the following move in the justification game as being won by the critic:
  • Claimant: The number one ticket has not won.

  • Critic: The number two ticket has not won.

The critic has produced an objection to the claim, since it is more reasonable to accept it on the assumption that the critic’s claim is false, than on the assumption that the critic’s claim is true.2 If the critic’s claim is false, then the original claim must be true. If the critic’s claim is true, then the probability of the original claim is reduced to 999,998/999,999 since the number of potential winners is reduced to 999,999. Consequently, the critic’s claim is not answered. Lehrer then asserts, without argument, that the critic’s claim cannot be neutralized either (Lehrer 2000, p. 147).

I questioned the assertion that the objection could not be neutralized. I suggested that the claim that both the number one ticket and the number two ticket have not won, while less probable than the claim that the number one ticket has not won, might be at least as reasonable as the skeptic’s claim, in virtue of its additional content. Since the conjunction is not itself an objection to the claim that the number one ticket will not win, the conjunction can neutralize the objection.

3 Responses to Lehrer’s replies and objections

Lehrer (2003) made the following points and replies:
  1. 1.

    It is an advantage of his account that it can explain conflicting intuitions regarding whether or not it is reasonable to accept that one’s ticket will lose, and these intuitions will depend upon the judgments of reasonability required for neutralization.

  2. 2.

    Symmetry (he called it “impartiality”) is a worthwhile epistemic goal. Though it might conflict with other epistemic goals, Lehrer counseled avoiding any strategy that guaranteed our being partial.

  3. 3.

    Lehrer questioned what he took to be my claim that with sufficient generality, acceptance is always a lottery, and thus we cannot avoid either inconsistency or partiality. He noted that he saw no argument for the claim that our evidence for any general description of the world is analogous to the merely probabilistic grounds we have for accepting a lottery proposition, nor that the empirical propositions themselves are lottery propositions.

To these replies, I have three responses. The elaboration of each response will form the basis for the remaining three sections of this paper.
  1. 1.

    I join with Lehrer in appreciating the versatility of his approach to rational acceptance. Whether we use it to support a skeptical or a modestly credulous solution to the lottery paradox will turn on our judgments of reasonability. In (Ross 2003), I took reasonable belief to be a kind of epistemic permissibility, but I argued that arbitrary reasons for belief could comfortably fit within Lehrer’s theory of personal justification. Though Lehrer’s personal justification is not the kind of justification that turns un-gettierized true belief into knowledge, it is auxiliary to such a notion of justification: undefeated justification, which is personal justification undefeated by error. So, the first of my remaining tasks is to investigate in what sense of reasonable it might be thought reasonable to believe that one’s lottery ticket will lose.

  2. 2.
    I concede that the goal of being non-arbitrary or impartial has prima facie epistemic worth. All else being equal, one should adopt no policy that guarantees that we fail to meet that objective. Nonetheless, in the lottery, we must choose a policy that guarantees that at least one of three objectives is unfulfilled:
    1. (a)

      Non-arbitrariness If one recognizes that one has equally good reason to accept one proposition as to accept the other, one should accept both or neither.

    2. (b)

      Consistency One should not accept all the members of a set of propositions that one recognizes to be inconsistent.

    3. (c)

      The Goal of Truth We ought to aim “at both believing only what is true and at believing all that is true” (Lehrer 1974, p. 202).

I shall consider below the goal of truth and its implications for reasonable belief.
  1. 3.

    I did not intend to argue that our evidence for all general empirical claims is analogous to our evidence that a lottery ticket will lose, nor that all empirical propositions are analyzable as lottery propositions. Instead, I meant only to suggest a disturbing analogy between lotteries and putative cases of statistical knowledge, such as Vogel’s case (Vogel 1990, p. 16) of knowing where one’s car is parked, and knowing it has not been stolen, despite one’s possessing merely statistical evidence that the car has not been stolen. I criticized attempts to find distinctions between lotteries and such Vogel-style cases that could make a difference in their epistemic status. A third task for below, then, will be to examine further whether we can know on the basis of mere statistical probability, and what sort of belief we can have in a lottery proposition, absent such knowledge.


4 How is it reasonable to believe that one’s lottery ticket will lose?

Clearly, it is more reasonable to believe that one’s lottery ticket will lose than it is reasonable to believe that one’s lottery ticket will win. Thus, if we define a minimal notion of reasonability—more reasonable to believe that p than to believe that not-p—it is reasonable in this sense to believe that one’s lottery ticket will lose. Yet, since this minimal notion of reasonability attaches to believing of each ticket that it will lose, it cannot be of use in explaining how one can be reasonably arbitrary: believing of some, though not all tickets, that they will lose.

It would seem that it is not reasonable to believe that one’s lottery ticket will lose in the sense of a proposition’s being epistemically justified in the manner required for knowledge. That is, it seems that recognition of the relevant chances cannot fulfill every epistemic condition required to turn un-gettierized true belief that one’s lottery ticket will lose into knowledge that one’s lottery ticket will lose. At least it will seem that one is not so justified if it is plain that one cannot know, on mere statistical evidence, that one’s lottery ticket will lose. For if one is justified in believing in the sense required for knowledge, then one can know that one’s lottery ticket will lose, should it, in fact, lose (so long as one is not gettierized).

Is it obvious that mere recognition of the chances cannot ground knowledge that a lottery ticket will lose? It is standard to think so, although some have recently expressed slight doubt, due to the difficulty of distinguishing lottery cases from statistical knowledge (cf. Kvanvig 2009, p. 155; Hawthorne and Lasonen-Aarnio 2009, p. 104). Here are some reasons to continue to deny the possibility of knowledge of lottery propositions:
  1. 1.

    Those impressed with conditions of sensitivity or safety as being required for knowledge have independent reasons to object to the idea that one can know, on mere statistical grounds, that one’s lottery ticket will lose. If one believes that one’s lottery ticket will lose solely on the basis of perceived chances, one would still have the same evidence and belief even if one were mistaken. So, the belief that one’s lottery ticket will lose is not sensitive to the evidence. Moreover, proponents of safety conditions can point to the close similarity of a situation in which one is mistaken in believing that one will lose in order to argue that it is unsafe to believe, prior to the drawing, that one’s lottery ticket will lose, and thus one does not then know that one’s ticket will lose.

  2. 2.

    Whether or not one agrees with those who maintain that knowledge is required for assertion, it seems that knowledge is sufficient for meeting all of the epistemic requirements of assertion. So, if one knows that one’s lottery ticket will lose, it should be epistemically unproblematic to assert that one’s lottery ticket will lose. Yet it is problematic. If someone asserts that a lottery ticket will lose, on the mere basis of the ordinary statistical evidence, it is legitimate for the audience to object that the speaker does not know what was asserted. Since the objection can legitimately be made in the absence of any knowledge of whether the ticket will lose or whether the speaker is in an unusual Gettier situation, it appears the objection points to the inadequacy of mere statistical evidence to constitute the propositional justification required for knowing that the ticket will lose.

  3. 3.

    The propositional justification required to know plausibly satisfies multiple premise closure. If one has justifications for believing each of the premises of an argument one recognizes to be valid, then one has a justification for believing the conclusion. The statistical evidence one has that any losing ticket will lose is the same evidence one has for any other losing ticket. Accordingly, if one has a justification for believing of any losing ticket that it will lose, and one has the same justification for all the other losing tickets, one has a justification for believing the highly improbable, but true, conjunctive proposition that is equivalent to the proposition that identifies the winning ticket (i.e., the proposition that identifies all but one of the tickets as losing). So long as one is not in a Gettier situation, one can thereby know a proposition that identifies the winning ticket. Yet, one plainly would know no such thing.


Apparently, any reasonability of arbitrarily believing that a lottery ticket will lose will not solely be due to the warrant or justification one has for the propositional content that a particular ticket will lose. The statistical evidence that any ticket will lose is the same evidence one has for any other ticket. While one has reason to believe of any ticket that it will lose, it does not follow that one can justifiably believe that any ticket will lose. For what one justifiably believes (doxastic justification) depends upon one’s set of beliefs. One can have the propositional justification for believing a proposition without believing it, but one cannot justifiably believe a proposition and not believe it. Moreover, one can have propositional justification for a proposition, and also believe it, but still not justifiably believe it, because one has not properly based one’s belief on other evidential beliefs. If we are to adopt a modestly credulously solution to the lottery paradox, and we take reasonability to be a kind of epistemic justification, then it would seem that we should treat that justificatory status as attaching to the believing of the proposition rather than to the proposition itself. So long as we are willing to give up the goal of non-arbitrariness, it may then be open for us to say that one can justifiably believe that a given lottery ticket will lose, but not to extend that status to believing the same of every other losing ticket, because of a resulting incoherence in one’s belief system. Justification of the sort Lehrer (2000) proposes, requiring coherence within a belief system, may be the kind of justification that could allow one justifiably, but arbitrarily, to believe of some tickets, though not of all, that they will lose. In any event, whether we attach the justificatory status to the believing of a lottery proposition or to the proposition itself, we had best not think of that justificatory status as being the sort that is sufficient to turn un-gettierized true belief into knowledge.

5 Arbitrariness in the service of our epistemic goal?

In “The Lottery Paradox and Our Epistemic Goal,” Douven (2008) also questions whether arbitrariness must be epistemically unreasonable. Douven’s case rests upon the idea that our overarching epistemic goal is truth. More precisely, Douven proposes these specifications of our epistemic goal:
  • (G1) We ought to aim at “[amassing] a large body of beliefs with a favorable truth-falsity ratio” (Alston 1985, p. 59).

  • (G2) We ought to aim “at both believing only what is true and believing all that is true” (Lehrer 1974, p. 202).

While conceding that (G2) is practically unattainable, he argues that the pursuit of (G2) will be no different from the pursuit of (G1), since the best we can do in pursuing (G2) is to accomplish (G1). Douven then proposes that other things being equal, which theory of justification one should prefer should be a matter of which theory best serves our epistemic goal.

Douven agrees with those who adopt a skeptical resolution of the lottery paradox that belief should satisfy a deductive closure condition. Thus, he rejects the inconsistency horn of the lottery dilemma: we should not believe of each ticket that it is a losing ticket while also believing that at least one ticket will win.

Douven argues that the pursuit of our epistemic goal requires that we be extraordinarily credulous in our lottery beliefs:

[Given a 10-ticket lottery with 1-winner] accept of nine tickets that they will lose, and believe of the remaining one that it will win. Clearly, there is no longer an inconsistency in your beliefs about the lottery. And there is a 90 % chance that you added eight true beliefs and two false ones to your stock of beliefs—still not a bad score (Douven 2008, p. 211).

So Douven and I are both committed to the reasonability of arbitrariness in pursuit of the truth. I contended that it was epistemically permissible to believe of my own ticket, and a few more besides, that they were losing tickets. Douven does not stop with just a few. He takes our epistemic goal to be served by believing all that is probable up to the brink of inconsistency. Thus, on his account, I can be justified in believing of a particular ticket that it will win, even though I have extraordinarily low confidence that this arbitrarily selected ticket will win. In effect, while I have recommended arbitrarily selected a small group of tickets, and believing of all of them that they will lose, Douven has arbitrarily selected a single ticket, and recommended that one should believe that it will win. Moreover, he takes us as having a justification in the sense required for knowledge for such a proposition. Should one luckily have picked the winner, and one was not gettierized, then one will have known the winner by a lucky guess.
Douven’s position seems extreme. Intuitively, it seems wrong to say that knowledge of such low-chance propositions is possible. The following simplified version of a principle considered by Hawthorne and Lasonen-Arnio in (2009, p. 96) is plausible:

[Low Chance] For all worlds w, times t, subjects s, and propositions P, if [the chance of P at t in w] is low, then s does not know P at t in w.3

Low Chance excludes knowledge of Douven’s lottery proposition, an arbitrarily selected complete state description of the outcome of the lottery, and thus implies it cannot be justified in the sense required for knowledge.
Yet, what are we to say of Douven’s argument that accepting such a maximally specific lottery proposition would serve our epistemic goal, since we would improve on the truth-falsity ratio of our belief system? Does this not give us a reason to deny Low Chance? Not if we have good reason to reject maximizing the truth-falsity ratio in our belief set as our overriding epistemic goal. And a reason for rejecting this as our overriding goal is provided by Marian David:

Take the admittedly unlikely case of someone who has a large body of true beliefs without having any false beliefs. Given Alston’s goal, one may wonder how a person could [possess] justification for an additional truth that she does not believe. After all, adding a further truth to her body of beliefs will not improve the already perfect truth-ratio (David 2001, p. 159).

Alternatively, Low Chance plus some suitably realistic refinement of the truth-goal proposed by Lehrer (1974) might provide a pro tanto reason for believing of a few, but only a few, lottery tickets that they will lose. If it is unreasonable to believe a proposition that one takes to be objectively improbable, then to avoid the skeptical horn, one must draw an arbitrary line that Douven need not draw. One must contend that with respect to some number of lottery tickets one can reasonably believe that they will lose, but not believe this of any other tickets, despite the fact that one has no principled reason for drawing a line at that particular place. The plausibility or implausibility of reasonably drawing such arbitrary lines should stand or fall with the judgment we make about the symmetry principle. If we are prepared to admit that we can have no better epistemic reason to make one judgment rather than another, yet still have reason enough, then such arbitrary lines need not be epistemically unreasonable.

6 Can one believe that a lottery ticket will lose on merely statistical evidence?

So far, I have argued that we can serve a truth goal by believing of a few lottery tickets that they will lose, while not believing this of the rest, despite the fact that we do not have a reason to distinguish these tickets from the others. Yet, even if arbitrariness of this sort is not unreasonable, there may still be a problem with believing that one’s lottery ticket will lose: not because of a difficulty with the arbitrariness of one’s reasons, but because of an intrinsically problematic feature of reflectively holding such a belief.

What might it mean to say that one believes that one’s lottery ticket will lose on the basis of the statistical evidence that it is a ticket in a very large lottery, with only one winner, and has a chance of winning equal to that of any other ticket? If we have something akin to a Bayesian credence function, we might assign a very high credence to the proposition that a particular ticket will lose. If we then adopt a threshold model of outright belief—that having a sufficiently high credence constitutes outright belief—then we could make sense of having an outright belief that a lottery ticket will lose. Yet, this will not help us understand how someone can believe of some but not all tickets that they will lose, since whatever credence one assigns to any ticket’s losing is equal to the credence that one assigns to another ticket’s losing. I take this to be a problem not for a credulous solution to the lottery dilemma, but with this threshold conception of belief. Those adopting a skeptical solution should agree. For it is not even possible to withhold from believing that any ticket will lose, if such a belief is no more than assigning a sufficiently high credence to its losing. So, outright belief should not be understood in terms of a threshold model of credence-level assignments.

Some might suggest that a belief that a given ticket will lose must be a kind of partial belief that is not understood in terms of Bayesian credence. Specifying such an alternative conception of partial belief would seem to be a difficult analytical challenge, but I remain skeptical that such a state, whatever it might be understood to be, will have enough of the features of belief to be considered a species of the same kind. In particular, I expect any notion of partial belief will not satisfy multiple premise closure, will not have a constitutive norm of consistency, will not ground sincerity in assertion, and will not aim at the truth in the (admittedly) obscure way that outright beliefs do.

So, if one reasonably, but arbitrarily, believes, that one’s lottery ticket is to lose, it would seem that the belief must be outright and not partial. For those of us who have considered the matter, and judged that we can take a modestly credulous approach to the lottery, such an outright belief could also be a fully reflective one. (Perhaps it will be something more akin to what Lehrer calls acceptance.) We would then take ourselves as having the belief that a particular ticket will lose. If such a reflective belief were to be analogous to internal speech, then the conditions appropriate to assertion would seem to carry over to such reflective outright belief. One plausible condition on assertion, supported by G. E. Moore and many others, is that to assert that p is to represent oneself as knowing that p. If this is right, then if I do not take myself to know that my lottery ticket will lose, then I cannot have the reflective outright belief that it will lose. So, whatever epistemic reasonability can reside in being arbitrary, the problem for a modestly credulous solution to the lottery dilemma may not reside in the arbitrariness itself, but in the appreciation of the fact that the reasons for such a lottery belief cannot constitute adequate grounds for knowledge.


Interestingly, Lehrer (1983) considers the possibility of reasonable arbitrariness in a novel interpretation of Wilfrid Sellars’s empirical acceptance rules. While not endorsing a rejection of a symmetry principle, Lehrer finds that such a position is “justified by systematic epistemic objectives and is in no way ad hoc (…)” (Lehrer 1983, p. 469).


In (Ross 2003), I responded to Lehrer’s earlier argument (of the first edition of Theory of knowledge), framed in the language of beating a competing claim proposed by a skeptic. Essentially the same argument is reformulated in (2000), and thus I update the argument to use his later theory here.


Hawthorne and Lasonen-Arnio (2009) have to complicate Low Chance considerably in order to accommodate the possibility of knowing contingent a priori truths having a low objective chance of being true. These details need not concern us here.


Copyright information

© Springer Science+Business Media B.V. 2012