Two-state solution to the lottery paradox


This paper elaborates a new solution to the lottery paradox, according to which the paradox arises only when we lump together two distinct states of being confident that p under one general label of ‘belief that p’. The two-state conjecture is defended on the basis of some recent work on gradable adjectives. The conjecture is supported by independent considerations from the impossibility of constructing the lottery paradox both for risk-tolerating states such as being afraid, hoping or hypothesizing, and for risk-averse, certainty-like states. The new proposal is compared to views within the increasingly popular debate opposing dualists to reductionists with respect to the relation between belief and degrees of belief.

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  1. 1.

    ‘Justified’ here and in all the remaining premises refers to propositional justification, not doxastic justification. See below.

  2. 2.

    We are imagining a case where the lottery has already been drawn, but the results have not yet been announced. And S has no insider knowledge apart from the fact that the lottery is fair and has one winner.

  3. 3.

    That is, the proposals that are not eliminitivist proposals with respect to epistemic justifiction (or rationality). A paradigmatic example of an eliminitivist reply to the lottery paradox (and other paradoxes) can be found in the later works of Quine. According to Roy Sorensen’s (cf. Sorensen 2017) reconstruction of Quine’s proposal, Quine is rejecting the very idea that ‘justified’ is a meaningful adjective. According to eliminitivism ‘justified’ is as meaningless as ‘zillion’. Giving up on the idea that ‘justification’ is a meaningful term is one quite radical way of dissolving the paradox. In what follows we are setting aside such a radical eliminitivist view, though.

  4. 4.

    Though see Smith (2016: 65) for a dissenting interpretation.

  5. 5.

    In what follows I rely mainly on the recent work by Chris Kennedy. See Kennedy and McNally (2005) and Kennedy (2007), forthcoming and further references therein. See also Cariani et al. (2017: ms.) for an alternative approach.

  6. 6.

    ‘?’ indicates infelicity.

  7. 7.

    So, for instance, the interpretation of the adjective tall will be (on some accounts) as follows: \(\llbracket \)tall\(\rrbracket \) = \(\lambda d\lambda x\).tall(x) = d (cf. Kennedy and McNally 2005: 349). Roughly, tall is a measure function that has an individual as an input and the individual’s degree of tallness as its output, that is, a degree on the scale of height (or, perhaps more precisely, on the scale of vertical extension). Measure functions are of the type \(\langle e, d \rangle. \) They take individuals and return degrees.

  8. 8.

    See Kennedy: “One feature that all analyses [of gradability] agree on, however, is that gradable adjectives are distinguished from their non-gradable counterparts in introducing (either lexically or compositionally) a parameter that determines a THRESHOLD of application, such that a predicate based on a gradable adjective holds of an object just in case it manifests the relevant property to a degree that is at least as great as the threshold. A predicate expression formed out of a gradable adjective therefore comes to denote a property only after a threshold has been fixed.” Leffel et al. (2017).

  9. 9.

    See Kennedy (2007), for instance, for further criteria, such as the behaviour of the adjectives in Sorites-paradox-style reasoning.

  10. 10.

    See Kennedy (2007: 37–38) for more details on contexts where ‘opaque’ and ‘transparent’ may have minimal interpretations and contexts where they may have maximal interpretations, e.g. cases where one is manipulating the degree of tint of a car window, going from completely transparent to completely opaque and vice versa.

  11. 11.

    One place where confident is assumed to be a relative adjective is in Unger (1975: 63–65). However, the problem there seems to be that Unger focuses exclusively on the combination of confident with the specific modifier rather and doesn’t consider other relevant data and possible interpretations of his proposal.

  12. 12.

    Alternatively, one might think that being confident that p is determined somewhat contextually (in a restricted sense of contextualism): in ‘minimal-standard’ contexts being confident that p requires any amount of confidence, whereas in ‘maximal-standard’ contexts being confident that p requires the maximal amount of confidence. One might think of being confident that p within this double standard contextualism by analogy to a similar view about being full. On some occasions, a glass containing an amount of liquid that fills, say, 2/3 of the glass (or perhaps even less) will count as a full glass. On other occasions, nothing less than a glass filled to the brim will count as a full glass. Let me stress that what counts as being confident, or being full for that matter, is not utterly context-dependent on such a view. There are only two possible standards: a minimal standard and a maximal one. And they are fixed in a sense. Details about the context will matter for determining which sort of standard applies in a given situation. For matters of clarity, I prefer the simpler view above. Nevertheless, I think that the solution to the Lottery Paradox that I will sketch below could also be transposed into the minimal-maximal restricted contextualist framework.

  13. 13.

    See the following context from Kennedy (2007), where ‘transparent’ can be interpreted as a minimal standard adjective: “Consider a context in which I am manipulating a device that changes the degree of tint of a car window from 0\(\%\) (completely transparent) to 100\(\%\) (completely opaque). (67a) can be felicitously uttered at the point at which I have almost reached 100\(\%\) of tint, demonstrating both that opaque can have a maximum standard (I am denying that the glass is completely opaque) and that transparent can have a minimum standard (partial transparency).

    1. (67)
      1. a

        The glass is almost opaque, but not quite. It’s still transparent.

      2. b

        The glass is almost transparent, but not quite. It’s still opaque.

  14. 14.

    Thanks to an anonymous referee for this journal for making me aware of this potential worry.

  15. 15.

    One way to further clarify the present claim is to see it as a claim about kinds of ‘being confident that p’ states. The idea is that one kind of state of being confident that p is such that it requires a maximal ‘amount’ of confidence whereas the other is such that it requires only some ‘amount’ of confidence (i.e. any ‘amount’ of confidence is enough for one to be in a state of this second sort, like any amount of transparency is enough for a window to still count as partially transparent in a relevant context: see footnote 13 above). Thus, the account does allow for variations in the ‘amount’ of confidence that one may have when one is somewhat (minimally) confident that p. In the above case of manipulating the tint of a window, the utterance 63(a) ‘The glass is almost opaque, but not quite. It’s still transparent’ comes out as true at more than one point during the process of changing the tint of the window. We can imagine that there is a continuum of states \(x_{1}\) to \(x_{n}\) such that state \(x_{m+1}\) is a bit closer to the state of the window’s being totally opaque than state \(x_{m}\) is, but such that window is still transparent in all of them. I think something similar is allowed by the present account of being confident that p. The claim that there are only two possible kinds of states of being confident that p, one subject to a minimal standard and the other subject to a maximal standard, doesn’t preclude the possibility of there being variation in how much confidence one actually has when one is somewhat confident. Thanks again to an anonymous referee for this journal for making me aware of the need to clarify this point.

  16. 16.

    Or, at any rate, to some condition on an underlying scale. Thanks to a referee for pointing to this specification.

  17. 17.

    One may object that strictly speaking the claim that belief and being confident are somehow connected is a weaker claim than the claim that belief just is being confident. Consequently, one might think that rejecting one need not mean rejecting the other, and while rejecting any connection between the two surely leads to some puzzling questions, merely rejecting the stronger claim need not. Moreover, it can be objected that the present proposal has to take up the stronger assumption in order to deliver on its promises. Again, I think this is a genuine worry that a full account of belief should address in detail. However, that task exceeds our present project, which is simply to put on the table a previously overlooked way of dealing with the lottery paradox and belief. With that being said, I nonetheless think that some of the questions posed below do appear puzzling even for less demanding views on the connection between belief and being confident. Now, even if these considerations are misguided, the belief–being confident identity assumption is not obviously implausible, and the fact that it makes possible a potential solution to the lottery paradox and that it provides a simple account of the epistemic justification of mental states in general (see below) surely speak in its favour. Finally, I would also like to point towards considerations from Engel (2012) against the idea that belief and acceptance are two genuinely distinct doxastic attitudes. Arguably, similar considerations may apply in the present context. Thanks to an anonymous referee for alerting me to this issue.

  18. 18.

    One might think that there is nothing puzzling here: utterances like ‘I think that p, but I am not at all confident that p’ seem fine in some contexts. Assuming that ‘I think that p’ expresses a state of belief, one might argue that this shows that belief and being confident should not be associated too closely. While I agree that this is something we can felicitiously assert, I am not sure that it actually speaks against my proposal here. This data, I want to suggest, is compatible with the idea that ‘think that p’ can express a minimal standard belief, or a state of being somewhat confident (see also below on the thesis that belief is ‘weak’, cf. Hawthorne et al. 2016), while ‘I am confident that p’, by default, expresses a maximal standard belief, or a state of being maximally confident (see above on Kennedy’s suggestion that for pragmatic reasons in the case of closed-scale adjectives the strongest interpretations are favoured over weaker ones, cf. Kennedy 2007: section 4.3). On this reading, the data in question doesn’t speak against the present proposal. Indeed, it fits very neatly within our framework. Thanks again to an anonymous referee for bringing this to my attention.

  19. 19.

    Cf. Hawthorne (2003: 48-49) and Smith (2016: 72–73) for this sort of example, which might appear as a variant of the famous Preface Paradox, cf. Makinson (1965). Note, however, that this example doesn’t give rise to the preface paradox within the present setting. For we are not suggesting that for any individual invitee, the host may be maximally confident that she/he will attend. And even if one were to have such a maximal confidence state, it would not comply with the relevant standards for justification of maximal confidence.

  20. 20.

    The same goes for other risky states (being afraid that p, suspecting that p, etc.). The rest of the argument should be read in the same sense: premises are not only about being worried that p, but about any risky state. Also, again, the focus is on propositional justification.

  21. 21.

    Again we are assuming that S knows that the lottery is fair, has one winner, and S has no other insider information. The lottery has been drawn but results have not yet been announced.

  22. 22.

    Other cases without impossibility are also available. Maybe Nancy is justified in fearing that her brother will be attacked, and she is justified in fearing that her brother will get into a road accident, and she is justified in fearing that her brother will fall off a cliff and so on. But, given the high number of such possible negative outcomes she is probably not justified in being worried that her brother will have them all (i.e. that he will be attacked and involved in a road accident, and fall off a cliff etc.). The conjunction of all of them, though possible, is so improbable that being worried about them all occurring at once might just be epistemically unjustified.

  23. 23.

    Thanks to an anonymous referee for this journal for making me aware of the need to clarify these points.

  24. 24.

    I think it makes sense to associate crispness with what some authors call ‘being given’ or ‘being taken for granted’. See Weisberg: “Full beliefs provide premises that are treated as givens in reasoning, while partial beliefs are used as weights.” Weisberg, forthcoming: 30

  25. 25.

    Having the maximal/minimal degree of confidence is having a degree of confidence. Admitting of a degree is being gradable. Compare to radically different, clearly non-gradable items. Being next, being left, being digital, for instance, has no degree whatsoever. On the other side, being tall, rich and so on, is gradable and vague: there is no inherent crispness in what counts as rich or tall. Being tall or rich depends entirely on the specificities of context. But to suppose that something is either non-gradable (akin to being next or being digital) or tall-like gradable, is a false dilemma. There is the middle ground of being transparent, being dry, being open and so on, that comes in degrees but has inherent standards and crispness.

  26. 26.

    See Weisberg: “Two characteristics distinguish this state [i.e. the state of full belief that is not ‘just being highly confident’] . First, we become disposed to rely on P—to use it as a premise in future reasoning, to assume it in decision-making, and to assert it. [...] Second, we become resistant to re-opening deliberation—we treat the question whether P as settled.” Weisberg, forthcoming: 6.

  27. 27.

    Which is not to say that evidential probability cannot play a significant role with respect to justification of the states of being confident.

  28. 28.

    I would also like to thank an anonymous referee for this journal for stressing this point.


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Many thanks to Richard Dub, Jeremy Goodman, John Hawthorne, Benjamin Kiesewetter, Thomas Kroedel, Tristram Oliver-Skuse, Edgar Phillips, Travis Timmerman, Alexis Wellwood, audiences at University of Geneva, University of Fribourg, SOPHA 2018 congress, and two anonymous referees for this journal for discussion and comments on earlier version of this paper. The research work that led to this article was supported by the Swiss National Science Foundation Grants Number 171464 and 169293.

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Logins, A. Two-state solution to the lottery paradox. Philos Stud 177, 3465–3492 (2020).

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  • The lottery paradox
  • Epistemic justification
  • Closure
  • Emotions
  • Justified belief
  • Confidence
  • Gradable adjectives
  • Paradox
  • Lockean thesis
  • Belief
  • Knowledge
  • Certainty
  • Risky states
  • Absolute adjectives
  • Credences
  • Dualism about belief
  • Reductionism about belief